Time and History: Proceedings of the 28. International Ludwig Wittgenstein Symposium, Kirchberg am Wechsel, Austria 2005 9783110333213, 9783110333022

Table of contents :
Preface
I. Th e Philosophy of Time / I. Philosophie der Zeit
Ordinary Th inking about Time
How Real Are Future Events?
Memory and the A-Series
Wishing It Were Now Some Other Time
Time and Self
Time and Reality of Phenomenal Becoming
Are Th ere ‘Tensed’ Facts (A-Series)?
Is Time an Abstract Entity?
Zur Entwicklung des Zeitbegriff s: Aristoteles und der Zeitbegriff in der relativistischen Kosmologie
II. Time in the Physical Sciences / II. Zeit in den physikalischen Wissenschaften
Can We Understand the Cosmic Evolution?
Time and the Deep Structure of Dynamics
In the Beginning, At the End, and All in Between: Cosmological Aspects of Time
Against Pointillisme about Geometry
Zeit im Gödelschen Universum
“Close to the Speed of Light”: Dispersing Various Twin Paradox Related Confusions
Time’s Arrow, Time’s Fly-Bottle
Th ree Concepts of Irreversibility and Th ree Versions of the Second Law
Are the Laws of Nature Time Reversal Symmetric? Th e Arrow of Time, or Better: Th e Arrow of Directional Processes
III. Time in the Social and Cultural Sciences / III. Zeit in den Sozial- und Kulturwissenschaften
Time and Communication
Perspektiven der Subjektivität: Das Verhältnis von Systemzeit und Eigenzeit in den perfektischen Tempusformen
Drei Pioniere der philosophisch-linguistischen Analyse von Zeit und Tempus: Mauthner, Jespersen, Reichenbach
Zeit, Performanz und die ontosemantische Struktur des Kunstwerks
Die Th eorie der somatisch-neuronalen Entstehung von Werten, die a-chronologische Gedächtniszeit und die Verschränkung von Zeit und Bewerten
Wittgenstein und Sraff a. Zeitproduktion durch Zeit
IV. Temporal Logic / IV. Zeitlogik
A Mini-Guide to Logic in Action
On the Problem of Defi ning the Present in Special Relativity: A Challenge for Tense Logic
A Deontic Logic with Temporal Qualifi cation
V. History / V. Geschichte
Kulturelle Zeitgestalten
Zeit und Geschichte in frühen Kulturen
‚Zukunft‘ als Schlüsselkategorie der Geschichtsphilosophie
Verantwortung und Geschichte
VI. Wittgenstein on Time / VI. Wittgenstein über die Zeit
Wittgenstein’s Times (And Ours)
Wittgenstein on Time (1929–1933)
Das „Jetzt“ bei Wittgenstein – Über Gegenwart und Wandel
Time, Music and Grammar. When Understanding and Performing What is Understood are Two Facets of the Same Action
List of Authors

Citation preview

Friedrich Stadler / Michael Stöltzner (Eds.) Time and History

Publications of the Austrian Ludwig Wittgenstein Society. New Series Volume 1

Friedrich Stadler / Michael Stöltzner (Eds.)

Time and History Proceedings of the 28. International Ludwig Wittgenstein Symposium Kirchberg am Wechsel, Austria 2005

ontos verlag Frankfurt I Paris I Ebikon I Lancaster I New Brunswick

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliographie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

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2006 ontos verlag & the Authors P.O. Box 15 41, D-63133 Heusenstamm www.ontosverlag.com ISBN10: 3-938793-17-1 ISBN 13: 978-3-938793-17-6 2006 No part of this book may be reproduced, stored in retrieval systems or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use of the purchaser of the work Printed on acid-free paper ISO-Norm 970-6 FSC-certified (Forest Stewardship Council) This hardcover binding meets the International Library standard Printed in Germany by buch bücher dd ag

Table of Contents Friedrich Stadler & Michael Stöltzner Preface

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I. The Philosophy of Time I. Philosophie der Zeit John Campbell Ordinary Thinking about Time

1

John Perry How Real Are Future Events?

13

Robin Le Poidevin Memory and the A-Series

31

L. Nathan Oaklander Wishing It Were Now Some Other Time

43

Katia Saporiti Time and Self

51

Sergio Galvan Time and Reality of Phenomenal Becoming

63

Edmund Runggaldier Are There ‘Tensed’ Facts (A-Series)?

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Jan Faye Is Time an Abstract Entity?

85

Peter C. Aichelburg Zur Entwicklung des Zeitbegriffs: Aristoteles und der Zeitbegriff in der relativistischen Kosmologie

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II. Time in the Physical Sciences II. Zeit in den physikalischen Wissenschaften Walter Thirring Can We Understand the Cosmic Evolution?

115

Julian Barbour Time and the Deep Structure of Dynamics

133

John Earman In the Beginning, At the End, and All in Between: Cosmological Aspects of Time

155

Jeremy Butterfield Against Pointillisme about Geometry

181

Heinz Rupertsberger Zeit im Gödelschen Universum

223

Miloš Arsenijevi “Close to the Speed of Light”: Dispersing Various Twin Paradox Related Confusions

233

Huw Price Time’s Arrow, Time’s Fly-Bottle

253

Jos Uffink Three Concepts of Irreversibility and Three Versions of the Second Law

275

Paul Weingartner Are the Laws of Nature Time Reversal Symmetric? The Arrow of Time, or Better: The Arrow of Directional Processes

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III. Time in the Social and Cultural Sciences III. Zeit in den Sozial- und Kulturwissenschaften Kristóf Nyíri Time and Communication

301

Richard Schrodt Perspektiven der Subjektivität: Das Verhältnis von Systemzeit und Eigenzeit in den perfektischen Tempusformen

317

Elisabeth Leinfellner Drei Pioniere der philosophisch-linguistischen Analyse von Zeit und Tempus: Mauthner, Jespersen, Reichenbach

337

Constanze Peres Zeit, Performanz und die ontosemantische Struktur des Kunstwerks

363

Werner Leinfellner Die Theorie der somatisch-neuronalen Entstehung von Werten, die a-chronologische Gedächtniszeit und die Verschränkung von Zeit und Bewerten

387

Peter Weibel Wittgenstein und Sraffa. Zeitproduktion durch Zeit

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IV. Temporal Logic IV. Zeitlogik Johan van Benthem A Mini-Guide to Logic in Action

419

Thomas Müller On the Problem of Defining the Present in Special Relativity: A Challenge for Tense Logic

441

Eduard F. Karavaev A Deontic Logic with Temporal Qualification

459

V. History V. Geschichte Aleida Assmann Kulturelle Zeitgestalten

469

Jan Assmann Zeit und Geschichte in frühen Kulturen

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Herta Nagl ‚Zukunft‘ als Schlüsselkategorie der Geschichtsphilosophie

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Hans Jürgen Wendel Verantwortung und Geschichte

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VI. Wittgenstein on Time VI. Wittgenstein über die Zeit Jaakko Hintikka Wittgenstein’s Times (And Ours)

539

Joachim Schulte Wittgenstein on Time (1929–1933)

557

Gabriele M. Mras Das „Jetzt“ bei Wittgenstein – Über Gegenwart und Wandel

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Antonia Soulez Time, Music and Grammar. When Understanding and Performing What is Understood are Two Facets of the Same Action 585

List of Authors

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Preface

This present volume contains primarily the invited papers of the 28th International Wittgenstein Symposium that was held in Kirchberg am Wechsel (Lower Austria) in August 2005. It was dedicated to the topic Time and History (Zeit und Geschichte) in an interdisciplinary perspective, ranging from the philosophy of time, in the narrower sense, the approaches of the single scientific disciplines, in so far as they are informed by foundational and philosophical issues, to culture and art. As usual, the contributed papers (Beiträge) were already published prior to the symposium.1 While the latter volume contains, in a special section, papers dedicated to all aspects of Wittgenstein’s work, the present volume focuses on his views about time. The editors are well aware that both time and history are prominently discussed within the phenomenological and hermeneutic traditions in philosophy. This was well reflected in the contributed papers, as can be seen in the Beiträge volume, and in some papers dealing with time and history from a cultural perspective. For reasons of thematic coherence and as a consequence of the general orientation of this book series, however, the editors have given priority to philosophers belonging to the analytic tradition in the broad sense. The editors nonetheless hope to have succeeded in presenting an equally focused and comprehensive picture of the contemporary debates.

We have either too little or too much time. Time goes by either too quickly or too slowly, or simply stands still. Time is omnipresent: in everyday life, philosophy, the sciences, the humanities and the arts. We are presently witnessing a real plethora of popular books dealing with time in cosmology or with the arrow of time against the backdrop of chaos, entropy and com-

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Zeit und Geschichte/Time and History. Beiträge der Österreichischen Ludwig Wittgenstein Gesellschaft/Contributions of the Austrian Ludwig Wittgenstein Society. Ed. by Friedrich Stadler and Michael Stöltzner. Band XIII/Volume XIII. Kirchberg am Wechsel 2005.

F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, v–xii.

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plexity.2 The fact that physics continues to be so influential on our picture of science in spite of the increasing dominance of the life sciences and of cognitive science indicates that issues of time are of high topicality on many fields of inquiry. Questions regarding the existence, reality or construction, origin and end, linearity and universality of time pervade all scientific disciplines and find manifold expressions in literature and the arts, among them the works of Marcel Proust and H. G. Wells. Outside the domain of the natural sciences, time typically appears in the guise of history, be it as a moment of artistic perception or as a historical process in its entirety. History involves memory, be it the memory of individuals or of entire cultures. But how is the order of memory related to the order of time? Is memory episodic or is it composed of different time Gestalten? What is the role of particular events and historical dates for our conception of time? Today we have almost forgotten to what large extent the year 2000, or Y2K, had become a field of intersection between digitaltechnological and cultural-apocalyptic visions.3 The key role of time as history — as opposed to time as a measurable quantity — within the humanities does, however, not compel one to adopt the notorious methodological divide between Geistes- und Naturwissenschaften, or the alleged dualism between Verstehen and scientific explanation. Most prominently, the French school of the Annales, in the form of their concept of longue durée des temps, has laid new foundations for an interdisciplinary kind of historical scholarship that is applicable across different epochs. At the last turn of the century, the so-called second scientific revolution brought the temporality of natural phenomena into the focus of physical 2

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Here are just a few titles that indicate what metaphors or rhetoric strategies are used to circumscribe the puzzle of time. Stephen Hawking 1988 A Brief History of Time: From the Big Bang to Black Holes, London: Bantam; Julian Barbour 1999 The End of Time: The Next Revolution in Our Understanding of the Universe, London: Weidenfeld and Nicolson (with the new subtitle The Next Revolution in Physics, Oxford: Oxford University Press, 2001); Paul Davies 1996 About Time: Einstein’s Unfinished Revolution, New York: Simon and Schuster; Igor D. Novikov 2001 The River of Time, Cambridge: Cambridge University Press; David S. Landes 2000 Revolution in Time: Clocks and the Making of the Modern World, Cambridge, MA: Harvard University Press; Peter Galison 2003 Einstein’s Clocks, Poincare’s Maps. Empires of Time, New York: Norton; Huw Price 1996 Time’s Arrow and Archimedes’ Point. New Directions for the Physics of Time, New York: Oxford University Press. „Das Jahr 2000 findet nicht statt“, Österreichische Zeitschrift für Geschichtswissenschaften 10 (no. 3), 1999.

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science, thus abandoning the Kantian a priori conception of time and the Laplacian ideal of causal explanation according to which each moment of the Universe stood on equal footing. It had in fact been an important motivation for the advocates of the Geisteswissenschaften to distance themselves from the mechanistic paradigm expressed in Laplace’s demon. The second law of thermodynamics, the increase of entropy in all physical processes made it possible — at least in the statistical interpretation developed by Ludwig Boltzmann and James Clerk Maxwell during the last decades of the 19th century — to pinpoint a physical reason of the unidirectionality of natural processes. Taken at face value, statistical mechanics allowed for local violations of the second law, even in the form of spontaneously recombining glasses, and eventually ended in a global heat death. The new physics also paved the way for a physical cosmology that was starkly different from the Laplacian clockwork universe. As it had been the case with atomism, the physics of time entered the sphere of philosophy and culture. Already Boltzmann compared the lapse of time with a movie — as would do Wittgenstein in his Philosophical Remarks. Interestingly, Boltzmann gave an estimate for the number of temporal atoms (or pictures) in a second of this movie. He and many contemporaries emphatically embraced Darwin’s theory of evolution, which neatly fitted into this dynamic picture — even though the precise nature of the hereditary mechanism remained open for half a century. Since the emergence of general relativity and big bang cosmology we know that time itself has a history. But the chronology we are familiar with from our daily life may cease to hold on the large scale. In a year commemorating Albert Einstein’s achievements this story has been told frequently, in brief or even briefer terms.4 And Kurt Gödel’s centenary in 2006 has given ample space to amuse oneself about, or contribute philosophical reflections on, the possibility of traveling into one’s own past. Furthermore, in nearly all scientific disciplines and in experimental research, from the neurosciences to linguistics, we can today admire the sometimes puzzling aspects of the interdisciplinary and (sometimes even transdisciplinary) “Matter of Time”.5 From a philosophical perspective, we should perhaps rather speak of a ‘long history of time’ that dates back to the pre-Socratics, to Plato and Aris4 5

Cf. Hawking, Stephen and Mlodinow, Leonard 2005 A Briefer History of Time, New York: Bantam Dell. “A Matter of Time” was the title of a special edition of the Scientific American (vol. 16, no. 1, 2006).

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totle. The most frequently cited reference as to the nature of time, however, stems from Saint Augustine, who wrote in the 11th Book of his Confessions: At no time then had You made anything, for itself You made. And no time is co-eternal with you, for You stand changeless; whereas if time stood changeless, it would not be time. What then is time? Is there any short and easy answer to that? Who can put the answer into words or even see it in his mind? Yet what commoner or more familiar word do we use in speech than time? Obviously when we use it, we know what we mean, just as when we hear another use it, we know what he means. What this is time? If no one asks me, I know; if I want to explain it to a questioner, I do not know. But at any rate this much I dare affirm I know: that if nothing passed there would be no past time; if nothing were approaching, there would be no future time; if nothing were, there would be no present time. But the two times, past and future, how can they be, since the past is no more and the future is not yet? On the other hand, if the present were always present and never flowed away into the past, it would not be time at all, but eternity. But if the present is only time, because it flows away into the past, how can we say that it is? For it is, only because it will cease to be. Thus we can affirm that time is only in that it tends towards not being.6 This classical passage leads straight to contemporary philosophy. Ludwig Wittgenstein deals with Augustine’s puzzle about time in his Blue Book, where argues that the problem of time is primarily a problem of language. Consider as an example the question “What is time?” as Saint Augustine and others have asked it. At first sight what this question asks for is a definition, but then immediately the question arises: “What should we gain by a definition, as it can only lead us to other undefined terms?” And why should one be puzzled just by the lack of a definition of time, and not by the lack of a definition of “chair”? Why shouldn’t we be puzzled in all cases where we haven’t got a definition? Now a definition often clears up the grammar of a word. And in fact it is the grammar of the word ”time“ 6

Westphal, Jonathan and Levenson, Carl (Eds.) 1993 Time, Indianapolis: Hackett, 15.

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which puzzles us. We are only expressing this puzzlement by asking a slightly misleading question, the question: “What is …?” This question is an utterance of unclarity, of mental discomfort, and it is comparable with the question “Why?” as children so often ask it. This too is an expression of a mental discomfort, and doesn’t necessarily ask for either a cause or a reason. (Hertz, Principles of Mechanics.) Now the puzzlement about the grammar of the word ”time“ arises from what one might call apparent contradictions in that grammar. It was such a “contradiction” which puzzled Saint Augustine when he argued: How is it possible that one should measure time? For the past can’t be measured, as it is gone by; and the future can’t be measured because it has not yet come. And the present can’t be measured for it has no extension.7 It was quite surprising for the editors that within the enormous and multifaceted body of Wittgenstein scholarship, there has been only little discussion of his views on the issue of time — even in comparison to other themes for which one finds only a small number of passages.8 In order to enhance the interactions of the topical part of the Kirchberg symposium and the annual section dedicated to Wittgenstein, the editors encouraged the main speakers of the Wittgenstein section to focus on this hitherto neglected aspect of Wittgenstein’s work. Remarkably, also a bunch of contributed papers were dwelling upon this issue, so that the symposium provided for the first time a broader view on “Wittgenstein and time”. (Chapter VI) Several core tenets of the present philosophical debates, as well as two important aspects of the physics of time, have emerged roughly a hundred years ago. In 1908, J. Ellis McTaggart set the philosophical stage by distinguishing an A-series, in which time actually flows and we have a clear sense of past, present, and future, and a B-series, in which these categories are unavailable and time resembles a spatial coordinate.9 Although McTaggart’s conclusion that time was unreal has found little approval, a lot of ink has since been spilt in arguing for or against the A- or the B-series. As can 7 8

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Wittgenstein, Ludwig 1960 The Blue and Brown Books, New York: Harper, 26. Two exceptions by prominent Wittgenstein scholars only confirm the rule; Hintikka, Jaakko 1996 “Wittgenstein on being and time”, Theoria 62 (1996), 3–18; Bouveresse, Jacques 2003 “L’ ‘énigmes du temps’”, in: Essais III. Wittgenstein & les sortilèges du langage, Paris: Agone, 189–234. J. Ellis McTaggart 1908 “The Unreality of Time”, Mind 18, 457–474.

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be seen from the contributions to Chapter I, present-day metaphysics of time has developed many nuances of the old classification and introduced important distinctions into the debate, among them David Lewis’s concepts of perdurance and endurance, and elaborated the problem of identity over time. In this modified form, the alternative between an irreducible temporality of the world and its permanence, for which time is only a fourth dimension, remains on the agenda. An important aspect of these genuinely philosophical debates is the relationship between a certain metaphysical stance and the results of modern science. This includes the attempts to provide a logical basis for the analysis of temporal phenomena. Chapter V shows that temporal logic is both influenced by game theory when modeling social actions and by special relativity as regards the temporal order of events. Chapter III deals with the physics of time. Einstein’s special theory of relativity in 1905 set out from a critical analysis of the concept of simultaneity. How could the simultaneity of distant events and of observers moving at high speed actually be established by real world instruments, that is, by clocks and by exchanging light signals? In the Minkoswki diagram, time plays the role of a fourth coordinate almost on a par with the three spatial ones. When Einstein, ten years later, published his general theory of relativity, space-time became intimately linked to the material events in a certain region and, at least in principle, in the whole universe. The Newtonian concept of space and time, as a container and an absolute order of causal interactions, was shattered. Based on earlier debates concerning the relationship between geometry and physics, relativity theory was now seen as a proof for the conventionality of geometry and as the final farewell to any Kantian synthetic a priori. To a whole generation of physicist-philosophers, most notably the Logical Empiricists, relativity theory became the touchstone of epistemology. And absolute simultaneity served as a paradigm case of an in principle unverifiable and, accordingly, meaningless concept. It took some time until relativistic cosmology really got off the ground. It began, on the one hand, after Hubble’s observation of a red-shift in the spectra of almost all distant galaxies and the discovery of the cosmic microwave background radiation, developments which eventually led to big bang cosmology. On the other hand, in 1949 Gödel published a solution of Einstein’s field equations that allowed one, at least in principle, to travel into one’s own past. Since then, many other chronology-violating scenarios have been devised, and together with the subsequent discussions about singulari-

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ties and black holes, we have come to realize how many possible worlds are consistent with Einstein’s general theory of relativity. In a sense, we might perhaps never find compelling evidence for preferring one solution over the other without accepting some philosophical principles.10 Interestingly, Gödel himself viewed his result as confirmation of a subjectivist notion of time in the Kantian sense.11 This is quite in line with recent claims that time is unreal. But even if one disagrees at this point, one problem raised by Gödel remains. Must the concept of time and its basic properties be anchored in the basic laws (axioms) of physical theory or is time a property that emerges from these laws or within the evolution of the Universe? This issue becomes particularly pressing in the discussions about the arrow of time in the context of statistical physics. After Planck in 1900 had applied Boltzmann’s statistical mechanics for the derivation of his radiation formula, the discussions obtained a new twist. Did quantum theory and atomism force us to accept indeterminism at the very bottom? Did the second law, rather than being a stranger in the mechanical cosmos, in fact express the most primary property of all natural phenomena, the fact that they are all directed? Remarkably, this was almost two decades before the advent of quantum mechanics. Yet the aspect of quantum mechanics most relevant to the issue of time consists in the question as to whether it is the measurement process itself that breaks the time invariance of the underlying (deterministic) Schrödinger evolution. Quantum field theory, finally, has contributed a new feature to the debates about the arrow of time because in high energy particle physics an inversion of temporal order can be compensated by other symmetries. The high topicality of the issue of time can be noted also in other areas of knowledge that refer, to a greater or lesser extent, to the philosophical traditions. The wide spectrum of inquiry covered in Chapters III and V includes the ethnography of time cultures, the different historical conceptions of time, the cultural constitution of time,12 the different Gestalten temporality 10 Cf. Ellis, George F. R. 1991 „Major Themes in the Relation between Philosophy and Cosmology“, Mem. Ital. Ast. Soc. 62, 553–605. 11 See Chapter (G) in Buldt, Bernd et al. 2002 Kurt Gödel: Wahrheit und Beweisbarkeit. Kompendium zum Werk Wien: öbv & hpt. For a contemporary defense of Gödel’s views, see Yourgrau, Palle 2005 A World Without Time: The Forgotten Legacy of Gödel and Einstein, New York: Basic Books. 12 For a broader discussion of time and history from the perspective of cultural studies, see Chvojka, Erhard et al. (eds.) 2002 Zeit und Geschichte. Kulturgeschichtliche Perspektiven, Wien/München: Oldenburg.

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can assume, the role of time in the philosophy of history, biological versus script time, and temporality in the arts (including the media of time/time of the media). Among the social aspects of time are the problem of communication, the personal vs. social nature of time, the temporal index of ethical judgments, the emergence of values and the economic aspects of time. It is quite in line with Wittgenstein’s approach to time — as exemplified in the above quotation — that two contributions address the issue from a linguistic perspective, both historically and systematically.

The editors are indebted to the board and the local staff of the Austrian Ludwig Wittgenstein Society as well as to the staff of the co-organizing Vienna Circle Institute (Robert Kaller, Karoly Kokai, Christoph Limbeck, Camilla Nielsen) for making possible a wonderful symposium. Christoph Limbeck has additionally contributed during the editorial process in many ways, and Mirca Szigat has assisted us in proofreading. We are greatly indebted to Thomas Binder for the editorial work and to Rafael Hüntelmann of ontos verlag. With this volume the Proceedings of the Wittgenstein-Symposia have moved to a new publisher, and we hope that this volume has become a respectable start for the new series.

Friedrich Stadler University of Vienna and Vienna Circle Institute Austria

Michael Stöltzner University of Wuppertal Germany

Ordinary Thinking about Time John Campbell, Berkeley 1. Why do it the ordinary way? I will describe two non-standard ways of thinking about time. The first is ubiquitous in animal cognition. I will call it ‘phase time’. Suppose for example you consider a hibernating animal. This animal might have representation of the various seasons of the year, and modulate its actions dependent on the season. But it need have no distinction between the winter of one year and the winter of another; it thinks of time only in terms of repeatable phases. The second non-standard way of thinking about time has been ascribed to children at an early stage in development. I will call it ‘script time’. A ‘script’ or ‘schema’ is representation of the structure of a repeated type of event, such as going to a restaurant, attending a lecture or visiting the doctor. You know what types of event happen in what order. And in script time, you identify temporal locations with respect to events in the script. Within each of these ways of thinking of time there is a recognizable “earlier than” relation. In phase time, the relation is not transitive. In script time, the “earlier than” relation only holds between times within the same script. Both these ways of thinking of time contrast with our ordinary conception. It is not immediately obvious just how to draw the contrast. You might say that we ordinarily think of time as linear, so that “earlier than” is transitive and connected. That is certainly how our use of calendars and clocks suggests we think. But does our ordinary way of thinking of time in autobiographical memory and in planning for the future demand the full strength of transitivity and connectedness? Why do we use our ordinary way of thinking of time, rather than the nonstandard ways? You might say, “This is simply what we do”, and argue that no explanation can be given. I shall argue, though, that we can explain why we think of time as we do by looking at the way we make sense of the transmission of causal influence from place to place by concrete objects. I begin by setting out the two non-standard ways of thinking of time in more detail. F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 1–12.

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2. Phase time Consider an animal using a circadian clock. The clock dictates when the animal sleeps and wakes, when it feels hungry, and so on, for a number of routine procedures through which it goes every day. So far, though, we might have here only what we might call biological time. It is one thing for an animal to have biological mechanisms that are time-sensitive, another for it to have temporal cognition. A sun-flower seed might germinate at a particular time of year; that does not of itself show that the seed is representing time. What does it take for an organism to have not merely biological, but cognitive time? A natural answer is that it has to do with whether the animal can be viewed as performing computations over temporal representations (Gallistel 1990). The sunflower seed does not perform computations relating to the time of year. On the other hand a foraging bird, for example, determining its rate of return for the time spent in a particular field, might be engaging in quite complex calculations concerning time. So if we consider an animal with a circadian clock, we can say that we have properly cognitive time if we have an ability to use its temporal knowledge in finding, for example, the optimum plan for the day, the optimum order in which to perform various tasks and how long to spend on them. The agent knows what phase the day is currently at: whether it is early, late or mid-morning, for example. And the agent may have discovered and stored information about what typically happens at various particular phases of the day — that breakfast is served at 10.00am on Forel’s balcony, for example. And the agent may put this stored information to use in guiding action, as honeybees used to gather at Forel’s balcony at breakfast time (Gallistel 1990, 243). So the agent arrives for food at the right time of day, and leaves shortly afterwards. Notice, though, that the agent so far has only the conception of time as (repeatable) phase. The agent does not draw, and makes no use of, the distinction between the morning of one day and the morning of another. The domain of times over which the agent’s temporal representations are defined is oriented: adjacent times are ordered by “earlier than”. Early morning precedes mid-morning. But the domain is cyclically ordered, somewhat as places on the equator are cyclically ordered by “to the east of ”. The “earlier than” relation here is not transitive, just as “to the east of ” is not transitive. To be able to use its circadian clock, there must be some sense in which the agent can use temporal indexicals: terms like “now” or “in a little while”.

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The agent has to be able to represent, “it’s mid-morning now”, or “in a little while it will be mid-morning”. The agent can do this by using indexicals which are governed by rules whose formal statement is exactly like that of the rules governing our ordinary temporal indexicals. Just as we have a term, “now”, whose reference is fixed by the rule I will call ‘Linear’: Linear: Any token of “now” refers to the time at which it was produced, so the agent will use a term “now” governed by the rule I will call ‘Phase’: Phase: Any token of “now” refers to the time at which it was produced. The similarity between the two rules, Linear and Phase, is I hope evident. The difference between them is in the domain of times over which they are defined. Linear is defined over the everyday domain of unrepeatable moments, linearly ordered by “earlier than”. Phase, on the other hand, is defined over the domain of phases, such as “early morning” “mid-morning” and so on, which are cyclically ordered by “earlier than”. Despite this difference, a system of indexicals governed by rules such as Phase, will serve the animal for practical purposes — planning, action and so on — somewhat as our ordinary indexicals, defined over a domain of linearly organized times, serve us. There surely is a sense in which indexicals are essential (Perry 1979), but indexicals governed by rules defined over domains of phase times will do; it is not essential that we have indexicals governed by rules defined over domains of linearly organized times. There seems, indeed, to be no reason why we could not have, within phase time, temporal operators for which we need Reichenbach’s distinction between time of utterance, reference time and event time (Reichenbach 1947). It is just that again, the domain of times over which the notions are defined will be a domain of phases. And the “earlier than” relation will be a relation defined over times as phases. So we could have representations such as “when X happens, Y will have happened already”. Here the opening clause defines the reference time, with respect to which the event Y is then temporally located. So it is not just that we can have rudimentary indexicals in phase time; we can have a relatively sophisticated set of tenses and temporal operators. One way to bring out the difference between an agent operating with phase times and an agent operating with linear time is to remark that there

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is a sense in which the agent operating with phase times does not have an authentic past tense. The agent can indeed have an operator, “in the past, ___”, governed by a rule: “In the past, ___” is true if at some time earlier than the time of utterance, ___. But there is a sense in which this is not an authentic past tense, as I will now explain. There are some types of event that I am powerless to affect: the weather, the tides, the rise of hip-hop, and so on. I am no more able to affect tomorrow’s weather than I am to affect yesterday’s weather. When you and I speak of events in the past, however, there is a sense in which their temporal location alone renders them insusceptible to being affected by us. Past events may be events of types that, in general, I am able to affect. What I have for breakfast is, within limits, something over which I have a lot of control each day. But I cannot now affect what I had for breakfast yesterday. This is not a matter of the event being of a type that I am in general unable to affect. It has to do entirely with the event being past; or, if you prefer, with the event being earlier than the time at which I am attempting to act. No such conception of the past is available to an agent with only the conception of time as phase. Suppose as before that I am able to affect what is for breakfast. Whether there is marmalade for breakfast at 10.00 am is then something that I can affect. Just after breakfast on one morning, I can think, “breakfast is just over”. Early the next morning I will be able to think, “Breakfast is just about to come up”. But the only conception I have of there being marmalade for breakfast is the conception of a state of affairs I am currently able to change. I can think of there being marmalade for breakfast at 10.00 am; but that is the state of affairs I am currently able to change. I can affect whether there is, in general, marmalade for breakfast at 10.00 am. I can’t now affect whether there was marmalade for breakfast yesterday at 10.00 am. But that conception of a particular, unrepeatable time, yesterday at 10.00 am, is just what is not available to an agent who has only the conception of time as phase. You might argue that the agent may have the episodic memory of yesterday’s marmalade at breakfast, a memory of that particular event, which is indeed unchangeable by the agent now. My present point, however, is that the agent does not have the conception of the temporal location of the event as that which makes the event incapable of being affected by the agent now;

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and that point remains even if the agent does have, say, a memory-image of yesterday’s event. However, once this point is grasped, it seems questionable whether the agent could be said to have a truly episodic memory of that individual event, since the agent does not have the conception of the event as having a particular temporal location. It is not just that the agent does not know exactly when the past event occurred. Rather, the agent does not even have the conception of there being a particular time at which the past event occurred. In what sense then does the agent have the conception of this as a particular event at all? So much, for the moment, for the conception of time as phase. As I said, this conception is ubiquitous in animal cognition. It is usually said that only humans have the conception of time as linear. But animal timing is commonplace. And one element in animal’s representation of time is the representation of phase time. This is what I have been trying to characterize.

3. Script time Young children seem able to form and retain information about the temporal structure of observed sequences. For example, there are cases of deferred imitation, in which children can repeat observed action sequences after a delay. McCormack and Hoerl (1999) make the point that the information retained here may be, as they say, generic, rather than relating to the temporal relations between the specific events observed. The information is generic in that it is “information about a temporal structure numerically different event sequences may have and which, therefore, does not distinguish between one event sequence and another.” (McCormack and Hoerl 1999, 158). So there is a sense in which the child has learned from observation of the past events, but what the child has may be generic information, rather than episodic memory of the particular events initially observed, and the temporal relations among them. McCormack and Hoerl describe this type of generic memory as involving the construction of scripts, in something like the sense of Schank and Abelson (1977); cf also Nelson (1986). A script represents what usually happens in a situation in which there is a well-established pattern to what happens, such as going to a seminar, visiting a doctor, or eating in a restaurant. Having a script is having a representation that can, as it were, talk you through the situation. McCormack and Hoerl propose that scripts function as ‘temporal frameworks’ for young children. They provide ways of representing the

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temporal locations of novel events. You may recognize a sequence of events as falling under a script even if it contains some unexpected elements; the script itself provides a framework of times with respect to which the temporal locations of the unexpected elements can be plotted. As McCormack and Hoerl stress, there is a sense in which the framework provided by a script is non-perspectival: scripts do not of themselves locate events with respect to one’s present temporal position. However, here as in the case of phase time, perspectival representations will be needed if the temporal framework is to be put to use in practice. How are we to find the temporal token-reflexives that we will need to express a perspectival representation of time? We might use terms such as “now”, governed by the rule: Script: Any token of “now” refers to the time at which it was produced. The domain of times over which this rule is defined will not, of course, be times drawn from our ordinary range of linearly organised times; they will themselves be times defined in terms of the temporal framework provided by the script. Within each script times are temporally related; but we cannot express temporal relations between times identified in different scripts. In virtue of what do we have our ordinary understanding of temporal indexicals, rather than a more primitive conception? McCormack and Hoerl suggest that the key difference here, the marker of a mature grasp of temporal indexicals, is a certain capacity for temporal decentring, “where decentring means being able to conceive of other times as affording alternative temporal perspectives.” (p. 171). They first describe a simple type of temporal decentring, which is not enough for the ordinary conception of time. This kind of decentring is put to work in understanding any use of the present tense to refer to a time other than the time at which the understanding takes place. This kind of decentring is used by any child who manages to follow a narrative beginning with the words, “Once …” or “One day …”, followed by quotations of present-tense utterances. For example, if I say, “Once, as I stood in line, someone said to me, ‘you’re standing on my foot’”, anyone following the narrative has to interpret the quoted present-tense utterance “You’re standing on my foot” as relating to a time other than the time at which they are hearing the narrative. McCormack and Hoerl suggest that this simple decentring can be described as form of perspective-switching, in that the child who can do this

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may not so much as be able to make sense of the question what temporal relations hold between the various perspectives themselves. At this stage all the child has is the ability to switch from one script-time to another scripttime, and may have no ability to inquire into the temporal relations between the various script times. They say: We wish to suggest that temporal decentring, if it is to be more than the perspective-switching described above, must similarly involve a conception of temporal perspectives as perspectives onto the same temporal reality and an understanding of the systematic relations that obtain between different temporal perspectives in virtue of this fact. Understanding that there can be multiple temporal perspectives onto the same event, for example, involves being able to reason that a current ongoing event will be in the past from the perspective of subsequent days and was in the future from the perspective of previous days. (p. 174) And it is in virtue of our possession of this richer capacity for temporal decentring, they suggest, that we can be said to have the mature conception of time. In the terms I suggested, their proposal is that it is this richer capacity for temporal decentring that constitutes operating with our ordinary understanding of temporal indexicals, rather than the more primitive indexicals I have been describing. It seems to me that there is a role for temporal decentring in an analysis of our ordinary understanding of time, but that this approach does not identify that role correctly. For example, consider an agent who does not have our ordinary conception of time, but does have the conception of phase time, as well as various scripts. It would be possible to have the conception of a single network of phases onto which all scripts could be plotted. For instance, suppose our agent has the conception of the various phases of the day: morning, afternoon, evening and night, and perhaps divides them into early morning, mid-morning, late morning and so on. Such an agent might be unable to distinguish the times of one day from the times of another day; this agent can only talk and think about those phases. In effect, this agent has a single script for the day, onto which all events are plotted. So this agent would not be operating with a range of more or less specialized scripts, just a single general-purpose script for a typical day. Such an agent would be capable of decentring in the simple sense: this agent could decentre to consider what happens in the early morning or in the mid-afternoon,

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for example, and this agent could interpret present-tense utterances considered as made at those times. But our agent could also consider the temporal relations between those various phase-times; our agent could reflect on the relation between early morning and mid-afternoon, for instance, reflecting on the intervening phases of mid- and late-morning, and early afternoon. So this subject would meet the McCormack/Hoerl characterisation of rich decentring, without yet having our ordinary understanding of temporal indexicals. It therefore does not seem that it could be the capacity for rich decentring that explains our ordinary understanding of temporal indexicals.

4. Causation and physical objects It is natural to suspect that what is missing here is an understanding of the relation between grasp of our ordinary conception of time and a grasp of causal concepts. I want to approach this question in terms of the interventionist approach to causation developed by a number of authors (Pearl 2000, Spirtes, Glymour and Scheines 1993, Woodward 2003). The idea here is that for a variable X to be a cause of Y is for intervening on X to be a way of intervening on Y. We can explain this more explicitly by introducing the notion of an ‘intervention’ variable (Woodward and Hitchcock 2003). For I to be an intervention variable for X with respect to Y, it must be that: I causally affects X. I does not causally affect Y otherwise than by affecting X. I is not correlated with any variable Z that is correlated with Y through a route that excludes X. I suspends any other variable from affecting the value of X. Suppose all these conditions are met by the intervention variable. Suppose that X remains correlated with Y under interventions on X. The correlation then cannot be due to some common cause of X and Y; the conditions above exclude that possibility. So it can only mean that there is a causal connection between X and Y. The interventionist says: for X to be a cause of Y is for X and Y to be correlated under interventions on X. Or to put it another way, for X to be a cause of Y is for intervening on X to be a way of intervening on Y. This characterization of causation itself makes free use of the notion of cause. That does not mean that the characterization is trivial. It does not

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merely explain “X causes Y ” in terms of the idea that X causes Y; on the contrary, it does not appeal at all to the idea of a causal connection specifically between X and Y. The fact that the characterization itself uses the notion of cause does, however, mean that there is a question whether it makes explicit all the ideas that are required for an ordinary understanding of causation. In particular, it leaves it open whether an understanding of temporal concepts is required for a grasp of causation; and if so, what kind of temporal concepts. I want to end with some remarks on this. The interventionist account is a counterfactual account; if it were a reductionist counterfactual account in the style of Lewis, it is hard to see how it could avoid using temporal concepts at some point. If we explain “X causes Y ” by saying that there are counterfactual circumstances in which a difference in X would make a difference to Y, we have to say just which kind of circumstances we have in mind. For a Lewis-style account, an appeal to time will be needed to specify just what we have to hold constant and what we are varying. So we might say that we will hold constant everything about the way the world is up until the time at which there is a difference in X. And here we really will need the ordinary notion of time, not merely phase time or script time. Because the interventionist account claims the right to make free use of the notion of cause in characterizing cause, however, the interventionist has another way of saying what is to be held constant and what varied when the value of X changes. That is exactly what is achieved by the four conditions I quoted above. And there is here no explicit appeal to temporal notions. In explaining what it is for an ordinary subject to have grasped these four conditions, it will be natural at some point to appeal to the subject’s own capacity for agency, a tendency in some situations to regard one’s own actions as interventions (Gopnik et. al. 2004). But this kind of approach needs to appeal only to the subject’s sensitivity to the causal relations in which her own actions stand to other phenomena. It does not, on the face of it anyhow, have to appeal to the subject’s grasp of time at all. Suppose, though, that we look at the details of our ordinary understanding of how causal influence is transmitted from place to place by physical objects. It is really basic to our understanding of ourselves as spatially located that what happens at one place can have repercussions for what happens at other places. Places are not causally insulated from one another. But neither are places promiscuously related by causation. It is not as if what happens at any one place affects any arbitrary range of other places. So how is causal influence transmitted from place to place? If you think about the

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surface of a pond, you might think of waves fanning out from one place to another set of locations, and causal influence being transmitted in that way. But there is another way of thinking that we often use. We think that causal influence is transmitted from place to place by the movement of objects. I light an oil heater in the garden and then find it’s cold indoors, so I bring the heater in. Because of the movement of the heater, my intervention, my lighting of the gas at one place, makes a difference to the temperature at another place. We would be baffled by the idea that my lighting the heater outside could have made a difference to the temperature of the room without the heater having been moved, unless of course we appeal to some other story about heat waves or the motion of molecules. The movement of the physical thing, the oil heater, explains how it is that my intervention outside is making a difference to the temperature inside. This implies: (a) there is more to the movement of a physical object than there being correlations under interventions between what goes on at one place and what goes on at another, and (b) this further sameness of physical object at one place with physical object at another place is what explains the existence of those correlations between what goes on at one place and what goes on at the other, under interventions at one of the places. We could sum all this up by saying that there is more to our ordinary concept of causation than the interventionist account allows. In addition to the fact that an intervention on one variable makes a difference to the value of another, we think there is, in many cases, a mechanism by which this happens. Ordinary physical objects are just the very simplest examples of such mechanisms. An intervention at one location can make a difference at another location; and movement of the object from one location to another is the mechanism by which this transmission of influence from one location to another is achieved. It is this dimension of our ordinary understanding of causation that demands the ordinary conception of time. Suppose we have an intelligent agent operating with phase time only, making no distinction between one morning and another. An object may be observed one morning to be G when the previous morning it had been observed to be F. And our agent can recognize a counterfactual dependence of the G-ness on the F-ness of that object. If there had been an intervention on the F-ness of the object, it

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would not be G. As I said, formulating this kind of counterfactual does not seem to require use of temporal ideas. What our agent cannot do is identify the ground of the counterfactual dependence. For the ground of the counterfactual is the persisting object itself, transmitting causal influence from its earlier place to its present place. Recognizing that temporal relation demands that our agent go beyond the confines of phase time, towards something more like our ordinary thinking about time. Consider now an intelligent agent using only script time. In running a number of scripts over a period of time, the agent may find that one and the same object is encountered in the course of many different scripts: a parent, for example, may figure in trips to the doctor, restaurant and so on. And the behavior of that thing in the running of one script may counterfactually depend on what happened in the running of another script. Merely stating the counterfactual dependence, if this is done in interventionist terms, does not require the use of temporal concepts. But recognizing the ground of the counterfactual connection, in the persistence of the object that figures in the running of one script to its figuring in the running of another script, requires recognizing temporal relations that go across scripts. So we have to move away from the series of temporal islands provided by script time, to a more connected conception of time. This is not to say that recognizing the role of persisting physical objects as the mechanisms that transmit causal influence from place to place will demand the full strength of the conception of time as linear. When the introduction of a calendar and clock system makes linearity explicit, this is a bold, simple stroke that clarifies our thinking about time. But the pressure to recognize transitivity and connectedness is already there when we consider the temporal structure that has to be recognized by a self-conscious agent. In general, we believe in the transitivity of the connectedness of the self, in this sense: if an intervention on the self ’s F-ness would have made a difference to its G-ness, and an intervention on the self ’s G-ness would have made a difference to its H-ness, then an intervention on the self ’s Fness would have made a difference to its H-ness. And we think of the self as a concrete object that transmits causal influence from every place remembered to every place remembered or anticipated. This is a special case, albeit a particularly striking one, of the general point that it is our conception of the causal roles of concrete objects that explains the structure of our ordinary understanding of time.

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Acknowledgement This paper benefited from a lively discussion at the Wittgenstein Symposium in 2005; thanks in particular to Jeremy Butterfield and Robin LePoidevin. I am also indebted to Naomi Eilan, Christoph Hoerl and Teresa McCormack for many discussions of these topics.

References Gallistel, C. R 1990 The Organization of Learning. Cambridge, MA: MIT Press. Gopnik, Alison, Clark Glymour, David Sobel, Laura Schulz, Tamar Kushnir and David Danks 2004 “A Theory of Causal Learning in Children: Causal Maps and Bayes Nets”, Psychological Review 111, 1–31. Lewis, David 1973 “Causation”, Journal of Philosophy, 70, 556–67. McCormack, Teresa and Christoph Hoerl 1999 “Memory and Temporal Perspective: the Role of Temporal Frameworks in Memory Development”, Developmental Review, 19, 154–182. Nelson, Katherine 1986 Event Knowledge: Structure and Function in Development. Hillsdale, N. J.: L. Erlbaum Associates. Perry, John 1993 “The Essential Indexical”. In: The Essential Indexical and Other Essays. New York: Oxford University Press. Reichenbach, Hans 1947 Elements of Symbolic Logic. New York: Macmillan. Pearl, Judea 2000 Causation. Cambridge: Cambridge University Press. Schank, Roger C., and Robert P. Abelson 1977 Scripts, Plans, Goals, and Understanding: An Inquiry into Human Knowledge Structures. Hillsdale, NJ: Lawrence Erlbaum. Spirtes, Peter, Clark Glymour and Richard Scheines 1993 Causation, Prediction and Search. New York: Springer-Verlag. Woodward, James 2003 Making Things Happen: A Theory of Causal Explanation. Oxford: Oxford University Press. Woodward, James and Christopher Hitchcock 2003 “Explanatory Generalizations, Part 1: A Counterfactual Account”, Nous 37, 1–24.

How Real Are Future Events? John Perry, Stanford 1. Fatalism My main aim in this talk is to discuss McTaggart’s argument for the unreality of time. I will find a flaw in his argument, but finding the flaw will lead us to the conclusion that although time is real, future events are not. I will begin, however, not with McTaggart, but with a version of an ancient argument for fatalism. By fatalism I do not mean the doctrine that we are fated, like Oedipus, to do something terrible at some point in the future, no matter what choices we make now, and no matter what happens in between. I mean the philosophical doctrine that we can do nothing at all to effect the future in any way. For most of us this will mean we are not only fated, in the first sense, to do terrible things, although perhaps not as terrible as Oedipus was fated to do, but that that even the route to these terrible deeds is not in our power to alter. I adopt as a working principle that we are entitled to the distinctions we need to avoid fatalism. I will see what these are, and then maintain that they show us the way to avoid McTaggart’s argument, and that, in doing that, we will see that although there is no reason to agree with McTaggart that time is unreal, the future is, in fairly clear sense, not real. The fatalistic argument goes like this: 1. The proposition that Hillary Clinton will be inaugurated President in 2009 is either true or false, and not both. 2. If Hillary Clinton will be inaugurated President in 2009, then that proposition is true. 3. Propositions do not change their truth-values. 4. So, if the proposition that Hillary Clinton will be inaugurated President in 2009 is true today, then it was true a year ago. 5. You cannot change the past. 6. So, if something was true a year ago, no one can do anything now, or at any time later than now, that will affect its truth-value. 7. So, if Hillary Clinton will be inaugurated President in 2009, there is nothing that she can do or Bill Clinton can do or Jeb Bush can do toF. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 13–30.

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day, or anything that anyone else can do today, or any day in the future, to prevent it. 8. As important as Hillary Clinton is, and as important as the issue of her inauguration is, there is nothing in principle that makes this event special in terms of this argument. 9. So, if something is going to happen in the future, there is nothing anyone can do now or at any time later than now to prevent it. The main problem with this argument, it seems to me, is that it does not recognize that many propositions, certainly including propositions about who wins elections and who is then inaugurated as president the following January, are made true or false by events, in this case, crucially, the elections that are held in November preceding the inauguration, and the events between then and the next January 20th, which may include Supreme Court decisions and the like. If it turns out that Hillary Clinton wins the 2008 U.S. presidential election and is then inaugurated in January 2009, this will be on account of events many of which will not happen until Election Day, the first Tuesday in November, 2008. Suppose that Hillary and Bill Clinton have a big fight in late September 2008. It would be pleasant to provide details concerning what this fight might be about, but I will leave that to the audience’s imagination. Feeling hurt and unappreciated, it occurs to Bill that he could make a speech in which he details every nasty thing Hillary has ever done and said, and asserts that he wouldn’t vote for her for dogcatcher, much less for President, and that by doing so, he could prevent her election. He can prevent her from being inaugurated. Remembering all the good times he had in the White House, he doesn’t do it, and she is elected and inaugurated. Still, if he had done it, he would have been affecting the future, not the past. He wouldn’t have changed the past at all. He will not have changed the past, for he will not do anything that makes false something that has already been made true, nor does he do anything that makes true something that has already been made false. We need to think about the truth of propositions in some way that at least does not rule out this common sense response to the fatalist argument. Even if we are hard determinists, that is, even if we believe that determinism is true and it rules out freedom, we should not be convinced by fatalism, for if fatalism is true, determinism is really quite beside the point. It seems to me that to allow this common sense response, we need to rec-

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ognize two concepts of truth in relation to propositions. The first is a property of propositions: a proposition be true or be false, or, if you prefer, be true or false of a world. It’s best to avoid tense in using this concept, hence the untensed “be”. Second, events up to a certain time make a proposition true, or make it false, or leave it’s truth or falsity open. There is an important connection between these concepts: If events up to a certain time make a proposition true (or false), then it be true (or be false). The converse principle does not hold. There are, or at least seem to be, propositions that are not made true by events, such as the propositions of logic and mathematics and other necessary truths. If we believe that the laws of nature are contingent, and also believe that they are not merely the empirical generalizations that remain true at the end of time, but somehow structural principles that shape what happens, then they too will not be made true by events; events will conform to them, but not make them true. From the connection, that propositions made true, be true, and what we know about being true, we know that if events up to a certain time make a proposition true (or false), events up to some other time, earlier or later, don’t make it false (or true). From this, I believe, we arrive at the correct understanding of the claim that you can’t change the past: You can’t do anything to make a proposition false that has already been made true, or to make a proposition true that has already been made false. However, the following principle is not correct: If proposition P be true, no one can do anything now that will affect its truthvalue. This is not correct, because the events that will make P true may lie in the future, and someone powerful, like Bill Clinton, may well be able to do something to prevent them. Stated carefully, the first steps of the fatalistic argument become: The proposition that Hillary Clinton is inaugurated President in 2009 be true, or be false, but not both.

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If Hillary Clinton will be inaugurated President in 2009, then that proposition be true. But from this, nothing too significant follows. From the supposition that the proposition be true, it does not follow that it has been made true by today, or yesterday, or will have been made true by, say, late September 2008. So it doesn’t follow that, if Bill Clinton were to prevent it from being made true, by a nasty speech in late September 2008, he would have in any sense changed the past. For discussions of determinism, a third concept is important. If a proposition is entailed by propositions that have been made true by what has happened by a certain time, together with the laws of nature, and the laws of nature are propositions that be true without having to be made true by events, then I will say that the proposition is settled by the time in question, although it has not yet been made true. So, if determinism is true, it may be settled by late September 2008, and indeed may have been settled by the time that Adam bit into the apple, that Hillary Clinton would be inaugurated as President in 2009. The question of the compatibility of human freedom and determinism is whether we can, at a given time, do things to prevent events whose occurrence is already settled. But that issue, I leave aside; I mention it only to emphasize that being settled, in this sense, is one thing, being made true is another. Let’s consider a manageable series of events, say the presidential inaugurations in the United States of the 20th century, from McKinley’s in 1901 to George Bush’s in 2001.1 Consider the domain of all the people who appeared on the ballot for U.S. president in the 20th century. Consider the property, being inaugurated as President of the U.S in D, where D is a year. This domain and property give us a set of atomic propositions. Some of these, like That Carter be inaugurated President in 1976 be true of the sequence, and others, like 1

Through 1933 inaugurations were held on March 4; since then they have been held on January 20. Vice-Presidents Theodore Roosevelt, Calvin Coolidge, Harry Truman, Lyndon Johnson and Gerald Ford all became president without being inaugurated, through the death or resignation of a president. Roosevelt, Coolidge, Truman and Johnson were subsequently elected to full terms and inaugurated.

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That Carter be inaugurated President in 1980 be false of the sequence. Each atomic proposition was made true, or was made false, by the events that occurred in the twentieth century up through March 4 of the year that is a constituent of the proposition. So, as of, say, April of 1975, a number of propositions that be true of the sequence were not yet made true or made false. As of April 1975, it had not been made true that Bush was inaugurated in 2001, nor had it been made false that Gore was inaugurated in 2001.

2. Chronological possibility Now I’d like to introduce a second class of propositions, of the form: X in D can prevent P where X is a candidate from our domain, D is a year in the twentieth century, and P is one of the original atomic propositions. Examples are: Carter in 1980 can prevent that Reagan be inaugurated 1981 This would seem to be true if in 1980 there was some set of basic bodily movements that Carter could have made (or refrained from making), which, had he made them (or refrained from making them), other circumstances being what they were, this would have led to his winning the election of November 1980, rather than Reagan. Many analysts think that had Carter been less obsessed about the hostage situation in Iran, and not tried to micromanage the issue, he would have defeated Reagan. If they are correct, this proposition is true. It is almost certainly true That Reagan in 1980 can prevent that Reagan be inaugurated 1981. Reagan could have withdrawn from the race. Or, just to put some more intriguing possibility before us, he could have taken off all of his clothes in the middle of a speech, proclaimed that he was at a nudist, and would never wear another stitch, even if elected President. Or he could have divorced his wife Nancy, and married a young Hollywood starlet. I could go on, but perhaps that suffices for me to make the point.

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At each time there are propositions that have been made true, and propositions that have been made false, and, in addition, many propositions that have not been made true or made false. There is a certain kind of impossibility involved with respect to propositions that have already been made true (or false) by a certain time. There is nothing anyone can do, and for that matter, nothing that can happen, whether done by a person or simply the result of non-human causes, that will make those propositions false (or true). The kind of possibility involved is not the same as logical possibility; that is, the falsity (or truth) of these propositions does not involve a contradiction. Nor is it a matter of pure metaphysical impossibility; the propositions involved are not guaranteed to be true by whatever deep structural facts there are about the properties, relations and objects involved. I’ll call this sort of impossibility and possibility, the sort of possibility and impossibility appealed to, but misused, in the fatalist argument, chronological. Propositions that, at a given time, might still be made true by events are chronologically possible at that time, those that can no longer be made true are chronologically impossible at that time.2

3. McTaggart’s B-series and C-series That said, I turn to John Ellis McTaggart and his famous argument, of almost one hundred years ago, that time is unreal. He describes three series of events, which he calls the C-series, the B-series and the A-series. We’ll put the A-series aside for the moment, and discuss the C-series and the Bseries. The C-series and the B-series both comprise all the events in history in order. They differ in that the B-series also includes the temporal direction of the events. Suppose that in the course of archival investigations you came across a list of leaders of some small country of which you had never heard, the Land of Woe. It looks like this: Year 100: the reign of Elwood the Unready begins Year 110: the reign of Gretchen the Inept begins Year 120: the reign of Ephraim the Ignorant begins

2

J. Ellis McTaggart, “The Unreality of Time”, Mind, N.S. 68, October 1908, pp. 457–474.

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You don’t know whether the years listed are B.C. or A.D., however. Thus you don’t know whether Elwood came before or after Gretchen. You know that Elwood’s reign was next to Gretchen’s, and not next to Ephraim’s, but you don’t know which came first. You have order, but no direction. Basically, if you can say which events came between which events, you know the order; if you can say which came first, you know the direction. McTaggart believed that both order and direction are part of our concept of time, and that the direction of time is the direction of change. So the Cseries, plus the direction of change, should give us the B-series. McTaggart can come up with no coherent account of the direction of change, so he concluded that time is unreal. Let us take a miniature B-series, a partial list of the history of the world, to have something manageable to think about. Let us limit ourselves again to events that are inaugurations of United States Presidents. This is, I realize, in many ways an extremely unpleasant subject to continue to think about, but I will plunge ahead nevertheless. I considered using the inaugurations of governors of California as an example for this talk; that would have provided a pleasant bond between Austria and California, since California’s present governor is the beloved Austrian Arnold Schwarzenegger. However, there is a great deal of jealousy among Americans from other states, some of whom are at the conference, because California has a European governor and no other state does. So I decided it was best to stick to the example of presidents. Since you are required to be 35 years old to be inaugurated as President, we can be sure that all of the presidents inaugurated up until 2037 are already alive. So let us limit ourselves to the succession of inaugurations that begins with Franklin Delano Roosevelt’s second inauguration in 1937, continues through George W. Bush’s second inauguration last January, and then continues with Hillary Clinton’s two inaugurations in 2009 and 2013, Laura Bush’s two inaugurations in 2017 and 2021, Chelsea Clinton’s two inaugurations in 2025 and 2029, Jenna Bush’s inauguration in 2033 and her twin sister Barbara’s in 2037. I’m not entirely certain of these results, of course, but that’s my best guess, and I am fairly confident. So: 100 years of inaugural events and 100 propositions we’ll take to be of the form: That X be inaugurated succeeding Y starting with

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That FDR be inaugurated succeeding himself and continuing to That Barbara Bush be inaugurated succeeding Jenna Bush. Let’s give this series of propositions another name, say, “Dismal”. If we think of Dismal as an ordered sequence of propositions then of course it cannot be changed; it is a set theoretical object, defined by its members. If all of the propositions in Dismal are true, then Dismal is a sequence of propositions that corresponds to a part of the B-series of events. I’ll call such a series a “B-P series”. Now, even if Dismal be a B-P series, and even if we are all powerless to do change the membership of Dismal, still someone can do something that will make it not a B-P series, by doing something that will prevent one of the propositions in it from being made true, so that the event, to which this proposition would have corresponded, does not occur. For example, Jenna and Barbara Bush might become nuns, in which case they will be prevented by Papal decree from taking part in electoral politics. That is not likely, but it could happen. Even if Dismal be a BP series, which requires that all the propositions in it be true, some of the events necessary to make it a B-P series have not yet happened. We, or at least powerful politicians, and their children, can do things that effect which propositions having to do with future elections; all we cannot do is affect the truth-values of the propositions that have already been made true or false. Still, we will assume that Dismal is a B-P series and plunge on.

4. The A-series and the D-series The first thing we need to do is to pull out of Dismal the information about the direction of time. This gives us a C-P series, so I will call it DismalC. This is a series of propositions of the form X be inaugurated next to Y. This series contains information about which inaugurations were between which other inaugurations, but not about which came earlier and later. Now, how do we put back in what we have just taken out? To get back to a B-

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series, to get from DismalC back to Dismal, we have to add information that imposes a direction on DismalC. McTaggart tells us three things about what we need to add. First, it is related to temporal change, which he takes to be something we experience — or rather seem to experience, for, according to him, it turns out to be an illusion. Second, temporal change is fundamentally different from the sort of change we talk about when we say, for example, of a stretch of Hiway 4 in the Sierra Nevada Mountains, “For each town from Angel’s Camp to Ebbett’s Pass, the elevation above sea-level increases as the distance from the Pacific Ocean increases”. I am inclined to agree with McTaggart about both of these things, although I do not think our experience of change is an illusion. The third thing McTaggart tells us is that what we need to add to the Cseries, to add direction, is what he calls the A-series, which is the series of events ordered by whether they are in the present, past or future, and if one of the latter two, how distant. The fact that Truman’s inauguration came after Roosevelt’s 4th inauguration, rather than merely being next to it, and before Eisenhower’s first, rather than just being next to it, consists in the fact that Roosevelt’s fourth inauguration is more distantly past than Truman’s, and Truman’s more distantly past than Eisenhower’s. This idea of McTaggart’s seems to me to have been rather unfortunate, a wrong turn in the philosophy of time. I will return to what is wrong with it later, but for now I will try to give what seems to me the correct solution. I believe that what we need to add to DismalC, in order to return to Dismal, is facts about chronological possibilities, which I will call the “D-P series”, or, in this case DismalD. That is, we need to add propositions about what the chronological possibilities were at the time of Roosevelt’s third and fourth inaugurations, Truman’s inauguration, and so forth, right down through the chronological possibilities at the time of Bush’s first inauguration, Bush’s second inauguration, Hillary Clinton’s first inauguration, and beyond. I am allowing myself the concept of simultaneity in time, for as I understand McTaggart, this allowed in the C-series. Events X and Y occur at the same time if there is no event between them, and we can have this information, even if we do not have the information about the direction of events. So we can add DismalD to DismalC, giving us a sequence of propositions about the occurrence of inaugurations and the chronological possibilities and impossibilities at the times those inaugurations occurred.

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During his third term Roosevelt’s vice-president was Henry Wallace, a Democrat from the left wing of the party, perhaps correctly called, as he was by many Republicans, a socialist. Had Roosevelt chosen Wallace for his running mate for the 1944 election, as he certainly could have, then it seems that either of two things would have happened. One is that Thomas Dewey, the Republican candidate in 1944, would have defeated Roosevelt. The other is that Roosevelt would have won, and Wallace rather than Truman would have succeeded to the Presidency in 1945 when Roosevelt died. If either of these courses of events had occurred, then it is virtually certain that Truman would never have been President, for he was a amiable Missouri Senator with no presidential ambitions. So, it seems that at the same time as Roosevelt was inaugurated for his third term in 1941, it was chronologically possible that Truman not be inaugurated in 1949, as he in fact was, having served out the rest of Roosevelt’s fourth term and having defeated Dewey in 1948. However, by the time of Eisenhower’s first inauguration in 1953, it was no longer chronologically possible that Truman not be inaugurated in 1949. When we add true propositions about chronological possibilities for other inaugurations to DismalC, a clear pattern emerges, that gives us two clear directions, one of which is the direction of temporal change. As we move one direction through augmented DismalC, possibilities will increase, and as we move another direction, possibilities will decrease. The possibility that Gore be inaugurated in 2001 is present at Roosevelt’s second inauguration in 1937, and at every subsequent inauguration through Clinton’s second inauguration in 1997. Then it disappears, sad to say. The direction of time, and of change, is the direction of decreasing possibilities. If we confine ourselves to any finite subset of the contingent propositions that be true, more and more will be made true, and more and more rendered impossible, as we consider events that are later in time, and fewer and fewer will have been made true, and fewer and fewer rendered impossible, as we consider events that are earlier in time.

5. Change and the D-series I now turn to the question of whether the D-series meets McTaggart’s requirements for what needs to be added to a C-series to get a B-series, namely, whether it is something that characterizes our experience of temporal change, and is not merely a matter of change along a dimension.

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Angels Camp, Murphys, and Arnold are three towns one drives through, as one travels east along California hiway 4, which runs from the Bay Area through the Sierras to Nevada. Consider the facts about the elevations above sea level of these three towns, roughly 1500 feet, 2500 feet, and 3500 feet. One experiences this change in elevation of towns along hiway 4, relative to the distance from the Pacific Ocean, as one travels along the hiway. Each of these facts, by itself, does not necessitate the others; Angels Camp could be at 1500 feet, even if Murphys were not at 2500 feet, and vice versa, for example. On the other hand, if we travel along hiway 4 from west to east, then by the time we get to Murphy’s we have eliminated the possibility that Angels Camp has any elevation other than 1500 feet, and that Murphy’s has any elevation other than 2500 feet, but not have eliminated the possibility that Arnold has some elevation other than 3500 feet. So there is a progression, a change, in what is possible, as we travel from east to west. This is, however, much different from chronological possibility. For one thing, the change in what is possible, in this case, has to do with what the subject knows; it is a matter of epistemic possibility. Related to this, the situation is reversible; if you travel from east to west, then when you get to Murphy’s it will not be possible that Arnold has any elevation other than 3500 feet, but it will be possible that Angels Camp has some elevation other than 1500 feet. Chronological possibility is not simply a matter of epistemic possibility, and it is not reversible in this way. Let’s return to our friend Bill Clinton, late September 2008, contemplating whether to make a speech that will destroy Hillary Clinton’s chances of being inaugurated the following January. He might think that it is possible for him to destroy her chances; it is something that he can do; it is a possibility for him; it is up to him. On the other hand, at that point in time, it is not possible for him to destroy her chances for being nominated, which will have happened in Summer of 2008. That was once a possibility, say in the early days of the Democratic Convention and before, but it is no longer a possibility by early September 2008. In other words, Clinton, is aware of chronological possibilities, and chronological impossibilities, as all of us are. As we experience change, we experience the decrease in chronological possibilities, and the increase in chronological impossibilities. Opportunities, once missed, are gone. The direction in which the chronological impossibilities increase and the chronological possibilities decrease is the direction of change as we experience it.

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As we experience change, it involves events occurring, which we perceive and participate in, which make it the case that various propositions are true or false, and leave no possibility of our doing anything, or anything happening, that will make them otherwise. Past events and future events play different roles in our lives, cognitive, emotional, and practical, all connected to our sense of chronological possibility. We regret the past, and try to make up for it, or cover it up, or change its consequences. But we do not try to change the past itself. We try to shape the future, to make some events more likely to occur, and others less likely to occur. We remember the past; we anticipate the future. Everyone seems to agree about this. People move from west to east, and east to west, and as they do they remember where they have been, and anticipate the experience of what they have yet to encounter. But all are anticipating the future, and remembering the past. No one moves from future to past, trying to affect the past, and remembering the future. So, I conclude, chronological change meets McTaggart’s requirements, and that by augmenting the DismalC with DismalD we have returned to our starting point, Dismal, a B-P series, and done what McTaggart thought could not be done. But you won’t be convinced yet, not until we have discussed the A-series, and some other issues about the B-series.

6. McTaggart and the A-series McTaggart thought that you had to add the A-series, the one that divides events into past, present, and future, to the C-series to get the B-series. He did not think that there was a coherent way to do this, and so concluded that time was unreal; that is, we have a clear concept of what time would have to be, something characterized by the combined C-series and A-series, but we see clearly that there can be no such thing. So according to McTaggart, the fact that Truman’s inauguration came after Roosevelt’s 4th inauguration, rather than before it, or merely being next to it, consists in the fact that Roosevelt’s 4th inauguration is further in the past than Truman’s. I cannot see much merit in McTaggart’s idea. Suppose I say now, truthfully if sadly, “George W. Bush’s 2nd inauguration is past”. What fact makes that true? A pretty reasonable theory says that my utterance, call it u, is true simply because George Bush’s 2nd inauguration precedes u, the utterance itself. So, if we want to put the fact that I seem to be getting at when I say

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“George Bush’s 2nd inaugural is past”, into DismalC, we ought to add the proposition to that George Bush’s 2nd inauguration precedes u. But that proposition is already a B-P series proposition. If we are going to add Bseries proposition to DismalC in order to get to Dismal, we may as well simply jump back to Dismal and be done with it. I should point out that one can agree with this, even if one thinks, as I do, that David Kaplan’s theory is correct, in saying the proposition that George Bush’s 2nd inauguration precedes u is not the proposition expressed by my utterance. According to his theory, the proposition expressed by my utterance of “George Bush’s inauguration is past”, is not a proposition about my utterance itself, but (roughly) a proposition about the time t, the time at which my utterance occurs, to the effect that George Bush’s inauguration precedes t. This proposition would be true, even if I had not made the utterance u. If we trace Kaplan’s analysis through, we see that my utterance will be true iff George Bush’s election precedes it, for only under those conditions will it express a proposition that is true. Moreover, the expressed proposition, that George Bush’s inauguration precedes t, is also a B-proposition, not a C-proposition. I don’t mean to imply that McTaggart somehow missed the fact that adding these propositions to the C-series would be begging the question. He did not think that the words “past”, “present”, and “future” were basically getting at relations to the utterances of sentences containing them, but that they get at properties of events; indeed, he thought that the passage of events from future to present, and present to past, was the essence of temporal change. He thought that to find real temporal change, we needed to find some temporal way in which the B-series changes, that corresponds to our subjective sense of events passing from the future, into the present, and then into the past. But, McTaggart thought, there can be no change in the B-series. The B-series contains all the events, and their temporal relations, which never change — as he puts it, If M is ever earlier than N, it is always earlier. So, our concept of time demands that the B-series is the A-series added to the C-series; but then the B-series will have to involve events changing from future to present to past; but this makes no sense. So we need to consider whether we have evaded this conundrum. Does

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my account of what we need to add to the C-series to get a B-series imply a sort of change in the B-series that makes no sense?

7. The future is unreal As I just said, early in his discussion, McTaggart says, If M is ever earlier than N, it is always earlier. It seems to me this is a mistake. Consider Bush’s 2nd inauguration. It has happened, while Hillary Clinton’s first inauguration has not happened yet. Is it correct to say that Bush’s 2nd inauguration is earlier than Hillary Clinton’s first inauguration? Certainly after Hillary Clinton’s first inauguration we can say that Bush’s inauguration was earlier than it. But was it earlier, before Hillary’s first inauguration happened? Is it now earlier than her inauguration? It seems to me that Hillary Clinton’s inauguration is not now later than Bush’s second inauguration, because Hillary Clinton’s inauguration does not yet exist. By Hillary Clinton’s inauguration I mean the concrete event, not a description of it, or an abstract object that characterizes it. The concrete event, it seems to me, has no reality at all until it happens, even if the propositions that say that it will happen be true, or for that matter, even if it is settled, in the sense I adumbrated in the brief discussion of determinism early on, before it happens. The concrete event has no existence, no reality, until it happens. My argument for this is that the status of the events that will occur, before they occur, is the same as the status of the events that might occur in their stead, but will not. All are possibilities, not realities, before one of them occurs; it becomes real, the others do not. According to the picture I have put forward, at each time there is a future which is characterized by a number of contingent possibilities or propositions, none of which have been made true at that time, and all of which might still be made true, but only one of which will be made factual: a garden of forking paths, as Borges puts it. Then, at a later time, some of those possibilities will be eliminated or made false. Now the simplest explanation, for the legitimacy of this picture, as far as I can see, is that at each time all of the possibilities for the future have the same status: they are mere possibilities. That is, until an event happens (and so makes the proposition

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according to which it happens, true), the event is a merely possible event, and not a real event. And by saying it is merely a possible event, I mean to say, basically, that it is not an event at all; there are descriptions, and abstract types, that will denote or characterize the event once it exists to be denoted or characterized. But there is no concrete event. Consider, for example, the two events, Hillary Clinton’s inauguration in 2009 and Joseph Biden’s inauguration in 2009. Those are both, as I speak, possibilities, as these seem to be the two leading contenders for the Democratic nomination. We assume however that Clinton will be inaugurated, and Biden will not be. If so, there is not now, and never will be, a concrete event correctly called “Biden’s inauguration in 2009”. The lack of such a concrete event does not impede us in any way in describing the possibilities that there are at the present time, including the possibility that Biden wins and is inaugurated. Now if Biden were to win the nomination, and the election, and be inaugurated, we would definitely feel the need to recognize a concrete event. The event in question would be visible, televised, and consequential. It would have effects, and these only concrete events can have. But we won’t miss the concrete event of his inauguration if he does not win. By the same token, we will not need the concrete event of Hillary Clinton’s inauguration until she wins the nomination and election and is inaugurated. Until then descriptions of it and abstract types that characterize it will serve all of our needs in language and thought. I conclude that future events are not real; they do not exist, until they occur. Until that time they, the concrete events themselves, are not after the other events occur before them. As Broad put it, Let us take McTaggart’s example of the death of Queen Anne, as an event which is supposed to combine the incompatible characteristics of pastness, presentness, and futurity. In the first place, we may say at once that, on our view, futurity is not and never has been literally a characteristic of the event which is characterised as the death of Queen Anne. Before Anne died there was no such event as Anne’s death, and “nothing” can have no characteristics. After Anne died the sum total of existent reality does contain Anne’s death, but this even then has the characteristic of pastness. No doubt I can say “Anne’s death was future to William III”. But I simply mean that, so long as William III was alive, there was no event characterised as the death of Anne; and that afterwards, as the sum total of exist-

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ence increased by becoming, it contained both the events of William’s life and the event of Anne’s death. Anne’s death succeeded William’s life so soon as Anne’s death existed at all, and it succeeds it henceforth for ever; but it does not succeed it while William was alive, because it had not become, it was not anything, and therefore could not have any characteristics or stand in any relations …”.3 And that means, I think, that as I speak, there are no concrete events at all that are referred to or denoted by the two descriptions “Hillary Clinton’s inauguration in 2009” and “Joseph Biden’s inauguration in 2009”. There are two event descriptions, or event types; that is, two linguistic objects and two abstract objects. Some day, if things turn out the way I predicted, one of these linguistic objects will refer or describe a concrete event, which will be of the type in question. So, as confident I am of her victory and inauguration, at the present time, still, there is, in the strict and metaphysical sense, no event that fits the description “Hillary Clinton’s 2009 inauguration” any more than there is an event that fits the description “Joseph Biden’s 2009 inauguration”. Both are chronologically possible, which is to say that it hasn’t been ruled as of 2005, when I write, that events that will meet these descriptions occur in 2009.

8. Does the B-series change? But does not this mean that the B-series changes? How can the B-series change, if all of the changes are in the B-series? If the B-series is an actual succession of concrete events, it does change; it grows, and we see it do so, when we observe change. If this is what we mean by “the B-series” then it is not an abstract set-theoretical object, but an actual process, the all-inclusive process, of things happening. If by “the B-series” we a set theorietical sequence of concrete events, then it seems to me our idea is incoherent. Right now, in 2005, lots of events that will happen have not yet happened. For example, Hillary Clinton’s inauguration has not yet happened. We can talk about such events using bits of language like “Hilary Clinton’s inauguration” or we can model them with suitably constructed abstract objects to serve as proxies. We can use these 3

C. D. Broad, Scientific Thought, London: Routledge and Kegan Paul, 1923, pp. 79–80.

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bits of language or abstract objects to build (or imagine building) models of various possible candidate B-series, and we can describe one of them as the unique one that contains a description of or a proxy for all that ever will occur. But the B-series of events itself doesn’t exists until all its constituents exist, and they don’t all exist yet. If by “the B-series” we mean the sequence of descriptions or abstract proxies that we can use to model events, or a set-theoretical sequence of propositions of the sort we have considered earlier, that will uniquely fit what happens as long as things keep happening, then it is fair to say the B-series does not change. But its status may change. At the present time, when the results of the 2008 election are still slightly in doubt, there are a number of such unchanging abstract objects, sequences of propositions say, that are possible B-series: they fit everything that has happened so far. Some of them contain the proposition that in 2009 Hillary Clinton is inaugurated, some of them contain the proposition that Joseph Biden is inaugurated, and I suppose still others contain the propositions that Condoleezza Rice, or Jeb Bush, or John McCain, or even, given the possibility of a constitutional amendment, Arnold Schwarzenegger, is inaugurated. None of these sequences will ever contain different propositions than it does, but most of them will change, as time passes, in losing the status of possible B-series. In concrete terms, our experience of change, and of the passage of time, is the experience new events coming into existence, and old possibilities being eliminated. In abstract terms, it is seeing propositions being made true and false, and sequences containing those propositions ceasing to be possible B-series.

Conclusion I have argued that to avoid fatalism we need to recognize a distinction between propositions that have been made true as of a given time, and propositions that be true, in the sense that they have been or will be made true. The “be” in the expression of the second concept is tenseless; it is the way we ought to talk about what goes on in “possible worlds”, including our own, when we are not describing them from a time within them, or abstracting from that time. The principle appealed to in the fatalistic argument I considered is properly couched with the first concept: we cannot now, or at any time in the future, make a proposition true that has already been made false,

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or make a proposition false that has already been made true. But this principle does not lead to fatalism. Given this distinction, it is natural to recognize chronological possibility as the sort of possibility that is appealed to in the principle just stated. Chronological possibilities change with time; they are eliminated as one among a number of possible events that could have happened, does happen. If we characterize the change that is peculiar to time, what is sometimes called “temporal becoming”, in terms of this elimination of chronological possibilities, rather than in terms of McTaggart’s “A-series”, then we can, I claim, answer the challenge that is at the heart of his argument for the Unreality of Time, and explain what must be added to the “C-series” to get to the “Bseries”. The A-series, for all the ink that has been spilled discussing it, is a bit of a red herring. Now a couple of final words. First, insofar as I have made any progress here it is due to conversations with Thomas Hofweber; we hope someday to publish a paper on the important differences between events, representations of them, and models of them which will explain and put in their proper places all of the many insights that have accumulated in discussions of McTaggart’s argument. But I have probably not gotten these matters quite clear, or quite straight, in this paper. Second, I was privileged to have Nathan Oaklander as the chair of the session at which this paper was presented. Oaklander holds a version of the B-theory, the theory I have believed to be correct most of my philosophical life, although I have never felt particularly comfortable with it. In conversations in person and by email Oaklander has helped me express my views more clearly, and tugged me a bit towards my old view. I am very thankful to him for his help, and for his fine writings on time. Although I do believe the view I have developed here to be correct, or at least on the right track, as I write, I am very conscious of how difficult the philosophy of time is. Even my hero Broad, Oaklander tells me, held five different theories of time over his career. This is my second. I am not all that confident it will be the last.

Memory and the A-Series Robin Le Poidevin, Leeds 1. The epistemology of episodic memory I am trying to find a book I have mislaid. It is not in its usual place on the bookshelf. It is not on the bedside table, nor on my desk. Frustrated, I set about making a pot of tea. Suddenly, an image comes to mind: I picture myself handing the book to a friend, recommending it as a good read. ‘No hurry about giving it back’, I hear myself saying in my mind’s ear, ‘I shan’t need it for some time’. I remember, then, lending the book to a friend. But I do not merely remember that I lent it to them. I remember the experience of doing so, what the book looked like, the expression on my friend’s face, the sound of my foolish utterance. What we have here is an example of what psychologists since Tulving have called episodic memory: remembering an event ‘from the inside’, recalling the experience itself (see e.g. Tulving 1983). It is often distinguished from remembering that, or semantic memory, for example remembering that Schliemann’s discovery of what he took to be the mask of Agamemnon occurred in 1876. What is required for an episodic memory of an event, but not for a semantic memory concerning it, is that the event in question was experienced by the rememberer. Of course, the semantic memory will have arisen from some relevant experience, but it does not need to have been a perceptual experience of the event itself, as opposed to hearing or reading a report of it. Of these two kinds of memory, it is episodic memory that seems to link us most intimately with the past. We can, in fact, characterise the difference between episodic and semantic memory as that between knowledge by acquaintance and knowledge by description, where the object of knowledge is the past. It is tempting to describe episodic memory as a re-experiencing of the past. That description should not be taken too literally, but it is a feature of episodic memory that it is not just a representation of a past event; it is also a representation of the experience of that event. So here is my proposed definition: an episodic memory of an event (i) arises from an original experience of that event and (ii) includes that experience (or a representation F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 31–42.

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of that experience) as part of its content. In what follows, I am not going to pay much attention to the second component of that definition, concentrating rather on the part of the content of the memory that concerns the event itself. The question I want to ask is this: does the link between episodic memory and the past tell us anything about the nature of time? I think it does. This may seem surprising, but what I want to argue is that what one might call the epistemology of time interacts in interesting ways with the metaphysics of time. The argument starts with an epistemological principle concerning episodic memory. It is this: a truly episodic memory cannot acquire a greater degree of closeness to the truth than the original experience. The memory is accurate only to the extent that the original experience was accurate. Call this principle (a). ( Just to clarify: I am treating experience and memory here as items with propositional content, and so as vehicles for truth and falsity, not simply a collection of sensations.) Now we might think it would be quite easy to come up with counterexamples to this principle. Suppose, walking in the park one day, you are an unseen witness to what is evidently a very emotionally charged meeting between two people. You recognise these two people as friends of yours: a married couple. Seeing evidence of intense affection between them, but also what seems to be anxiety, you do not announce yourself, but walk quietly away. Later that day you learn, quite fortuitously, that the woman you thought you had seen in the park is in fact in some remote location, visiting an aunt. In an instant you recognise the significance of what you saw: not a meeting between husband and wife, but … an affair! You now see the scene with different eyes (or different mind’s eyes). You see the woman, but not as the woman you know. Perhaps you now see her as younger, taller, with darker hair. Is this not still an episodic memory? The experience is representing itself, but it is now a reinterpreted experience. The memory has become closer to the truth than the original experience. This is an entirely coherent story, but it does not, I think, impugn the epistemological principle. The additional information turns the memory into a rather different kind of memory, no longer a purely episodic memory of the original experience. An honest report of the matter would be this: ‘I recall seeing two people, and thinking at the time that they were a married couple of my acquaintance. I now know that one of them was not the person I took her to be’. The genuinely episodic element of the memory is not changed by the later information, only its significance is. What I am pro-

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posing is that the epistemological principle is less a substantive fact about episodic memory than something that follows from our definition of it. The reason why episodic memories cannot be more veridical than the original experience, I suggest, is that the content of the episodic component of a memory is determined by the memory’s link to the original experience, and in addition that the well-groundedness of the memory derives from that of the original experience. Call this principle (b). The representational content, and hence closeness to the truth, of that component is what it is by virtue of the content of the experience, and contains nothing that was not part of the content of the experience. (Of course, given that memories fade, the representational content of the memory may be somewhat less than the content of the experience. But it cannot be more.) So (b) entails (a). Once (a) is made immune from counterexample in this way, we can continue to employ the episodic/semantic distinction in the face of warnings from psychologists that the distinction is largely conventional. Here, for instance, is the psychologist Alan Baddeley’s verdict: An example of semantic memory might be knowing the chemical symbol for salt, while episodic memory would be exemplified by remembering a personally experienced event, such as meeting a retired sea captain while on holiday. There clearly are differences between these two situations, but it is questionable whether a distinction based on anything as subjective and phenomenological as personal reference is either viable or appropriate. Since all memory is surely based ultimately on personal experience, it is hard to see what is gained by assuming different memory stores depending on whether the personal reference is or is not recalled. An alternative way of conceptualizing the difference between remembering personal incidents and recalling information is in terms of the degree of abstraction involved … long term memory has a strong abstractive component; we tend to minimize memory load by stripping away inessential details and encoding new material in terms of existing schemata, keeping only enough to allow us to reconstruct the event if recall is required … It is … not only the case that semantic memory is built from personal experience by a process of abstraction; it is also the case that what appears to be a direct record of personal experience is itself a reconstruction based on an abstraction … Hence, although Tulving’s distinction between episodic and semantic memory may provide a useful reminder of the range of semantic memory, it remains very doubtful whether this reflects a clear

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dichotomy between separate storage systems, as Tulving suggests. (Baddeley 1976, 317–18.) What these remarks show is that, for psychologists, the issue is about encoding mechanisms. But philosophers, who tend to operate at higher levels of abstraction, can remain agnostic about this. What I have suggested is that, insofar as we are concerned with content and causal history, the distinction remains entirely viable and appropriate. The episodic element of a memory is simply defined as that to which principles (a) and (b) apply. Once (b) is in place, a further thesis emerges, one about truth-makers: whatever in reality made true, or veridical, the original experience, corresponds in some way to whatever makes true, or veridical, the later memory (principle (c)). The simplest picture represents experience and memory as having exactly the same truth-maker. The memory, however, unlike the original experience, represents the event as past. This is not adding to the content of the memory in such a way as to come into conflict with principle (a), for if the purpose of memory is to preserve information about the past, then it can only do so by truly reflecting the fact that time has passed since the event took place. The passage from ‘this is now happening’ to ‘this happened’ is therefore required if the memory is to be as close to the truth as the original experience. What this requires in turn is that there be, at the least, a necessary connection between the truth-makers of the earlier and later mental state. Only if there is this necessary connection can the memory be said to be as well-grounded as the experience. So we now have a number of connected principles characteristic of episodic memory: (a) The memory is no more veridical than the original experience. (b) The content and well-groundedness of the memory is determined by that of the original experience. (c) The truth-maker of the memory is necessarily connected to that of the original experience. (b), I have suggested, implies both (a) and (c). Let us take a look at (c), the thesis about truth-makers, and inquire into the nature of those truth-makers.

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2. The A-series and B-series: passage and order Consider the event I recalled earlier of my lending a book to a friend. Is that event receding into the past or not? We can think of events in time as constituting what McTaggart (1908, 1927) called the ‘A-series’, that is, as being more or less past, present, or more or less future ⎯ positions which, of course, would always be changing. Or we can think of events as constituting only the ‘B-series’, that is, as standing in the unchanging relations of temporal precedence or simultaneity to each other. Now, if there really is an A-series, if time really is passing, then B-series relations are simply a product of A-series position. If, for instance, x is past and y is future, then x is earlier than y. To hold that there is an A-series is not to assert the unreality of the B-series. But there is no plausible way to invert this and represent the A-series as supervening on the B-series in such a simple way. Passage does not simply emerge from order. Those that hold that the B-series does not supervene on the A-series typically hold that the A-series is unreal, that time does not pass: there is only temporal order. It would, perhaps, be possible to hold that A-series passage and B-series order were completely independent of each other, but this would be a deeply unattractive position. So those who, for whatever reason, hold that events form only a B-series, have something to explain: when in ordinary speech we appear to allude to the A-series, as when we say ‘I’m flipping the switch now’, or ‘My aunt arrived yesterday’, or ‘The lunar eclipse will occur tonight’, do we speak truly? It is here that the metaphysician of time has to engage with the semantics of tensed discourse. On one account (there are others), if I say ‘I’m flipping the switch now’ at time t, where t is a position in the B-series, then what I say is true if and only if it is a fact that I flip the switch at t. Let us call this semantics, and the associated assertion that there is in reality no A-series, the B-theory. Contrasted with this is what we might think of as the natural semantics for tensed statements: ‘I’m flipping the switch now’ is true if and only if I am flipping the switch now, in the objective present. Call this semantics, and the associated assertion that A-series terms reflect corresponding divisions in reality, the A-theory. A-theorists do not always want to be associated with the idea of events literally receding in time, as if pastness is something an event can acquire more of. Some A-theorists, indeed, are presentists: they hold that only what is present is real. Outside the present, there are for them no events to be past. Believing in the A-series is for them a matter of ‘taking

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tense seriously’: taking the tensed structure of language to reflect a real and transient aspect of reality. We need not concern ourselves here with presentist versus non-presentist interpretations of the A-theory: what is relevant for the argument I want to develop is the semantics of tensed discourse and belief. The crucial difference between the A-theory and the B-theory is their view of the kinds of thing that make temporal statements and beliefs true, namely the facts of the matter. Here I am taking ‘facts’ to be constituents of reality. For the B-theorists, facts do not change. For the A-theorist, they do. The facts which are constitutive of the present, for instance that this talk is now going on, that Mars is such-and-such a distance from the Earth, are in due course replaced by other facts, for instance that this talk is over. (See Mellor 1998, Chapters 2 and 3, for a full account of the A-theory and the B-theory in terms of facts and truth-makers for tensed statements.) These two views of reality correspond to two models of memory. For the A-theorist, our changing beliefs about a given event track the changing facts of reality (the A-model). Your belief that this talk is going on will be replaced by the belief that it is over. On this model of memory, the mind tracks changing states of affairs, viewed from the same perspective (the present). What we have, in effect, is two parallel processes: the external changing facts, and the internal changing beliefs, which run in pre-established harmony (that is, evolutionary-established harmony): A-model Past Experience

Present Memory

e is occurring

e occurred

For the B-theorist, in contrast, our changing beliefs about an event do not track changing facts, but rather the same fact from different perspectives (the B-model):

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B-model t(0 means that qo < 0. Equation (4) then implies that either there is a positive Λ or else a strange form of matter with w < − ⅓, violating the strong energy condition (SEC). To discuss the latter possibility in more detail, rewrite the first term on the rhs of (4) as the sum of two terms, one for normal matter (‘N ’) satisfying the weak and strong energy conditions, and one for strange matter (‘S ’) satisfying the weak energy condition but violating the SEC: q=

1 M [Ω (1 + 3wN) + ΩM (1 + 3wS)] − ΩΛ 2 N

S

(5)

Since at the present time wN 0 or a strange form of matter-energy in sufficient quantity to dominate normal matter. The term “dark energy” has been coined to cover both possibilities. Because of the radical nature of these conclusions it is important to note that they are supported by multiple independent lines of evidence, of which I will mention a few: X-ray observations of galactic clusters confirms the conMN MS Λ clusion that ao >0; CMB-measurements give Ωtot o := Ωo + Ωo + Ω o ≈ 1, whereas observations of the dynamics of galaxies gives ΩoM N ≈ 0.3, which together imply that 70% of matter-energy is in the form of dark energy; and measurements of the integrated Sacks-Wolf effect confirm that dark energy is a large fraction of the total.

2.4 Possibilities for explaining accelerating expansion Going beyond orthodox GTR opens a Pandora’s box of possibilities for explaining the accelerating expansion of the universe. For example, adding an 1/R term to the Hilbert action for GTR leads to field equations that allow accelerating expansion without dark energy (see Carroll, Duvvuri, Trodden, and Turner 2003). While this particular possibility appears to be ruled out by solar system tests (see Chiba 2003), there are no doubt more subtle modifications of GTR that give accelerating expansion without dark energy and that are in accord with other extant observational constraints. Going further afield to speculative theories of quantum gravity, an effective cosmological constant can result from the extra spatial dimension postulated by M-theory (see Gu and Hwang 2002); and the causal sets approach to quantum gravity leads to a fluctuating value for Λ (see Ahmed at al. 2004). I will not open this Pandora’s box here but will confine myself to orthodox GTR. If the possibilities are confined to those that can be described within orthodox GTR and if the FRW cosmologies are used to model the universe, then as explained above, the cause of the accelerating expansion can be identified as dark energy. But the second “if ” necessitates a note of caution since the actual cosmos is not exactly homogeneous and isotropic (as is assumed in the FRW models), although suitable averaging over large volumes will produce a model with these features. Kolb et al. (2005) claim that “back reaction” effects of smoothing inhomogeneities over a volume on the order of the size of the current Hubble volume can produce accelerated expansion. Let gab and Tab be respectively the actual, exact metric and stressenergy tensor for an inhomogeneous universe, and let g ab and Tab denote respectively the smoothed-out metric and the smoothed-out stress-tensor.

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The Einstein field equations may not be satisfied by the pair consisting of the Einstein tensor Gab (computed from g ab ) and Tab . If not, the difference Gab − Tab can be interpreted as an extra source term, which may violate the SEC even though the exact Tab does satisfy it. There is no doubt that this phenomenon is mathematically possible (as was first noted by Ellis 1984). But care needs to be exercised in order to separate out real effects from gauge-dependent artifacts since the computation of the back reaction relies on the choice of gauge (see Ellis and Buchert 2005). Then it needs to be demonstrated that the magnitude of a gauge-independent back reaction effect can be great enough to explain the actually observed accelerated expansion while being consistent with other observational constraints. Ishibashi and Wald (2005) argue that it is implausible that such a demonstration will be forthcoming. These issues can also be investigated by studying features of inhomogeneous cosmological models satisfying Einstein’s field equations. Hirata and Seljak (2005) show that within this class of models, if the vorticity vanishes (as indicated by observations and as required by inflationary cosmology), the deacceleration parameter qˆ cannot be negative unless the SEC is violated, where qˆ (which is now a function of spatial position as well as time) is defined in terms of the time rate of change of the local expansion of matter. It could be objected that what matters to the supernovae Type Ia observations is not the deacceleration parameter qˆ but rather the deacceleration parameter q that appears in the luminosity distance(dL)-redshift (z) relation. 1 - q 2 z Expanding dL in a Taylor series in z gives dL = + z + O(z3), where 2 H H  q and the Hubble parameter H are now functions of the direction of observation as well as of the spacetime location of the observer. While q can be negative without a violation of the SEC, Hirata and Seljak (2005) argue that q < 0 requires anisotropic expansion and, thus, that if there are accelerating directions (q < 0) then there are also deaccelerating directions ( q > 0). There is no observational support for the latter. Furthermore, it is a mystery how such a variation in q can be reconciled with the observed isotropy of the CMB. While the controversy over whether dark energy is needed to explain accelerating expansion is apt to continue for the foreseeable future, at this juncture it seems that the weight of the evidence strongly favors the presumption that dark energy is the cause, and I will discuss the fate of the universe under this presumption.

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2.5 The fate of the universe Three categories of dark energy can be distinguished in terms of the equation of state: a true cosmological constant corresponds to the case wΛ = −1; so-called quintessence corresponds to the case −1 < wQ < − ⅓; and so-called phantom matter corresponds to the case wP < −1. The implications for the fate of the universe can be discussed by noting that if the equation of state wX does not change with time, then the matter-energy density ρX scales as a −3(1 + w ). The major possibilities can be divided as follows. 1) A true Λ is entirely responsible for ao >0. Then, regardless of whether k = 0, −1, or +1, the universe will expand forever. However, since the scale factor increases faster than the horizon distance, an observer comoving with the expansion will see a universe that becomes increasingly empty, dark, and cold. In sum, we get Eternity with a Big Chill. It is dubious that critters such as ourselves can survive the Big Chill to experience Eternity (see Krauss and Starkman 2000). 2) Λ = 0 and quintessence is entirely responsible for ao >0. Then if the equation of state wQ does not change we again get Eternity with a Big Chill. If wQ does change as the universe expands, then the fate of the universe has to be discussed in terms of particular models for quintessence. 3) Λ = 0 and phantom matter is entirely responsible for ao >0. If the equation of state wP does not change, ρP (t) → +∞ in a finite time, and the universe ends in a Big Smash. Before the end, there is a Big Rip in which gravitationally bound systems are ripped apart (see Caldwell et al. 2002; Caldwell et al. 2003). Again if wP does change as the universe expands, then the fate of the universe has to be discussed in terms of particular models for phantom matter. It is far from clear how seriously to take phantom matter since it violates not only the SEC but it also the dominant energy condition4, leading to the possibility of acausal propagation and making it difficult to build stable models in QFT (see Carroll, Hoffman, and Trodden 2003). X

4

The stress-energy tensor Tab satisfies the dominant energy condition iff for any future directed timelike vector Va, −TabVa is a future directed timelike or null vector. This condition conjoined with the conservation of Tab (i.e. aTab = 0, which is entailed by EFE) implies that matter-energy is not transmitted faster than light: if Tab vanishes in some achronal spacetime region, then it vanishes throughout the domain of dependence D(S), which consists of all spacetime points p such that every endless non-spacelike curve that passes through p meets D(S). See Hawking and Ellis (1973, Lemma.4.3.1).

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4) There are many mix-and-match possibilities. For example, suppose that quintessence is entirely responsible for ao >0 and that there is small negative Λ, so small that it cannot be detected by current observations. Then as the universe expands, the attractive force due to the negative Λ will eventually dominate the repulsive force of quintessence, and the universe will begin to contract. On the other hand, a negative Λ will never come to dominate phantom matter and prevent a Big Rip followed by a Big Smash. A major challenge for observational and theoretical astrophysics is to sort through these possibilities to determine the equation of state of dark energy and, thus, the fate of the universe.

3. Does time have a beginning? Specifying what it means for a general relativistic spacetime to be finite in the past turns out to be a more difficult task than might be imagined. To illustrate the point, one might take the following as a necessary and sufficient condition for such finiteness: for any global time slice S (i.e. connected spacelike hypersurface without edges) — which may be thought of as representing “now”— any past directed timelike curve with future endpoint on S and no past endpoint has a proper length less than or equal b(S) < ∞. This condition suffers from multiple deficiencies. First, there are spacetimes, such as Gödel spacetime, which do not intuitively count as being temporally finite in the past but which vacuously satisfy the condition because they possess no global time slices. Second, unless restrictions are put on the choice of S the condition in question would yield incorrect results. For example, with (x, t) an inertial coordinate system for (1 + 1)-dim Minkowski spacetime, delete all the points on or below t = 0. This spacetime should count as finite in the past direction, but the condition in question fails if S is given by t =

1+ x 2 . Third, the condition wrongly counts the surgically

mutilated Minkowski spacetime shown in Fig. 2 as being temporally finite in the past.

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t etc. ...

−∞ ← → +∞ −∞ ← → +∞ −∞ ← → +∞ ... etc.

Figure 2: (1 + 1)-dim Minkowski spacetime with spacelike strips (heavy lines) removed. Light cones are at 45°. Every timelike curve has finite proper length. Coming up with a more adequate analysis that is applicable to any general relativistic spacetime is a ticklish problem. But the problem need not be tackled here since the cosmological models under consideration belong to a special class of models having features that make it obvious how to decide whether time in a model in the class is finite or infinite in the past/future. All of these models have a global time function t whose level surfaces t = const are Cauchy surfaces (i.e. each is intersected exactly once by every timelike curve without endpoint). Moreover, level surfaces t = const are orthogonal to a congruence of timelike geodesics, and the lengths of geodesic segments between any two levels are all the same; these geodesics are the longest timelike curves between the levels. Such a t induces a natural metric of time, and one can say that time is finite in the past (respectively, the future) iff the range of t is bounded from below (respectively, the future). The class of FRW models is a subclass of the class whose members admit a natural time metric, and the cosmic time t of the line element (1) has the properties enumerated above. If this t is bounded (from above or below), then the scale factor a(t) approaches 0 as the t approaches the bound. For the standard FRW Big Bang model that best fits the actual universe, the

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lower bound is around 14 billion years. The bound rises by only a few seconds for the simple inflationary models that insert an inflationary era after the Big Bang. Thus, assuming the validity of the models, the actual universe had a beginning, not in the sense that there is a first instant of time, but in the sense that time in the actual universe is finite in the past. Within orthodox classical GTR speculation about what happened before the Big Bang is physically meaningless since there is no way to extend the FRW models as solutions to EFE even in the distributional sense of solution. Of course, this didn’t stop some cosmologists from speculating. For example, R. C. Tolman (1934) speculated about oscillating models in which a spatially closed universe cycles through expanding and contracting phases. He admitted that “our differential equations for the motion of the model are not sufficient to describe the mechanism of passage through the lower limit of contraction” (1934, p. 428). Nevertheless, he thought that “the existence of which [i.e. the passage through a = 0] is physically inevitably necessary” (ibid.). His confidence was based on the notion that the initial and final singularities were artifacts of the idealizations of the models. The singularity theorems of Hawking and Penrose undermine this confidence by showing that past timelike or null geodesic incompleteness is to be expected under quite generic conditions, at least if energy conditions discussed in the preceding section are assumed to hold. It is interesting to note that Tolman thought that in his Λ = 0 oscillating model the entropy of successive cycles would increase. Reading this backwards in time, Steinhardt and Turok (2002, p. 126003–1) argue that the lengths of the cycles decreases sufficiently rapidly in the past direction that the sum of the cycles lengths is finite, giving a beginning of time. Steady state cosmology, despite the name and despite the fact that it does not involve a Big Bang5, does imply that time is finite in the past. This can be seen from the fact that the line element used in steady state cosmology is just the de Sitter metric written in a coordinate system that is adapted to flat space slices of de Sitter spacetime. These flat slices cover only a portion of the spacetime, a portion that is timelike geodesically incomplete in the past (see Fig. 3).

5

This phrase was invented by the steady state cosmologists as a term of derision.

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t flat space slices

Figure 3: De Sitter spacetime as an hyperboloid embedded in a flat space. The upper portion can be covered by a family of flat spacelike slices. This portion is past timelike geodesically incomplete. In addition to the simple inflationary models referred to above, there are also more elaborate models that entail “eternal inflation” in the future direction. The picture presented by the latter is that of thermalized regions set in an inflating background. As time goes on, the thermalized regions expand and new ones come into existence; but since the background expands so much faster than the thermalized regions, the universe is never completely thermalized and inflation continues forever. Could it be that inflation is also eternal in the past direction and that the initial Big Bang singularity and, indeed, all cosmological singularities in the past are avoided? Some hope for a positive answer comes from the fact that some of the Hawking-Penrose singularity theorems use the SEC, which is violated by inflationary mechanisms. However, a negative answer has been claimed by inflationary cosmologists who have extended the Hawking-Penrose theorems by proving past timelike or null geodesic incompleteness using only the weak energy condition or no energy condition at all (see Borde and Vilenkin 1994, 1997; Borde, Guth, and Vilenkin 2003). On the other side, Ellis and Maartens (2002, 2004) have argued that these theorems are based on overly restrictive assumptions, in particular that k = 0 or −1 and/or that H > 0 for all past times. As for the first assump-

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tion, it is not excluded by inflation that k = +1 even though inflation does drive k/a2 towards 0 and the density parameter towards 1. And in any case, k = +1 or Ωtot o > 0 is not excluded by present observations which give tot Ωo = 1.02 ± 0.02. Ellis and Maartens show that if k = +1, eternal inflation to the past can be achieved in that, without having to invoke negative energy densities, there is no past singularity. They explicitly exhibit three models of this type, whose scale-factor behavior is illustrated in Fig. 4. In two of these models the scale factor has a minimum value a* > 0, which can be set large enough that Planck-scale quantum gravity effects are avoided. Just how physically plausible these models are remains to be seen since some of them seem to require fine tuning of initial conditions. And in any case these models do not conform to the most recent astronomical observations that indicate that before entering the present accelerating phase, the universe was expanding ( a >0 ) but deaccelerating ( a 0, SB(t + Δ) > SB (t). But what is the justification for the posit of initially low entropy? And even if this posit is granted, what justifies the normal expectation that at t − Δ, SB (t − Δ) < SB (t)? The latter question becomes pressing when it is realized that if the laws governing the microdynamics are time reversal invariant, then the very apparatus that leads to the prediction that SB (t + Δ) > SB (t) also leads to the prediction that SB (t − Δ) > SB (t ). Thus, if we are told that a thermally isolated system consists at t of an ice cube in a glass of lukewarm water, we would normally predict that a few minutes after t the ice cube will have partly melted and the temperature of the water will have decreased, and also that a few minutes prior to t the ice cube would have been less melted and the temperature of the water would have been higher. But according to what was just said, the Boltzmann apparatus does not underwrite this asymmetry of inference. The current dogma is that modern cosmology comes to Boltzmann’s rescue in that the answers to the initial state puzzle and the asymmetry puz8

This section summarizes some of the conclusions of Earman (2006), to which the interested reader is referred for details.

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zle lie in the fact that the very early universe was in an extraordinarily low entropy state. Here “very early” means shortly after the Big Bang, if one is using a standard hot Big Bang model, or shortly after the universe reheats, if one is using an inflationary model. It might seem counterintuitive to say that a thermalized state of a homogeneous and isotropic universe has low entropy, but (the story goes) intuitions are misleading because they neglect the gravitational contribution to entropy — since gravity tends to clump matter, a smooth state is very improbable.9 I have two objections to this dogma. The first is that it is very likely that it is (to echo Pauli) not even false. The Boltzmann entropy for a deterministic dynamical system is defined by choosing a coarse graining and then setting the value of the entropy for a coarse-grained state to be proportional to the log of the measure μ(V ) of the volume V of the micro-state space compatible with the coarse-grained state in question, where μ is a normed measure on the micro-state space that is invariant under the deterministic flow. In the cosmological setting, however, it is doubtful that this formalism yields coherent results. An intimation of the difficulties can be gleaned from features of the model that Hawking and Page (1988) used to try to answer the question of how probable it is that inflation initiates. They investigated a FRW-φ model in which the matter content is given by a minimally coupled massive scalar field φ. The state space X for this system is four-dimensional and can be coordinatized using the scale factor a of the FRW model, the field φ, and their respective conjugate momenta pa and pφ. As the discussion from the previous section would lead one to expect, the equations of motion have the form of a constrained Hamiltonian system with the one and only constraint being the vanishing of the Hamiltonian H = 0. The threedimensional subspace C ⊂ X where the constraint is satisfied is called the constraint surface. A reduced phase space free of gauge redundancy can be formed by choosing a two-dimensional surface Σ that is transverse to the dynamical trajectories on C. Then the pullback of the (degenerate) symplectic form ω = dpa ∧ da + dpφ ∧ dφ defines a volume measure (2)μ on Σ that is invariant under dynamical evolution.10 9

This idea has been promulgated by Penrose (1979, 1989, 2004). It is accepted by physicists who worry about the foundations of statistical mechanics, e.g. Lebowitz (1993, 1999) and by philosophers of science, e.g. Price (2004). 10 Alternatively, as noted by Hollands and Wald (2002), an invariant volume measure (3) μ can be defined on C by the volume element (3)ε given by the condition dH ∧ (3)ε = (4)ε, where (4)ε := dpa ∧ da ∧ dpφ ∧ dφ is the Liouville volume element for X.

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Now a crucial fact is that (2)μ(Σ) does not normalize.11 Since this result holds for the k = +1 FRW model, the non-normalizability cannot be blamed on the infinity of space implied by the k = 0 and k = −1 models. One could still try to assign finite measures to coarse grained states as follows. Let M ⊆ Σ be the region corresponding to a coarse grained state m. Set (2) Pr(m) = 0 in case (2)μ(M) < ∞; set (2) Pr(m) = 1 in case (2)μ(M ) = ∞ and (2) μ(Σ − M) < ∞; and set declare (2) Pr(m) undefined otherwise. Then it is easy to show that (2) Pr is a finitely additive, partial probability measure on the coarse graining. Hawking and Page found that the third case of ill-definedness holds for the probability of inflation. And one should be prepared to find that (2) Pr(mi ) is ill-defined for the contemplated smooth, thermalized initial coarse grained state mi of the universe. Even if (2) Pr(mi ) is welldefined, the best that one can hope for as an explication of the notion that the universe begins with a low entropy state is that (2) Pr(mi ) = 0, which makes conditional probabilities (2) Pr(ml /mi ) of later coarse-grained states ml ill-defined if conditional probability is given its usual interpretation. My second objection is that even if there were an appropriate normalized measure and even if the Boltzmann entropy of the very early universe is well-defined and has a low value, it seems dubious that the latter explains the thermodynamic asymmetries of the kinds of systems typically encountered by critters like us. It is, of course, a truism that if some asymmetry does not follow from the laws of physics, then its origins must be sought in initial/boundary conditions. But it is a fallacy to reason: The thermodynamic asymmetries we observe today are traceable to the conditions of the early universe; the early universe was in a low entropy state; therefore, the low entropy of the early universe is the key ingredient in explaining the thermodynamic asymmetries we observe today. I will mention two among many reasons for thinking that there is a gap in the argument. First, Boltzmann entropy is not a spatially localized quantity but rather a global property of the system — in this case the system consisting of the entire universe — and the value of this entropy places only weak constraints on the entropy of the small subsystems of the kind we are interested in (e.g. the ice cube in the glass of water), and this is especially so since these subsystems are not dynamically isolated. Second, if the argument worked, it should work even better if the entropy of the early universe were even lower than it is in the actual universe because, for example, the state is completely homogeneous 11 The same holds for the (3)μ(Σ) of Hollands and Wald (2002).

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and isotropic. But in this case the galaxies and solar systems would not have formed and the universe would not develop localized systems that are in thermal disequilibrium and that display the familiar thermodynamic arrow of time. In sum, I think that, despite the rosy pronouncements to the contrary, we are far from an understanding the role that cosmology plays in the local physics of thermal systems.

4.2 Black holes and black hole evaporation The FRW models describe only the large scale structure of the universe — the homogeneity and isotropy assumed by the models appears only at scales above 10 mega parsecs. At smaller scales many interesting and disturbing things can happen — in particular, the formation and evaporation of black holes. Classical GTR predicts the formation of black holes through the process of gravitational collapse of stars of up to 10 M . In addition, the cores of galaxies can collapse to form black holes with masses ranging from millions to billions of solar masses. There is very strong evidence that our universe is well populated by both solar mass and supermassive black holes. It is also possible that tiny black holes with masses of the order of 10−19 M could have formed in the early universe. There is currently no observational confirmation of these primordial black holes, but if they do exist they would provide a test for predictions about black hole evaporation described below. An observer who falls through the horizon of a black hole has his fate sealed: twist and turn how he will, his world line can be extended for only a finite amount of proper time (see Fig. 5)12, and even before the theoretical bound on his time expires, he will expire by being torn apart by the tidal forces of the black hole. Thus, even though time itself may be infinite, the fact that the universe is well populated with black holes means that even observers idealized to escape the normal effects of aging can have an inprinciple bound to their future existence.

12 The standard conventions of conformal diagrams are in effect, e.g. light cones are at 45°. I+ denotes future null infinity, the terminus of outgoing light rays; and ι° denotes spatial infinity.

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black hole horizon γ

ℑ+ 0

ι collapsing matter

−

ℑ

center of symmetry

Figure 5: Conformal diagram of black hole formation in spherically symmetric gravitational collapse. γ is the worldline of an observer who falls through the black hole horizon. Suppose that the absence of evidence for white holes — the time reverses of black holes — is evidence for their absence. Then the combination of this evidence with the evidence for the existence of black holes would be evidence for a pervasive temporal asymmetry. It is currently a matter of speculation as to whether this black hole-white hole asymmetry is purely de facto or whether it is indicative of some deep lawlike asymmetry.13 The discussion to this juncture was limited to classical GTR. But now we must consider quantum effects. Stephen Hawking showed that in the presence of a black hole a quantum field will thermalize so that, effectively, a black hole radiates with a thermal spectrum. This prediction of black hole radiation does not depend on any nascent quantum theory of gravity but simply uses quantum field theoretical calculations for a quantum field propagating on a spacetime background supplied by classical GTR. The next step does take a step towards quantum gravity by calculating the back reaction of the radiation on the spacetime geometry of the black hole. The prediction is that the black hole will lose mass and its horizon area will shrink. Presumably these semi-classical calculations will break down when the Planck 13 Penrose (2004, Sec. 30.9) takes the latter alternative.

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regime is reached. But if the black hole does evaporate completely and if the result can be described to good approximation in classical GTR, then the upshot will be something like that pictured in Fig. 6.14 The time required for a solar mass black hole to evaporate is very long indeed — on the order of 1067 years; but if the universe expands forever, there is no shortage of time.

Σ2

singularity

ℑ+ ι

black hole interior

B

Σ1

0

ℑ−

center of symmetry

Figure 6: Conformal diagram of conventional treatment of black hole evaporation. An observer who falls through the horizon of an evaporating black hole is still doomed. But two even more ominous things emerge. First, the evaporation results in a naked singularity, i.e. a singularity visible from future null infinity I+. Second, a quantum field propagating on this spacetime undergoes a transition from a pre-evaporation pure state to a mixed post-evaporation state, at least assuming that the pre-evaporation state was pure. (Consider the subalgebra of observables A(Σ2) associated with a thin neighborhood of the post-evaporation time slice Σ2 of Fig. 6. Correlations between observables belonging to A(Σ2) and observables belonging to the subalgebra A(B) associated with a region B in the interior of the black hole are established by the presence of a common cause in the form of the Hawking radiation. Since B and a thin sandwich of Σ2 are relatively spacelike, [A(Σ2), A(B)] = 0. 14 See Wald (1994, Ch. 7) for an overview.

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Consider a state ω on the global algebra of observables that encodes the correlations between A(Σ2) and A(B). It follows that the restriction of ω to A(Σ2) is a mixed state.) Such a pure-to-mixed transition is necessarily non-unitary, which is the precise content of what is called the Hawking “information loss paradox”.15 So desperate are some commentators to resolve this paradox that they are prepared to believe six contradictory things before lunch, such as “black hole complementarity”. Part of the desperation is based on the misguided notion that loss of unitarity is a disaster that must be avoided at all costs. On the contrary, unitarity is not essential to conservation of probability, and a respectable amount of QFT can be done without it.16 One result of coming to terms with the pure-to-mixed state transition is that black hole evaporation involves a violation of time reversal invariance and, thus, is a possible source for time’s asymmetries.17 It must be emphasized, however, that the preceding discussion is based on semi-classical quantum gravity and that a full theory of quantum gravity may present quite a different picture of black hole evaporation. And in fact, the proponents of LQG have recently claimed that on the basis of the resolution of black hole singularities in LQG, it is reasonable to think that the depiction of black hole evaporation in Fig. 6 will be replaced by something more like of Fig. 7 (see Asktekar and Bojowald 2005). It is argued that, analogously to the FRW case, the quantum evolution continues through the classical singularity and that a pure state remains pure and, in this sense, no information is lost.18 Note, however, that as in the FRW case, the classical singularity is not resolved in the sense of being replaced by smooth classical relativistic spacetime structure; rather it is replaced by quantum spin-foam where a description in terms of classical relativistic spacetime does not have even approximate validity. It is but cold comfort to tell an observer who falls into a black hole that, within a finite time, he will dissolve into quantum spin-foam rather than splatting on a classical spacetime singularity. On the other hand, it is real comfort to tell an external observer that the strange things he is seeing are the results of quantum spin-foam rather than a naked singularity of classical GTR; for the latter is a source of genuine unpredict15 16 17 18

See Wald (1994, pp. 178–181) and Belot et al. (1999). See Wald (1994, pp. 181–183) for a defense of this viewpoint. See Wald (1980) and Earman (2002). As in the FRW case, “quantum evolution” means that some suitable clock variable is chosen and that the Hamiltonian constraint equation is used the track the quantum state through discrete steps of the clock variable.

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ability since classical GTR places no restrictions on the pathologies that can devolve from such singularities, whereas the former is subject to the laws of LQG.

ℑ+ deep Planck regime

0

ι

H

ℑ−

center of symmetry

Figure 7: Loop quantum gravity paradigm for black hole evaporation. H is the dynamical horizon. From Ashtekar and Bojowald (2005).

5 . Conclusion For what it is worth, I list my bets regarding cosmological aspects of time. 1. I would offer better than even odds that time is infinite in the future because the universe will expand forever; but I would also bet that no one will be around to collect the stakes since critters like us will perish in the Big Chill. 2. I would bet that time in the sense of classical special and general relativity does have a beginning because classical spacetime dissolves into quantum foam in the finite past. I would not be surprised to learn — but would not

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bet — that the correct quantum theory of gravity resolves the initial Big Bang singularity. But until the shape of this sought-after theory is more clearly discerned, I would decline bets about the sense in which the resolution of the Big Bang singularity legitimates talk of what happened “before” the Big Bang. 3. I would offer high odds that the explanation of the local temporal asymmetries of concern to us is not to be found in a low entropy state for the very early universe. 4. I would bet that our universe is populated with black holes and that these objects are not really black but radiate with a thermal spectrum, as Hawking predicted. But I would try to beg off bets on the upshot of black hole evaporation, except insofar as I could bet against such expediencies as black hole complementarity. Bets aside, it is satisfying to reflect on how much we have learned over the past few decades about cosmological aspects of time. It is humbling to realize how much we still don’t know. And it is inspiring to learn that we can discern pathways that will lead — we can hope — to a resolution of our ignorance.

References Ahmed, M., Dodelson, S., Greene, P. B., and Sorkin, R. 2004 “Everpresent Λ”, Physical Review D 69: 103523-1-8. Ashtekar, A. and Bojowald, M. 2005 “Black Hole Evaporation: A Paradigm”, gr-qc/0504029. Ashtekar, A., Bojowald, M., and Lewandowski, J. 2003 “Mathematical Structure of Loop Quantum Cosmology”, Advances in Theoretical and Mathematical Physics 7: 233–268; gr-qc/0304074. Belot, G., Earman, J., and Ruetsche, L. 1999 “The Hawking Information Loss Paradox: The Anatomy of a Controversy”, British Journal for the Philosophy of Science 50: 189–229. Bojowald, M. 2001 “Absence of a Singularity in Loop Quantum Cosmology”, Physical Review Letters 86: 5227–5230. Borde, A. and Velinkin, A. 1994 “Eternal Inflation and the Initial Singularity”, Physical Review Letters 72: 3305–3308. — 1997 “Violation of the weak energy condition in inflating spacetimes”, Physical Review D 56: 717–723.

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Borde, A., Guth, A., and Vilenkin, A. 2003 “Inflationary spacetimes are not past-complete”, gr-qc/0110012 v2. Caldwell, R. R. 2002 “A Phantom Menace? Cosmological consequences of a dark energy component with super-negative equation of state”, Physics Letters B 545: 23–29. Caldwell, R. R., Kamionkowski, M., and Weinberg, N. N. 2003 “Phantom Energy and Cosmic Doomsday”, astro-ph/0302506. Carroll, S. M., Duvvuri, V., Trodden, M., and Turner, M. S. 2003 “Is Cosmic Speed-Up Due to New Gravitational Physics?” astro-ph/0306438. Carroll, S. M., Hoffmann, and Trodden, M. 2003 “Can the Dark Energy Equation-of-State Parameter w be less than −1?” astro-ph/0301273. Chiba, T. 2003 “1/R Gravity and Scalar-Tensor Gravity”, astroph/0307338. Earman, J. 2001 “Lambda: The Constant That Refuses to Die”, Archive for History of Exact Sciences 55: 189–220. — 2002 “What Time Reversal Invariance Is and Why It Matters”, International Journal for the Philosophy of Science 16: 245–264. — 2006 “Note Even False: The ‘Past Hypothesis’”, Studies in History and Philosophy of Science, in press. Eddington, A. S. 1933 The Expanding Universe. New York: Macmillan. Einstein, A. 1917 “Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie”, Königlich Preussische Akademie der Wissenschaften (Berlin). Sitzungsberichte, 142–152. English translation as “Cosmological Considerations on the General Theory of Relativity”, in W. Perrett and G. B. Jeffrey (eds.), The Principle of Relativity, 177–188. New York: Dover Books, 1952. Ellis, G. F. R. 1984 “Relativistic Cosmology”, in B. Bertotti, F. de Felice, and A. Pascolini (eds.), General Relativity and Gravitation, 215–288. Dordrecht: D. Reidel. Ellis, G. F. R. and Buchert, T. 2005 “The universe at different scales”, grqc/0506106. Ellis, G. F. R. and Maartens, R. 2002 “Eternal inflation without quantum gravity”, gr-qc/0211082. Ellis, G. F. R. and Maartens, R. 2004 “The emergent universe: inflationary cosmology with no singularity”, Classical and Quantum Gravity 21: 223– 232. Gamov, G. 1958 “The Evolutionary Universe”, in The Universe: A Scientific American Book. London: G. Bell and Sons.

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Green, D. and Unruh, W. G. 2004 “Difficulties with Re-collapsing Models in Closed Isotropic Loop Quantum Cosmology”, Physical Review D 70: 103501-1-7. Gu, Je-An and Hwang, W-Y. P. 2002 “Accelerating Universe as From the Evolution of Extra Dimensions”, Physical Review D 66: 024003-1-6. Hawking, S. W. and Ellis, G. F. R. 1973 The Large Scale Structure of SpaceTime. Cambridge: Cambridge University Press. Hawking, S. W. and Page, D. N. 1988 “How Probable Is Inflation?” Nuclear Physics B 298: 789–809. Hirata C. M. and Seljak, U. 2005 “Can surperhorizon cosmological perturbations explain the acceleration of the universe?” astro-ph/0503582. Hollands, S. and Wald, R. M. 2002 “Comment on Inflation and Alternative Cosmology”, hep-th/0210001. Ishibashi, A. and Wald, R. M. 2005 “Can the Acceleration of Our Universe Be Explained by the Effects of Inhomogeneities?” gr-qc/0509108. Isham, C. J. 1992 “Canonical Quantum Gravity and the Problem of Time”, in L. A. Ibot and M. A. Rodríguez (eds.), Integrable Systems, Quantum Groups, and Quantum Field Theories, 157–287. Boston: Kluwer Academic. Khoury, J., Ovrut, B. A., Steinhardt, P. J., and Torok, N. 2001 “Ekpyrotic universe: Colliding branes and the origin of the hot big bang”, Physical Review D 64: 123522-1-24. Khoury, J., Ovrut, B. A., Seiberg, N., Steinhardt, P. J., and Torok, N. (2002) “From big crunch to big bang”, Physical Review D 65: 086007-1-8. Kolb, E. W., Matarrese, S. and Riotto, A. 2005 “On cosmic acceleration without dark energy”, astro-ph/0506534. Krauss, L. M. and Starkman, G. D. 2000 “Life, the universe, and nothing: Life and death in an ever-expanding universe”, Astrophysical Journal 531: 22–30. Krauss, L. M. and Turner, M. S. 1999 “Geometry and Destiny”, General Relativity and Gravitation 31: 1453–1459. Kuchař, K. 1992 “Time and the Interpretation of Quantum Gravity”, in G. Kunsatter, D. Vincent, and J. Williams (eds.), Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, 211– 314. Singapore: World Scientific. Lebowitz, J. L. 1993 “Macroscopic laws, microscopic dynamics, time’s arrow and Boltzmann’s entropy”, Physica A 194: 1–27. — 1999 “Statistical mechanics: A selective review of two central issues”, Reviews of Modern Physics 71: S346–S357.

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Penrose, R. 1979 “Singularities and time asymmetry”, in S. W. Hawking and W. Israel (eds.), General Relativity: An Einstein Centenary, 581–638. Cambridge: Cambridge University Press. — 1989 The Emperor’s New Mind: Concerning Computers, Minds, and the laws of Physics. Oxford: Oxford University Press. — 2004 The Road to Reality: A Complete Guide to the Laws of the Universe. London: Jonathan Cape. Price, H. 2004 “On the Origins of the Arrow of Time: Why There is Still a Puzzle about the Low-Entropy Past”, in C. Hitchcock (ed.), Contemporary Debates in Philosophy of Science, 219–239. London: Blackwell. Steinhardt, P. J. and Turok, N. 2002 “Cosmic evolution in a cyclic universe”, Physical Review D 65: 126003-1-20. Tolman, R. C. 1934 Relativity, Thermodynamics and Cosmology. Oxford: Oxford University Press. Wald, R. M. 1980 “Quantum Gravity and Time Reversibility”, Physical Review D 21: 2742–2755. — 1984 General Relativity. Chicago, IL: University of Chicago Press. — 1994 Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago, IL: University of Chicago Press.

Against Pointillisme about Geometry J. Butterfield, Oxford 1

Introduction

This paper forms part of a wider campaign: to deny pointillisme. That is the doctrine that a physical theory’s fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there; so that properties of spatial or spatiotemporal regions and their material contents are determined by the point-by-point facts.1 I will first describe this wider campaign (Section 2). Then I will argue against pointillisme as regards the structure of space and/or spacetime itself (Sections 3 and 4). A companion paper (2006) argues against pointillisme in mechanics, especially as regards velocity. I will argue that the geometrical structure of space, and/or the chronogeometrical structure of spacetime, involves extrinsic properties of points, typically properties that I shall call ‘spatially extrinsic’. The main debate here is whether properties of a point that are represented by vectors, tensors, connections etc. can be intrinsic to the point; typically, pointillistes argue that they can be. After formulating this debate in Section 3, I will in Section 4 focus on Bricker’s (1993) discussion. For it is an unusually thorough pointilliste attempt to relate vectors and tensors in modern geometry to the metaphysics of properties. But Bricker exemplifies a tendency I reject: the tendency to reconcile pointillisme with the fact that vectorial etc. properties seem extrinsic to points and point-sized objects, by proposing some heterodox construal of the properties in question. Thus Bricker proposes that we should re-found geometry in terms of Abraham Robinson’s non-standard analysis, which rehabilitates the traditional idea of infinitesimals (Robinson 1996). I reply that once the spell of pointillisme is broken, such proposed heterodox foundations of geometry are unmotivated. In saying this, I do not mean to be dogmatic. I of course agree that the nature of the continuum is an active research area, not only historically 1

I think David Lewis first used the art-movement’s name as a vivid label for this sort of doctrine: a precise version of which he endorsed.

F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 181–222.

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Mancosu (1996, Chapters 4f.), Leibniz (2001), Arthur (2006) but also in mathematics and philosophy. Indeed there are several heterodox mathematical theories of the continuum that are technically impressive and philosophically suggestive. In this paper we will only make contact with one of them, viz. non-standard analysis, as invoked by Bricker. This is for the simple reason that all the other theories offer no support for my target, pointillisme. More precisely: so far as I know, these theories do not suggest that fundamental quantities represent intrinsic properties of points or pointsized bits of matter; because either they do not attribute such quantities to points, or they even deny that there are any points.2 But it is worth glimpsing at the outset the philosophical interest of these theories; so I here list the main ideas of some of them. (1) Two theories that are essentially revisions of analysis (calculus) are non-standard analysis, and a different rehabilitation of infinitesimals (smooth infinitesimal analysis; McClarty 1988, Bell 1998). (2) Two other approaches are based on the idea of a space with no points, and so are no friends of pointillisme. That is: the collection of the space’s parts, ordered by parthood, has no atoms, i.e. no elements that themselves have no parts. (i) The first is essentially a revision of measure theory, and is mainly motivated by its avoidance of the measure-theoretic paradoxes, like the BanachTarski paradox. (It was pioneered by Carathéodory (1963); for philosophical introductions, cf. Skyrms (1993), Arntzenius (2000, Section 5, pp. 201–205; 2004, Section 11); we will touch on the measure-theoretic paradoxes in Section 3.3.2, but for a full account cf. Wagon (1985).) (ii) The second is essentially a revision of topology: topology is characterized by relations between regions taken as primitive. (Cf. Menger (1978), Roeper (1997); for a philosophical introduction, cf. Arntzenius (2004, Sections 8–10).) Finally, three comments about the connections between, and signficance of, such theories. (a) These theories have various connections, which this quick list does not bring out. For example, Nelson (1987) shows that a modicum of nonstandard analysis greatly simplifies a rigorous development of the theories of measure and probability. 2

Broadly speaking, the second option seems more radical and worse for pointillisme; though in such theories, the structure of a set of points is often recovered by a construction, e.g. on a richly structured set of regions.

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(b) All the theories mentioned have been developed largely without regard to applications in physics. But Arntzenius discusses the prospects for doing physics, even quantum physics, in these spaces (ibid.; and for (2i), his 2003). (Of course, within quantum physics there is a tradition of speculation about discrete space or time (Kragh and Carazza 1994): for a rigorous non-relativistic quantum theory on a discrete space that is empirically equivalent to the conventional theory, cf. Davies (2003).) (c) As regards philosophy rather than physics, the main topic connected to the above theories is mereology: which has been discussed especially in connection with the measure-theoretic paradoxes, and (2i). Recent work includes Arntzenius and Hawthorne (2006, especially Sections II, IV) and Forrest (2004, especially Sections 3–6; 2002, especially Sections 5–10). So I am very open to suggestions about heterodox treatments of the continuum. It is just that I find the philosophical doctrine of pointillisme an insufficient reason for rejecting the orthodox treatment. Similarly in my companion paper (2006) about mechanics; though with the difference that the proposals by the targeted authors, Tooley, Robinson and Lewis, do not invoke any well-established mathematical theory. That is: I again find pointillisme an insufficient reason for rejecting orthodoxy. I will conduct the discussion almost entirely in the context of “Newtonian” ideas about space and time. This restriction keeps things simple: and at no cost, since both the debate and my arguments carry over to the treatment of space and time in relativistic, and even quantum, physics.

2

The wider campaign

As I mentioned, this paper is part of a wider campaign, which I now sketch. I begin with general remarks, especially about the intrinsic-extrinsic distinction among properties (Section 2.1). Then I state my main claims; first in brief (Section 2.2), then in more detail (Section 2.3).

2.1 Connecting physics and metaphysics My wider campaign aims to connect what modern classical physics says about matter with two debates in modern analytic metaphysics. The first debate is about pointillisme; but understood as a metaphysical doctrine rather than a property of a physical theory. So, roughly speaking, it is the debate whether the world is fully described by all the intrinsic properties of all the points and/or point-sized bits of matter. The second debate is whether an

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object persists over time by the selfsame object existing at different times (nowadays called ‘endurance’), or by different temporal parts, or stages, existing at different times (called ‘perdurance’). Endeavouring to connect classical physics and metaphysics raises two large initial questions of philosophical method. What role, if any, should the results of science have in metaphysics? And supposing metaphysics should in some way accommodate these results, the fact that we live (apparently!) in a quantum universe prompts the question why we should take classical physics to have any bearing on metaphysics. I address these questions in my (2004: Section 2, 2006a: Section 2). Here I just summarize my answers. I of course defend the relevance of the results of science for metaphysics; at least for that branch of it, the philosophy of nature, which considers such notions as space, time, matter and causality. And this includes classical physics, for two reasons. First, much analytic philosophy of nature assumes, or examines, so-called ‘common-sense’ aspects and versions of these notions: aspects and versions which reflect classical physics, especially mechanics, at least as taught in high-school or elementary university courses. One obvious example is modern metaphysicians’ frequent discussions of matter as point-particles, i.e. extensionless point-masses moving in a void (and so interacting by actionat-a-distance forces), or as continua, i.e. bodies whose entire volume, even on the smallest scales, is filled with matter. Of course, both notions arose in mechanics in the seventeenth and eighteenth century. Second, classical physical theories, in particular mechanics, are much more philosophically suggestive, indeed subtle and problematic, than philosophers generally realize. Again, point-particles and continua provide examples. The idea of mass concentrated in a spatial point (indeed, different amounts at different points) is, to put it mildly, odd; as is action-at-a-distance interaction. And there are considerable conceptual tensions in the mechanics of continua; (Wilson (1998) is a philosopher’s introduction). Unsurprisingly, these subtleties and problems were debated in the heyday of classical physics, from 1700 to 1900; and these debates had an enormous influence on philosophy through figures like Duhem, Hertz and Mach — to mention only figures around 1900 whose work directly influenced the analytic tradition. But after the quantum and relativity revolutions, foundational issues in classical mechanics were largely ignored, by physicists and mathematicians as well as by philosophers. Besides, the growth of academic philosophy after

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1950 divided the discipline into compartments, labelled ‘metaphysics’, ‘philosophy of science’ etc., with the inevitable result that there was less communication between, than within, compartments.3 Setting aside issues of philosophical method, pointillisme and persistence are clearly large topics; and each is the larger for being treatable using the very diverse methods and perspectives of both disciplines, metaphysics and physics. So my campaign has to be selective in the ideas I discuss and in the authors I cite. Fortunately, I can avoid several philosophical controversies, and almost all technicalities of physics.4 But I need to give at the outset some details about how I avoid philosophical controversy about the intrinsic-extrinsic distinction among properties, and about how this distinction differs from three that are prominent in mathematics and physics. 2.1.1 Avoiding controversy about the intrinsic-extrinsic distinction My campaign does not need to take sides in the ongoing controversy about how to analyse, indeed understand, the intrinsic-extrinsic distinction. (For an introduction, cf. Weatherson (2002, especially Section 3.1), and the symposium, e.g. Lewis (2001), that he cites.) Indeed, most of my discussion can make do with a much clearer distinction, between what Lewis (1983, p. 114) dubbed the ‘positive extrinsic’ properties, and the rest. This goes as follows. Lewis was criticizing Kim’s proposal, to analyze extrinsic properties as those that imply accompaniment, where something is accompanied iff it coexists with some wholly distinct contingent object, and so to analyze intrinsic (i.e. not extrinsic) properties as those that are compatible with being 3

4

Thus I see my campaign as a foray into the borderlands between metaphysics and philosophy of physics: a territory that I like to think of as inviting exploration, since it promises to give new and illuminating perspectives on the theories and views of the two communities lying to either side of it — rather than as a no-man’s-land well-mined by two sides, ignorant and suspicious of each other! Though persistence is not this paper’s topic, I note that among the philosophical issues my campaign avoids are several about persistence, such as: (a) the gain and loss of parts (as in Theseus’ ship); (b) the relation of “constitution” between matter and object (as in the clay and the statue); (c) vagueness, and whether there are vague objects. Agreed, there are of course connections between my claims and arguments, and the various issues, both philosophical and physical, that I avoid: connections which it would be a good project to explore. But not in one paper, or even in one campaign!

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unaccompanied, i.e. being the only contingent object in the universe (for short: being lonely). Lewis objected that loneliness is itself obviously extrinsic. He also argued that there was little hope of amending Kim’s analysis. In particular, you might suggest that to be extrinsic, a property must either imply accompaniment or imply loneliness: so Lewis dubs these disjuncts ‘positive extrinsic’ and ‘negative extrinsic’ respectively. But Lewis points out that by disjoining and conjoining properties, we can find countless extrinsic properties that are neither positive extrinsic nor negative extrinsic; (though ‘almost any extrinsic property that a sensible person would ever mention is positive extrinsic’ (1983, p. 115)). This critique of Kim served as a springboard: both for Lewis’ own preferred analysis, using a primitive notion of naturalness which did other important work in his metaphysics (Lewis 1983a); and for other, metaphysically less committed, analyses, developed by Lewis and others (e.g. Langton and Lewis 1998, Lewis 2001). But I will not need to pursue these details. As I said, most of my campaign can make do with the notion of positive extrinsicality, i.e. implying accompaniment, and its negation. That is, I can mostly take pointillisme to advocate properties that are intrinsic in the weak sense of not positively extrinsic. So this makes my campaign’s claims, i.e. my denial of pointillisme, logically stronger; and so I hope more interesting. Anyway, my campaign makes some novel proposals about positive extrinsicality: in this paper, I distinguish temporal and spatial (positive) extrinsicality; and in the companion paper against pointillisme in mechanics, I propose degrees of (positive) extrinsicality. 2.1.2 Distinction from three mathematical distinctions Both the murky intrinsic-extrinsic distinction, and the clearer distinction between positive extrinsics and the rest, are different distinctions from three that are made within mathematics and physics, especially in those parts relevant to us: viz. pure and applied differential geometry. The first of these distinctions goes by the name ‘intrinsic’/’extrinsic’; the second is called ‘scalar’/’non-scalar’, and the third is called ‘local’/’non-local’. They are as follows. (i) The use of ‘intrinsic’ in differential geometry is a use which is common across all of mathematics: a feature is intrinsic to a mathematical object (structure) if it is determined (defined) by just the object as given, without appeal to anything extraneous — in particular a choice of a coordinate system, or of a basis of some vector space, or of an embedding of the object into

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another. For example, we thus say that the intrinsic geometry of a cylinder is flat; it is only as embedded in R3 that it is curved. (ii) Differential geometry classifies quantities according to how they transform between coordinate systems: the simplest case being scalars which have the same value in all coordinate systems. (Nevermind the details of how the other cases — vectors, tensors, connections, spinors etc.— transform.) (iii) Differential geometry uses ‘local’ (as vs. ‘global’) in various ways. But the central use is that a mathematical object (structure) is local if it is associated with a point by being determined (defined) by the mathematical structures defined on any neighbourhood, no matter how small, of the point. In this way, the instantaneous velocity of a point-particle at a spacetime point, and all the higher derivatives of its velocity, are local since their existence and values are determined by the particle’s trajectory in an arbitrarily small neighbourhood of the point. Similarly, an equation is called ‘local’ if it involves only local quantities. In particular, an equation of motion is called ‘local in time’ if it describes the evolution of the state of the system at time t without appealing to any facts that are a finite (though maybe very small) time-interval to the past or future of t. I will not spell out seriatim some examples showing that the two philosophical distinctions are different from the three mathematical ones. Given some lessons in differential geometry (not least learning to distinguish (i) to (iii) themselves!), providing such examples is straightforward work. Suffice it to make three comments, of increasing relevance for this paper. (1) It would be a good project to explore the detailed relations between these distinctions. In particular, the mathematical distinction (i) invites comparison with Vallentyne’s (1997) proposal about the intrinsic-extrinsic distinction. Besides, there are yet other distinctions to explore and compare: for example, Earman (1987) catalogues some dozen senses of ‘locality’. But in this paper and its companion, two of the various differences amongst these distinctions are especially relevant. (2) The first is the difference between mathematical locality, (iii) above, and philosophical intrinsicality. The difference is clear for the case of instantaneous velocity. This is the main topic of my (2006); but the idea is that velocity has implications about the object at other times, for example that it persists for some time. So most philosophers say that instantaneous velocity is an extrinsic property. I agree. But emphasising its extrinsicness tends to make one ignore the fact that it is mathematically local, i.e. determined by the object’s trajectory in an arbitrarily small time-interval. And in pure and

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applied differential geometry, it would be hard to over-estimate the importance of — and practitioners’ preference for! — such local quantities and local equations involving them. (It is this locality that prompts me to speak of instantaneous velocity (and other local quantities) as ‘hardly extrinsic’.) (3): In this paper, we will also note the difference between being a mathematical scalar, (ii) above, and being philosophically intrinsic. Thus philosophers tend to think that any scalar quantity represents an intrinsic property of the points on which it is defined; (so that the pointilliste has only to worry about whether vectors, tensors etc. can represent intrinsic properties). But as we shall see in Section 4.4.1, that is wrong. For the scalar curvature R at a point p is surely extrinsic in the philosophical sense, since it gives information about the geometry of neighbourhoods of p. (R is also local and mathematically intrinsic; i.e. on the “intrinsic side“ of all three mathematical distinctions, (i)–(iii).)

2.2 Classical mechanics is not pointilliste, and can be perdurantist 2.2.1 Two versions of pointillisme To state my campaign’s main claims, it is convenient to first distinguish a weaker and a stronger version of pointillisme, understood as a metaphysical doctrine. They differ, in effect, by taking ‘point’ in pointillisme to mean, respectively, spatial, or spacetime, point. Taking ‘point’ to mean ‘spatial point’, I shall take pointillisme to be, roughly, the doctrine that the instantaneous state of the world is fully described by all the intrinsic properties, at that time, of all spatial points and/or pointsized bits of matter. As I said in Section 2.1, my campaign can mostly take ‘intrinsic’ to mean ‘lacking implications about some wholly distinct contingent object’; in other words, to mean the negation of Lewis’ ‘positive extrinsic’ (i.e. his ‘implying accompaniment’). But for this version of pointillisme, I will take ‘intrinsic’ to mean ‘spatially intrinsic’. That is, attributing such a property to an object carries no implications about spatially distant objects; but it can carry implications about objects at other times. (Such objects might be other temporal parts of the given object.) So I shall call this version, ‘pointillisme as regards space’. On the other hand: taking ‘point’ to mean ‘spacetime point’, I shall take pointillisme to be, roughly, the doctrine that the history of the world is fully described by all the intrinsic properties of all the spacetime points and/or

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all the intrinsic properties at all the various times of point-sized bits of matter (either point-particles, or in a continuum). And here I take ‘intrinsic’ to mean just the negation of Lewis’ ‘positive extrinsic’. That is, it means ‘both spatially and temporally intrinsic’: attributing such a property carries no implications about objects at other places, or at other times. I shall call this stronger version, ‘pointillisme as regards spacetime’. So to sum up: pointillisme as regards space vetoes spatial extrinsicality; but pointillisme as regards spacetime also vetoes temporal extrinsicality. On either reading of pointillisme, it is of course a delicate matter to relate such metaphysical doctrines, or the endurance-perdurance debate, to the content of specific physical theories. Even apart from Section 2.1’s questions of philosophical method, one naturally asks, for example, how philosophers’ idea of intrinsic property relates to the idea of a physical quantity. For the most part, I shall state my verdicts about such questions case by case. But one main tactic for relating the metaphysics to the physics will be to formulate pointillisme as a doctrine relativized to (i.e. as a property of ) a given physical theory (from Section 2.3 onwards). Anyway, I can already state my main claims, in terms of these two versions of pointillisme. More precisely, I will state them as denials of two claims that are, I think, common in contemporary metaphysics of nature. 2.2.2 Two common claims Though I have not made a survey of analytic metaphysicians, I think many of them hold two theses, which I will dub (FPo) (for ‘For Pointillisme’) and (APe) (for ‘Against perdurantism’); as follows. (FPo): Classical physics — or more specifically, classical mechanics — supports pointillisme: at least as regards space, though perhaps not as regards spacetime. There are two points here:— (a) Classical physics is free of various kinds of “holism”, and thereby antipointillisme, that are suggested by quantum theory. Or at least: classical mechanics is free. (With the weaker claim, one could allow, and so set aside, some apparently anti-pointilliste features of advanced classical physics, e.g. anholonomies in electromagnetism and the non-localizability of gravitational energy in general relativity: features rich in philosophical suggestions (Batterman 2003, Belot 1998, Hoefer 2000) — but not for this paper!) (b) The concession, ‘perhaps not as regards spacetime’, arises from the endurance-perdurance debate. For it seems that pointillisme as regards spacetime must construe persistence as perdurance; (while pointillisme as

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regards space could construe it as endurance). And a well-known argument, often called ‘the rotating discs argument’, suggests that perdurance clashes with facts about the rotation of a continuum (i.e. a continuous body) in classical mechanics. So the argument suggests that classical mechanics must be understood as “endurantist“. Besides, whether or not one endorses the argument, in classical mechanics the persistence of objects surely can be understood as endurance — which conflicts with pointillisme as regards spacetime. (The considerations under (a) and (b) are usually taken as applying equally well to non-relativistic and relativistic classical mechanics: an assumption I largely endorse.) I also think that many metaphysicians would go further and hold that: (APe) Classical mechanics does indeed exclude pointillisme as regards spacetime: their reason being that this pointillisme requires perdurance and that they endorse the rotating discs argument. So they hold that in classical mechanics the persistence of objects must be understood as endurance, and that this forbids pointillisme as regards spacetime. 2.2.3 My contrary claims I can now state the main position of my wider campaign. Namely, I deny both claims, (FPo) and (APe), of Section 2.2.2. I argue for two contrary claims, (APo) (for ‘Against Pointillisme) and (FPe) (for ‘For perdurantism’), as follows. (APo): Classical mechanics does not support pointillisme. By this I do not mean just that: (a) it excludes pointillisme as regards spacetime. Nor do I just mean: (b) it allows one to construe the persistence of objects as endurance. (But I agree with both (a) and (b).) Rather, I also claim: classical mechanics excludes pointillisme as regards space. That is: it needs to attribute spatially extrinsic properties to spatial points, and/or point-sized bits of matter. (But this will not be analogous to the kinds of “holism” suggested by quantum theory.) (FPe) Though (as agreed in (APo)) classical mechanics excludes pointillisme as regards spacetime (indeed, also: as regards space): classical mechanics is compatible with perdurance. That is: despite the rotating discs argument, one can be a “perdurantist” about the persistence of objects in classical mechanics. The reason is that once we reject pointillisme, perdur-

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ance does not need persistence to supervene on temporally intrinsic facts. In fact, perdurantism can be defended by swallowing just a small dose of temporal extrinsicality. So to sum up my wider campaign, I claim that:— (APo) Classical mechanics denies pointillisme, as regards space as well as spacetime. For it needs to use spatially extrinsic properties of spatial points and/or point-sized bits of matter, more than is commonly believed. (FPe) Classical mechanics permits perdurantism. It does not require temporally extrinsic properties (of matter, or objects), in the sense of requiring persistence to be endurance: as is commonly believed. A mild dose of temporal extrinsicality can reconcile classical mechanics with perdurance. To put the point in the philosophy of mind’s terminology of ‘wide’ and ‘narrow’ states, meaning (roughly) extrinsic and intrinsic states, respectively: I maintain that classical mechanics: (APo) needs to use states that are spatially wide, more than is commonly believed; and (FPe) does not require a specific strong form of temporal width, viz. endurance. With a small dose of temporal extrinsicality, it can make do with temporally quite narrow states — and can construe persistence as perdurance.

2.3 In more detail … So much by way of an opening statement. I will now spell out my main claims in a bit more detail: first (APo), and then, more briefly, (FPe). 2.3.1 Four violations of pointillisme I will begin by stating pointillisme as a trio of claims that apply to any physical theory; and making two comments. Then I list four ways in which (chrono)-geometry and classical mechanics violate pointillisme: three will form the main topics of this paper and its companion. The trio of claims is as follows: (a) the fundamental quantities of the physical theory in question are to be defined at points of space or of spacetime; (b) these quantities represent intrinsic properties of such points; (c) models of the theory — i.e. in physicists’ jargon, solutions of its equations, and in metaphysicians’ jargon, possible worlds according to the theory — are fully defined by a specification of the quantities’ values at all such points.

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So, putting (a)–(c) together: the idea of pointillisme is that the theory’s models (or solutions or worlds) are something like conjunctions or mereological fusions of “ultralocal facts”, i.e. facts at points. Two comments. First: the disjunction in (a), ‘at points of space or of spacetime’, corresponds to Section 2.2’s distinction between pointillisme as regards space, and as regards spacetime. Nevermind that it does not imply the convention I adopted in Section 2.2, that pointillisme as regards spacetime is a stronger doctrine: since it vetoes temporally extrinsic properties, as well as spatially extrinsic ones. The context will always make it clear whether I mean space or spacetime (or both); and whether I mean spatially or temporally extrinsic (or both). Second: Though I have not made a systematic survey, there is no doubt that pointillisme, especially its claims (a) and (b), is prominent in contemporary metaphysics of nature, especially of neo-Humean stripe. The prime example is David Lewis’ metaphysical system, which is so impressive in its scope and detail. One of his main metaphysical theses, called ‘Humean supervenience’, is a version of pointillisme: I will return to it in Section 3.2. When we apply (a)–(c) to classical mechanics, there are, I believe, four main ways in which pointillisme fails: or, more kindly expressed, four concessions which pointillisme needs to make. The first three violations (concessions) occur in the classical mechanics both of point-particles and of continua; the fourth is specific to continua. And the first two are addressed in this paper; the third is discussed in the companion paper (2006). (1) The first violation is obvious and minor. Whether matter is conceived as point-particles or as continua, classical mechanics uses a binary relation of occupation, ‘… occupies …’, between bits of matter and spatial or spacetime points (or, for extended parts of a continuum: spatial or spacetime regions). And this binary relation presumably brings with it extrinsic properties of its relata: it seems an extrinsic property of a point-particle (or a continuum, i.e. a continuous body) that it occupy a certain spatial or spacetime point or region; and conversely. Agreed, there is more to be said about this claim (as always in philosophy!): both about (a) the connections between the intrinsic-extrinsic distinction among properties and the classification of relations, and (b) how the individuation of spatial or spacetime points or regions might depend on matter (the “relational conception“ of space or spacetime). I will discuss (a) and (b), albeit briefly in Sections 3.3. But anyway, I will there endorse the claim. That is: the concession will remain in force: the pointilliste about

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classical mechanics should accept this binary relation of occupation, and the modicum of extrinsicality it involves. (2) Classical mechanics (like other physical theories) postulates structure for space and/or spacetime (geometry or chrono-geometry); and this involves a complex network of geometric relations between, and so extrinsic properties of, points. This concession is of course more striking as regards space than time: three-dimensional Euclidean geometry involves more structure than does the real line. This will be the main topic of this paper. (3) Mechanics needs of course to refer to the instantaneous velocity or momentum of a body; and this is temporally extrinsic to the instant in question, since for example it implies the body’s existence at other times. (But it is also local in the sense of (iii), Section 2.1.2.) So this second violation imposes temporal, rather than spatial, extrinsicality; i.e. implications about other times, rather than other places. This is the main topic of Butterfield (2006). But I should stress here that this third violation is mitigated for point-particles, as against continua. For a pointilliste can maintain that the persistence of point-particles supervenes on facts that, apart from the other violations (i.e. about ‘occupies’ and (chrono)geometry), are pointillistically acceptable: viz. temporally intrinsic facts about which spacetime points are occupied by matter. In figurative terms: the void between distinct point-particles allows one to construe their persistence in terms of tracing the curves in spacetime connecting points that are occupied by matter. I develop this theme in my (2005). On the other hand: for a continuous body, the persistence of spatial parts (whether extensionless or extended) does not supervene on such temporally intrinsic facts. This is the core idea of the rotating discs argument, mentioned in Section 2.2.2. To sum up: the rotating discs argument means that pointillisme fits better with point-particles than with continua. To put the issue in terms of Section 2.2’s two forms of pointillisme: the strong form of pointillisme, pointillisme as regards spacetime, fails for the classical mechanics of continua, even apart from the other concessions mentioned. (4) Finally, there is a fourth way that the classical mechanics of continua violates pointillisme: i.e., a fourth concession that pointillisme needs to make. Unlike the rotating discs argument, this violation seems never to have been noticed in recent analytic metaphysics; though the relevant physics goes back to Euler. Namely, the classical mechanics of continua violates (the weaker doctrine of ) pointillisme as regards space, because it must be formulated in terms of spatially extended regions and their properties and rela-

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tions. But in this paper, I set this fourth violation aside entirely; my (2006a) gives details. So to sum up these four violations, I claim (APo): classical mechanics violates pointillisme. This is so even for the weaker doctrine, pointillisme as regards space. And it is especially so, for the classical mechanics of continua rather than point-particles. 2.3.2 For perdurantism I turn to Section 2.2.3’s second claim, (FPe): that once pointillisme is rejected, perdurantism does not need persistence to supervene on temporally intrinsic facts, and can be defended for classical mechanics provided it swallows a small dose of temporal extrinsicality. About (FPe) I can be much briefer, since this paper will not need details. I will just identify this small dose: it is the extrinsicality of the third violation of pointillisme above — in particular, the presupposition of persistence by the notion of a body’s instantaneous velocity. Thanks to the rotating discs argument, ‘body’ here means especially ‘point-sized bit of matter in a continuum’; since for point-particles we can construe persistence as perdurance without having to take this dose. Elsewhere (2004, 2004a, 2006) I argue that for a “naturalist” perdurantist, this dose is small enough to swallow.

3. Can properties represented by vectors be intrinsic to a point? 3.1 Prospectus I turn to the geometrical structure of space, and/or the chrono-geometrical structure of spacetime. I will argue that this structure involves extrinsic properties, especially spatially extrinsic properties. I will undertake three specific tasks, in Sections 3.2, 3.3, 4 respectively. In Section 3.2, I present Lewis’ version of pointillisme. Though this version is in some ways logically stronger than I need, it is important to present it. Not only has it been a focus of recent metaphysical discussion; it is also needed for Section 4. In Sections 3.3 and 4, I argue in detail that pointillisme needs to be qualified to accommodate the structure of space and/or spacetime. I think the need for this qualification is uncontentious; in particular, it is agreed by

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Lewis. But how exactly to state the qualification is a matter that is both important and unresolved. It is important for three reasons. First, all physical theories of course appeal to space (and/or spacetime). Second, they all represent the properties that encode the structure of space or spacetime, with mathematical entities such as vectors, tensors, connections etc. So the question arises: can properties that are so represented be intrinsic to a point? The third reason is taken up in the companion paper (2006): it is that physical theories also represent the other properties they mention, i.e. properties of matter such as velocity, momentum etc., by such mathematical entities as vectors, tensors, connections etc. So the question — can properties represented by vectors, tensors etc. be intrinsic to a point? — is at the centre of this paper (and its companion). First, in Section 3.3, I will lead up to this question by discussing, in a broadly metaphysical way, how to represent the structure of space or spacetime. (I will concentrate on the notion of length, and so on space rather than spacetime; but this discussion carries over intact to the case of spacetime.) Once the question is posed, Section 4 addresses it in detail, using as a foil Bricker’s (1993). As I said in Section 1, Bricker’s paper illustrates how strongly some contemporary metaphysicians are attracted by pointillisme. For recognizing that they must accept vectorial properties in physical theories, and that these seem not to be intrinsic to points, they propose to save pointillisme by advocating a heterodox construal of the property. Thus in Section 4, Bricker will construe the metric tensor of differential geometry in terms of non-standard analysis. (And in the companion paper, Tooley and others will construe instantaneous velocity as intrinsic.) My own view will of course be that there is no need for such heterodoxy: instead, we can and should reject pointillisme. My discussion will be simplified by a restriction. I will consider only properties represented by vectors and tensors, which I will for short call vectorial properties and tensorial properties: not those represented by other mathematical entities such as connections. This restriction will be natural, in that: (i) vectors and tensors are about the simplest of the various mathematical entities that physical theories use to represent properties and relations — so they are the first case to consider; (ii) the restriction is common in the literature; in fact most of the authors I discuss (here and in the companion paper) consider only vectorial properties.

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3.2 Humean supervenience I will assume familiarity with the main ideas of Lewis’ metaphysical system, above all his notions of possible world and natural property. Central to this system is Lewis’ version of pointillisme, which he says (1994, p. 494) is inspired by classical physics. He calls this doctrine ‘Humean supervenience’. It is stronger than pointillisme as defined in Section 2.3, in that it is not relative to a theory. Roughly, it is relative to a possible world; (of course a metaphysician like Lewis who accepts the idea of a law of nature can link relativizations to a theory and to a possible world using the idea of the “complete“ theory of a world, say as an axiomatization of all its laws of nature). And Lewis claims that it holds at the actual world. The idea of Humean supervenience is that all truths supervene on truths about matters of local particular fact: where ‘matters of local particular fact’ is to be understood in terms of Lewis’ metaphysics of natural properties, with the properties having spacetime points, or perhaps point-sized bits of matter, as instances. Thus he writes that Humean supervenience … says that in a world like ours, the fundamental relations are exactly the spatiotemporal relations: distance relations, both spacelike and timelike, and perhaps also occupancy relations between point-sized things and spacetime points. And it says that in a world like ours, the fundamental properties are local qualities: perfectly natural intrinsic properties of points, or of point-sized occupants of points. Therefore it says that all else supervenes on the spatiotemporal arrangement of local qualities throughout all of history, past and present and future. (1994, pp. 225–226.)5 Humean supervenience, so defined, is not widely believed — few philosophers sign up to all the notions deployed in its statement. But it has been a natural focus of metaphysicians’ attention in the last twenty years, not least because Lewis has been the pre-eminent neo-Humean. In the literature, we can distinguish three broad groups of topics: (i) Issues about whether to analyse law, causation, chance etc., and “higher-level“ concepts about mind and language, in terms of the notions of Lewis’ framework. Lewis (1986, p. xi–xiv) sketches how his work on all these topics provides a “battle-plan”: i.e. roughly, a sequence of supervenience claims for these concepts. More generally, much literature of neo5

Cf. also his (1986, pp. ix–x).

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Humean stripe is concerned with how truths using familiar central concepts of common-sense knowledge and belief — concepts such as law, causation, the persistence of objects and mental and semantic concepts, such as belief and reference — might supervene on a basis acceptable to Humeans, though perhaps not exactly the basis proposed by Lewis. (For example: for laws, cf. Earman and Roberts (2006).) And some of these truths pose a challenge in that they seem not to thus supervene; cf. (iii) below. (ii) General metaphysical issues about the notions of Lewis’ framework, in particular possible worlds and natural properties, and/or about related notions. For example, one well-known issue is: can possible worlds and natural properties be construed less “realistically” than Lewis proposes (e.g. Taylor 1993), and yet do the philosophical work they are meant to do? More relevant to us will be Lewis’ view of the intrinsic-extrinsic distinction, viz. that it can be analysed in terms of perfectly natural properties; (details in Section 4.2). But as discussed in Sections 2.1.1 and 2.2.1, I can for the most part use only a much clearer distinction, viz. between the positive extrinsic properties and the rest, sub-divided in terms of temporal and spatial implications (or lack of them). (iii) Direct threats to Humean supervenience. There are two main examples. First, chance; which Lewis addresses in detail in (1986, pp. xiv–xvi, 121–131), and to his greater satisfaction in (1994). Second, persistence. For Lewis as a Humean wants to be perdurantist, as well as pointilliste in the sense of Humean supervenience: this means that he faces the rotating discs argument. In this paper, I can set aside all of (i) and (iii), and all of (ii) except for the intrinsic-extrinsic distinction.

3.3 Accommodating space and spacetime 3.3.1 An agreed concession As I mentioned, pointillisme’s need to accommodate the structure of space and/or spacetime is agreed by all parties: in particular, by Lewis. In both the quotations above, Lewis includes relations of spatiotemporal distance (spacelike and timelike) in the supervenience basis. So his Humean supervenience is not so pointilliste, at least as regards the structure of space and/or spacetime, as it might at first seem. But no doubt even the most ardent pointilliste will find the inclusion sensible. That is: no one will hold that the structure of space and/or spacetime,

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in particular spatial and/or spatiotemporal metrical relations, is to supervene on intrinsic properties of points.6 The natural thing to say is, instead, that the points and these metrical relations (and maybe also some spatiotemporal but non-metrical relations) form collectively a background, or canvas, on which other physical quantities taking various values get “painted”. And it is to these latter that pointillisme’s doctrines are to apply. But there is no consensus (indeed, not much discussion) about how precisely to state the concession. More’s the pity, since apart from the concession’s own importance, it leads to the more general question (taken up in the following Subsections) how pointillisme can accommodate vectorial and tensorial properties. I begin with a preliminary issue. The concession obviously relates to the debate between relationist and substantivalist views of space and spacetime; and though I will not pursue this debate, I should register that this concession is not meant to prejudge it. There are two points here. (i) Though I spoke like a substantivalist, about a canvas of points, with various metrical and non-metrical relations between them, it is safe to assume that a relationist would appeal to similar relations holding between items of their preferred ontology, i.e. bodies. (I set aside whether Leibniz’s monads with only their intrinsic properties might be enough to subvene all spatial and spatiotemporal facts!) (ii) Similarly, my talk of a canvas of points, with metrical and non-metrical relations between them, was not meant to deny that the metric (or the other relations) could be dynamical, i.e. influenced by matter, in the way they are in general relativity. So our question is what exactly metrical (and other geometrical) structures require. As physical geometry has developed in the last two hundred years, these requirements have not only become subtler but have also become bound up with other properties and relations, especially of matter, in 6

Here I recall this paper’s restriction to classical mechanics. So I of course set aside speculations in quantum gravity that classical spacetime structure emerges somehow from a “quantum pre-geometry“: speculations which, I agree, might have this structure emerge from (or even supervene on) intrinsic properties of some pointlike objects. But I doubt it: most schemes for quantum pre-geometry are thoroughly “relational“ rather than pointilliste. For surveys of such speculations, cf. e.g. Monk (1997), Butterfield and Isham (1999, Sections 1–4; 2001, Section 5); for a brief discussion in relation to Lewis’ Humean supervenience, cf. Oppy (2000, p. 88, 91–94).

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ways which threaten pointillisme. This issue will extend to Section 4. But let us start by raising the issues involved in as simple a context as possible: the length of a straight line in elementary geometry. 3.3.2 The length of a line The length of a line is the topic of a venerable paradox. The length of a straight line should surely be the sum of the lengths of a decomposition, i.e. an exhaustive set of mutually non-overlapping parts; and it seems legitimate to take as these parts the line’s constituent points; but the length of each point is zero, and the sum of all these zeroes is presumably (though a continuously large sum) zero — what else could it be? So the length of the line is zero! I stated this paradox in its most familiar form, as about summing lengths. But of course it can also be stated in philosophical terms, as about supervenience: the length of a line surely does not supervene on the lengths of its points, on pain of being zero. That is no doubt why, as discussed in Section 3.3.1, no pointilliste holds that lengths (or other metrical properties of lines, or indeed metrical relations between points) supervene on intrinsic properties of points.7 This paradox is of course one of many that eventually prompted the development of measure theory. And as noted in (2i) and (c) of Section 1, measure theory invites philosophical scrutiny because (i) it has some wellnigh paradoxical results of its own, like the Banach-Tarski paradox, and (ii) it is connected to mereology. But I shall not need details about these topics. I only need to present: (a) a philosophical reply to this paradox; though it does not block the paradox, it introduces an important metaphysical trichotomy among relations; (b) the main idea of the technical measure-theoretic reply to this paradox. 3.3.2.A The philosophical reply. The philosophical reply is just that length is a property of the line as a whole, where ‘line as a whole’ can be taken to mean either the set, or the mereological fusion, of its points (or of 7

There is of course a similar failure of supervenience for an extensive quantity such as mass, applied to a continuous body. For the point-sized parts of such a body have zero mass, so that the mass of the body is not the (uncountable) sum of the masses of those parts. Hawthorne notes this, while assessing how Lewis’ metaphysics treats quantities (2006); and we will return to it in Section 4.6.

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its extended parts). That is, the length does not supervene on the properties of the points (or other parts). This is surely true, so far as it goes. But it is not enough to block the paradox, since it does not pinpoint what is wrong with the premise that the length of a straight line is the sum of the lengths of an exhaustive set of mutually non-overlapping parts. (The technical reply will do this.) However, this reply prompts a trichotomy among relations corresponding to the intrinsic-extrinsic dichotomy among properties: a trichotomy that will be useful in what follows. Lewis states the trichotomy clearly (1983a, p. 26 fn. 16; 1986a, p. 62); and I shall adopt his proposed terminology (which has become widespread). Though he explains it in terms of his preferred understanding of intrinsic and extrinsic properties (viz. defined in terms of his natural properties), the trichotomy can be explained in the very same words, using other understandings of intrinsic and extrinsic. In particular, it can be thus explained using Section 2.1.1’s suggested understanding of ‘extrinsic’ as ‘positive extrinsic’ and ‘intrinsic’ as ‘not positive extrinsic’; (or using Lewis’ “second favourite” analysis developed by Langton and him (1998, p. 129)). The trichotomy also uses the idea of the mereological fusion, or composite, of objects: an idea I am happy to accept, and for which there is a powerful argument (Lewis 1986a, pp. 212–213, developed by Sider 2001, pp. 121–139). (1) An internal relation is determined by the intrinsic properties of its relata. So if xRy, and x′ matches x in all intrinsic properties, and y′ matches y in all intrinsic properties, then we must have x′Ry′. So any relation of similarity or difference in intrinsic respects is internal; for example, if height is an intrinsic property, then ‘being taller than’ is an internal relation. (2) On the other hand, there are relations, notably relations of spatiotemporal distance, that are not internal, but do supervene on the intrinsic nature of the composite (mereological fusion) of the relata. Thus suppose x, y are point-particles 1 metre apart. Then it seems reasonable to say both of the following:— (i) There could be point-particles x′, y′ that intrinsically match x and y respectively, and that are 2 metres apart — so that distance is not internal. But on the other hand: (ii) Any object intrinsically matching the fusion or composite of x and y would have two parts intrinsically matching x and y, 1 metre apart. Accordingly, Lewis calls relations that supervene on the intrinsic nature of the fusion of the relata, external; and he takes (ii) to show that spatiotemporal relations are external.

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(3) Finally, there are relations that do not supervene even on the intrinsic nature of the composite of the relata; i.e. relations that are neither internal nor external. Lewis’ example is the relation having the same owner: x and y could intrinsically match x′ and y′ respectively, and their composites might also match; and yet x and y might have the same owner, while x′ and y′ do not. But more relevant to us than ownership, geometry and mechanics provide many examples of such relations. The objects x, y, x′, y′ could be solid bodies, again with each pair, x, x′ and y, y′, intrinsically matching, and the composites x + y, x′ + y′ also matching — and yet the centre of mass of x and y might be a certain distance from a body of some kind Z, while the centre of mass of x′ and y′ is not. 3.3.2.B The technical reply. The technical reply to the paradox of length comes from measure theory. It blocks the paradox by denying the premise that the length of a straight line is the sum of the lengths of any decomposition (exhaustive set of mutually non-overlapping parts) of the line. It upholds this only for certain decompositions. The main idea is to consider only decompositions containing points and intervals, and to accept the additivity of length for at most denumerably large sums. These ideas give a rich theory which can be extended to cover area and volume, as well as length; and which underpins the theory of integration. But we do not need further details of measure theory. For us the point is that, even in the elementary geometry of Euclidean space (R, R2 etc.), we cannot say all that is true in terms just of intrinsic properties of points. For we need to assign lengths to spatial intervals. And — to use Lewis’ terms — the length of an interval is surely not an internal relation between the interval’s end-points, since any two points seem to match in intrinsic properties. Besides, the length of an interval does not seem to be an internal relation between all the interval’s uncountably many points, since: (i) the points of any two intervals seem to match pairwise in intrinsic properties, and (ii) additivity of length fails for an uncountable set. On the other hand, it seems reasonable to say, as Lewis does, that: (a) intervals are composites or fusions of their points; and (b) intervals matching in their intrinsic properties are congruent; (cf. (2) (ii) above). If we say (a) and (b), then it follows that the length of a straight interval is an external relation among the interval’s points. (So far, this is a relation of uncountable polyadicity: the next Subsection will ask whether the relation can

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be taken to be just a dyadic relation between the interval’s two end-points.) To sum up: to describe length, even the length of a straight interval in Euclidean geometry, pointillisme must concede that it needs to go beyond intrinsic properties of points, and even relations that are internal in Lewis’ sense. But so far, it seems (cf. (2)(ii) in Section 3.3.2.A and (b) above) it can manage with what Lewis calls external relations.8

3.3.3 Accommodating more geometry The further development of geometry, including more general geometries than the Euclidean line or plane, reinforces the point that we cannot say all that is true in terms just of intrinsic properties of points. But as we shall see, it is doubtful that we can manage with just Lewis’ category of external relations. That is, it is doubtful that all pointillisme needs to do, in order to accommodate spatial and spacetime structure, is to admit the network of external relations of spatial and spatiotemporal distance. Talk of ‘spatial and spatiotemporal relations’ tends to suggest that space or spacetime is a metric space, in the usual mathematical sense that (given a unit of length) there is a real-valued function on pairs of points: to any pair of points x, y is assigned a distance d(x, y) ∈ R. (So each real number determines a binary relation on points; and for a relativistic spacetime, d need not be positive-definite.) But the development of geometry and physics has shown that this is much too limited a conception of spatial (or spatiotemporal) structure. One needs to distinguish various subtly related levels of structure: for example, geometers distinguish topological structure, differential structure, metrical structure, and many more. Besides, most of these kinds of structure include definitions of irreducibly global features of the space concerned; for example, a space can have the global topological feature of being simply connected, i.e. such that all closed curves can be continuously deformed to a point. 8

Incidentally, returning to the original paradox of length: the practice of measure theory seems indifferent between the following options (and surely metaphysicians can be as well): a) to allow ab initio that each point has a length, viz. zero; and avoid paradox by denying uncountable additivity of length; b) to ascribe length primarily only to sets of points in a certain well-behaved family of sets. But the technical need for the family to have certain closure properties is likely to lead to singleton sets of points being included — as in the usual choice of family, the Borel sets.

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But here I will focus only on “local” metrical structure.9 Even without developing the formal details (which go back to Gauss and Riemann), we will be able to see that geometry (and more generally physics) needs to attribute to points both vectorial and tensorial properties — raising the question, pursued in the next Subsection, whether pointillisme can accommodate such properties. Gauss and Riemann proposed that we take as the primary notion, the length of a curve between two points; (so since a pair of points is in general connected by infinitely many curves, any such pair is associated with infinitely many lengths, not just one). This proposal is adopted by modern spacetime theories, in particular by the most successful such theory, general relativity. So an advocate of pointillisme (or some similar doctrine of local supervenience, such as Humean supervenience) would do well to formulate their doctrine so as to incorporate, or at least allow for, this proposal. At first sight, it seems that the pointilliste can manage just fine. She only needs to apply Lewis’ idea of external relations (or perhaps, some similar notion), not to the endpoints of a straight interval or to all the points of a straight interval, but to all the points of an arbitrary curve. Thus she can take an arbitrary curve as the fusion of its points, and the length of the curve as an external relation, albeit of uncountable polyadicity, among the points: an external relation which determines an intrinsic property of the fusion. I presume that Lewis, who was well aware of the Gauss-Riemann conception of geometry, would have said this. That is, he would have taken ‘spatiotemporal distance relations’ in his definition of Humean supervenience to allow for this conception — and not to be committed to the idea of a metric space.10 Besides, the structure required to define the lengths of all curves is given “locally” in a way that at first sight seems congenial to a pointilliste. In particular, it seems that the pointilliste does not need to postulate continuously many external relations, each of uncountable polyadicity, one for each congruence-class of curves. For: 9

I put ‘local’ in scare-quotes, since I here intend a vaguer meaning than that of (iii) of Section 2.1.2. But in what follows, the meaning will always be clear from the context. Note also that though I will discuss only space, all I say carries over intact to spacetime. 10 But Lewis seems never to have pursued the question exactly what relations he should propose as fundamental for modern geometry and topology; and we shall shortly see trouble for him, as for other pointillistes.

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(i) measure theory can be extended to apply to this kind of geometry, so that the length of a curve is the sum of the lengths of a countable decomposition of it; (ii) similarly, calculus can be extended to apply to this kind of geometry, so that the length of a curve is given by an integral along it — intuitively, an uncountable sum of infinitesimal contributions one for each infinitesimal element of the curve. These features, (i) and (ii), reflect the fact that the Gauss-Riemann conception of metrical structure presupposes topological and differential structure: which make sense, respectively, of the notions of continuous function, and differentiable function. But there is a devil in the details. The details of how to define the length of a curve require us to attribute vectors and tensors to a point. A bit more precisely: we need to attribute: (i) to any point on any curve, the tangent vector to the curve at that point; and so (ii) to any point, the set of all such tangent vectors at it (which form a vector space, called the tangent space); (iii) to any point, a metric tensor which maps pairs of tangent vectors at the point to real numbers — generalizing the elementary scalar product of two vectors. (More details about (i)–(iii) in the next Subsection.) Thus the pointilliste has to face — already in geometry, even before considering physics’ description of matter — the question announced in Section 3.1: can a property represented by a vector or a tensor be intrinsic to a point?

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Accommodating tangent vectors and the metric tensor

4.1 Bricker and others As I announced in Section 3.1, my main effort in this Section will be to report and criticise Bricker’s (1993) discussion of this question, for metrical properties. I choose him for two reasons. (i) His paper is an unusually thorough and perceptive attempt to relate vectors and tensors, as they are treated in modern geometry, to the modern metaphysics of properties. So it repays detailed scrutiny. (ii) His paper illustrates the tendency that, as I said at the end of Section 3.1, I want to reject: the tendency of some contemporary metaphysicians to reconcile pointillisme with physical theories’ use of vectorial properties,

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which seem extrinsic to points, by proposing some heterodox construal of the properties in question. Bricker proposes that in order to understand space’s or spacetime’s metrical structure as intrinsic, we should appeal to non-standard analysis. I will deny this heterodoxy: instead, we can and should reject pointillisme. But before going into the details of Bricker’s discussion, I should register that other metaphysicians have also addressed our question; though (so far as I know) more briefly and with less attention to technicalities than Bricker. (Besides, they are not all attracted by pointillisme, or by the above tendency.) For example, Robinson maintains that the directionality of a vector forbids it from representing an intrinsic property: “direction seems to me an inherently relational matter” (1989, p. 408). And he would presumably say the same about tensorial properties. (His paper is about the rotating discs argument; I discuss its proposals in 2006.) Robinson gives an argument for this, using Lewis’ notion of duplicates, i.e. objects that share all their intrinsic properties. He also credits Lewis for the argument; so presumably Lewis himself thought at the time (ca. 1988) that vectorial properties could not be intrinsic. The argument combines two intuitions: (a) It seems that a vectorial property could not be instantiated in a zerodimensional world consisting of a single point; though since arbitrarily close points define a direction, there is of course no lower limit to the “size“ of a world in which a point instantiates a vectorial property. (b) But it also seems that, since a point in an extended world that instantiates a vectorial property is indeed a point, it could have a duplicate that existed on its own, i.e. was the only object in its world. Taken together, (a) and (b) imply that duplicate points might differ in their vectorial properties; so that any such property is not intrinsic.11 But for anyone who is attracted by pointillisme, and is aware of physical theories’ use of vectorial properties, this is a very uncomfortable conclusion. Lewis himself is a case in point. Indeed, he seems to have come round to 11 Other metaphysicians also maintain that vectorial properties are extrinsic to points. For example, Black (2000, p.103) holds that vectorial properties can be intrinsic only for the special case of vectors on a manifold with a flat connection; i.e. roughly, a manifold in which there is a unique preferred way to compare vectors located at different points. (His discussion is briefer than Robinson’s; since the topic is less relevant to his paper’s main aims, than it is to Robinson’s.) And Zimmerman (1998, p. 277–278) and Oppy (2000, pp. 79–82) are similarly inclined; though they also discuss sympathetically the opposing view.

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believing that vectors can represent intrinsic properties of points, sometime between ca. 1988 and ca. 1993. For in a discussion of Humean supervenience (1994, p. 474), he says he is inclined to think that vectorial properties are, or at least can be, intrinsic: “any attempt to reconstrue them as relational properties seems seriously artificial”. But, so far as I know, that is all Lewis says by way of defending the idea; (though in his (1999) he used the idea to try and reply to the rotating disc argument — unsuccessfully I maintain (2006)). In any case, I now turn to Bricker’s extended struggle to avoid the uncomfortable conclusion.

4.2 Bricker’s three claims about metrical structure 4.2.1 Bricker’s metaphysical framework Bricker’s (1993) overall aim is metaphysical understanding of spatial (or spatiotemporal) relations. He adopts a metaphysical framework very close to Lewis’— with of course all due acknowledgement (1993, pp. 273–5). The ingredients we need are:— (i) He speaks of possible worlds and perfectly natural properties and relations. He applies mereology freely to points of space and spacetime; (in fact, substantivalism about space and spacetime is widespread among analytic metaphysicians). And so he takes worlds and parts of worlds as possibilia. (ii) He says that any two possibilia X, Y are duplicates iff there is a one-toone correspondence between their parts that preserves all perfectly natural properties and relations. He calls any such correspondence an (X, Y ) counterpart relation, and corresponding parts are (X, Y )-counterparts of each other. (So in this Subsection, ‘counterpart’ is tied to ‘duplicate’ and so will not have the usual Lewisian connotations of allowing vagueness and extrinsicness.) (iii) He says that a property is intrinsic iff any two duplicates both have it or both lack it. (Otherwise the property is extrinsic.) It follows that: (a) the class of all possibilia is partitioned by the equivalence relation of being duplicates; and (b) an intrinsic property corresponds to a union of cells of this partition; and (c) all perfectly natural properties are intrinsic. He extends the notion of intrinsic to relations by saying that a relation is intrinsic iff it is either internal or external in the senses of Lewis (defined in Section 3.3.2.A); otherwise the relation is extrinsic.

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(iv) He assumes (following Lewis 1986a, pp. 86–92) a principle of recombination for spatial or spacetime points. This is a principle of modal plenitude, inspired by a Humean denial of necessary connections between distinct existences: “anything can follow anything”. Stated for spatial points, it holds: for any points p and q, perhaps from spaces of different worlds, there is a world whose space is a duplicate of the space of p, except that it contains a duplicate of q where the duplicate of p would be (1993, p. 290); and similarly for spacetime. I do not endorse this framework. But in discussing Bricker, I will use it (and a variant of it considered by him). Though it would be a good project to ascertain how well Bricker’s arguments fare under a different framework (in particular, under weaker assumptions about the intrinsic-extrinsic distinction), it is not a project for this paper. Here it must suffice to note that if we used my distinction between positive extrinsics and the rest, advocated in Section 2.1.1, the main points of my critique of Bricker below, would carry over intact. But I shall not spell this out point by point, from now on. I just note here that: (a) Since my distinction takes ‘intrinsic’ to mean ‘not positively extrinsic’, it yields more intrinsic properties than does Bricker’s (or Lewis’) framework; and so a logically stronger notion of duplicatehood as sharing of all intrinsic properties. (b) Bricker’s argument for the spacetime metric being extrinsic to points (Section 4.3) remains valid on my distinction’s construal of ‘extrinsic’ as ‘positive extrinsic’. For Bricker’s argument implicitly appeals to positive extrinsicality. (c): My anti-pointilliste reply to Bricker (Section 4.6) is unaffected by adopting my distinction. Bricker goes on to connect his framework with the Gauss-Riemann conception of distance, as endorsed by general relativity. His discussion includes aspects (1993, pp. 275–286) which we can skip, in particular: (a) a comparison with two other conceptions of distance (which he dubs the ‘naive’ and ‘intrinsic’ conceptions); and (b) a discussion of how the principle of contact-action (denial of action-at-a-distance) bears on the the Gauss-Riemann conception. Setting these aside, I read Bricker as connecting his metaphysical framework with the Gauss-Riemann conception, as follows. He assumes that: (i) the perfectly natural properties and relations, that are instantiated at a possible world that has laws of nature, figure in that world’s laws (however the notion of a law of nature is to be analysed);

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(ii) general relativity is a logically possible theory, giving the gravitational and metrical laws of some possible worlds; (iii) general relativity can be “formulated locally”; which is taken to imply, as regards metrical structure, formulated in terms of local metrical relations. Taking these assumptions together, he concludes that the property of having such-and-such a local metric tensor is a perfectly natural property, and is instantiated at points in general relativistic worlds. So far, Bricker is in a position like the one we articulated at the end of Section 3.3, in which the pointilliste seemed well able to manage local metrical structure. Bricker also notes (as we did) that since even topology brings in irreducibly global properties of space like being simply connected, there can be no sweeping supervenience of the global on the local. So he formulates a doctrine he calls Einsteinian supervenience, on analogy with Lewis’ Humean supervenience: there is “a manifold of spacetime points … and a distribution of perfectly natural local properties (including local metrical properties) over those points; all else supervenes on that” (1993, p. 288). 4.2.2 Bricker’s three claims Bricker then notices what I called ‘the devil in the details’, i.e. the fact that local metrical structure attributes vectorial and tensorial properties to points; and he goes on to address the question whether such properties are intrinsic, in terms of his metaphysical framework (1993, pp. 288f.). He argues for the following three claims (in order, with my added mnemonic labels). (MetrExtr) The metric tensor, as standardly conceived in differential geometry, represents an extrinsic property of a point. (VetoExtr) The obvious (and anti-pointilliste) response to the conflict between this and the metric being perfectly natural — viz. that some but not all perfectly natural properties are intrinsic — does not work. For, Bricker argues, it clashes with the Humean principle of recombination for spacetime points. That is, Bricker rejects this response as engendering necessary connections between distinct existences, viz. a point and its surrounding space. So Bricker claims we do better to revise our conception of the metric tensor, as follows. (Heterodox) We should take the metric to represent an intrinsic property of an infinitesimal neighbourhood of a point. Bricker cites Robinson’s non-standard analysis as justifying taking such neighbourhoods as genuine

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mathematical objects, rather than as a facon de parler for calculus’ standard notion of limit as “∀∃∀” (e.g. for a real sequence {an}: ∀ ε > 0 ∃N ∀ m, n > N | am − an | < ε). So Bricker’s overall conclusion is radical: that in order to save pointillisme, we should revise the foundations of differential geometry. In the next three Subsections, I will report his arguments for (MetrExtr) to (Heterodox). Then in the last Subsection (Section 4.6), I will deny his conclusion. Since I hold no brief for pointillisme, I see no reason to pay his price of revising the foundations of differential geometry.

4.3 The standard metric is extrinsic Bricker’s argument for (MetrExtr) — the metric tensor, as standardly conceived, represents an extrinsic property of a point — is not absolutely precise. But it uses more technicalities about local metrical structure than I have introduced so far, in particular differential geometry’s idea that the tangent vectors at a point be taken to be directional derivative operators. So I need to review this; I shall give rather more detail than Bricker does. (i) First, the set of spatial or spacetime points is assumed to form a manifold M. The definition of ‘manifold’ is elaborate, and was only given in its modern formal guise in the 1930s — and fortunately I can skip it! It suffices to say that the definition gives sense to various crucial ideas such as the dimension of a manifold, its boundary (if any), its global topological structure (e.g. being simply connected), the idea of a smooth scalar function i.e. a smooth real-valued function defined on a subset of the manifold — and most important for us, the idea of a smooth curve in the manifold, which is taken as a map q from an interval of real numbers I ⊂ R to M. (Here ‘smooth’ refers to differentiability a specified number of times.) As I said in Section 3.3, the pointilliste will be hard pressed to account for this manifold structure: but I will not labour this point. (ii) Any curve q thus includes in its definition its real-number parameter, λ say. So, understanding the tangent vector to the curve at the point q(t), t ∈ I, in an intuitive way: the tangent vector specifies a directional derivative of any scalar function f defined on a neighbourhood, N say, of the point q(t), f : N ⊂ M → R. (For the direction of the curve at q(t), together with the “rate at which λ ticks away”, defines an “instantaneous rate of change” of f.) (iii) It is convenient to identify the tangent vector to the curve q at the point q(t) with the directional derivative operator acting on the set, R say,

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of all scalar functions defined on some neighbourhood of the point q(t): d df | :f∈R | ∈ R. d λ q(t) d λ q(t) Why is it convenient? In short: because the directional derivative operators behave just like tangent vectors. For example, for an n-dimensional manifold M, the directional derivative operators at any point p ∈ M form an n-dimensional vector space, just as one would want the tangent vectors to do: think of the 2-dimensional tangent plane at a point p on the surface of a sphere. This vector space is called the tangent space at p, Tp. (Other equivalent identifications are also used: some presentations identify a tangent vector at p ∈ M with an equivalence class of curves through p — intuitively, curves that are all tangent to each other at p and with parameters “ticking” at the same rate.) (iv): To define the length of a curve requires still further structure: structure which is not fixed by the postulation of a manifold, with all its tangent vectors V ∈ Tp at each point p. Namely, it requires a metric tensor g, which is an assignment to each point in p ∈ M of a mapping from pairs of vectors 〈U, V 〉 with U, V ∈ Tp to R: a mapping of a certain sort that generalizes the elementary scalar product of vectors. So g : 〈U, V 〉 g(〈U, V 〉) ∈ R. This metric tensor applied to the pair 〈V, V 〉, where V is the tangent vector to a curve q passing through p, gives in effect the squared length of the “infinitesimal part” of the curve at p. Now, if we let p vary from one point of the curve to another and add up the corresponding contributions, we are performing an integration. So integrating (the square-root of ) g(〈V, V 〉) gives the length of the curve. One can prove that (as one would want) the length of a curve depends on the metric tensor used, but not on how the curve is parameterized. To connect (i)–(iv) with Bricker’s claim (MetrExtr), one needs some “bridge-principles” between the mathematical constructions and philosophical notions such as that of an intrinsic property. For this, Bricker proceeds as follows. He defines (1993, p. 289) a property P of points to be local iff for any points p, q, any neighbourhood N of p and any neighbourhood M of q: if N is a duplicate of M, and p is an (N, M)-counterpart of q, then P holds either of both p and q or of neither (i.e. p and q match as regards P). So, roughly speaking, Bricker calls a property P of points ‘local’ if whether a point p possesses P is wholly determined by the intrinsic nature of any arbitrarily small neighbourhood of p. So, modulo the use of metaphysical ideas of intrinsic property, duplicatehood etc., this usage clearly corresponds to mathematicians’ use of ‘local’ (cf. (iii) in Section 2.1.2).

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It follows that for Bricker any intrinsic property of points is local, since counterpart points, being duplicates of each other, share all their intrinsic properties. But, Bricker maintains, the converse fails: there are local but extrinsic properties of points. These he dubs neighbourhood-dependent. He briefly discusses as examples from elementary calculus, derivatives of functions, in particular instantaneous velocity. He says the instantaneous velocity of a point-particle at position x at time t, i.e. at a spacetime point p, depends on where the particle is at other times; and so is a neighbourhood-dependent, but not intrinsic, property of p. The ‘so’ here is not spelt out precisely, i.e. by justifying the implicit premise about duplicate spacetime regions containing the particle (or its counterpart). But Bricker’s intuition is clear enough: as we emphasised already in Section 2.1.2 and 2.3.2, instantaneous velocity and momentum are temporally extrinsic since for example they imply the object’s existence at other times. Besides, the intuition is shared by others — as we will see when we return to instantaneous velocity in the next Section. Bricker goes on to claim by analogy that in differential geometry all the tangent vectors at a point p ‘give information not just about p, but about the space immediately surrounding p … in short … neighbourhood-dependent information about p’. To which he adds: “since the local metric at p is an operator on tangent vectors, it inherits neighbourhood-dependence from its operands” (1993, p. 289). Again, Bricker’s argument here is not entirely precise. He cannot really prove that any property represented by an element of tangent space is extrinsic; for his metaphysical apparatus does not tie its notions of perfectly natural property, and so duplicate, and so (X, Y )-counterparthood, sufficiently tightly to the notions of differential geometry. A footnote admits that (as in my (iii) above), tangent vectors are directional derivative operators; but again there is no justification for the implicit premise about duplicate spacetime regions. But fair enough, I say: his intuition is again both clear and shared by others. And the intuition is enough to deliver Bricker his problem. That is: the metric’s being neighbourhood-dependent contradicts the previous claim that it is perfectly natural (i.e. perfectly natural because mentioned in the laws of general relativity) — once we recall that according to his metaphysical framework, all perfectly natural properties are intrinsic. In response, Bricker considers two tactics for escape from contradiction: an obvious one which he rejects in the second stage of his argument

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(Section 4.4); and an unobvious one which he endorses in the last stage (Section 4.5).

4.4 Vetoing perfectly natural extrinsics Bricker now argues for: (VetoExtr): The obvious anti-pointilliste response to the contradiction between the metric being neighbourhood-dependent and perfectly natural — viz. that some but not all perfectly natural properties are intrinsic — clashes with the Humean principle of recombination for spacetime points. That is: it engenders necessary connections between a point and its surrounding space. Bricker first considers saying that some but not all perfectly natural properties are intrinsic. So the idea is that the perfectly natural but extrinsic properties of points include vectorial and tensorial properties, like having a metric tensor with such and such features. Bricker notes that this response implies that his previous definition of ‘duplicate’ bifurcates into a weaker and a stronger notion. The weaker notion is that of intrinsic duplicates: this requires only that the one-one correspondence between the parts of objects X and Y preserve the intrinsic perfectly natural properties and relations. (Recall that Bricker calls a relation ‘intrinsic’ iff it is internal or external in Lewis’ sense, given in Section 3.3.) The stronger notion, which Bricker calls local duplicates, has the same definition, word for word, as the previous definition of duplicates: X and Y are local duplicates iff there is a one-one correspondence between their parts preserving all perfectly natural properties and relations. Bricker proposes that we now define a local property as one that never differs between local duplicates. So it is now built in to the definitions that perfectly natural properties are local — just as previously it was built in that they were intrinsic. Returning to geometry, the idea will be that such perfectly natural, and so local, properties include vectorial and tensorial properties, like having a metric tensor with such and such features. So far, so good. But there is a clash with Bricker’s Humean principle of recombination for points ((iv) of Section 4.2.1): that for any points p and q, there is a world whose space is a duplicate of the space of p, except that it contains a duplicate of q where the duplicate of p would be. More precisely: Bricker says there is a dilemma. For this principle must now refer either to (A) local duplicates, or to (B) intrinsic duplicates: and on either interpretation, Bricker sees trouble. I will reply that the second interpretation, (B), is fine — provided one is not a pointilliste.

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4.4.1 Trouble with local duplicates If the principle of recombination refers to local duplicates, then it will yield contradictory worlds when p and q have contrary perfectly natural, extrinsic (but of course local) properties. Bricker gives as his example positive and negative curvature. He writes: “suppose that p is surrounded by positively curved space, q by negatively curved space. Then a world whose space is a duplicate of the space of p but with a local duplicate of q in p’s place must be both positively and negatively curved in the immediate neighbourhood of q” (1993, p. 290; the last phrase of course means ‘immediate neighbourhood of the duplicate of q’). Here I should amplify Bricker’s example — and point out a problem raised by it. Given a metric, one can define a scalar function, in the usual mathematical sense of ‘scalar’ (viz. a function from the manifold M to R, so that its value at a point p ∈ M is the same, independently of any choice of coordinate system), called the scalar curvature R, that has the following remarkable property: although it is a scalar, at each point p its value R(p) is a numerical measure of how curved is the geometry in a neighbourhood of p. (In fact a metric is sufficient but not necessary to define R: a connection also allows one to define scalar curvature.) So Bricker is no doubt here assuming that: (i) p and q have positive and negative scalar curvature, respectively, i.e. R(p) > 0 and R(q) < 0; (and if we like, we can take him to assume that all points in their respective neighbourhoods have positive and negative scalar curvature); (ii) the scalar curvature R is perfectly natural but extrinsic: (more precisely, it is a determinable whose determinates, given by specific values R(.) = 5 etc., are perfectly natural but extrinsic). Assumption (ii) raises a problem. Hitherto, we have implicitly assumed that scalar functions represent intrinsic properties of points: our worries have concerned only vectorial and tensorial properties. Now we see there is also a gap between: (a) the mathematical notion of a scalar, which is a matter of how a quantity transforms (viz. trivially: it takes the same value in all coordinate systems); and (b) the metaphysical idea of intrinsicness. That is: some scalars can “give information not just about p, but about the space immediately surrounding p”— to quote Bricker’s words from his discussion of tangent vectors (quoted in Section 4.3’s discussion of Bricker’s

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(MetrExtr)). So Bricker owes us a discussion of how exactly being a scalar, and being intrinsic, relate. But this is not to say that the onus is only on Bricker. So far as I know, this is a lacuna in the whole metaphysical literature. (Cf. comment (a) at the end of Section 2.1.2.) To sum up: the metaphysical literature assumes that any scalar represents an intrinsic property of points, so that the pointilliste need “only“ worry whether vectors, tensors etc. do as well. But now we see that pointillistes should also worry about scalars such as the scalar curvature. 4.4.2 Alleged trouble with intrinsic duplicates On the other hand, suppose the principle of recombination refers to intrinsic duplicates. Then contradictory worlds are avoided; but, says Bricker, the principle is now too weak to capture the spirit of the Humean denial of necessary connections between distinct existences. For the principle now rules out necessary connections between the intrinsic natures of distinct objects. But on the present response, an object’s “nature” can include more than its intrinsic nature, viz. its perfectly natural extrinsic properties. So, says Bricker, the principle’s free combinability of intrinsic natures is not enough to prevent unwanted necessary connections. I reply that this second horn of Bricker’s dilemma has force only for a pointilliste. To see the point, let us take an example. Bricker does not give one: but he could add to the above example of p and q, as follows. If one scalar function, say temperature θ , represents an intrinsic property, while the scalar curvature R represents a perfectly natural extrinsic property (and for simplicity, there are no other scalar, vector or tensor functions to consider), the principle of recombination yields a world that has in p’s place an intrinsic duplicate of q — i.e. a point with: (a) the same temperature θ that q has (in its world), but (b) the scalar curvature R that p has (in its world), i.e. a positive value, not q’s negative value. In short, the fact that the neighbourhood of this point is a duplicate of p’s neighbourhood forces the duplicate of q going into p’s place to “shed” its negative scalar curvature. I take it that Bricker would see this as an unacceptable un-Humean necessary connection between the point p and its surrounding space. But I claim that the wise Humean has no worries here: the necessary connection merely reflects the extrinsicality of scalar curvature, so that the value of the scalar curvature in the surrounding space can constrain its value in p’s

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place.12 Besides, I would say that only someone in the grip of pointillisme — an explicit advocate like Lewis, or someone feeling its lure — would be uneasy at having fundamental (if you like, in Lewisian terms: perfectly natural) quantities that are extrinsic to a point. And so much the worse for pointillisme! But to return to Bricker: he believes that both horns of the dilemma are unacceptable — and so his own preference is …

4.5 A heterodox but intrinsic metric Bricker thinks we should retain his original metaphysical framework, with its claim that all perfectly natural properties are intrinsic; and we should escape the contradiction at the end of Section 4.3, by giving up the idea that the metric is a perfectly natural property of a point. That is, he proposes: (Heterodox): We should take the metric tensor to represent an intrinsic property of an infinitesimal neighbourhood of a point, taking such neighbourhoods as genuine mathematical objects. More precisely, we should hold that the metric, an extrinsic and not perfectly natural property of a point, is “grounded” in another intrinsic, perfectly natural property of a neighbourhood (Bricker’s scare-quotes). Since Bricker presents this preferred solution briefly, and I shall object to it, it is both clearest and fairest to quote him at length. He writes To illustrate the sort of grounding I have in mind, consider mass density. If one assumes that each neighbourhood of a point has some determinate (finite) mass and volume, then the mass density at a point can be characterized as the limit of the ratio of mass to volume, as volume shrinks to zero. So characterized, mass density is an extrinsic property of points. But it is customary in physics, when considering a continuous matter field, to instead take mass density to be a primitive scalar field: a function that assigns to each point a real number representing (given appropriate units) 12 A mollifying side-remark:— On the other hand, I see no worries, even for Bricker and other advocates of a principle of recombination, lurking in the fact that laws typically require the values of quantities, including scalar intrinsic quantities (such as temperature in my example), to be continuous, or even differentiable a specified number of times. For the principle says only that a world given by recombination is logically possible — not that it obeys the laws of either of the worlds of the “recombined ingredients”.

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the intrinsic mass density at the point. Given intrinsic mass density, and an assumption about its smooth distribution, mass can be defined by integration. Extrinsic mass density thus supervenes upon intrinsic mass density. And, thanks to a fundamental theorem of integral calculus, the values of extrinsic and intrinsic mass density coincide …. The suggestion, then, is to say something analogous about the local metric: the extrinsic local metric supervenes on an intrinsic local metric (plus manifold structure).(1993, p. 290–1.) But, he then says, there is a problem. How can a tensor be intrinsic to a point? Points are spatially simple. Tensors, being operators on vector spaces, are spatially complex. It is repugnant to the nature of a point to suppose that a local metric, which is a tensor, could be intrinsic to a point … [the intrinsic local metric] had better be intrinsic not to a point, but to something spatially complex. (1993, p. 291.; with a footnote endorsing Robinson’s argument which I reported in Section 4.1, that vectorial properties must be extrinsic to a point.) He immediately goes on No sooner said than done. If we are willing to postulate perfectly natural properties on theoretical grounds, we should be willing to posit appropriate entities to instantiate those properties: in this case, entities that are spatially complex. I propose that we reify talk of the “infinitesimal neighbourhood” of a point. The tangent space at a point is now conceived as the infinitesimal neighbourhood of the point “blown large” … it no longer depends for its existence on the manifold structure. Tensor quantities are intrinsic … to the infinitesimal neighbourhoods of points. … space (or spacetime) has a “non-standard” structure. There are “standard” points, and there are “nonstandard” points that lie an infinitesimal distance from standard points. The points along a path in space are ordered like the nonstandard continuum of Abraham Robinson’s non-standard analysis (ibid.)

4.6 Anti-pointilliste reply My reply is clear from what I said in Section 4.4.2. Namely, I think Bricker’s principle of recombination is a poor reason for proposing non-standard

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analysis. Though of course non-standard analysis is impressive and fascinating, the fact that vectorial and tensorial properties are extrinsic to a point gives no good reason to adopt non-standard analysis as a metaphysical foundation for differential geometry: only pointillisme makes one think so. The errors of pointillisme also show up in what Bricker says about his motivating example, mass density; in particular, his saying “it is customary in physics, when considering a continuous matter field, to instead take mass density to be a primitive scalar field”. I reply that this is a mistake. That is: the classical mechanics of continua (whether fluids or deformable solids) conceives mass density exclusively as a limit of a ratio of mass to volume, and so as extrinsic — in just the way Bricker says at the start of the first quotation. And it is right to do so. For use of a primitive mass density scalar field leads to conceptual conundrums. (Agreed, under suitable conditions of smoothness, such a field meshes as regards the mathematics with the usual definition as a limit — as Bricker mentions.) As a very simple example of such a conundrum, imagine that the unit square [0, 1]2 ⊂ R2 is a sheet of continuous material, with a uniform mass density ρ (x, y) = 1 (so that the total mass is also 1). Now suppose the material is expanded to four times its original area, by a uniform stretch, so as to cover the set [0, 2]2. That is, there is a stretching function f: f : (x, y) ∈ [0, 1]2

(2x, 2y) ∈ [0, 2]2

(4.1)

The conservation of mass requires that after the expansion ρ (x, y) = 0,25 for all (x, y). But if as Bricker suggests, the mass density ρ is primitive, it is natural to ask: how does the point-sized bit of matter at a point (x, y) “know” how to decrease its value of ρ between the initial and final times, say t 0 and t 1: ρ (x, y; t 1) = 14 ρ (x, y; t 0)? After all, each point is mapped by f to just one point, not to four points! On the other hand, there is no such conundrum (in this example, or countless others) if we first state the conservation of mass in terms of extended regions, and then treat mass density as a derived concept. In fact, we have here come full circle: we have returned to Section 3.3.2’s paradox of the length of a line, which launched our discussion of whether pointillisme can accommodate the structure of space or spacetime. For rigorous presentations of continuum mechanics (e.g. Truesdell 1991, pp. 16–19, 92–94) treat mass and mass density in exactly this way. That is: they postulate a mass measure that assigns values of mass to (an appropriate subset

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of all) spatial regions. Mass density is then introduced as a derived concept (essentially the limit of the ratio of mass to volume, as mentioned above), subject to certain conditions that ensure that its integral yields back the original mass-measure for regions. The full details of this treatment require modern measure theory: (for example, the conditions for the density’s integral to equal the original measure are given essentially by the RadonNikodym theorem; for details, cf. e.g. Kingman and Taylor (1977, Theorem 6.7)). But I do not need to rehearse these details: here it is enough to give a non-rigorous statement of how this treatment, applied to the unit square example, avoids the conundrum of how ρ can “know” how to decrease.13 In this example, we postulate that all regions R of a suitable kind K are assigned a mass m(R, t) at a time t, which is conserved under the stretching in the sense that ∀ R ∈ K, m(R, t 0) = m( f (R), t 1).

(4.2)

We also postulate that each region R is assigned an area a(R); and that the kind K is rich enough in the sense that for each point (x, y), there is a sequence of regions {Rn } which all contain (x, y) but whose areas descend to 0 — which we write as Rn → (x, y). Then we define the mass density at (x, y) as the corresponding limit of the ratio of mass to area: we assume here that this limit exists, for all (x, y). That is:

ρ (x, y; t) := limRn → (x, y)

m (R, t ) . a (R )

(4.3)

The conservation of mass, represented fundamentally by eq. 4.2, can then be re-expressed in terms of the integral of the density

∫

R

ρ (x, y; t0) dxdy =

∫

f (R)

ρ (x, y; t1) dxdy

(4.4)

And from this, it follows that ρ must decrease uniformly by a factor of 4: i.e. ρ (x, y; t0) =4ρ (x, y; t1). That is “how ρ knows” how to decrease!14 13 I also admit that (as mentioned in Section 1 and 3.3.2) measure theory has some well-nigh paradoxical results of its own. But neither swallowing those results, nor avoiding them by revising measure theory, gives any support to pointillisme. 14 Of course, conundrums like this about the unit square can be formulated not only about mass and mass density, but about arbitrary measures and their densities. And

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Acknowledgements:— I am grateful to audiences at Florence, Kirchberg, Leeds, London, Oxford, and Princeton; and to A. Elga, G. Belot, P. Forrest, J. Hawthorne, S. Leuenberger, L. Lusanna, M. Pauri, H. Price, J. Uffink, and B. van Fraassen for conversations and comments.

References Arntzenius, F. 2004 ‘Gunk, topology and measure’, available at: http:// philsciarchive.pitt.edu/archive/00001792. Arntzenius, F. and Hawthorne, J 2006 ‘Gunk and continuous variation’, forthcoming. Arthur, R. 2006 ‘Leibniz’s syncategorematic infinitesimals, smooth infinitesimal analysis and Newton’s proposition 6’, forthcoming. Batterman, R. 2003 ‘Falling cats, parallel parking and polarized light’, Studies in History and Philosophy of Modern Physics 34B, pp. 527–558. Bell, J. 1998 A Primer of Infinitesimal Analysis, Cambridge University Press. Belot, G. 1998 ‘Understanding electromagnetism’, British Journal for the Philosophy of Science 49, pp. 531–555. Black, R. 2000 ‘Against Quidditism’, Australasian Journal of Philosophy 78, pp.87–104. Bricker, P. 1993 ‘The fabric of space: intrinsic vs. extrinsic distance relations’, in P. French et al. (eds.), Midwest Studies in Philosophy 18, University of Minnesota Press, pp. 271–294. Butterfield, J. 2004 ‘On the Persistence of Homogeneous Matter’, available at:physics/0406021: and at http://philsci-archive.pitt.edu/archive/ 00002381/ — 2004a ‘The Rotating Discs Argument Defeated’, forthcoming in British Journal for the Philosophy of Science; available at: http://philsci-archive. pitt.edu/archive/00002382/ — 2005 ‘On the Persistence of Particles’, in Foundations of Physics 35, pp. 233–269, available at: physics/0401112; and http://philsci-archive. pitt.edu/archive/00001586/. — 2006 ‘Against Pointillisme about mechanics’, forthcoming in British Journal for the Philosophy of Science; available at http://philsci-archive.pitt.edu. the solution provided by modern measure theory is the same: take the measure, with its assignment to extended regions, as primary, and take the density as a derived concept, viz. as a limit of the ratio of the measure to a volume or area measure.

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— 2006a ‘Against Pointillisme: a call to arms’, in preparation. Butterfield, J. and Isham, C. 1999 ‘On the Emergence of Time in Quantum Gravity’, in ed. J Butterfield, The Arguments of Time, British Academy and Oxford University Press, pp. 111–168; available at: gr-qc/ 9901024. — 2001 ‘Spacetime and the Philosophical Challenge of Quantum Gravity’, in ed.s C. Callender and N. Huggett, Physics meets Philosophy at the Planck Scale, Cambridge University Press pp. 33–89; available at: gr-qc/9903072. Carathéodory, C. 1963 Algebraic Theory of Measure and Integration, trans. F. Linton, New York: Chelsea Publishing Company. Davies, E. 2003 ‘Quantum mechanics does not require the continuity of space’, Studies in History and Philosophy of Modern Physics 34B, pp. 319–328. Earman, J. 1987 ‘Locality, non-locality and action-at-a-distance: a skeptical review ofsome philosophical dogmas’, in Kelvin’s Baltimore Lectures and Modern Theoretical Physics, eds. R. Kargon and P. Achinstein, Cambridge Mass: MIT Press. Earman, J. and Roberts, J.T. 2006 ‘Contact with the Nomic: a challenge for deniers of Humean supervenience about laws of nature’, Philosophy and Phenomenological Research, forthcoming. Forrest, P. 2002 ‘Non-classical mereology and its application to sets’, The Notre Dame Journal of Formal Logic 43, pp. 79–94. — 2004 ‘Grit or gunk: implications of the Banach-Tarski paradox’, The Monist 87, pp. 351–384. Hawthorne, J. 2006 ‘Quantity in Lewisian Metaphysics’, forthcoming in his Metaphysical Essays, Oxford University Press. Hoefer, C. 2000 ‘Energy Conservation of in GTR’, Studies in History and Philosophy of Modern Physics, 31B pp. 187–200. Kingman, J. and Taylor, S. 1977 Introducton to Measure and Probability, Cambridge University Press. Kragh, H. and Carazza, B. 1994 ‘From time atoms to spacetime quantization: theidea of discreet time 1925–1926’, Studies in History and Philosophy of Modern Physics 25, pp. 437–462. Langton, R. and Lewis, D. 1998 ‘Defining ‘intrinsic“, Philosophy and Phenomenological Research 58, pp. 333–345; reprinted in Lewis (1999a), page reference to reprint. Leibniz, G. 2001 The Labyrinth of the Continuum: writings on the continuum

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problem 1672–1686, ed. sel. and transl. R. Arthur, New Haven: Yale University Press. Lewis, D. 1983 ‘Extrinsic properties’, Philosophical Studies 44, pp. 197–200; reprinted in Lewis (1999a); page references to reprint. — 1983a ‘New Work for a Theory of Universals’, Australasian Journal ofPhilosophy 61, pp. 343–77; reprinted in Lewis (1999a), page reference to reprint. — 1986 Philosophical Papers, volume II, New York: Oxford University Press. — 1986a On the Plurality of Worlds, Oxford: Blackwell. — 1994 ‘Humean Supervenience Debugged’, Mind 103, p 473–490; reprinted in Lewis (1999a), pp. 224–247; page reference to reprint. — 1999 ‘Zimmerman and the Spinning sphere’, Australasian Journal of Philosophy 77, pp. 209–212. — 1999a Papers in Metaphysics and Epistemology, Cambridge: University Press. — 2001 ‘Redefining ‘intrinsic“, Philosophy and Phenomenological Research 63, pp. 381–398. McLarty, C. 1988 ‘Defining sets as sets of points of spaces’, Journal of Philosophical Logic 17, pp. 75–90. Mancosu, P. 1996 Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, Oxford: University Press. Menger, K. 1978 ‘Topology without points’, in Selected papers in Logic and Foundations, Didactics, Economics, Dordrecht: Reidel, pp. 80–107. Monk, N. 1997 ‘Conceptions of spacetime: problems and possible solutions, Studies in History and Philosophy of Modern Physics 28, pp. 1–34. Nelson, E. 1987 Radically Elementary Probability Theory, Princeton University Press. Oppy, G. 2000 ‘Humean supervenience?’, Philosophical Studies 101 pp. 77– 105. Robinson, A. 1996 Non-standard Analysis, Princeton University Press. Robinson, D. 1989 ‘Matter, Motion and Humean supervenience’, Australasian Journal of Philosophy 67, pp. 394–409. Roeper, P. 1997 ‘Region-based topology’ Journal of Philosophical Logic 26, pp. 251–309. Sider, T. 2001 Four-Dimensionalism, Oxford University Press. Skyrms, B. 1993 ‘Logical atoms and combinatorial possibility’, Journal of Philosophy 90, pp. 219–232.

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Taylor, B. 1993 ‘On natural properties in metaphysics’, Mind 102, pp. 81– 100. Truesdell, C. 1991 A First Course in Rational Continuum Mechanics, volume 1; second edition; Academic Press. Vallentyne, P. 1997 ‘Intrinsic properties defined’, Philosophical Studies 88, pp. 209–219. Wagon, S. 1985 The Banach-Tarski Paradox, Cambridge: University Press. Weatherson, B. 2002 ‘Intrinsic vs. extrinsic properties’, Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/intrinsic-extrinsic. Wilson, M. 1998 ‘Classical mechanics’; in The Routledge Encyclopedia of Philosophy. Zimmerman, D. 1998) ‘Temporal parts and supervenient causation: the incompatibility of two Humean doctrines’, Australasian Journal of Philosophy 76, pp. 265–288.

Zeit im Gödelschen Universum Heinz Rupertsberger, Wien Die Existenz einer globalen Zeit wurde bereits 1905 durch die spezielle Relativitätstheorie von Albert Einstein eingeschränkt. Mit seiner allgemeinen Relativitätstheorie 1915 legte er die Grundlage für die moderne Kosmologie. Es dauerte jedoch 34 Jahre bis Gödel durch seine kosmologische Lösung von Einsteins Feldgleichungen der Gravitation bewies, dass Lösungen mit akausalem Verhalten nicht ausgeschlossen sind. Das Gödelsche Universum wird daher als erstes Beispiel einer Kosmologie, in der es keine globale Zeitordnung der Ereignisse gibt, beschrieben und mit experimentellen Daten verglichen.

1. Einleitung Die Vorstellung der Existenz einer absoluten Zeit, durch die alle RaumZeitpunkte, bzw. Ereignisse in diesen, eindeutig zeitlich geordnet werden können, wie sie u.a. Newton vertrat, wurde bereits durch die spezielle Relativitätstheorie eingeschränkt. Eine ihrer beiden Grundlagen sind das Postulat der Konstanz der Lichtgeschwindigkeit, d.h. jeder Beobachter misst im Vakuum für Licht (stellvertretend für elektromagnetische Wellen) die gleiche Geschwindigkeit. Zusammen mit dem Postulat, dass es kein ausgezeichnetes Inertialsystem gibt, also physikalische Gesetze für alle Beobachter, die sich relativ zueinander mit konstanter Geschwindigkeit und damit kräftefrei (erstes Newtonsches Axiom) bewegen, gleich aussehen, ergibt sich als Konsequenz, dass Raum und Zeit nicht mehr getrennt voneinander betrachtet werden können. In diesem 4-dimensionalen sogenannten RaumZeit Kontinuum gibt es allgemeines Einverständnis über die zeitliche Aufeinanderfolge zweier Ereignisse nur dann, falls ein Signal, das höchstens mit Lichtgeschwindigkeit läuft, zumindest prinzipiell vom früheren zum späteren gesendet werden kann. Der frühere Raum-Zeitpunkt kann dann, zumindest im Prinzip, den späteren beeinflussen, sie sind kausal verknüpfbar. Es gibt also keine Umkehr von Ursache und Wirkung, die Kausalität bleibt gewahrt. Der Begriff der Gleichzeitigkeit verliert jedoch seine Bedeutung, da in einem Bezugssystem gleichzeitige Ereignisse wegen der Endlichkeit F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 223x–232.

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der Lichtgeschwindigkeit nie kausal verknüpft sein können. Was für den einen Beobachter gleichzeitig ist, wird von einem relativ dazu bewegten Beobachter als nicht gleichzeitig gesehen und die Aufeinanderfolge von kausal prinzipiell nicht verknüpfbaren Ereignissen kann sich für verschiedene, relativ zueinander gleichförmig bewegte Beobachter, sogar umkehren. Weiterhin misst jedoch jeder Beobachter in seinem System räumliche Abstände mit einem Maßstab und zeitliche mit einer Uhr. Die zeitlichen und räumlichen Beiträge zum Abstand zweier Ereignisse in der 4-dimensionalen Raum-Zeit, der für alle Beobachter gleich ist, unterscheiden sich durch ein Vorzeichen und einen relativen Faktor der Lichtgeschwindigkeit zum Quadrat. Es ist daher keineswegs so, dass Raum und Zeit nicht voneinander unterschieden sind, aber die absolute Zeit und der von dieser unabhängige absolute Raum eines Newton existieren seit der speziellen Relativitätstheorie zumindest in der Physik nicht mehr.

2. Raum-Zeitstruktur in der allgemeinen Relativitätstheorie In der speziellen Relativitätstheorie ist die Struktur der Raum-Zeit, in der Ereignisse beschrieben werden, vorgegeben und daher die in ihr herrschende Kausalität bzw. eingeschränkte zeitliche Ordnung problemlos global beschreibbar. Das ändert sich wesentlich in der allgemeinen Relativitätstheorie. Einstein postulierte, dass lokal, d.h. in kleinen Raum-Zeit Bereichen, die Bewegung in einem beschleunigten Bezugssystem nicht von der in einem Gravitationsfeld unterschieden werden kann (Äquivalenzprinzip). Ein im Gravitationsfeld frei fallender Beobachter stellt daher ein Inertialsystem dar, da alle Körper im Schwerefeld gleich schnell fallen und daher keine Geschwindigkeitsunterschiede durch dieses hervorgerufen werden können. Das gilt aber nur lokal, da im Allgemeinen das Schwerefeld vom betrachteten Raum-Zeitpunkt abhängt. Davon ausgehend gelangte er zu der Vorstellung, dass die Massenverteilung oder allgemeiner die Energie-Impuls-Dichte die Quelle der Raum-Zeit Struktur ist, genauso wie in der Elektrodynamik die Ladungen als Quellen das zugehörige elektromagnetische Feld erzeugen. Damit wird aus der pseudo-Euklidischen, ebenen Raum-Zeit der speziellen Relativitätstheorie die pseudo-Riemannsche, gekrümmte der allgemeinen Relativitätstheorie. Dieser Übergang ähnelt in zwei Dimensionen dem von einer Ebene zur Oberfläche einer Kugel oder einer anderen gekrümmten Fläche, wobei „pseudo“ noch den diffizilen Un-

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terschied zwischen Raum- und Zeitkoordinaten andeutet. Der kräftefreien Bewegung auf Geraden im ebenen Raum entspricht die Bewegung auf geodätischen Linien im gekrümmten Raum, im speziellen Fall der Kugeloberfläche auf Großkreisen. Raum und Zeit können nun nicht mehr getrennt von der Materieverteilung bzw. Energie-Impuls-Dichte betrachtet werden, sie sind durch die Einsteinschen Feldgleichungen der allgemeinen Relativitätstheorie miteinander verbunden. Die beschleunigte Bewegung in der Newtonschen Gravitationstheorie im flachen Raum wird zur kräftefreien Bewegung auf einer geodätischen Linie im gekrümmten Raum, auf ihr bewegen sich frei fallende Beobachter. Gleichzeitig wird damit das Problem der instantanen Newtonschen Fernwirkungstheorie der Gravitation aufgelöst. Dafür geht die Überschaubarkeit der globalen Raum-Zeit Struktur, wie sie in der speziellen Relativitätstheorie noch gegeben ist, verloren.

3. Vereinfachungen für kosmologische Lösungen der allgemeinen Relativitätstheorie Es lag natürlich von Anfang an nahe, auch unser Universum mit Hilfe der Einsteinschen Feldgleichungen zu beschreiben. Die Galaxien mit ihren Sternen, schwarzen Löchern usw. erzeugen dann über ihre Energie-Impuls-Dichte unsere Raum-Zeit Geometrie. Die Lösung der Feldgleichungen würde dann sowohl die Vergangenheit wie auch die Zukunft unseres Universums angeben. In dieser Allgemeinheit ist das jedoch aussichtslos. Erstens müsste die Verteilung der Energie-Impuls-Dichte des Universum als Quelle global und für alle Zeiten bekannt sein und zweitens wären die Gleichungen für diese ungeheure Zahl von Objekten, die sich unter dem Einfluss der Gravitation bewegen, viel zu kompliziert. Es sind daher drastische Vereinfachungen bezüglich der Energie-Impuls-Dichte Verteilung des Universums notwendig. Jede auf diese Weise erhaltene Lösung wird als kosmologische bezeichnet und ist von Interesse, da aufgrund der nichtlinearen Struktur der Feldgleichungen selbst bei einfachsten Annahmen keineswegs alle Lösungen im Allgemeinen bekannt sind. Daher kann jede mögliche Effekte andeuten, oder Hinweise dazu liefern, welche zusätzlichen Forderungen zu verlangen sind, um eine dem realen Universum entsprechende kosmologische Lösung zu finden, die mit den experimentellen Daten übereinstimmt. Zunächst werden Einflüsse der drei anderen fundamentalen Wechselwirkungen, der elektro-schwachen und der starken, vernachlässigt, da sie entweder nur kurzreichweitig oder wegen der

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in Übereinstimmung mit experimentellen Daten angenommenen Ladungsneutralität des Universums zu keinen langreichweitigen Kräften führen. Die Mittelung über Distanzen, die groß im Vergleich zum Abstand der Galaxien des Universums sind, ersetzt dessen zunächst diskrete Struktur durch eine stetige Energie-Impuls-Dichte Verteilung. Es entsteht das Bild einer strömenden Flüssigkeit mit entsprechender Druck- und Dichteverteilung, die durch die Gleichungen der relativistischen Hydrodynamik beschrieben wird. Es gibt keinen Grund dafür, einen Punkt des Universums, also auch nicht unser Sonnensystem, als irgendwie ausgezeichnet anzusehen. Daher wird zusätzlich angenommen, dass das Weltall homogen ist, das heißt es sieht überall gleich aus. Diese Annahme wird als kosmologisches Prinzip bezeichnet. Eine inselförmige Materieverteilung mit Vakuum rundherum würde z.B. dieses Prinzip verletzen.

4. Das Gödelsche Universum Aufgrund dieser Vereinfachungen wurden im Rahmen der allgemeinen Relativitätstheorie verschiedenste kosmologische Modelle entwickelt, so zuerst schon 1917 von Einstein selbst (Einstein 1917). Doch erst 1949 gelang es Gödel (Gödel 1949) durch seine exakte kosmologische Lösung mit einer vollständigen Beschreibung ihrer globalen Struktur zu beweisen, dass die allgemeine Relativitätstheorie im Allgemeinen keine Aussage über die Existenz einer absoluten Zeitkoordinate macht, die eine eindeutige Bestimmung der zeitlichen Ordnung von im Prinzip kausal verknüpfbaren Ereignissen ermöglicht. Im Gödelschen Universum sind Zeitreisen in die Vergangenheit möglich, d.h. man kann ausgehend von einem Raum-Zeitpunkt zu diesem oder vor diesem zurückkommen, sozusagen seine eigene Geburt sehen bzw. beeinflussen. Vergangenheit und Zukunft können nicht mehr eindeutig definiert werden. Die Grundlagen für diese kosmologische Lösung kann man folgendermaßen beschreiben. Die Materie wird durch eine ideale, druckfreie Flüssigkeit, eine Art kosmischer Staub, die rotiert aber nicht expandiert, dargestellt. Diese Rotation sieht für jeden Beobachter, der sich mit der Materie des Universums bewegt, die in diesem Modell geodätischen Linien folgt, gleich aus, alle sehen die gleiche konstante Winkelgeschwindigkeit. Sie kann mit Hilfe eines Foucaultschen Pendels, das sich mit der Materie sozusagen frei fallend mitbewegt, gemessen werden. Aufgrund des Trägheitsgesetzes kann sich seine Pendelebene nicht ändern und damit die Drehung der Pendele-

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bene relativ zur lokalen Umgebung beobachtet werden. Die Situation ist ähnlich der eines Beobachters in einem Satelliten mit Ellipsenbahn um die Erde. Die Ellipsenbahn entspricht einer geodätischen Linie im Schwerefeld der Erde, der Beobachter ist daher in einem frei fallenden Zustand und damit in einem lokalen Inertialsystem. Aber die Schwingungsebene eines Pendels wird sich während eines Umlaufs um die Erde für diesen Beobachter um 360° drehen. Durch die Rotation ist eine Richtung, die Drehachse, ausgezeichnet und gleichzeitig Bewegung in Form der Drehung vorhanden. Daher ist dieses Universum anisotrop, d.h. es sieht nicht nach allen Richtungen gleich aus, sondern eben nur homogen und auch nicht statisch, aber stationär, es bietet für alle Zeiten den gleichen Anblick. Es kann also nicht unser Weltall darstellen, da es im Widerspruch zur experimentell beobachteten Expansion der Raum-Zeit steht. Das war auch der Anlass für Gödels zweite allgemeinere Arbeit über rotierende Universen (Gödel 1950), in der jedoch keine weiteren expliziten Lösungen angegeben werden. Die Vorstellung liegt nahe, dass eine Rotation, die auch die Zeitkoordinate betrifft, die kausale Struktur einer Raum-Zeit im Großen zerstören kann. Um so erstaunlicher ist es, dass in vorhergehenden Arbeiten, die den Einfluss rotierender Massen auf die Raum-Zeit Geometrie untersuchten, dieser Aspekt nicht diskutiert wurde (Thirring 1918, Stockum 1937). Dieser sogenannte Thirring-Lense Effekt ist erst vor kurzem experimentell durch Beobachtung der Bahnen zweier Satelliten um die Erde bestätigt worden (Ciufolini 2004). Es gab auch schon einige Zeit vor Gödel zumindest eine exakte kosmologische Lösung mit Rotation (Lanczos 1924), die aber weder die globale Raum-Zeitstruktur noch die Rotation diskutierte. Gödel hat vermutlich höchstens von den Arbeiten Thirrings gewusst, da er ja zur relevanten Zeit in Wien studierte und arbeitete. In seiner Vorlesung in Princeton 1949 wird allerdings nur Gamow mit dem Vorschlag, dass das gesamte Universum sich in einem Zustand der gleichförmigen Rotation befindet, namentlich erwähnt (Gödel 1949b). Die Rotation wirkt gegen die Gravitationsanziehung. Beide zusammen reichen jedoch noch nicht für einen Gleichgewichtzustand aus, sodass Gödel noch eine kosmologische Konstante hinzufügen musste. Diese wurde bereits von Einstein (Einstein 1917) für seine kosmologische Lösung eines statischen, räumlich endlichen aber unbegrenzten Universums benötigt und war für ihn ein Teil der Raum-Zeit Geometrie. Bei Gödel steht sie jedoch auf der selben Seite der Feldgleichungen wie die Energie-Impuls-Dichte des Universums und wird damit zu einer Quelle der Raum-Zeit. Das entspricht auch dem derzeitigen Ver-

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ständnis, dass die kosmologische Konstante u.a. die quantenmechanische Vakuumenergie und/oder so genannte dunkle Materie repräsentiert. Wird die Energie-Impuls-Dichte eines Universums durch die relativistische Hydrodynamik beschrieben, dann kann eine kosmologische Konstante stets durch Umdefinition der Dichte- und Druckverteilung in diesen Größen absorbiert werden. Im Fall von Einstein bedeutet das einen konstanten negativen Druck, was jedoch noch nichts über seine Auswirkungen aussagt, da es zunächst nur auf Druckunterschiede ankommt. In der allgemeinen Relativitätstheorie bewirkt ein konstanter negativer Druck eine Expansion der Materie, also eine Art Antigravitation, die im Einsteinschen statischen Kosmos der anziehenden Schwerkraft entgegen wirkt. Also genau das Gegenteil einer Implosion, die man zunächst mit negativem Druck verbinden würde. Bei Gödel hat die kosmologische Konstante im Gegensatz zu Einstein einen negativen Wert, entspricht also einem positiven konstanten Druck und wirkt daher anziehend, verstärkt also die Schwerkraft. Sie ist offensichtlich zusätzlich notwendig, um das durch die Rotation hervorgerufene Auseinanderstreben der Materie zu stabilisieren. Als der expandierende Charakter des Universums an Hand der Rotverschiebung der Spektrallinien entfernter Galaxien experimentell nachgewiesen wurde, hat Einstein diese ad hoc eingeführte kosmologische Konstante wieder verworfen. Theoretisch kann über ihr Vorzeichen kaum etwas ausgesagt werden. Nimmt man an, dass sie durch die quantenmechanische Vakuumenergie entsteht, so tragen Fermionen, Teilchen mit halbzahligem inneren Drehimpuls, negativ und Bosonen, Teilchen mit ganzzahligem inneren Drehimpuls, positiv zu ihr bei. Lange Zeit dominierte die Annahme einer verschwindend kleinen kosmologischen Konstanten, worin man auch eine weitere Begründung für die Existenz der Supersymmetrie für Elementarteilchen sah, d.h. jedes bekannte Fermion hat einen supersymmetrischen bosonischen Partner und umgekehrt. Daher erhält man in einer supersymmetrischen Theorie insgesamt keinen Beitrag der quantenmechanischen Vakuumenergie zur kosmologischen Konstanten. Allerdings hat man bis heute keinerlei Anzeichen für diese durch die Supersymmetrie geforderten neuen Teilchen gefunden. Die derzeitigen experimentellen Daten über eine starke Expansion der Raum-Zeit haben jedoch zu einer Wiedereinführung einer positiven kosmologischen Konstanten, der Antigravitation von Einstein, geführt. Unabhängig vom konkret betrachteten Universum wird jedem Ereignis sein zugehöriger Raum-Zeitpunkt zugeordnet. Physikalisch mögliche Bewegungen entsprechen dann der kontinuierlichen Aufeinanderfolge solcher

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Raum-Zeitpunkte, den sogenannten zeitartigen Weltlinien, mit der Eigenschaft, dass in jedem Punkt die Geschwindigkeit kleiner als die, oder für masselose Teilchen gleich der Lichtgeschwindigkeit sein muss. Im Spezialfall der kräftefreien Bewegung in der gekrümmten Raum-Zeit erfolgt die Bewegung auf den entsprechenden zeitartigen geodätischen Linien. Da die Lichtgeschwindigkeit die größtmögliche Signalgeschwindigkeit darstellt, beschränken Lichtstrahlen lokal in einem Raum-Zeitpunkt den Bereich, der diesen beeinflussen kann, also die Vergangenheit, sowie auch den, der von diesem beeinflusst werden kann und damit seine Zukunft. Das gilt jedoch in der allgemeinen Relativitätstheorie im Allgemeinen nur mehr lokal, da Lichtstrahlen zwar speziellen geodätischen Linien entsprechen, aber eben in einer gekrümmten Raum-Zeit. Im Gödelschen Universum sieht nun zunächst alles relativ harmlos aus. Der Raum besitzt in einer Richtung eine ebene, euklidische Struktur, daher kann diese bei der Untersuchung der kausalen Struktur vernachlässigt werden. Zur Beschreibung dieses Universums wird das bereits von Gödel angegebene Koordinatensystem, über das in der allgemeinen Relativitätstheorie ja frei verfügt werden kann, verwendet. In diesem verlaufen die geodätischen Materieweltlinien parallel zur gewählten Zeitkoordinate, es wird der Standpunkt eines mit der Materie mitschwimmenden Beobachters gewählt. Es sind alle zeitartigen geodätischen Linien bekannt, sie sind unendlich ausgedehnt und schneiden sich selbst nie im selben Raum-Zeitpunkt. Es stellt sich heraus, dass jede dieser Geodäten, die von einem Raum-Zeitpunkt der Materieweltlinie ausläuft, nach endlicher Zeit wieder zu dieser zurückkehrt, nachdem sie ein endliches Raumgebiet durchlaufen hat. Am weitesten können sich Lichtstrahlen entfernen, die damit die maximale räumliche Distanz bestimmen, die man sich auf einer geodätischen Bahn von der Materieweltlinie entfernen kann. Alle Lichtstrahlen ausgehend von einem Punkt der Materieweltlinie laufen später wieder in einem Punkt dieser zusammen. Sie erzeugen damit für den Beobachter auf der Materieweltlinie einen kreisförmigen optischen Horizont, von außerhalb diesem Gebiet können von ihm keine Lichtsignale empfangen werden. Innerhalb dieses Bereichs gibt es eine eindeutige, durch die Materieweltlinie des Beobachters bestimmte Zeitrichtung für alle Bewegungen. Untersucht man nun wie Lichtstrahlen ausgehend von Raum-Zeitpunkten außerhalb des optischen Horizonts für den Beobachter auf der Materieweltlinie aussehen (die er natürlich nie sehen kann), so stellt man fest, dass plötzlich Bewegungen möglich sind, die relativ zu seiner Zeitrichtung in die Vergangenheit zeigen. Nur beschleunigte Bewegungen, z.B. mit einer Rakete,

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können den optischen Horizont durchbrechen. Dann kann es zu überraschenden Effekten kommen. Startet man mit einer Rakete von einer Materieweltlinie und überschreitet den optischen Horizont, so kann man zu einer benachbarten gelangen, bevor diese ein gleichzeitig vom selben Punkt ausgesendeter Lichtstrahl erreicht. In gewissem Sinn eine Reise mit Überlichtgeschwindigkeit in die Zukunft, Informationen können rascher als das Licht übermittelt werden (Pfarr 1981). Diese Art von Zukunftsreise hat nichts mit dem Effekt der Zeitdilatation durch eine beschleunigte Bewegung bzw. in einem Gravitationsfeld zu tun, führt aber noch immer nicht zu akausalen Verhältnissen. Entfernt man sich jedoch von einer Materieweltlinie auf einer beschleunigten Bahn weiter als dem Durchmesser des optischen Horizonts, so erscheinen die dort lokal ausgestrahlten Lichtstrahlen derart verzerrt, dass zeitartige Weltlinien möglich werden, die zum selben oder sogar zu einem früheren Zeitpunkt der ursprünglichen Materieweltlinie zurückkehren. Damit ist bewiesen, dass die allgemeine Relativitätstheorie Universen zulässt, in denen aufgrund der globalen Geometrie der Raum-Zeit die Kausalität verletzt wird. Zeitreisen in die Vergangenheit werden möglich mit allen damit zusammenhängenden Problemen, wie u.a. der schon erwähnten Frage, ob man seine eigene Geburt verhindern kann. Diese geometrischen Verhältnisse werden z.B. in (Rupertsberger 2002) veranschaulicht.

5. Größenordnungen und Schlussbemerkungen Um eine Vorstellung über die involvierten Größenordnungen zu bekommen, kann man für den einzigen freien Parameter dieses Universums den gegenwärtigen Wert der Materiedichte unseres Universums einsetzen. Für eine volle Umdrehung wird etwa das zehnfache des gegenwärtigen Alters des Universums benötigt, der optische Horizont hat etwa die Größe des Durchmessers des Universums, falls man nur das Alter desselben zur Abschätzung heranzieht. Die kosmologische Konstante entspricht dem etwa 10−15-fachen des normalen Luftdrucks auf der Erde. Man kann sich nun für Zeitreisen die Erde als Rakete vorstellen, deren Materie als Treibstoff dient, der mit Lichtgeschwindigkeit ausgestoßen wird. Dann ergibt eine äußerst grobe Abschätzung, dass für eine Reise von einer Materieweltlinie um 100 Jahre in die Vergangenheit derselben bei einer Reisedauer von 100 Jahren mindestens soviel Materie verbraucht wird, dass die Erde am Schluss der Reise auf eine Kugel mit etwa 6m Radius geschrumpft ist. Um die astronomischen Entfernungen in der kurzen Zeit von nur 100 Jahren zurücklegen

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zu können, sind zusätzlich Reisegeschwindigkeiten extrem nahe der Lichtgeschwindigkeit notwendig (Pfarr 1981). Diese Größenordnungen zeigen die Schwierigkeiten real funktionierende Zeitmaschinen, falls sie überhaupt möglich sind, zu konstruieren, ändern jedoch nichts an der Tatsache, dass sie innerhalb der Voraussetzungen der allgemeinen Relativitätstheorie nicht ausgeschlossen werden können. So konnte auch die Antwort Einsteins auf die kausalen Probleme in Gödels Lösung nur lauten: „Es wird interessant sein zu erwägen, ob diese nicht aus physikalischen Gründen auszuschließen sind “ (Einstein 1949). Ein wesentlicher Fortschritt zu dieser Bemerkung ist derzeit nicht ersichtlich. Gegenwärtig werden akausale kosmologische Lösungen durch die chronology protection conjecture Anfang 1990 von Hawking (Hawking 1992) ausgeschlossen. Sie besagt, dass die Gesetze der Physik das Auftreten geschlossener zeitartiger Bahnen verhindern. Das wird versucht mit Hilfe von speziellen Beispielen zu belegen, bleibt aber trotzdem nur eine Vermutung. So entsteht im Rahmen der Physik aus der absoluten, universellen Zeit Newtons der Zeitbegriff der speziellen Relativitätstheorie, der nur mehr für im Prinzip kausal verknüpfbare Ereignisse eine eindeutige Zeitrichtung festlegt, die dann ohne weitere Zusatzannahmen in der allgemeinen Relativitätstheorie global nur mehr abhängig vom speziell betrachteten Universum existiert oder verschwindet.

Literatur Ciufolini, Ignazio und Pavlis, Erricos C. 2004 „A confirmation of the general relativistic prediction of the Lense-Thirring effect“, Nature, 431, 958–960. Einstein, Albert 1917 „Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie“, Sitzungsberichte der Preußischen Akademie der Wissenschaften 142–152. — 1949 „Bemerkungen zu den in diesem Band vereinigten Arbeiten“, in: Paul Arthur Schilpp (Hrsg.), Albert Einstein als Philosoph und Naturforscher, Stuttgart (Kohlhammer) 1955 (engl. Original Evanston IL (Northwestern UP) 1949). Gödel, Kurt 1949 „An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation“, Reviews of Modern Physics 21, 447–450.

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Gödel, Kurt 1949b „Lecture on rotating universes“, in: Solomon Feferman (Hrsg.), Kurt Gödel, Collected Works, vol.3, 269–287. — 1950 „Rotating Universes in General Relativity Theory“, in: L. M. Graves et al (Hrsg.), Proceedings of the International Congress of Mathematicians, Providence Rh.I.: American Mathematical Society, vol.1, 175–181. Hawking, Stephen W. 1992 „Chronology protection conjecture“, Physical Review D46/2, 603–611. Lanczos, Kornel 1924 „Über eine stationäre Kosmologie im Sinne der Einsteinschen Gravitationstheorie“, Zeitschrift für Physik 21, 73–110. Pfarr, Joachim 1981 „Time Travel in Gödel‘s Space“, General Relativity and Gravitation 13/11, 1073–1091. Rupertsberger, Heinz 2002 „Das Gödelsche Universum“, in: Bernd Buldt et al (Hrsg), Kurt Gödel - Wahrheit & Beweisbarkeit, Wien: öbv & hpt, vol.2, 219–229. Stockum, W.J. van 1937 „The Gravitational Field of a Distribution of Particles Rotating about an Axis of Symmetry“, Proceedings of the Royal Society of Edinburgh, A 57, 135–154. Thirring, Hans 1918 „Über die Wirkung rotierender ferner Massen in der Einsteinschen Gravitationstheorie“, Physikalische Zeitschrift, 19, 33.

“Close to the Speed of Light”: Dispersing Various Twin Paradox Related Confusions Miloš Arsenijevi, Belgrade …I noticed that mathematical clarity had in itself no virtue for Bohr. He feared that the formal mathematical structure would obscure the physical core of the problem, and in any case, he was convinced that a complete physical explanation should absolutely precede the mathematical formulation. (Heisenberg 1967, p. 98)

1. Introduction One century after the appearance of the first of Einstein’s several articles about what was later called Special Relativity Theory (SRT), one still cannot say that SRT has been sufficiently conceptually clarified to render concern about the meaning of its key concepts, or about the way in which it explains phenomena, a matter of mere ignorance. Some recent articles can be taken as evidence for this. Let me quote the diagnosis given in one of them, which concerns what Langevin called “The Twin Paradox” (Langevin 1911, p. 31), referring to Einstein’s original two clocks thought experiment described in “Die Relativitätstheorie” (Einstein 1911, pp. 12ff.): “…Students often inquire as to ‘why’ the accelerated twin ages less and ‘when’ the extra aging of the home twin occurs. These questions are not well defined in the scientific sense but have promoted a variety of analyses (many can be found on the pages of this journal [American Journal of Physics]) which for most part have been useful additions to the pedagogy of special relativity theory” (Boughn 1989, p. 792). I join Boughn’s appreciation of the analyses prompted by the two cited students’ questions (see, for instance, Romer 1959, Perrin 1970, Unruh 1981, Good 1982), but, contrary to him, I hold the students’ questions themselves perfectly well formulated, directly answerable, and more than vindicated due to the great diversity of answers (or at least of the ways in which “the F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 233–252.

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right answer” is formulated), including those given by some great physicists (among them Einstein himself ). In dealing with the two students’ questions, we shall come across more explanations of the Twin Paradox that are at least misleadingly formulated if not straightforwardly wrong, and the attempt to disperse the conceptual confusion they cause, or can cause, is the main task of this paper.

2. A purely phoronomic re-description of the paradox Nearly all discussions of the Twin Paradox begin with the problem of symmetry: how can one of the twins turn out to be younger if becoming further from and becoming closer to are symmetrical processes? Bergson solved the problem of symmetry in the way Alexander the Great had solved the problem of the Gordian knot: there will be no age difference at the end of the trip because the difference in aging during twins’ getting apart form each other is compensated by the inverse process during their approaching each other (see Bergson 1976, pp. 434ff.). The standard reaction to Bergson’s “solution” is that there is an asymmetry since one of the twins has to slow down and speed up in order to meet his brother. The problem with this answer is that deceleration and acceleration are neither necessary nor sufficient for the explanation of the phenomenon, as we shall see in what follows. Let us describe the situation purely phoronomically, by stipulating that A’s and B’s “becoming closer to each other” and “becoming further from each other” have no commitments of “approaching” and “getting apart” in the sense that it is neither implied that it is only A or only B that is moving, nor that A and B are moving in opposite directions, nor that A is chasing B, nor that B is chasing A. In this way, it is also left indeterminate whether it is only not said (for whatever reason) what the case de re is or it makes no sense to differentiate de re among the possible cases mentioned because the distinction is not observer-invariant. Let us suppose that A and B were initially becoming closer to each other at a uniform relative speed, and then, at the meeting point, synchronized their clocks. When later B met C after supposedly becoming closer to him at a speed that was uniform but greater than the speed of A’s becoming closer to B, C synchronized his clock with B’s. After some time, A and C must have met. Were their clocks synchronized at the meeting point or not, and if not, is it C’s clock or B’s clock that was slow relative to the other one?

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3. Overcoming the underdetermination of the phoronomic re-description in SRT The underdetermination of the situation described purely phoronomically follows from the indeterminateness of the directions of motion of B and C. Is this indeterminateness overcome in SRT even under the assumption that all motion has been uniform since ever? Let A and B be light sources that, by passing each other, emit two light rays in both directions of their relative motion. Unexpectedly for the Newtonian physicist, each two of the four rays that move in the same direction will simultaneously reach any screen put orthogonally to their propagation. The fact that light behaves in this way has been known at least since 1849, when Fizeau performed the famous experiments that Einstein liked to cite (see Einstein 1977, pp. 39–41). The three well-known theories explain this behavior of light in the following three ways. According to the Maxwell-Lorentz’s theory, every lightwave propagates in any direction at the same speed with respect to the ether, which means that all the four rays propagate at the same speed with respect to the same point in absolute space from which they were emitted. Since the ether theory is incompatible with the deeply rooted idea of classical physics that all inertial states of motion are equivalent, Einstein had tried, sometime before 1905, to modify electrodynamics by supposing that the speed of the light source is to be added to the propagating effect, which was later elaborated in Ritz’s 1908 emission theory (see Norton 2004, pp. 58ff.). According to the emission theory, one cannot refer to a point left behind the moving sources but may say only that each two of the four rays that propagate in the same direction propagate differently with respect to A and with respect to B. Re-considering constantly the thought experiment he had been allegedly obsessed with since he was sixteen, which concerns the possibility of chasing a beam of light till the point at which it becomes frozen for the chaser, Einstein finally gave up the emission theory for the three reasons (see Norton 2004, Sections 5–6), one of them being that it does not only allow for the possibility that the light beam be seen as frozen but makes something like this likely to happen, which, however, does not happen. In SRT, this is impossible to happen by definition: light is postulated to propagate, with respect to any source, at the same speed in all directions, which implies, automatically, that it is not possible to catch up with a light beam by chasing it.

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If light propagates at the same speed in all directions, what about the case in which the light sources are moving relative to each other? The only way to avoid contradiction is to accept that the spacetime metric, and so also space-like and time-like metrics, are different in the referential systems of the two sources. This consequence is crucial for overcoming the underdetermination of the phoronomic description of the Twin Paradox, but it is not obvious how it is to be used. Let us call a property Shakespearean (recalling the famous verse on the rose), be it non-relational or relational, if and only if its true ascription to something is observer-invariant (see Geach 1972, p. 139). For instance, if John is shorter than Peter, he will be shorter even if he looks taller (for whatever reason) to an observer, and he will remain shorter even if we start to praise him as taller than Peter. The properties being shorter than and being taller than are relational, but Shakespearean relational properties. Is direction of motion a Shakespearean property? According to Descartes’ metaphysics, motion is always only relative because God, by permanently bringing the world into being, is also the ultimate cause of any change in the world (cf. Descartes 1986, Part II § 36, 25, 27). So, as in a movie, if the position between two bodies is changed, it is not changed because one of them or both moved but because God put them in a different spatial relation. Consequently, though different world stages are real and not just an appearance, motion as such is just an appearance. This is why for the Cartesians the direction of motion can be, and in fact is, a non-Shakespearean relational property. In SRT, the light propagation is absolute not only in view of the assumed constancy of its speed, which is independent of the motion of light sources, but also because its direction is a Shakespearean property according to the main postulate of SRT: it propagates in the direction in which it is impossible to catch up with it by chasing it. If there were two bodies positioned at the line along which a light beam propagates but at the opposite sides of the light beam, the beam would propagate necessarily in the direction in which it could come across one of the two bodies, and necessarily not in the other direction. But what about the direction of motion of the bodies themselves (as potential or actual light sources)? The main problem in answering this question follows from the assumption that light propagates always and in all directions at the same speed relative to any light source, so that each light source should be considered to be at rest relative to any ray emitted, while, at the same time, the light sources

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can move, as they often do, relative to each other. Is it then possible to say, or perhaps even necessary to conclude, that the direction of motion of two light sources moving relative to each other is yet a non-Shakespearean property, in spite of the fact that the direction of the light propagation itself is a Shakespearean property? For the negative answer to the last question it would be sufficient to show that, in some cases at least, the electromagnetic effects will differ according to what we suppose to be the directions of motion of the two light sources involved. Now, the Twin Paradox shows just this, which is one of its two main roles that has never been explicitly mentioned as such. Namely, the situation described in the Twin Paradox is the exemplum crucis for showing that in SRT the direction of motion of the light sources themselves must be a Shakespearean property, which is a good reason for agreeing with Michael Redhead that “the terminology ‘relativity’ for Einstein’s theory is arguably misleading” (Redhead 1993, p. 120). The second important role of the Twin Paradox, obvious and recognized, is that it exemplifies the essential difference between SRT and the other two rival theories through some striking consequences that are in principle testable (and have already been tested indirectly — see, for instance, Hafele and Keating 1970). (a) If B and C (in the above phoronomic description) are supposed to move in opposite directions, the average speed of the composite motion of B and C must be greater than the speed of A independently of whether A and B move in the same or in opposite directions. Namely, if A and B move in the same direction, then B is obviously faster than A, and B and C are supposed to move at a relative speed that is greater than the relative speed at which A and B move. So, the path traversed by B and C will be longer than that of A, and so their average speed greater than that of A. If A and B move in opposite directions, then, however small the relative speed of B may be, C will more than balance it by chasing and supposedly reaching A. So, the average speed of the composite motion of B and C will be greater again. This means that, because B and C were faster than A in the absolute sense, their average speed was closer to the speed of light when compared with the speed of A. This finally means (by taking into account how the contradiction threatening from the assumption that the speed of light is the same in all directions relative to any source has been avoided) that between the meeting of A and B and the meeting of A and C the composite time of B and C must have been metrically shorter than the time of A, which means in effect that, if A, B, and C emitted light signals periodically according to identical clocks

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synchronized in the described way, the number of signals emitted by B and C must have been smaller than the number of signals emitted by A. (b) It is easy to show, by applying the analogous reasoning, that the outcome will be inverse if we suppose that B and C move in the same direction: the time of A will be shorter than the composite time of B and C, and the number of signals emitted by A will be smaller than the number of signals emitted by B and C. It follows from (a) and (b) that there will be differences concerning the electrodynamic effects depending solely on whether B and C move in opposite directions or in the same direction, which means that, according to SRT, the direction of motion of the light sources must be a Shakespearean property (Q.E.D.).

4. Einstein’s original description of the Clock Paradox and the role of the turning point In his description of the Two Clocks Thought Experiment Einstein mentioned a “speed close to the speed of light” twice, the first time when assuming that one of the clocks “moves at a high speed (close to the speed c) at a uniform motion”, the second time by deriving the general conclusion that, in the end, it will turn out that “the position of the hand of the clock almost didn’t change during the entire journey” and that, in the analogous case of an organism, “the long journey lasted for just a moment because it moved at a speed close to the speed of light” (Einstein 1911, pp. 12–13 [my translation]). The natural question about Einstein’s description is why we need a roundtrip for concluding that the clock that moves back and forth will be slow, given that the conclusion is derived from nothing else but the closeness of its speed to the speed of light and that we have assumed at the very beginning that it is this clock that moves at a speed close to the speed of light. One could be tempted to say that the round-trip is introduced with the sole intention to shock by its strange result, and this is the first reason why it is better to omit Einstein’s assumption about the initial speed of any of the two clocks. The second reason for omitting the assumption that one of the clocks (or twins) moves at a speed close to the speed of light already by becoming further from the other clock (twin) concerns the generality of the situation described. There is an infinite number of ways in which the round-trip can

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take place with necessarily the same outcome concerning the question of how the clocks are desynchronized at the end. Recalling our phoronomic description, the only additional assumption necessary and sufficient for the conclusion that C’s clock will turn out to be slow is that B and C move in opposite directions. So, Einstein’s description is just one very special case. The third reason for omitting any assumption concerning the initial speed of any of the clocks relative to the speed of light concerns the impossibility of testing any such assumption either during the clocks’ receding further from each other alone or during their getting closer to each other alone. As we shall see below, it is only the change in frequency of the received signals that can enable the twins (or A and C, with the help of B, in our re-description) to calculate what will be the ratio of the number of the sent signals to the number of the received signals, which is necessary for concluding how each traveler has been moving and aging. In view of the last point, however, it is important to note that the fact that it is not testable how the aging processes are developing while the twins recede further from each other alone and get closer to each other alone does not mean that the final age difference is not the result of the difference in aging. In spite of the fact that Einstein’s assumption about the initial speed of a clock is unnecessary, it is certainly not wrong in the sense that it makes no sense to assume something like this. The suggestion that it cannot be said “when” the difference in aging is coming into being, based on the necessity of a turning point for saying anything about the age difference, led to the “mysterious” explanation (we shall come across below) that the age difference takes place instantaneously.

5. Invoking dynamical factors All misleadingly formulated, incomplete or directly wrong solutions to the Twin Paradox are based on a misinterpretation of the role of the turning point on the round-trip. Lecturing about the Twin Paradox, Feynman said that those who think it is a real paradox do so because they “believe that the principle of relativity means that all motion is relative” (Feynman 1963. p. 16–3). What he actually wanted to say is not that they think just that all motion is relative but rather that the direction of motion is an observer-dependent, non-Shakespearean property. Feynman’s answer to this Cartesian challenge is that “the man who has felt the acceleration, who has seen things fall against the walls, and so on,

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is the one who would be younger” (loc. cit.). Since Feynman, after having indicated an asymmetry between the twins, didn’t explain in which way exactly it is relevant for the difference in aging, his explanation is incomplete. But it is also misleading because deceleration and acceleration are neither necessary nor sufficient for the difference in aging. Deceleration and acceleration are not necessary because the direction of motion is a Shakespearean relational property according to the main postulate of the SRT, which is sufficient, as we have seen, for the explanation of the difference in aging even in the case in which all motions involved in the experiment are supposed to be uniform. Deceleration and acceleration are not sufficient for the occurrence of the phenomenon because a twin that has changed direction can still age faster, namely, if the relative speed after he has changed his direction of motion is not sufficient to reach his brother. In brief, possible dynamical factors related to the turning point are as such not causally responsible for the difference in aging, being at best an indication of which twin will be younger given that he moves at a speed that enables him to reach his brother. However, arguing against “the orthodox relativists”, Whitehead claimed that “acceleration and deceleration (as distinct from uniform velocity) express an essential fact of the life of any body” (Whitehead 1923, p. 41), and that it is just “the diverse history” of the body of the chronologer on the earth and of the traveler’s body, that is, “the real diversity of relations of their bodies to the universe”, that “is the cause of their discordance in time-reckoning” (ibid., p. 34). Similarly, by trying to explain why it is the traveler’s clock U′ that will be slow relative to the earth clock U and not vice versa, Reichenbach invoked “the theory of gravitation”, which “shows that the special theory of relativity is applicable only because the distant masses of the fixed stars determine a particular metric field”, so that a retardation of U′ is “the effect of the moving fixed stars, which produce a gravitational field at the instant of the reversal of the motion” (Reichenbach 1956, p. 193). To refute this “solution”, it is sufficient to imagine two journeys of the same person, with the same clock, moving at the same speed and turning himself around in the same way at the same cosmic point, with the only difference being that one of them lasts longer (see Mermin 1968, p. 144). The two journeys bring about different age differences, since the age difference depends on the length of the journey, and not on anything that happened at the turning point.

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Finally, Richard Tolman, the most famous of Feynman’s predecessors at the California Institute of Technology, claimed that due to the possibility of the symmetrical analyses the paradox “can arise when the behavior of clocks is treated in accordance with the principles of special relativity without making due allowance for the principles of the general theory” (Tolman 1962, p. 194), while it is “readily solved with the help of the general theory of relativity, if we do not neglect the actual lack of symmetry between the treatment given to the clock A which was at no time subjected to any force, and that given to the clock B which was subjected to the successive forces F1, F2 and F3 when the relative motion of the clocks was changed” (ibid., p. 195).

6. The light signals exchange and the geometric representation of the thought experiment As shown in Section 3, the Twin Paradox exemplifies the situation in which we are vindicated in saying that one of the twins (or B and C in our re-description of the situation) moved at an average speed that was closer to the speed of light than was the speed of the other twin (A in our re-description). Ontologically speaking, this fact is ratio essendi for claiming that the number of light signals emitted by the back and forth traveling twin (B and C in our re-description) must have been smaller than the number of the signals emitted by the other twin (A in our description) in spite of the fact that the signals were supposedly emitted after the same time intervals had elapsed according to the identically made and synchronized clocks of the twins (of A, B, and C in our re-description). But then, the ratio of the total numbers of the signals emitted and received must be ratio cognoscendi for establishing, by the twins (A, B, and C) themselves, how their trip actually looked in view of the questions of the direction of their motion and of when and how close their motion was to the speed of light. Let us suppose that, at the time they meet, A and B have arranged to send light signals to each other every year. Let them arrange, in addition, that if B meets or reaches a cosmic wanderer C moving at a uniform speed, it will be C who will instead of B continue to receive and send signals in the arranged way (according to B’s clock). Now, due to the Doppler shift, both A and B will receive signals from each other with a frequency that is less than the frequency of the signals they send. But, the frequency with which the signals are received along with A’s and B’s knowledge of the relative speed

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at which they recede further from each other will not enable them to say anything about whose speed, if anyone’s, is closer to the speed of light. Mutatis mutandis, the same holds for A and C. However, the fact that neither A and B nor A and C can get any information about their speed relative to the speed of light on the basis of what was happening during the time A and B were receding further from each other alone, and during the time A and C are getting closer to each other alone, it does not follow that A and C (the latter with the help of B) cannot infer what was and what is going on in the absolute sense by calculating, on the base of their frequency, the total number of signals that will be sent and the total number of those that will be received, and finaly by comparing the two. Namely, if B and C de facto moved in opposite directions (independently of whether there is any dynamical indication of that or not), then, by the time A and C meet, the total number of signals that A received must be less than the total number of the signals he sent, while the total number of the signals received first by B and then by C must be greater than the total number of the signals they sent. If, on the other hand, B and C de facto moved in the same direction, the situation concerning the total numbers of sent and received signals will be the inverse. The just said can be illustrated through the following two Minkowskian diagrams (borrowed from Bohm 1996, pp. 169, 171).

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It is important to note that, by drawing these diagrams, Bohm followed Einstein’s original description of the thought experiment, according to which A (whose world-line is OH) is taken to be at rest while B (whose world-line is OE) and C (whose world-line is EH) are taken to be moving in opposite directions at the same uniform speed relative to A. That’s why OE is equal to EH and why, on the left diagram, the intervals between signals M1, M2, M3, and M4, sent by B, are equal to the intervals between signals N1, N2, N3, and N4, sent by C. But the point we are interested in would not be lost if we changed these special assumptions, obtaining a triangle of any other shape. The only necessary assumption, without which we wouldn’t obtain a triangle at all as the representation of what is going on, is that the speed at which B and C move relative to each other is greater than the speed at which A and B do so. The additional assumption, not necessary for getting a triangle but necessary for getting E to be the turning point, is that B and C move in opposite directions, for the purely phoronomic description allows for the possibility that, by meeting C, B actually overtakes C. In this latter case, due to the fact that, after B and C meet, it is C whose time starts being compared with that of A, the turning point would occur at A’s world-line, and not at the point where B and C meet, for it is now A who (without doing and feeling anything new!) actually starts chasing C. Now, it is clear from the left diagram that A will start receiving signals with a greater frequency (than the frequency of the signals he sends) only at E′, that is, relatively late, so that the total number of received signals will be less than the total number of sent signals. On the other hand, as it is clear from the right diagram, only a few signals sent by A will be received by B with a lesser frequency (than the frequency of the signals B sent), while the rest of signals, sent during a much longer time, will be received by C with a frequency that is greater than the frequency of the signals sent by C. So, the total number of the signals received by B and C will evidently be greater than the total number of the signals sent by them. Is it at all possible that A and C meet finding their clocks synchronized? No, because there is only one turning point. But if we imagine A meeting some fourth person D under conditions that are completely symmetrical relative to those in which B meets C, then, when C and D met, their clocks would be synchronized. The Minkowskian diagram representing this situation would contain two broken world-lines symmetrical in view of the line connecting the meeting points of A and B, and C and D. So, the Bergsonian “solution” is true only in a special case, where there are two turning points.

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7. When does the difference in aging come into being? However nice a device the Minkowski diagram method is for representing geometrically what is going on physically, an understanding of what is going on physically must be the guide for “reading off the diagram”. Since we have explained in physical terms in which way the final age difference of the twins is the result of the difference in aging, which can be but needn’t be the same during the whole trip though it must always be in favor of the twin who traveled faster in the absolute sense, we may use Minkowski diagrams to show students when and how exactly the difference in aging came into being. For instance, according to the two diagrams above, which represent Einstein’s original description, the final age difference is the result of the difference in aging that was coming into being during the whole trip. Namely (by using again our three persons A, B, and C instead of two twins), it can be directly read off from the left diagram that, if A, B, and C arranged to send signals every year, the composite motion of B and C lasted 10 years (these 10 years being OM1, M1M2, M2M3, M3M4, M4E, EN1, N1N2, N2N3, N3N4, N4H), while, as can be read off from the right diagram, A lived 15 years (these 15 years being OM ′1, M′1M′2, M′2N′1, N′1N′2, N′2N′3,…, N′11N′12, N′12H). Let us turn to other, less specific triangles, but still with the meeting point of B and C as the turning point (E on the diagram), meaning that both A and B must be supposed to be moving, and moving in opposite directions. In this case (in contrast to the example of Einstein and Bohm), the length of B’s and C’s years must be different, because C has to chase and catch up with A. The final age difference must again be in favor of B and C taken together, but though it can still be true that it is in favor of both B and C relative to A, it can be also in favor of A over B, and C over A. The former would be the case if B were supposed to move, when compared to A, at a speed closer to the speed of light, while the latter would be the case if the speed of A were closer to the speed of light.

8. A “mysterious” geometric explanation of the age difference Though one can find what has just been said evident and even trivial, the students’ question of when the difference in aging takes place has been often suggested, as in Boughn’s case, not to be well defined. But worse than that,

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Minkowski diagrams are often used to “show” that the faster aging is based on the existence of some “extra time”, which another twin saved “instantaneously”, which is at best a very misleading façon de parler. In his famous book Space, Time, and Spacetime, Lawrence Sklar explains the age difference using the following diagram: x O′

e′′ e

e′

O

He says: “The accelerated observer calculates that the inertial clock runs slow according to his from O to e′, and also from e″ to O′; but he sees himself as moving instantaneously from one inertial path to the other (at event e, his acceleration being ‘almost instantaneous’), yet e is simultaneous with e′ in his first inertial frame, and with e″ in the second. It is the life of the inertial clock from e′ to e″ which makes the inertial clock read a greater time interval from O to O′ than does the accelerated clock” (Sklar 1977, p. 270 [my italics]). The crucial thing is that Sklar says that the life of the inertial clock is longer due to its life from e′ to e″ (which is the interval between events N′2 and N ′9 on the right Bohm diagram above, where OI, and its parallels, are the simultaneity lines from A’s reference system, and OP and OQ, and their parallels, simultaneity lines from B’s and C’s referential systems respectively). Complementary to this, Sklar says that the accelerated clock saved time at turning point e, which is “simultaneous with e′ in his first inertial frame, and with e″ in the second”. Speaking about a similar diagram, Redhead says that it “shows β’s [the traveling twin’s] clock running ahead of α’ s [the oth-

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er one’s] along sections Oe′ and e″O, but mysteriously standing still along e′e″ allowing the differential aging of α to take place” (Redhead 1993, p. 123 [letters substituted to fit Sklar’s diagram]). I find the adverb “mysteriously” used by Redhead quite appropriate. By dealing with the Sklarian-style explanation based on the Minkowskian flat spacetime geometry, we are faced with a situation that is strikingly similar to that we were already faced with in Section 5, when we were dealing with the explanations based on dynamical factors. In both cases, the turning point is correctly associated with the twin who will finally turn out to be younger, but some effects of the change of the inertial system, independently of whether accidental or necessary, are wrongly allowed to figure in the explanation, which should be based instead on the difference in metrics alone, as a consequence of the difference in (at least average) speed relative to the speed of light. If we want to use geometric representation to explain what is going on physically, we have first to use two Minkowski diagrams, in the way in which Bohm did it, for it is only so that we can directly read off, and then compare, the time metric of the accelerated twin shown in terms of the total number of time units during the whole trip (the left Bohm diagram) and the time metric of the other twin shown through the total number of time units of the round-trip (the right diagram). Each of the two diagrams taken per se shows only in which way one of the twins judges the time of the other one. It is of great importance to notice that the time-distance between e′ and e″, which Sklar speaks of as the time saved by the change of the inertial system, depends on both speeds, the one between O and e and the other between e and O′, which determine the angle at e on the diagram. So, at e, the age difference that will show up at the end of the trip is still to come into being through the difference in aging caused by the difference in metrics, which itself is caused by the difference in speed relative to the speed of light. So, it is at least a very misleading façon de parler to say that the accelerated twin saved some time (represented on the diagram as e′e″) by the very act of changing the inertial system. “The discrepancy between the simultaneity relations” that Wesley Salmon takes to be “the key to the whole problem” (Salmon 1980, p. 98) is something that itself should be, and can be, explained through the spacetime metrics of the whole situation. It would be mysterious indeed if the fact that “there is no moment between e′ and e″ that the traveling twin could find simultaneous with any moment of his trip” were to mean that the age difference at the

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end of the trip is the result of an extra life that the home twin lived while his brother was (only for an instant!) at the turning point of his trip. But there is nothing mysterious in saying that due to the difference in spacetime metrics, and consequently in metrics of space-like and time-like intervals, the twins not only age differently but also cannot use any simultaneity relation that would be the equivalence relation — from which the “discrepancy” caused by the change of the referential system, which Salmon speaks about, can be derived as a corollary.

9. Conclusion and consequences: The Twin Paradox in the flat but closed spacetime Once it is clearly explained why the twin who turned out younger did (because the metric difference between his and his brother’s world-lines favored him due to his speed, which was, on average at least, closer to the speed of light) and when the favoring age difference was coming into being (which can be either during the whole trip or during a part of it only, but in a more than balancing way), the question about the necessity of the turning point becomes answerable without confusion. In the standard representation, we need the turning point for epistemological and operational reasons. Namely, in the open flat Minkowski spacetime, it is only the round-trip that enables us to prove that one of the twins must have been faster in the absolute sense and so “closer to the speed of light”. At the same time, it is only the change of one of the twins’ inertial systems that brings about the change in the frequency of the received light signals, enabling the twins themselves to calculate metrical differences between their world-lines. After dispersing all the confusions caused by the insufficiently clarified role of the turning point, we are free to turn to various closed spacetime models and describe the Twin Paradox situation without a turning point, as it is done, for instance, in (Brans and Stewart 1973), (Dray 1982), (Low 1990), (Uzan, Luminet, Lehoucq and Peter 2000). Let me give a nice example that shows what happens in one of these cases. It is shown above why in the open flat Minkowski spacetime the existence of one turning point implies the asymmetry that is necessary and sufficient for the final difference in age between the twins, whereas, on the other hand, the existence of two symmetric turning points implies that there will be no difference in age between them. What will happen if the twins move at a uniform speed in spacetime that is topologically so structured that it is flat

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but closed, so that it is unnecessary that there is a turning point before the twins meet? According to the account given above, there can be an infinite number of cases lying between the two extremes. One of these extreme cases is that both twins move at a speed that is equally close to the speed of light. In this case, there will be no difference in aging and, consequently, no difference in the final difference in age between the twins. In the other extreme case, one of the twins is at rest. In this case, he will turn out older at the meeting point. In all other cases, the twin who is moving at a speed that is closer to the speed of light will finally turn out younger. How is the second extreme case to be represented? The first thing to notice is that the topological structure is supposed to be flat in the sense that its Riemann tensor vanishes identically, so that we can pick a coordinate system in which the metric is constant. This means that we can have a case in which neither twin undergoes acceleration. x=±1

t=0

(0,0)

At the same time, however, the topological spacetime structure is supposed to be closed in the sense that there are pairs of space points — say, the point x = 1 and the point x = −1 — which are identical for all values of t. This means in effect that, as the geometric Minkowski spacetime representation, we get a cylinder (see the diagram above) with the t-axis as the vertical one. Now, let us suppose that one of the twins remains at rest at x = 0, while the other starts off at (0, 0), moving to the right with relative speed v, undergoing no acceleration at any time. This means that, between (0, 0) and (0, ±1),the former will travel along the t-axis only, while the latter will travel along a helix winding round the cylinder with slope determined by v.

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When t = 2/v (in the frame of the first twin), the two twins will meet at (0, ±1). It is clear that, in the given case, no resolution in Sklar’s style is possible (cf. Section 8 above), since there is no turning point at which the twin traveling along the helix could allegedly save some extra time. Moreover, the original paradox seems to re-emerge because one could argue that both twins can apply Lorentz transformations in the same way during the whole trip and get the same result at the end of the trip (see Low 1990, § 2). In other words, if one of them calculates the other’s measure of elapsed time by multiplying 2/v by (1−v 2 ), so does the other. Low says that “the entire technology of using Lorentz transformations to calculate is suspect in this context” because “the transformations make sense locally” but not “on the whole” of Minkowski spacetime (ibid., § 3). Namely, due to the fact that “spacetime in this instance, although flat, is not simply connected”, the twins’ receding from each other is, at the same time, their approaching each other. After that, Low claims that it is only the twin who remains at x = 0 who is right in calculating the other twin’s measure of the whole elapsed time by multiplying 2/v by (1−v 2 ) , because by unwrapping the cylinder and finding various positions of the twins we see that in the frame of the twin who is at rest, for any value of t, the departure of the other twin from the corresponding position is always simultaneous, while it is not so in the frame of the twin who travels along the helix. What Low says is, of course, correct from the point of view of the spacetime geometry. But again, following Bohr’s requirement, cited by Heisenberg — that a physical explanation should absolutely precede the mathematical formulation — there is more that needs to be said in order to give a full explanation in physical terms. Firstly, from the physical point of view, a part of the reason why only one of the twins is right in calculating the measure of the elapsed time of the other twin by multiplying 2/v by (1−v 2 ) is that it is supposed, in the given extreme case, that it is he who is at absolute rest relative to the light propagation, so that it is he who is in the absolutely privileged position to calculate any time saved by anybody moving relative to him. It is so, once again, only because in SRT it is not only true, as in the emission theory, that light propagates at the same speed in relation to any light source, but also that it propagates so in an absolute sense, independently of whether one can detect it or not.

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Secondly, another part of the reason why one of the twins is right in calculating the measure of the elapsed time of the other twin by multiplying 2/v by (1−v 2 ) is that it is supposed, in the given extreme case, that light itself propagates in such a way that it will finally reach the same space point from which it started off and that the other twin is chasing such a light beam. In other words, the cylindrical spacetime representation, according to which one of the twins travels along a helix, is correct only because light is supposed to propagate in the described way. Thirdly, the supposed fact that light propagates at the same speed both in an absolute sense as well as in relation to any light source, implies that there must be a difference in both space-like and time-like metrics, which means that traveling along the spacetime helix must be longer in view of space and shorter in view of time. In other words, the difference between the simultaneity relations, which Low draws our attention to, is only the consequence of the difference in metrics and not the other way round. Finally, the most delicate question, which Low does not address in his article, is how the twins themselves can find out by exchange of light signals, if they can do it at all, what their clocks will show at the end of the trip. They can do that in essentially the same way in which it has been shown that they can do it by traveling in the open Minkowski spacetime. Namely, though in the closed Minkowski spacetime there is no turning point, so that the twins cannot use the difference in frequencies between the signals sent periodically before and after the turning point, they can send and receive signals sent periodically in opposite directions (because the spacetime is supposedly closed) and use the difference in their frequencies for calculating the time that has elapsed and the time that is to elapse before they meet. So, both can apply Lorentz transformations for calculating the whole elapsed time. All in all, from the physical point of view, the resolution of the Twin Paradox formulated in the closed flat Minkowski spacetime is the same as the resolution of it when formulated in the open flat Minkowski spacetime.

References Bergson, H. 1976 “Discussion With Becquerel of the Paradox of the Twins”, in: M. Čapek (ed.), The Concepts of Space and Time, 433–439. Bohm, D. 1996 The Special Theory of Relativity, Routledge.

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Boughn, S. P. 1989 “The Case of the Identically Accelerated Twins”, Am. J. Phys. 57, 791–793. Brans, C. H and Stewart D. R. 1973 “Unaccelerated Returning Twin Paradox in Flat Spacetime”, Phys. Rev. D8, 1662–1666. Descartes, R. 1988 Principles of Philosophy, in The Philosophical Writings of Descartes, Vol. I, Cambridge University Press. Dray, T. 1990 “The Twin Paradox Revisited”, Am. J. Phys. 58, 822–825. Einstein, A. 1911 “Die Relativitätstheorie”, Vierteljahrsschrift der naturforschenden Gesellschaft in Zürich. — 1977 Relativity —The Special and the General Theory, Methuen and Co. Ltd, London (first edition 1920). Geach, P.T. 1972 Logic Matters, Basil Blackwell. Good, R. H. 1982 “Uniformly Accelerated Reference Frame and Twin Paradox”, Am. J. Phys. 50, 232–238. Hafele, J. C. and Keating, R. E. 1972 “Around-the-World Atomic Clocks: Observed Relativistic Time Gains”, Science 177, 168–170. Heisenberg, W. 1967 “Quantum theory and its interpretation”, in: S. Rozental (ed.), Niels Boh — His Life and Work as Seen by His Friends and Colleagues, North-Holland, Amsterdam. Langevin, P. 1911 “L’évolution de l’espace at du temps”, Scientia X (31). Low, R.J .1990: “An acceleration-free version of the clock paradox”, Eur. J. Phys. 11, 25–27. Mermin, N. D. 1968 Space and Time in Special Relativity, Waveland Press. Norton, J. D. 2004 “Einstein’s Investigations of Galilean Covariant Electrodynamics prior to 1905”, Archive for History of Exact Sciences 59, 45–105. Perrin, R. 1970 “Twin Paradox: A Complete Treatment from the Point of View of Each Twin”, Am. J .Phys. 44, 317–319. Redhead, M. 1993 “The Conventionality of Simultaneity”, in: J. Earman, A. I. Janis, G. J. Massey, N. Rescher (ed.), Philosophical Problems of the Internal and External Worlds, University of Pittsburgh Press, 103–128. Reichenbach, H. 1956 Philosophy of Space and Time, Dover. Romer, R. H. 1959 “Twin Paradox in Special Relativity”, Am. J. Phys. 27, 131–135. Salmon, W. C. 1980, Space, Time, and Motion (2nd edition, revised), University of Minnesota Press. Sklar, L. 1977 Space, Time, and Spacetime, University of California Press, Berkeley, Los Angeles, London.

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Tolman, R. 1962 Relativity, Thermodynamics and Cosmology, Clarendon Press, Oxford (first edition 1934). Unruh, W. G. 1981 “Parallax Distance, Time, and the Twin Paradox”, Am. J. Phys. 49, 589–592. Uzan, J.-P. , Luminet, J.-P., Lehoucq, R. L. and Peter, P. 2000 “Twin Paradox and Space Topology”, Ar. Phys., June, 1–6. Whitehead, A. N. 1923 “The problem of simultaneity”, Aristotelian Society, Suppl. Vol. 3, 34–41.

Time’s Arrow, Time’s F ly-Bottle Huw Price, Sydney For more than a century, physics has known of a puzzling conflict between the time-asymmetry of thermodynamic phenomena and the time-symmetry of the underlying microphysics on which these phenomena depend. In the spirit of “philosophy as therapy”, this paper examines the current status of this puzzle, distinguishing the central issue from various issues with which it is commonly confused. In particular, I argue that there are two competing conceptions of what is needed to resolve the puzzle of the thermodynamic asymmetry, which differ with respect to the number of distinct T-asymmetries they take to be manifest in the physical world. On the preferable one-asymmetry conception, the remaining puzzle concerns the ordered distribution of matter in the early universe. The puzzle of the thermodynamic arrow thus becomes a puzzle for cosmology.

1. The puzzle of temporal bias Late in the nineteenth century, on the shoulders of Maxwell, Boltzmann and many lesser giants, physicists saw that there is a deep puzzle behind the familiar phenomena described by the new science of thermodynamics. On the one hand, many such phenomena show a striking temporal bias. They are common in one temporal orientation, but rare or non-existent in reverse. On the other hand, the underlying laws of mechanics show no such temporal preference. If they allow a process in one direction, they also allow its temporal mirror image. Hence the puzzle: if the laws are so even-handed, why are the phenomena themselves so one-sided? What has happened to this puzzle since the 1890s? Many contemporary physicists appear to regard it as a dead issue, long since laid to rest. Didn’t it turn out to be just a matter of statistics, after all? However, while there are certainly would-be solutions on offer — if anything, too many of them — it is far from clear that the puzzle has actually been solved. Late in the twentieth century, in fact, one of the most authoritative writers on the conceptual foundations of statistical mechanics could still refer to an understanding of F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 253–273.

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the time-asymmetry of thermodynamics as ‘that obscure object of desire’. (Sklar 1995) One of the obstacles to declaring the problem solved is that there are several distinct approaches, not obviously compatible with one another. Which of these, if any, is supposed to be the solution, now in our grasp? Even more interestingly, it turns out that not all these would-be solutions are answers to the same question. There are different and incompatible conceptions in the literature of what the puzzle of the thermodynamic asymmetry actually is — about what exactly we should be trying explain, when we try to explain the thermodynamic arrow of time. What the problem needs is what philosophers do for a living: drawing fine distinctions, sorting out ambiguities, and clarifying the logical structure of difficult and subtle issues. In this paper, my aim here is to bring these methods to bear on the puzzle of the time-asymmetry of thermodynamics. In particular, I want to distinguish the true puzzle from some of the appealing false trails, and hence to make it clear where physics stands in its attempt to solve it.1 What does this have to do with Wittgenstein? Mainly that it provides an example of “philosophy as therapy”. To put it in terms of Wittgenstein’s famous metaphor, the goal is to free a fly who has been trapped for more than a century, in a fly-bottle constituted by the strange time-asymmetry of the second law of thermodynamics. True, Wittgenstein had in mind the fly-bottles in which philosophers entrap themselves, whereas our present prisoners are physicists, as much as philosophers. And the result of the exercise does not seem to be that the “problems … completely disappear” (PI, § 133) — some genuine puzzles remain, in my view. Nevertheless, as we’ll see, this is certainly a case in which philosophy is therapeutic, helping to purge us of pseudo-puzzles that arise, at least in part, from bad habits of thought.

2. The true puzzle — a first approximation and a popular challenge Everyone agrees, I think, that the puzzle of the thermodynamic arrow stems from the conjunction of two facts (or apparent facts — one way to dissolve 1

The present treatment is brief, but I discuss these topics at greater length elsewhere — see Price (1996, 2002a, 2002b, 2004).

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the puzzle would be to show that one or other of the following claims isn’t actually true): 1. There are many common and familiar physical processes, collectively describable as cases in which entropy is increasing, whose corresponding time-reversed processes are unknown or at least very rare. 2. The dynamical laws governing such processes show no such T-asymmetry — if they permit a process to occur with one temporal orientation, they permit it to occur with the reverse orientation. As noted, some people will be inclined to object at this point that the conjunction is merely apparent. In particular, it may be objected that we now know that the dynamical laws are not time-symmetric. Famously, T-symmetry is violated in weak interactions, by the neutral K meson. Doesn’t this eliminate the puzzle? No. If the time-asymmetry of thermodynamics were associated with the T-symmetry violation displayed by the neutral K meson, then anti-matter would show the reverse of the normal thermodynamic asymmetry. Why? Because PCT-symmetry guarantees that if we replace matter by anti-matter (i.e., reverse P and C) and then view the result in reverse time (i.e., reverse T), physics remains the same. So if we replaced matter by anti-matter but didn’t reverse time, any intrinsic temporal arrow or T-symmetry violation would reverse its apparent direction. In other words, physicists in anti-matter galaxies find the opposite violations of T-symmetry in weak interactions to those found in our galaxy. So if the thermodynamic arrow were tied to the T-symmetry violation, it too would have to reverse under such a transformation. But now we have both an apparent falsehood, and a paradox. There’s an apparent falsehood because (of course) we don’t think that anti-matter behaves anti-thermodynamically. We expect stars in anti-matter galaxies to radiate just like our own sun (as the very idea of an anti-matter galaxy requires, in fact). And there’s a paradox, because if this were the right story, what would happen to particles which are their own anti-particles, such as photons? They would have to behave both thermodynamically and antithermodynamically! Here’s another way to put the point. The thermodynamic arrow isn’t just a T-asymmetry, it is a PCT-asymmetry as well. There are many familiar process whose PCT-reversed processes are equally compatible with the underly-

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ing laws, but which never happen, in our experience. We might be tempted to explain this asymmetry as due to the imbalance between matter and antimatter, but the above reflections show that this is not so. So instead of the puzzle of the T-asymmetry of thermodynamics, we could speak of the puzzle of the PCT-asymmetry of thermodynamics. Then it would be clear to all that the strange behaviour of the neutral K meson isn’t relevant. Knowing that we could if necessary rephrase the problem in this way, we can safely rely on the simpler formulation, and return to our original version of the puzzle.

3. Four things the puzzle is not Some of the confusions common in debates about the origins of the thermodynamic asymmetry can be avoided distinguishing the genuine puzzle from various pseudo-puzzles with which it is liable to be confused. In this section I’ll draw four distinctions of this kind.

The meaning of irreversibility The thermodynamic arrow is often described in terms of the ‘irreversibility’ of many common processes — e.g., of what happens when a gas disperses from a pressurised bottle. This makes it sound as if the problem is that we can’t make the gas behave in the opposite way — we can’t make it put itself back into the bottle. Famously, Loschmidt’s reversibility objection rested on pointing out that the reverse motion is equally compatible with the laws of mechanics. Some responses to this problem (e.g., Ridderbos and Redhead 1998) concentrate on the issue as to why we can’t actually reverse the motions (at least in most cases). This response misses the interesting point, however. The interesting issue turns on a numerical imbalance in nature between ‘forward’ and ‘reverse’ processes, not case-by-case irreversibility of individual processes. Consider a parity analogy. Imagine a world containing many left hands but few right hands. Such a world shows an interesting parity asymmetry, even if any individual left hand can easily be transformed into a right hand. Conversely, a world with equal numbers of left and right hands is not interestingly Pasymmetric, even if any individual left or right hand cannot be reversed. Thus the interesting issue concerns the numerical asymmetry between the two kinds of structures — here, left hands and right hands — not the question whether one can be transformed into the other.

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Similarly in the thermodynamic case, in my view. The important thing to explain is the numerical imbalance in nature between entropy-increasing processes and their T-reversed counterparts, not the practical irreversibility of individual processes.

Asymmetry in time versus asymmetry of time Writers on the thermodynamic asymmetry often write as if the problem of explaining this asymmetry is the problem of explaining ‘the direction of time’. This may be a harmless way of speaking, but we should keep in mind that the real puzzle concerns the asymmetry of physical processes in time, not an asymmetry of time itself. By analogy, imagine a long narrow room, architecturally symmetrical end-to-end. Now suppose all the chairs in the room are facing the same end. Then there’s a puzzle about the asymmetry in the arrangement of the chairs, but not a puzzle about the asymmetry of the room. Similarly, the thermodynamic asymmetry is an asymmetry of the ‘contents’ of time, not an asymmetry of the container itself. It may be helpful to make a few remarks about the phrase ‘direction of time’. Although this expression is in common use, it isn’t at all clear what it could actually mean, if we try to take it literally. Often the thought seems to be that there is an objective sense in which one time direction is future (or ‘positive’), and the other past (or ‘negative’). But what could this distinction amount to? It’s easy enough to make sense of the idea that time is anisotropic — i.e., different in one direction than in the other. For example, time might be finite in one direction but infinite in the other. But this isn’t enough to give a direction to time, in the above sense. After all, if one direction were objectively the future or positive direction, then in the case of a universe finite at one end, there would be two possibilities. Time might be finite in the past, and or finite in the future. So anisotropy alone doesn’t give us direction. Similarly, it seems, for any other physical time-asymmetry to which we might appeal. If time did have a direction — an objective basis for a privileged notion of positive or future time — then for any physical arrow or asymmetry in time, there would always be a question as to whether that arrow pointed forwards or backwards. And so no physical fact could answer this question, because for any candidate, the same issue arises all over again. Thus the idea that time has a real direction seems without any physical meaning. (Of course, we can use any asymmetry we like as a basis for a conventional labelling — saying, for example, that we’ll regard the

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direction in which entropy is increasing as the positive direction of time. But this is different from discovering some intrinsic directionality to time itself.) For present purposes, then, I’ll assume that it is a conventional matter which direction we treat as positive or future time. Moreover, although it makes sense to ask whether time is anisotropic, it seems clear that this is a different issue from that of the thermodynamic asymmetry. As noted, the thermodynamic asymmetry is an asymmetry of physical processes in time, not an asymmetry of time itself.

Entropy gradient not entropy increase If it is conventional which direction counts as positive time, then it is also conventional whether entropy increases or decreases. It increases by the lights of the usual convention, but decreases if we reverse the labelling. But this may seem ridiculous. Doesn’t it imply, absurdly, that the thermodynamic asymmetry is merely conventional? No. The crucial point is that while it’s a conventional matter whether the entropy gradient slopes up or down, the gradient itself is objective. The puzzling asymmetry is that the gradient is monotonic — it slopes in the same direction everywhere (so far as we know). It is worth noting that in principle there are two possible ways of contrasting this monotonic gradient with a symmetric world. One contrast would be with a world in which there are entropy gradients, but sometimes in one direction and sometimes in the other — i.e., worlds in which entropy sometimes goes up and sometimes goes down. The other contrast would be with worlds in which there are no significant gradients, because entropy is always high. If we manage to explain the asymmetric gradient we find in our world, we’ll be explaining why the world isn’t symmetric in one of these ways — but which one? The answer isn’t obvious in advance, but hopefully will fall out of a deeper understanding of the nature of the problem.

The term ‘entropy’ is inessential A lot of time and ink has been devoted to the question how entropy should be defined, or whether it can be defined at all in certain cases (e.g., for the universe as a whole). It would be easy to get the impression that the puzzle of the thermodynamic asymmetry depends on all this discussion — that whether there’s really a puzzle depends on how, and whether, entropy can be defined, perhaps.

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But in one important sense, these issues are beside the point. We can see that there’s a puzzle, and go a long way towards saying what it is, without ever mentioning entropy. We simply need to describe in other terms some of the many processes which show the asymmetry — which occur with one temporal orientation but not the other. For example, we can point out that there are lots of cases of big difference in temperatures spontaneously equalising, but none of big differences in temperature spontaneously arising. Or we can point out that there are lots of cases of pressurised gas spontaneously leaving a bottle, but none of gas spontaneously pressurising by entering a bottle. And so on. In the end, we may need the notion of entropy to generalise properly over these cases. However, we don’t need it to see that there’s a puzzle — to see that there’s a striking imbalance in nature between systems with one orientation and systems with the reverse orientation. For present purposes, then, we can ignore objections based on problems in defining entropy. (Having said that, of course, we can go on using the term entropy with a clear conscience, without worrying about how it’s defined. In what follows, talk of entropy increase is just a placeholder for a list of the actual phenomena that display the asymmetry we’re interested in.)

Summary For the remainder of the paper, then, I take it (i) that the asymmetry in nature is a matter of numerical imbalance between temporal mirror images, not of literal reversibility; (ii) that we are concerned with an asymmetry of physical processes in time, not with an asymmetry in time itself; (iii) that the objective asymmetry concerned is a monotonic gradient, rather than an increase or a decrease; and (iv) that if need be the term ‘entropy’ is to be thought of as a placeholder for the relevant properties of a list of actual physical asymmetries.

4. What would a solution look like? Two models With our target more clearly in view, I now want to call attention to what may be the most useful distinction of all, in making sense of the many things that physicists and philosophers say about the thermodynamic asymmetry. This is a distinction between two very different conceptions of what it would take to explain the asymmetry — so different, in fact, that they disagree on how many distinct violations of T-symmetry it takes to explain the observed

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asymmetry. On one conception, an explanation needs two T-asymmetries. On the other conception, it needs only one. Despite this deep difference of opinion about what a solution would look like, the distinction between these two approaches is hardly ever noted in the literature — even by philosophers, who are supposed to have a nose for these things. So it is easy for advocates of the different approaches to fail to see that they are talking at cross-purposes — that in one important sense, they disagree about what the problem is.

The two-asymmetry approach Many approaches to the thermodynamic asymmetry look for a dynamical explanation of the second law — a dynamical cause or factor, responsible for entropy increase. Here are some examples, old and new: 1. The H-theorem. Oldest and most famous of all, this is Boltzmann’s development of Maxwell’s idea that intermolecular collisions drive gases towards equilibrium. 2. Interventionism. This alternative to the H-theorem, apparently first proposed by S. H. Burbury in the 1890s (Burbury 1894, 1895), attributes entropy increase to the effects of random and uncontrollable influences from a system’s external environment. 3. Indeterministic dynamics. There are various attempts to show how an indeterministic dynamics might account for the second law. A recent example (Albert 1994, 2000) is a proposal that the stochastic collapse mechanism of the GRW approach to quantum theory might also explain entropy increase. I stress two points about these approaches. First, if there is something dynamical which makes entropy increase, then it needs to be time-asymmetric. Why? Because otherwise it would force entropy to increase (or at least not to decrease) in both directions — in other words, entropy would be constant. In the H-theorem, for example, this asymmetry resides in the assumption of molecular chaos. In interventionism, it is provided by the assumption that incoming influences from the environment are ‘random’, or uncorrelated with the system’s internal dynamical variables. The second point to be stressed is that this asymmetry alone isn’t sufficient to produce the observed thermodynamic phenomena. Something which forces entropy to be non-decreasing won’t produce an entropy gra-

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dient unless entropy starts low. To give us the observed gradient, in other words, this approach also needs a low entropy boundary condition — entropy has to be low in the past. This condition, too, is time-asymmetric, and it’s a separate condition from the dynamical asymmetry. (It is not guaranteed by the assumption of molecular chaos, for example.) So this approach is committed to the claim that it takes two T-asymmetries — one in the dynamics, and one in the boundary conditions — to explain the observed asymmetry of thermodynamic phenomena. If this model is correct, explanation of the observed asymmetry needs an explanation of both contributing asymmetries, and the puzzle of the thermodynamic arrow has become a double puzzle.

The one-asymmetry model The two-asymmetry model isn’t the only model on offer, however. The main alternative was first proposed by Boltzmann in the 1870s (Boltzmann 1877), in response to Loschmidt’s famous criticism of the H-theorem. To illustrate the new approach, think of a large collection of gas molecules, isolated in a box with elastic walls. If the motion of the molecules is governed by deterministic laws, such as Newtonian mechanics, a specification of the microstate of the system at any one time uniquely determines its entire trajectory. The key idea of Boltzmann’s new approach is that in the overwhelming majority of possible trajectories, the system spends the overwhelming majority of the time in a high entropy macrostate — among other things, a state in which the gas is dispersed throughout the container. (Part of Boltzmann’s achievement was to find the appropriate way of counting possibilities, which we can call the Boltzmann measure.) Importantly, there is no temporal bias in this set of possible trajectories. Each possible trajectory is matched by its time-reversed twin, just as Loschmidt had pointed out, and the Boltzmann measure respects this symmetry. Asymmetry arises only when we apply a low entropy condition at one end. For example, suppose we stipulate that the gas is confined to some small region at the initial time t0 . Restricted to the remaining trajectories, the Boltzmann measure now provides a measure of the likelihood of the various possibilities consistent with this boundary condition. Almost all trajectories in this remaining set will be such that the gas disperses after t0 . The observed behaviour is thus predicted by the time-symmetric measure, once we conditionalise on the low entropy condition at t0 . On this view, then, there’s no time-asymmetric factor which causes

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entropy to increase. This is simply the most likely thing to happen, given the combination of the time-symmetric Boltzmann probabilities and the single low entropy restriction in the past. More below on the nature and origins of this low entropy boundary condition. For the moment, the important thing is that although it is time-asymmetric, so far as we know, this is the only time-asymmetry in play, according to Boltzmann’s statistical approach. There’s no need for a second asymmetry in the dynamics.

5. Which is the right model? It is important to distinguish these two models, but it would be even more useful to know which of them is right. How many time-asymmetries should we be looking for, in trying to account for the thermodynamic asymmetry? This is a big topic, but I’ll mention two factors, both of which seem to me to count in favour of the one-asymmetry model. The first factor is simplicity, or theoretical economy. If the one-asymmetry approach works, it simply does more with less. In particular, it leaves us with only one time-asymmetry to explain. True, this would not be persuasive if the two-asymmetry approach actually achieved more than the oneasymmetry approach — if the former had some big theoretical advantage that the latter lacked. But the second argument I want to mention suggests that this can’t be the case. On the contrary, the second asymmetry seems redundant. Redundancy is a strong charge, but consider the facts. The two-asymmetry approach tries to identify some dynamical factor (collisions, or external influences, or whatever) that causes entropy to increase — that makes a pressurised gas leave a bottle, for example. However, to claim that one of these factors causes the gas to disperse is to make the following ‘counterfactual’ claim: If the factor were absent, the gas would not disperse (or would do so at a different rate, perhaps). But how could the absence of collisions or external influences prevent the gas molecules from leaving the bottle? Here’s a way to make this more precise. In the terminology of Boltzmann’s statistical approach, we can distinguish between normal initial microstates (for a system, or for the universe as a whole), which lead to entropy increases much as we observe, and abnormal microstates, which are such that something else happens. The statistical approach rests on the fact that normal microstates are vastly more likely than abnormal microstates, according to the Boltzmann measure.

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In these terms, the above point goes as follows. The two-asymmetry approach is committed to the claim that the universe begins in an abnormal microstate. Why? Because in the case of normal initial microstates, entropy increases anyway, without the mechanism in question — so the required counterfactual claim isn’t true. It is hard to see what could justify this claim about the initial microstate. At a more local level, why should we think that the initial microstate of a gas sample in an open bottle is normally such that if it weren’t for collisions (or external influences, or whatever), the molecules simply wouldn’t encounter the open top of the bottle, and hence disperse? Thus it is doubtful whether there is really any need for a dynamical asymmetry, and the one-asymmetry model seems to offer the better conception of what it would take to solve the puzzle of the thermodynamic asymmetry. But if so, then the various two-asymmetry approaches — including Boltzmann’s own H-theorem, which he himself defended in the 1890s, long after he first proposed the statistical approach — are looking for a solution to the puzzle in the wrong place, at least in part. For present purposes, the main conclusion I want to emphasise is that we need to make a choice. The one-asymmetry model and the two-asymmetry model represent two very different views of what it would take to explain the thermodynamic arrow — of what the problem is, in effect. Unless we notice that they are different approaches, and proceed to agree on which of them we ought to adopt, we can’t possibly agree on whether the old puzzle has been laid to rest.

6. The Boltzmann-Schuetz hypothesis — a no-asymmetry solution? If the one-asymmetry view is correct, the puzzle of the thermodynamic arrow is really the puzzle of the low entropy boundary condition. Why is entropy so low in the past? After all, in making it unmysterious why entropy doesn’t decrease in one direction, the Boltzmann measure equally makes it mysterious why it does decrease in the other — for the statistics themselves are time-symmetric. Boltzmann himself was one of the first to see the importance of this issue. In a letter to Nature in 1895, he suggests an explanation, based on an idea he attributes to ‘my old assistant, Dr Schuetz’ (Boltzmann 1895). He notes that although low entropy states are very unlikely, they are very likely to

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occur eventually, given enough time. If the universe is very old, it will have had time to produce the kind of low entropy region we find ourselves inhabiting simply by accident. “Assuming the universe great enough, the probability that such a small part of it as our world should be in its present state, is no longer small”, as Boltzmann puts it. Entropy

B

A

C

Time

Figure 1: Boltzmann’s entropy curve It is one thing to explain why the universe contains regions like ours, another to explain why we find ourselves in such a region. If they are so rare, isn’t it more likely that we’d find ourselves somewhere else? But Boltzmann suggests an answer to this, too. Suppose, as seems plausible, that creatures like us couldn’t exist in the vast regions of near-equilibrium between such regions of low entropy. Then it’s no surprise that we find ourselves in such an unlikely place. As Boltzmann himself puts it, “the … H curve would form a representation of what takes place in the universe. The summits of the curve would represent the worlds where visible motion and life exist”. Figure 1 shows what Boltzmann calls the H curve, except that this diagram plots entropy rather than Boltzmann’s quantity H. Entropy is low when H is high, so the summits of Boltzmann’s H curve are the troughs of the entropy curve. The universe spends most of its time very close to equilibrium. But occasionally — much more rarely than this diagram actually suggests — a random re-arrangement of matter produces a state of low entropy. As the resulting state returns to equilibrium, there’s an entropy slope, such as the one on which we (apparently) find ourselves, at a point such as A.

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Why do we find ourselves on an uphill rather than a downhill slope, as at B? In another paper (Boltzmann 1897), Boltzmann offers a remarkable proposal to explain this, too. Perhaps our perception of past and future depends on the entropy gradient, in such a way that we are bound to regard the future as lying ‘uphill’. Thus the perceived direction of time would not be objective, but a product of our own orientation in time. Creatures at point B would see the future as lying in the other direction, and there’s no objective sense in which they are wrong and we are right, or vice versa. Boltzmann compares this to the discovery that spatial up and down are not absolute directions, the same for all observers everywhere. For present purposes, what matters about the Boltzmann-Schuetz hypothesis is that it offers an explanation of the local asymmetry of thermodynamics in terms which are symmetric on a larger scale. So it is a no-asymmetry solution — the puzzle of the thermodynamic asymmetry simply vanishes on the large scale.

7. The big problem Unfortunately, however, this clever proposal has a sting in its tail, a sting so serious that it now seems almost impossible to take the hypothesis seriously. The problem flows directly from Boltzmann’s own link between entropy and probability. In Figure 1, the vertical axis is a logarithmic probability scale. For every downward increment, dips in the curve of the corresponding depth are exponentially more improbable. So a dip of the depth of point A or point B is much more likely to occur in the form shown at point C — where the given depth is very close to the minimum of the fluctuation — than in association with a much bigger dip, as at A and B. Hence if our own region has a past of even lower entropy, it is much more improbable than it needs to be, given its present entropy. So far, this point seems to have been appreciated already in the 1890s, in exchanges between Boltzmann and Zermelo. What doesn’t seem to have been appreciated is its devastating consequence, namely, that according to the Boltzmann measure it is much easier to produce fake records and memories, than to produce the real events of which they purport to be records. Why does this consequence follow? Well, imagine that the universe is vast enough to contain many separate fluctuations, each containing everything that we see around us, including the complete works of Shakespeare, in all their twenty-first century editions. Now imagine choosing one of these fluc-

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tuations at random. It is vastly more likely that we’ll select a case in which the Shakespearean texts are a product of a spontaneous recent fluctuation, than one in which they were really written four hundred years earlier by a poet called William Shakespeare. Why? Simply because entropy is much higher now than it was in the sixteenth century (as we normally assume that century to have been). Recall that according to Boltzmann, probability increases exponentially with entropy. Fluctuations like our twenty-first century —‘Shakespearean’ texts and all — thus occur much more often in typical world-histories than fluctuations like the lower-entropy sixteenth century. So almost all fluctuations including the former don’t include the latter. The same goes for the rest of history — all our ‘records’ and ‘memories’ are almost certainly misleading. To make this conclusion vivid we can take advantage of the fact that in the Boltzmann picture, there isn’t an objective direction of time. So we can equally well think about the question of ‘what it takes’ to produce what we see around us from the reverse of the normal temporal perspective. Think of starting in what we call the future, and moving in the direction we call towards the past. Think of all the apparently miraculous accidents it takes to produce the kind of world we see around us. Among other things, our bodies themselves, and our editions of Shakespeare, have to ‘undecompose’, at random, from (what we normally think of as) their future decay products. That’s obviously extremely unlikely, but the fact that we’re here shows that it happens. But now think of what it takes to get even further back, to a sixteenth century containing Shakespeare himself. The same kind of nearmiracle needs to happen many more times. Among other things, there are several billion intervening humans to ‘undecompose’ spontaneously from dust. So the Boltzmann-Schuetz hypothesis implies that our apparent historical evidence is almost certainly unreliable. So far as I know, this point was first made in print by von Weizsäcker (1939). Von Weizsäcker notes that “improbable states can count as documents [i.e., records of the past] only if we presuppose that still less probable states preceded them”. He concludes that “the most probable situation by far would be that the present moment represents the entropy minimum, while the past, which we infer from the available documents, is an illusion”. Von Weizsäcker also notes that there’s another problem of a similar kind. The Boltzmann-Schuetz hypothesis implies that as we look further out into space, we should expect to find no more order than we already have reason to

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believe in. But we can now observe vastly more of the universe than was possible in Boltzmann’s day, and there seems to be low entropy all the way out. So the Boltzmann-Schuetz hypothesis faces some profound objections. Fortunately, as we’re about to see, modern cosmology goes at least some way to providing us with an alternative.

8. Initial smoothness We have seen that the observed thermodynamic asymmetry requires that entropy was low in the past. Low entropy requires concentrations of energy in useable forms, and presumably there are many ways such concentrations could exist in the universe. On the face of it, we seem to have no reason to expect any particularly neat or simple story about how it works in the real world — about where the particular concentrations of energy we depend on happen to originate. Remarkably, however, modern cosmology suggests that all the observed low entropy is associated with a single characteristic of the early universe, soon after the big bang. The crucial thing is that matter is distributed extremely smoothly in the early universe. This provides a vast reservoir of low entropy, on which everything else depends. In particular, smoothness is necessary for galaxy and star formation, and most familiar irreversible phenomena depend on the sun. Why does a smooth arrangement of matter amount to a low entropy state? Because in a system dominated by an attractive force such as gravity, a uniform distribution of matter is highly unstable (and provides a highly useable supply of potential energy). However, about 105 years after the big bang, matter seems to have been distributed smoothly to very high accuracy. One way to get a sense how surprising this is, is to recall that we’ve found no reason to disagree with Boltzmann’s suggestion that there’s no objective distinction between past and future — no sense in which things really happen in the direction we think of as past-to-future. Without such a distinction, there’s no objective sense in which the big bang is not equally the end point of a gravitational collapse. Somehow that collapse is coordinated with astounding accuracy, so that the matter involved manages to avoid forming large agglomerations (in fact, black holes), and instead spreads itself out very evenly across the universe. In my view, this discovery about the cosmological origins of low entropy is one of the great achievements of late twentieth century physics. It is a remarkable discovery in two quite distinct ways, in fact. First, it is the only

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anomaly necessary to account for the low entropy we find in the universe, at least so far as we know. So it is a remarkable theoretical achievement — it wraps up the entire puzzle of the thermodynamic asymmetry into a single package, in effect. Second, it is astounding that it happens at all, according to existing theories of how gravitating matter should behave — which suggests, surely, that there is something very important missing from those theories. (True, it is easy to fail to see how astounding the smooth early universe is, by failing to see that the big bang can quite properly be regarded as the end point of a gravitational collapse. But anyone inclined to deny the validity of this way of viewing the big bang faces a perhaps even more daunting challenge: to explain what is meant by, and what is the evidence for, the claim that time has an objective direction.)

9. Open questions Why is the universe smooth soon after the big bang? This is a major puzzle, but — if we accept that the one-asymmetry model — it is the only question we need to answer, to solve the puzzle of the thermodynamic arrow. So we have an answer to the question with which we began. What has happened to the puzzle noticed by those nineteenth century giants? It has been transformed by some of their twentieth century successors into a puzzle for cosmology, a puzzle about the early universe. It is far from clear how this remaining cosmological puzzle is to be explained. Indeed, there are some authors (e.g., Callender 1997, 1998, 2004; Sklar 1993) who doubt whether it needs explaining. If so, then the upshot of our therapy is truly Wittgensteinian: “the philosophical problems” do indeed “completely disappear.” (PI, §133); but there are grounds for scepticism, in my view (Price 2004). However, these issues are beyond the scope of this paper. I want to close by calling attention to some open questions associated with this understanding of the origins of the thermodynamic asymmetry, and by making a case for an unusually sceptical attitude to the second law. One fascinating question is whether whatever explains why the universe is smooth after the big bang would also imply that the universe would be smooth before the big crunch, if the universe eventually recollapses. In other words, would entropy would eventually decrease, in a recollapsing universe?2 2

The recognition of this possibility is commonly attributed to Thomas Gold (1962). However, as I am grateful to Larry Schulman for pointing out to me, the attribution may well depend on an extrapolation beyond anything explicit in Gold’s own work.

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This possibility is often been dismissed on the grounds that a smooth recollapse would require an incredibly unlikely ‘conspiracy’ among the components parts of the universe, to ensure that the recollapsing matter did not clump into black holes. However, as we have already noted, this incredible conspiracy is precisely what happens towards (what we usually term) the big bang, if we regard that end of the universe as a product of a gravitational collapse. The statistics themselves are time-symmetric. If something overrides them at one end of the universe, what right do we have to assume that the same does not happen at the other? Until we understand more about the origins of the smooth early universe, then, it seems best to keep an open mind about a smooth late universe. Some people dismiss the question whether entropy would reverse in a recollapsing universe on the grounds that the current evidence suggests that the universe will not recollapse. However, it seems reasonable to expect that when we find out why the universe is smooth near the big bang, we’ll be able to ask a theoretical question about what that reason would imply in the case of universe which did recollapse. Moreover, as a number of writers have pointed out (see Hawking 1985; Penrose 1979), much the same question arises if just a bit of the universe recollapses — e.g., a galaxy, collapsing into a black hole. This process seems to be a miniature version of the gravitational collapse of a whole universe, and so it makes sense to ask whether whatever constrains the big bang also constrains such partial collapses.

10. Scepticism about the second law In my view, the moral of these considerations is that until we know more about why entropy is low in the past, it is sensible to keep an open mind about whether it might be low in the future. The appropriate attitude is a kind of healthy scepticism about the universality of the second law of thermodynamics. The case for scepticism goes like this. What we’ve learnt about why entropy increases in our region is that it does so because it is very low in the past (for some reason we don’t yet know), and the increase we observe is the most likely outcome consistent with that restriction. As noted, however, the statistics underpinning this reasoning are time-symmetric, and hence the predictions we make about the future depend implicitly on the assumption that there is no corresponding low entropy boundary condition in that direction. Thus the Boltzmann probabilities don’t enable us to predict without quali-

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fication that entropy is unlikely to decrease, but only that it is unlikely to decrease, unless there is the kind of boundary condition in the future that makes entropy low in the past. In other words, the second law is likely to continue to hold so long as there isn’t a low entropy boundary condition in the future. But it can’t be used to exclude this possibility — even probabilistically! Sceptics about the second law are unusual in the history of thermodynamics, and I would like to finish by giving some long-overdue credit to one of the rare exceptions. Samuel Hawksley Burbury (1831–1911) was not one of the true giants of thermodynamics. However, he made an important contribution to the identification of the puzzle of the time-asymmetry of thermodynamic phenomena. And he was more insightful than any of his contemporaries — and most writers since, for that matter — in being commendably cautious about declaring the puzzle solved. Burbury was an English barrister. He read mathematics at Cambridge as an undergraduate, but his major work in mathematical physics came late in life, when deafness curtailed his career at the Bar. In his sixties and seventies, he thus played an important role in discussions about the nature and origins of the second law. In a review of Burbury’s monograph The Kinetic Theory of Gases for Science in 1899, the reviewer describes his contribution as follows: [I]n that very interesting discussion of the Kinetic Theory which was begun at the Oxford meeting of the British Association in 1894 and continued for months afterwards in Nature, Mr. Burbury took a conspicuous part, appearing as the expounder and defender of Boltzmann’s H-theorem in answer to the question which so many [had] asked in secret, and which Mr. Culverwell asked in print, ‘What is the H-theorem and what does it prove?’ Thanks to this discussion, and to the more recent publication of Boltzmann’s Vorlesungen über Gastheorie, and finally to this treatise by Burbury, the question is not so difficult to answer as it was a few years ago. (Hall 1899) It is a little misleading to call Burbury a defender of the H-theorem. The crucial issue in the debate referred to here was the source of the time-asymmetry of the H-theorem, and while Burbury was the first to put his finger on the role of assumption of molecular chaos, he himself regarded this assumption with considerable suspicion. Here’s how he puts it in 1904:

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Does not the theory of a general tendency of entropy to diminish [sic]3 take too much for granted? To a certain extent it is supported by experimental evidence. We must accept such evidence as far as it goes and no further. We have no right to supplement it by a large draft of the scientific imagination. (Burbury 1904) Burbury’s reasons for scepticism are not precisely those which seem appropriate today. Burbury’s concern might be put like this. To see that the dynamical processes routinely fail to produce entropy increases towards the past is to see that it takes an extra ingredient to ensure that they do so towards the future. We’re then surely right to wonder whether that extra ingredient is sufficiently universal, even towards the future, to guarantee that the second law will always hold. As the first clearly to identify the source of the time-asymmetry in the H-theorem, Burbury was perhaps more sensitive to this concern than any of his contemporaries. At the same time, however, Burbury seems never to have distanced himself sufficiently from the H-theorem to see that the real puzzle of the thermodynamic asymmetry lies elsewhere. The interesting question is not whether there is a good dynamical argument to show that entropy will always increase towards the future. It is why entropy steadily decreases towards the past — in the face, note, of such things as the effects of collisions and external influences, which are ‘happening’ in that direction as much as in the other! As we’ve seen, this re-orientation provides a new reason for being cautious about proclaiming the universal validity of the second law. Once we regard the fact that entropy decreases towards the past as itself a puzzle, as something in need of explanation, then it ought to occur to us that whatever explains it might be non-unique — and thus that in principle, there might be a low entropy boundary condition in the future, as well as in the past.4

3 4

Burbury is apparently referring to Boltzmann’s quantity H, which does decrease as entropy increases. An earlier version of this paper is due to appear as (Price 2006). I am grateful to the editor of the volume concerned for permission to reprint much of that material here.

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Literature Albert, David 1994 “The Foundations of Quantum Mechanics and the Approach to Thermodynamic Equilibrium”, British Journal for the Philosophy of Science 45, 669–677. — 2000 Time and Chance, Cambridge, Mass.: Harvard University Press. Boltzmann, Ludwig 1877 “Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung” (“On the Relation of a General Mechanical Theorem to the Second Law of Thermodynamics”), Sitzungsberichte der kaiserlichen Akademie der Wissenschaften, Wien 75, 67–73. Reprinted in translation in Stephen Brush (ed.), Kinetic Theory. Volume 2: Irreversible Processes, Oxford: Pergamon Press, 1966. — 1895 “On Certain Questions of the Theory of Gases”, Nature, 51, 413–15. — 1897 “Zu Hrn. Zermelo’s Abhandlung ‘Über die mechanische Erklärung irreversibler Vorgänge’”, Annalen der Physik 60, 392–398. Burbury, Samuel 1894 “Boltzmann’s Minimum Function”, Nature 51, 78. — 1895 “Boltzmann’s Minimum Function”, Nature 51, 320. — 1904 “On the Theory of Diminishing Entropy”, Philosophical Magazine, Series 6, 8, 43–49. Callender, Craig 1997 “Review of H. Price, Time’s Arrow and Archimedes’ Point”, Metascience 11, 68–71. — 1998 “The View from No-when”, British Journal for the Philosophy of Science 49, 135–59. — 2004 “There is No Puzzle About the Low Entropy Past”, in: Christopher Hitchcock (ed.), Contemporary Debates in Philosophy of Science, Oxford: Blackwell, 240–255. Gold, Thomas 1962 “The Arrow of Time,” American Journal of Physics 30, 403–410. Hall, E. H. 1899 “Review of S. H. Burbury, The Kinetic Theory of Gases (Cambridge: Cambridge University Press, 1899)”, Science, New Series 10, 685–688. Hawking, Stephen 1985 “Arrow of Time in Cosmology”, Physical Review, D 33, 2489–2495. Penrose, Roger 1979 “Singularities and Time-asymmetry”, in: S. W. Hawking and W. Israel (eds.), General relativity: an Einstein Centenary Cambridge: Cambridge University Press, 581–638.

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Price, Huw 1996 Time’s Arrow and Archimedes’ Point, New York: Oxford University Press. — 2002a “Boltzmann’s Time Bomb”, British Journal for the Philosophy of Science 53, 83–119. — 2002b “Burbury’s Last Case: the Mystery of the Entropic Arrow”, in: Craig Callender (ed.), Time, Reality and Experience, Cambridge: Cambridge University Press, 19–56. — 2004 “On the Origins of the Arrow of Time: Why There is Still a Puzzle About the Low Entropy Past”, in: Christopher Hitchcock (ed.), Contemporary Debates in Philosophy of Science, Oxford: Blackwell, 219–239. — 2006 “The Thermodynamic Arrow: Puzzles and Pseudo-puzzles”, in: Ikaros Bigi (ed.), Proceedings of ‘Time and Matter — An International Colloquium on the Science of Time’, Venice, 2002, Singapore: World Scientific. Ridderbos, T. M. and Redhead, M. 1998 “The Spin-echo Experiments and the Second Law of Thermodynamics”, Foundations of Physics 28, 1237– 1270. Sklar, Lawrence 1993 Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics, Cambridge: Cambridge University Press. — 1995 “The Elusive Object of Desire: in Pursuit of the Kinetic Equations and the Second Law”, in: S. Savitt (ed.), Time’s Arrows Today, Cambridge: Cambridge University Press, 191–216. von Weizsäcker, Carl 1939 “Der zweite Haupsatz und der Unterschied von der Vergangenheit und Zukunft”, Annalen der Physik (5. Folge) 36, 275– 283. Reprinted in translation in The Unity of Nature (New York: Farrar Straus Giroux, 1980).

Three Concepts of Irreversibility and Three Versions of the Second Law Jos Uffink, Utrecht This paper aims to clarify the relation between the second law of thermodynamics and the notion of irreversibility by distinguishing three different meanings of the latter and to study how they figure in three versions of the second law of thermodynamics. A more extensive discussion is given in (Uffink 2001).

1. Three concepts of (ir)reversibility Many physical theories employ a state space Γ containing all possible states s of a system. A process is then represented as a parameterised curve: P = {st ∈ Γ : ti ≤ t ≤ tf }. Usually a theory allows only a subclass, say W, of such processes (e.g. the solutions of the equations of motion). Let R be an involution (i.e. R 2 s = s) that turns state s into its ‘time reversal’ Rs. In classical mechanics, for example, R is the transformation which reverses the sign of all momenta and magnetic fields. In a theory like classical thermodynamics, where the state does not contain velocity-like parameters, one may take R to be the identity. Further, define the time reversal P* of process P by: P* = {(Rs)–t : −tf ≤ t ≤ −ti }.

(1)

The theory is called time-reversal invariant (TRI) if the class W is closed under time reversal, i.e. iff: P ∈ W ⇒ P* ∈ W. According to this definition the form of the laws (and a choice for R) determines whether a theory is TRI or not. Note that it is irrelevant here whether the processes P* actually occur, but only that the theory allows them. Thus, F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 275–287.

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the fact that the sun never rises in the West does not mean that celestial mechanics is non-TRI. Is time-reversal (non)invariance related to the second law? An application of the criterion to thermodynamics is not a matter of routine. In contrast to mechanics, thermodynamics does not possess equations of motion. This is because thermodynamical processes only occur after external intervention on the system (e.g.: removing a partition, pushing a piston, etc.) and do not reflect autonomous behaviour of a free system. This is not to say that time plays no role. Classical thermodynamics, in the formulation of Clausius, Kelvin or Planck, is concerned with processes, and its second law is clearly not TRI. However, in other formulations, e.g. by Gibbs, Carathéodory, or Lieb and Yngvason, this is less clear. Now, the term ‘(ir)reversible’ is usually attributed to processes rather than theories or laws. But in philosophy of physics, it is intimately connected with time-reversal invariance. Indeed, one calls a process P allowed by a given theory irreversible iff the reversed process P* is excluded by this theory. Obviously, such a process P exists only if the theory in question is not TRI. Conversely, every non-TRI theory admits irreversible processes in this sense. Therefore, discussions about (ir)reversibility and (non)-TRI in philosophy of physics mostly coincide. However, the thermodynamics literature often uses the term ‘irreversibility’ to denote an aspect of processes which one might also call irrecoverability. In many processes, the transition from an initial state si to a final state sf , cannot be fully ‘undone’, once the process has taken place. In other words, there is no process which starts off from state sf and restores the initial state si completely. Wear and tear, erosion, etc. are the obvious examples. This meaning of the term goes back to Kelvin (1852), actually the first author to employ the term in a thermodynamical context. It is also this sense of irreversibility that Planck intended, when he called it the essence of the second law. Many writers have emphasised irrecoverability in connection with the second law. Indeed, Eddington introduced his famous phrase of ‘the arrow of time’ while discussing the ‘running-down of the universe’, and illustrated it by examples involving ‘irrevocable changes’, including the case of Humpty-Dumpty who, allegedly, could not be put together again after his great fall. In retrospect, one might say that a better expression for this theme is the ravages of time rather than its arrow. (Ir)recoverability differs from (non)-TRI in at least two respects. First, the only thing that matters here is the retrieval of the initial state si. It is not

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necessary to specify a process P* retracing the intermediate stages of the original process in reverse order. A second difference is that we are dealing with a complete recovery. This means that all auxiliary systems that may have been used in the original process are also returned to their initial state. A schematic expression of the idea is this. Let s be a state of the system and Z a (formal) state of its environment. Let P be some process that brings about the transition: P

〈si , Zi 〉 → 〈sf , Zf 〉

(2)

Then P is reversible in Planck’s sense iff there exists another process P ′ that produces P′

〈sf , Zf 〉 → 〈si , Zi 〉

(3)

The term ‘reversible’ is also used in a third sense, to denote processes which proceed so slowly that the system remains in equilibrium ‘up to a negligible error’ during the entire process. This is the meaning embraced by Clausius, and it appears to be the most common usage of the term in the physicalchemical literature; see e.g. Hollinger and Zenzen (1985), Denbigh (1989). A more apt name for this kind of processes is quasi-static. Of course, the above characterisation is vague, and has to be amended by criteria specifying what ‘errors’ are intended and when they are ‘small’. These criteria take the form of a limiting procedure so that, strictly speaking, reversibility is here not an attribute of a particular process but of a series of processes. Quasi-static processes need not be the same as those called reversible in the previous two senses. E.g., an ideal harmonic oscillator is reversible in Planck’s sense, but not quasi-static. Conversely, the discharge of a condenser through a high resistance can be made to proceed quasi-statically, but even then it remains irreversible in Planck’s sense. Comparison with the notion of TRI is hampered by the fact that ‘quasistatic’ is not strictly a property of a process. Consider a process PN in which a system, originally at temperature θ 1 is consecutively placed in thermal contact with a sequence of N heat baths, each at a slightly higher temperature than the previous one, until it reaches a temperature θ 2. By making N large, and the temperature steps small, such a process becomes quasi-static, and we can represent it by a curve in the space of equilibrium states. However, for any N, the time-reversal of the process is impossible.

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Nevertheless, many authors call such a curve ‘reversible’, because one can consider another process QN, in which the system, originally at temperature θ 2, is placed into contact with a series of heat baths, each slightly colder than the previous one. Again, each process QN is non-TRI. A forteriori, no QN is the time reversal of any PN. Yet, if we now take the quasi-static limit, the state change of the system will follow the same curve in equilibrium space as in the previous case, traversed in the opposite direction. The point is, of course, that precisely because this curve is not itself a process, the notion of time reversal does not apply to it.

2. The second law according to Planck Planck’s position has always been that the second law expresses irrecoverability of all processes in nature. However, it is not easy to analyse his arguments for this claim. Various editions of his book (Planck 1897) differ in many decisive details. Also, the English translation is unreliable. It uses the term ‘reversible’ indiscriminately, where Planck distinguishes between umkehrbar, which he uses in Clausius’ sense, i.e. meaning ‘quasi-static’, and reversibel, in the sense of Kelvin (1852) meaning ‘recoverable’. Moreover, he presented a completely different argument from the eighth edition onwards. I shall only mention Planck’s latter argument, published first in Planck (1926). He starts from the statement that “friction is an irreversibel process”, which he considers to be an expression of Kelvin’s principle.1 He then considers an adiabatically isolated fluid capable of exchanging energy with its environment by means of a weight at height h. Planck asks whether it is possible to bring about a transition from an initial state s of this system to a final state s′, in a process which brings about no changes in the environment other than the displacement of the weight. If Z denotes the state of the environment and h the height of the weight, the desired transition can be represented as ? (s, Z, h) → (s′, Z, h′).

1

This may need some explanation because, at first sight, this statement does not concern cyclic processes or the perpetuum mobile at all. But for Planck, the statement means that there exists no process which ‘undoes’ the consequences of friction, i.e., a process which produces no other effect than cooling a reservoir and doing work. The condition ‘no other effect’ here allows for the operation of any type of auxiliary system that operates in a cycle.

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He argues that, by means of ‘reversibel-adiabatic’2 processes, one can always achieve a transition from the initial state s to an intermediary state s* in which the volume equals that of state s′ and the entropy equals that of s. That is, one can realise a transition (s, Z, h) → (s*, Z, h* ),

with

V (s*) = V (s′)

and

S(s*) = S(s).

Whether the desired final state s′ can now be reached from the intermediate state s* depends on the value of the only independent variable in which s* and s′ differ. For this variable one can choose the energy U. There are three cases: (1) h* = h′. In this case, energy conservation implies U(s*) = U(s′). Because the coordinates U and V determine the state of the fluid completely, s* and s′ must coincide. (2) h* > h′. In this case, U(s*) < U(s), and the state s′ can be reached from s* by letting the weight perform work on the system, e.g. by means of friction, until the weight has dropped to height h′. According to the above formulation of Kelvin’s principle, this process is irreversible (i.e. irrecoverable). (3) h* < h′ and U(s*) > U(s). In this case the desired transition is impossible. It would be the reversal of the irreversible process just mentioned, and thus realise a perpetuum mobile of the second kind. Now, Planck argues that in all three cases, one can also achieve a transition from s* to s′ by means of heat exchange in an umkehrbar (i.e. quasi-static) process in which the volume remains fixed. For such a process he writes dU = TdS.

(4)

Using the assumption that T > 0, it follows that, in the three cases above, U must vary in the same sense as S. That is, the cases U(s*) < U(s′), U(s*) = U(s′) or U(s*) > U(s′), can also be characterised as S(s*) < S(s′), S(s*) = S(s′) and S(s*) > S(s′) respectively. An analogous argument can be constructed for a system consisting of several fluids. Just as in earlier editions of his book, Planck generalises the 2

Apparently, Planck’s pen slipped here. He means: umkehrbar-adiabatic.

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conclusion (without a shred of proof ) to arbitrary systems and arbitrary physical/chemical processes: Every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased. In the limiting case, for reversible processes this sum remains unchanged. […] This provides an exhaustive formulation of the content of the second law of thermodynamics (Planck 1926, p. 463) Planck’s argument can hardly be regarded as satisfactory for the bold and universal formulation of the second law. It applies only to systems consisting of fluids, and relies on several implicit assumptions which can be questioned outside of this context. In particular, this holds for the assumption that there always exist functions S and T (with T > 0) such that đQ = TdS, where đ denotes an inexact differential; and the assumption of a rather generous supply of quasi-static processes.

3. The second law according to Carathéodory Carathéodory (Carathéodory 1909) construed thermodynamics as a theory of equilibrium states rather than (cyclic) processes. A thermodynamical system is described by a state space Γ, represented as a (subset of a) n-dimensional manifold with the state variables serving as coordinates. He assumes that Γ is equipped with the standard Euclidean topology. But metrical properties do not play a role, and there is no preference for a particular system of coordinates. The fundamental concept is a relation called adiabatic accessibility, which represents whether state t can be reached from state s in an adiabatic process,3 and the second law is formulated as follows: Carathéodory’s Principle: In every open neighbourhood Us ⊂ Γ of every state s there are states t such that for some open neighbourhood Ut of t: all states r within Ut cannot be reached adiabatically from s. 3

Carathéodory calls a container adiabatic if the system contained in it remains in equilibrium, regardless of what occurs in the environment, as long as the container is not moved nor changes its shape. Thus, the only way of inducing a process of a system in an adiabatic container is by deformation of the walls. (E.g. a change of volume or stirring.) A process is said to be adiabatic if it takes place while the system is in an adiabatic container.

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He then introduces the so-called ‘simple systems’ (defined by four additional conditions) and obtains Carathéodory’s Theorem: For simple systems, Carathéodory’s principle is equivalent to the proposition that the differential form4 đQ :=dU − đW possesses an integrable divisor, i.e. there exist functions S and T on the state space Γ such that đQ = TdS.

(5)

Thus, for simple systems, every equilibrium state can be assigned a value for entropy and absolute temperature. Curves representing quasi-static adiabatic changes of state are characterised by the differential equation đQ = 0, and by virtue of (5) one can conclude that (if T ≠ 0) entropy remains constant. Obviously S and T are not uniquely determined by the relation (5). Carathéodory discusses further conditions to determine the choice of T and S up to a constant of proportionality. I want to mention a number of strong and weak points of the approach. An advantage is that it provides a mathematical formalism for thermodynamics, comparable to relativity theory. There, Einstein’s original approach, which starts from empirical principles (the light postulate and relativity principle), was replaced by an abstract Minkowski spacetime, where the empirical principles are incorporated in local properties of the metric. Similarly, Carathéodory constructs an abstract state space where an empirical statement of the second law is converted into a local topological property. Furthermore, all coordinate systems are treated on the same footing (as long as there is only one thermal coordinate, and they generate the same topology). Note further that the environment of the system is never mentioned explicitly in his treatment. This too is a conceptual advantage. But Carathéodory’s work has also provoked objections. Many complain that the absence of an explicit reference to a perpetuum mobile obscures the physical content of the second law. Other problems in Carathéodory’s approach concern the additional assumptions needed to obtain the result (5), i.e. the restriction to simple systems. Further, Carathéodory’s proof actually establishes merely the local existence of functions S and T obeying (5). This 4

Here the notation đ is used to denote that đQ and đW are not exact differentials, in contrast to dU and dS.

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does not guaranteee the existance of a pair of functions, defined globally on Γ, that obey (5). For the present purpose, we need to investigate whether and how this work relates to ‘irreversibility’. Carathéodory conceives of thermodynamics as a theory of equilibrium states, rather than processes. But his concept of ‘adiabatic accessibility’ does refer to processes between equilibrium states. In order to judge the time-reversal invariance of the theory of Carathéodory one must specify a time reversal transformation R. It seems natural to choose this in such a way that Rs = s and R(≺) = . Then Carathéodory’s principle is not TRI. Indeed, the principle forbids that Γ contains a state s from which one can reach all states in some neighbourhood of s. It allows models where a state s exists from which one can reach no other state in some neighbourhood. Time reversal of such a model violates Carathéodory’s principle. However, this non-invariance manifests itself only in rather pathological cases. If we exclude them, Carathéodory’s theory becomes TRI. It is easy to show that Carathéodory’s approach fails to capture the content of the second law à la Planck, namely by exhibiting models of his formalism in which this version of the second law is invalid. An example is obtained by swapping the meaning of terms in each of the three pairs ‘heat / work’, ‘thermal / deformation coordinate’ and ‘adiabatic’ / ‘without any exchange of work’. The validity of Carathéodory’s formalism is invariant under this operation for fluids. Indeed, we obtain, as a direct analog of (5): đW = pdV for all quasistatic processes of a fluid. Thus pressure and volume here play the role of temperature and entropy respectively. Further, irreversibility makes sense here too. For fluids with positive pressure, one can increase the volume of a fluid without doing work, but one cannot decrease volume without doing work. But still, the analog of the principles of Clausius of Kelvin are false: A fluid with low pressure can very well do positive work on another fluid with high pressure by means of a lever or some hydraulic mechanism. And, thus, the sum of all volumes of a composite system can very well decrease, even when no external work is provided.

4. The second law according to Lieb and Yngvason Lieb and Yngvason (1999) have provided a rigorous approach to the second law. I can only sketch those main ideas that are relevant to my topic.

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Formally, their work builds upon the approach of Carathéodory (1909) and Giles (1964). (In its physical interpretation, however, it is more closely related to Planck.) A system is represented by a state space Γ on which a relation ≺ of adiabatic accessibility is defined. Further, one may combine two systems in state s and t into a composite system in state (s, t), and there is an operation of ‘scaling’, i.e. the construction of a copy in which all its extensive quantities are increased by a factor α. This is denoted by a multiplying the state with α. The main axioms read: A1. Reflexivity: s ≺ s A2. Transitivity: s ≺ t and t ≺ r imply s ≺ r A3. Consistency: s ≺ s′ and t ≺ t′ implies (s, t) ≺ (s′, t′) A4. Scale invariance: If s ≺ t then αs ≺ αt for all α > 0 A5. Splitting and recombination: For all 0 < α < 1 : s ≺ (αs, (1 − α)s) ≺ s A6. Stability: If there are states t0 and t1 such that (s, εt0) ≺ (r, εt1) holds for a sequence of ε’s converging to zero, then s ≺ r. 7.

Comparability hypothesis: For all states s, t in the same Γ:s ≺ t or t ≺ s.5

The comparability hypothesis is intended to characterise a particular type of ‘simple’ systems.6 A substantial part of their paper is devoted to to derive this hypothesis from further axioms. I will, however, not go into this. The central aim is to derive the following result, which Lieb and Yngvason call 5

6

The clause ‘in the same Γ’ means that the hypothesis is not intended for the comparison of states of scaled systems. Thus, it is not demanded that we can either adiabatically transform a state of 1 mole of oxygen into one of 2 moles of oxygen or conversely. Beware that the present meaning of the term does not coincide with that of Carathéodory. For simple systems in Carathéodory’s sense the comparability hypothesis need not hold.

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The Entropy Principle: There exists a function S defined on all states of all systems such that when s and t are comparable then s ≺ t iff S(s) ≤ S(t).

(6)

The question whether this result actually follows is somewhat involved. They show that the entropy principle follows from axioms A1–A6 and the comparability hypothesis under some special conditions which, physically speaking, exclude mixing and chemical reactions. To extend the result, an additional ten axioms are needed. And even then, only a weak form of the above entropy principle is actually obtained, where ‘iff ’ in (6) is replaced by ‘implies’. Note that the theorem is obtained without appealing to Carathéodory’s principle. In fact the axioms and hypothesis mentioned above allow models which violate the principle of Carathéodory (Lieb and Yngvason 1999, p. 91). For my purpose, the question is what connection with irreversibility is in this formulation of the second law. As before, there are two aspects to this question: irrecoverability and time-reversal (in)variance. We have seen that Lieb and Yngvason interpret the relation (6) as saying that entropy must increase in irreversible processes. At first sight, this interpretation is curious. Adiabatic accessibility is not the same thing as irreversibility. So how can the above axioms have implications for irreversible processes? This puzzle is resolved by looking at the physical interpretation Lieb and Yngvason propose for ≺: Adiabatic accessibility: A state t is adiabatically accessible from a state s, in symbols s ≺ t, if it is possible to change the state from s to t by means of an interaction with some device (which may consist of mechanical and electric parts as well as auxiliary thermodynamic systems) and a weight, in such a way that the auxiliary system returns to its initial state at the end of the process whereas the weight may have changed its position in a gravitational field (Lieb and Yngvason 1999, p. 17). This view differs from Carathéodory’s, or indeed, anybody else’s: clearly, this term is not intended to refer to processes occurring in a thermos flask. Even processes in which the system is heated are adiabatic, in the present sense, when this heat is generated by an electrical current from a dynamo driven by a descending weight. Actually, the condition that the auxiliary systems

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return to their initial state in the present concept is strongly reminiscent of Planck’s concept of ‘reversible’! This is not to say, of course, that they are identical. As we have seen before, a process P involving a system, an environment and a weight at height P h, which produces the transition 〈s, Z, h〉 → 〈s′, Z′, h′〉 is reversible for P′ Planck iff there exists a ‘recovery’ process P′ which produces 〈s′, Z′, h′〉 → 〈s, Z, h〉. Here, the states Z and Z′ may differ from each other. For Lieb and P Yngvason, a process 〈s, Z, h〉 → 〈s′, Z′, h′〉 is adiabatic iff Z = Z′. But in all his discussions, Planck always restricted himself to such reversible processes ‘which leave no changes in other bodies’, i.e. obey the additional requirement Z = Z′. These processes are adiabatic in the present sense. A crucial consequence is that, in the present sense, it follows that if a process P as considered above is adiabatic, any recovery process P′ is automatically adiabatic too. Thus, we may conclude that if an adiabatic process is accompanied by an entropy increase, it cannot be undone, i.e., it is irreversible in Planck’s sense. This explains why the result (6) is seen as a formulation of a principle of entropy increase. Thus we obtain the conclusion implying the existence of irrecoverable processes by means of a satisfactory argument! However, note that this conclusion is obtained from the comparability hypothesis and by excluding mixing and chemical processes. The weak version of the entropy principle, which is derived when we drop the latter restriction, does not justify this conclusion. Moreover, note that it would be incorrect to construe (6) as a characterisation of processes. The relation ≺ is interpreted in terms of the possibility of processes. Thus, when S(s) < S(t) for comparable states, this does not mean that all processes from s to t are irreversible, but only that there exists an adiabatic irreversible process between these states. So the entropy principle here is not the universal proposition of Planck. The fact that it is not necessary to introduce timereversal non-invariance into the formalism to obtain the second law, is very remarkable. However, there remains one problematical aspect of the proposed physical interpretation. It refers to the state of auxiliary systems in the environment of the system. Thus, we are again confronted by the old question, when shall we say that the state of such auxiliary systems has changed, and when are we fully satisfied that their initial state is restored. This question remains intractable from the point of view of thermodynamics, as long as the states of arbitrary auxiliary systems (e.g. living beings) are not representable in the

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thermodynamical formalism. Thus, the question when the relation ≺ holds cannot be decided in thermodynamical terms.

5. Discussion There is a large variety in the views on irreversibility and the second law. On one end, there is Planck’s view that the second law expresses the irreversibility of all processes in Nature. A convincing derivation of this bold claim has, however, never been given. On the other extreme is Gibbs’ approach, which completely avoids any connection with time. But even for approaches in the middle ground, the term ‘reversible’ is used in various meanings: time-reversal invariant, recoverable, and quasistatic. In the debate on the question how the second law relates to statistical mechanics, however, most authors have taken irreversibility in the sense of time-reversal non-invariance. The point that in thermodynamics the term usually means something very different has been almost completely overlooked. The formal approaches by Carathéodory and Lieb and Yngvason show that it is possible to build up a precise formulation of the second law without introducing a non-TRI element. The resulting formalism implies only that an entropy function can be constructed consistently, i.e. as either increasing between adiabatically accessible states of all simple systems, or decreasing. At the same time, the Lieb-Yngvason approach does imply that entropy increasing processes between comparable states are irreversible in Planck’s sense. This shows once more the independence of the two notions. Finally, I would like to point out an analogy between the axiomatisation of thermodynamics in the Carathéodory and Lieb-Yngvason approach and that of special relativity in the approach of Robb (1921). In both cases, we start out with a particular relationship ≺ which is assumed to exist between points of a certain space. In relativity, this is this is the relation of connectability by a causal signal. In both cases, it is postulated that this relation forms a pre-order. In both cases, important partial results show that the forward sectors Cs = {t : s ≺ t } are convex and nested and that s is on the boundary of Cs. And in both cases the aim is to show that the space is ‘orientable’ (Earman 1967) and admits a global function which increases in the forward sector. If this analogy is taken seriously, the Lieb-Yngvason entropy principle has just as much to do with TRI as the fact that Minkowski spacetime admits a global time coordinate.

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References Carathéodory, Constantin 1909 “Untersuchungen über die Grundlagen der Thermodynamik”, Mathematische Annalen, 67, 355–386, English translation by Joseph Kestin, “Investigation into the foundations of thermodynamics” in: Joseph Kestin (ed.), 1976 The second law of thermodynamics, Stroudsburg, PA: Dowden, Hutchinson and Ross, 229–256. (This translation is not quite accurate.) Denbigh, Kenneth G. 1989 “The Many Faces of Irreversibility”, British Journal for Philosophy of Science, 40, 501–518. Earman, John 1967 “Irreversibility and temporal asymmetry”, Journal of Philosophy, 64, 543–549. Eddington, Arthur 1935 The Nature of the Physical World, London: J. M. Dent & Sons. Giles, Robin 1964 Mathematical Foundations of Thermodynamics, Oxford: Pergamon. Hollinger, Henry B, and Zenzen, Michael J. 1985 The Nature of Irreversibility, Dordrecht: Reidel, 1985. Kelvin 1852 “On a universal tendency in nature to the dissipation of mechanical energy”, in Joseph Kestin (ed.), 1976 The second law of thermodynamics, op.cit., 194–198. Lieb, Elliott and Yngvason, Jakob 1999 “The physics and mathematics of the second law of thermodynamics”, Physics Reports, 310, 1–96, erratum, 314, (1999), 669. Planck, Max 1897 Vorlesungen über Thermodynamik, Leipzig: Veit & Comp. — 1926 “Über die Begründung des zweiten Hauptsatzes der Thermodynamik”, Sitzungsberichte der Preussischen Akademie der Wissenschaften, 453–463. Robb, Alfred A. 1921 The Absolute Relations of Time and Space, Cambridge: Cambridge University Press. Uffink, Jos 2001 “Bluff your way in the second law of thermodynamics”. Studies in History and Philosophy of Modern Physics, 32, 305–394.

Are the Laws of Nature Time Reversal Symmetric? The Arrow of Time, or Better: The Arrow of Directional Processes Paul Weingartner, Salzburg The paper discusses time reversibility (“arrow of time“) and shows that: (i) it is not satisfied on the microlevel; (ii) irreversibility should be replaced by very improbable recurrence; (iii) the “arrow” is in process not in time.

1. Introduction Max Planck hoped that all statistical laws, especially the law of entropy could (and should) be ultimately reducible to dynamical laws: I believe and hope that a strict mechanical significance can be found for the second law along this path, but the problem is obviously extremely difficult and requires time.1 Many fundamental laws of physics (of Classical Mechanics, of Special Relativity, of Quantum Mechanics) are invariant w.r.t time reversal; i.e. in a differential equation like that of Newton´s second law of motion (for Classical Mechanics) or that of Schrödinger for Quantum Mechanics, one can replace the sign t (for time) by −t without making the law invalid.2 The underlying view about dynamical laws is illustrated by the famous quotation from Laplace: We ought to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it — an intelligence sufficiently vast to submit these data to analysis — it would embrace in the same formula the movements of the greatest 1 2

Planck in a letter to his friend Leo Graetz. Cited in Kuhn (1978), p. 27. For Quantum Mechanics, this was shown by Wigner (1932) and Dirac (1937).

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bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.3 The dynamical law describes the time development of a physical system S in such a way that the following condition D1 is satisfied: D1 The state of the physical system S at any given time ti is a definite function of its state at an earlier time ti-1. A unique earlier state (corresponding to a unique solution of the differential equation) leads under the time evolution to a unique final state (again corresponding to a unique solution of the equation). But D1 is not satisfied in statistical laws, like those of thermodynamics or those describing processes of radiation: A unique later state S2 at t2 is not a definite function of an earlier state S1 at t1. The same initial state may lead to different successor states (branching). According to Prigogine this is a sign that the laws of physics are still incomplete since many processes are irreversible in time.4 Feynman expresses his view quite directly: Next we mention a very interesting symmetry which is obviously false, i.e., reversibility in time.5 The world view underlying Laplace’s quotation was based on the belief that all physical systems are — if analysed in their inmost structure — ultimately mechanical systems. Since a clock was understood as a paradigm example of a mechanical system, the main thesis of the mechanistic world view could be expressed by saying that all complex systems (things) of the world — even most complicated ones like gases, swarms of mosquitoes, or clouds — are ultimately (i.e., if we would have enough knowledge of the detailed interaction of the particles ) clocks. Or, put in the words Popper used in his A. H. Compton Memorial Lecture: “All clouds are clocks”6 After the discovery of statistical laws in thermodynamics and later in other areas, there was a general doubt with respect to the mechanistic and 3 4 5 6

Laplace (1814), ch. 2 Cf. Prigogine (1993). Feynman (1997), p. 28. Popper (1965).

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deterministic interpretation of the world. One of the first philosophers who noticed that a certain imperfection in all “clocks” allows to enter chance and randomness was Charles Sanders Peirce.7 The question was now: Could it not be the case that all laws are statistical and the deterministic outlook is only on the surface of macroscopic phenomena? That is, all complex systems (things) of the world are in fact — in their inmost structure, i.e. on the atomic level — like gases or swarms of mosquitoes or clouds. This led to another extreme picture discussed by Popper: “All clocks are clouds”.8 But neither of these extreme pictures — reduction to dynamical laws “all clouds are clocks” or reduction to statistical laws “all clocks are clouds”— proved satisfactory as an explanation of everything. The heroic ideal to explain everything by one (or one kind of ) principle had to be replaced by the aim to find relatively few (kinds of ) principles (laws) for relatively many facts.

2. Experimental facts Keeping the laws (time-reversal) symmetric and putting the responsibility for the time asymmetric phenomena into the initial or boundary conditions leads to explanations like the following ones: the thermodynamic asymmetry presupposes progenitor states far from the equilibrium; the CP asymmetry presupposes a spontaneous symmetry breaking of the Hamiltonian; the expansion of the universe presupposes a special singularity (big bang), etc. However, many authors have also discussed time-symmetric models of the universe.9 In this case the universe undergoes expansion and contraction in a symmetric way such that we have periodicity. However there are at least two difficulties: The improbable recurrence and the T-violation in weak interactions. The first is expressed by the following quotation: The difficulty with the time-symmetric models is their implausibility. They require a very finely tuned set of boundary conditions, for which no explanation is offered.10 The second is that since CPT (charge-parity-time) invariance (of laws) is 7 8 9 10

Peirce (1960) ch. 6.47. Popper (1965). Gell-Mann, Hartle, (1994), ch. 22.5. Haliwell (1994) p. 374.

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generally satisfied — all fundamental field equations are CPT invariant — but CP invariance is slightly violated in weak interactions, T has to outbalance the difference. Therefore T invariance (time-reversal invariance) is not completely satisfied for the fundamental laws of nature.11 CPT invariance — one of the most important symmetries of Quantum Field Theory — says that physical laws seem to be symmetric with respect to the complex exchange of particle-antiparticle, right-left and past-future. This CPT symmetry has remarkable consequences: It implies that the mass of (any) particle must be the same as that of its respective antiparticle. The same holds for their lifetimes. Their electric charges must have the same magnitude but opposite signs, their magnetic moments must agree. Moreover as it appears from recent experiments T-reversal symmetry seems to be violated directly, too, and not only via CPT symmetry and CP violation. The violation concerns weak interactions. But since weak interaction concerns all elementary particles except photons the experimental result appears to be very important. There have been two different series of experiments independently made at CERN12 and FERMILAB 13 which seem to prove the violation. The experiment made at CERN concerns time dependent rates for the strangeness-oscillation process with neutral kaons which are different for K° → −K° and its inverse −K° → K°. The experiment made at FERMILAB is of a more complicated structure.

3. Non recurrence versus irreversibility Skiing in fresh powder snow is a great pleasure. But if the slope is small and one is skiing down frequently, the slope will be filled with traces and after some time no new space (powder snow) is left and thus one has to use one’s own traces again (recurrence). This illustration tells us already some important conditions: The motion has to be area-preserving (the skier is not supposed to leave the slope) and in a finite region. Observe now that just by raising the complexity of the system recurrence becomes very improbable: Imagine that there are thousands of skiers on the slopes (the cable 11 This holds, provided there are no other ways out; for instance that the T-violation is not due to an asymmetry in the cosmological boundary conditions or to an asymmetry of our particular epoch and spatial location. Cf. Gell-Mann, Hartle (1994), p. 329. 12 Angelopoulos et al. (1998). 13 Schwarzschild (1999).

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cars and lifts of a big ski region in Austria can take up about 60,000 people per hour). The probability that at some later time all skiers would be in the same position again such that the whole state of this system would recur has much lower probability.

Boltzmann’s example Instead of living organisms take the molecules in a litre of gas (air) at temperature T = 0°C (273 K) and atmospheric pressure (1.033·103 g/cm2). A litre of air (at temperature and pressure mentioned) consists of 2.688·1022 molecules. It will be understandable that this system of 2.688·1022 molecules can be in a huge number of different (micro-)states. The number is about 105∙10 so as to realise the macrostate “litre of air under the conditions mentioned”. Thus the same (for our lungs the same) macrostate can be realised by a huge number of different microstates. Boltzmann’s discovery was that the probability of such a macrostate can be defined as the number of microstates which can realise the macrostate and that this number (more accurately the logarithm of it) is the entropy. We might ask the following questions: Will all the 105∙10 microstates of the litre of air be realised at all? And in what time? This leads to an interesting cosmological question: Assume that we are asking how many possible microstates are in the whole universe in order to calculate the entropy of the whole universe. Then the question arises whether every microstate can be realised within the life time of the universe, if the life time is finite. Since the number of microstates is extremely huge they probably will not all be realisable within the life time calculated by the Standard (Big Bang) Theory. If this is so, then there are more possible universes than the actual universe, which obey the same laws of nature and differ from each other only w.r.t. some microstates. In other words the laws of nature have more (possible) models than the one actually realised. 22

22

Irreversibility locally violated Schrödinger raised the question: How can we understand a living system in terms of Boltzmann’s theory?14 Or, how can these systems manage to keep, or even to increase, a low entropy level despite of the validity of the law of entropy? The answer, which was partially already given by Schödinger, includes the following points: 14

Schrödinger (1944).

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(i) Living systems (organisms) are not thermodynamically closed; they are open systems. (ii) Living systems receive high-grade energy (energy with low entropy) from their environment via metabolism, but they pass on low-grade energy (energy with high entropy). (iii) By process (ii) the living systems are capable of achieving orthogenesis (maturation), i.e. increasing order, quality, and differentiation. Cosmological investigations show that also planets, especially the earth, behave in a similar way as living systems. It receives high grade energy as electromagnetic radiation (with Planck-temperature of 5600 Kelvin) from the sun and passes on low-grade energy as heat radiation (with Planck-temperature of only about 300 Kelvin) into its environment. From the above considerations it will be understandable that we want to avoid the term ‘irreversibility’ for three reasons: (i) What thermodynamic processes — and many others like radiation, cosmological expansion, processes of measurement, biological and psychological processes — really show is that recurrence of the state of the whole system is very improbable but not that recurrence or time reversal is impossible. (ii) From the last example described, it is plain that thermodynamic processes can be reversed locally without violating the second law. (iii) Moreover one can show independently that non-recurrence and timeirreversibility are not equivalent notions: since we have cases of nonrecurrent phase density and time reversibility in chaotic motion of dynamical chaos.

Summing up The difference between dynamical and statistical laws which is usually viewed as the most striking one — time-reversal invariance of the laws versus irreversibility of the laws — has to be taken with care. Strict time-reversal symmetry is no longer valid on the microlevel and irreversibility on the macrolevel should be better replaced by very improbable recurrence. However, the differences in this respect are sufficiently large to forbid reducibility of one type of law to the other. But the more careful interpretation paves the way for the compatibility of both types of laws.

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4. Is the “arrow of time” compatible with dynamical laws? We shall discuss this question w.r.t. three different descriptions of time (a)–(c).

(a) Time flows That time flows we grasp from change, mutation and movement (i) w.r.t. to an ordered sequence (ii). (i) Without any change time would “stand still” such that change is a necessary condition of time — at least for our understanding: “It is utterly beyond our power to measure the changes of things by time. Quite the contrary, time is an abstraction, at which we arrive by means of the changes of things.”15 That time presupposes change was already pointed out by Aristotle in his definition of time.16 Cf. also Kant in the Critique of Pure Reason.17 That there is no past to future direction of time in regions that are at equilibrium was pointed out by Boltzmann.18 He compared this with gravitation: As there is no downward direction in regions of space where there is no (net) gravitational force, there is also no past to future direction of time in regions that are at equilibrium. (ii) Thus it seems better to speak of the asymmetry of a flowing process of a sequence of successive states which are ordered by a partial ordering instead of a “flowing time”. In such a sequence we distinguish past and future states and we measure the distance between them with the help of time units (produced by another physical periodic process in a clock).19 The presupposed ordered sequence is best described by the chronology of time or by the basic axioms of tense logic which assume partial ordering, transitivity, antisymmetry, irreflexibility and density. Newton’s interpretation: Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external … 20 15 16 17 18 19 20

Mach, E. (1933), p. 273. Phys., 219b1. KRV, B233. Boltzmann (1897), p. 583. Cf. also the critical remarks by Paul Davies in his (1994). Newton, Principia, Scholium.

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Einstein’s interpretation in GR: The relative time of an observer plus reference frame is measured by standard clocks. By stipulation it flows equably but it depends on the distribution of matter, fields, and boundary conditions. Every observer plus reference frame has its own time scale and there is no universal time scale that is relevant for all observers. In analogy to Newton we may paraphrase Einstein’s interpretation thus: “Relative, true and physical time … flows equably only locally, but in general it flows unequably with relation to something external i.e. to the distribution of matter, fields and boundary conditions.”

(b) Time flows only in one direction The subsequent considerations in (b) and (c) will show that the frequently used expression of the “arrow of time” is misleading; the arrow is intrinsic in process (not “in” time). Concerning the question of how to distinguish a (particle’s) movement on a spatial coordinate (in GR: space geodesic) from a (particle’s) movement on a time coordinate (in GR: either time-like geodesic or null geodesic21), we may formulate two subquestions: (α) Can the coordinates (geodesics) be distinguished by their directions (vectors)? (β) Can the coordinates (geodesics) be distinguished by their closure conditions? Both questions can be answered with: Yes. The answer to the first subquestion is very well expressed by the following quotation from Wigner: “The difference between the two cases arises from the fact that a particle’s world line can cross the t = constant line only in one direction (in the direction of increasing t); it can cross the x = constant line in both directions.”22 The answer to the second subquestion is the following: According to GR, the space of the universe is closed (even if the universe is expanding); that is, there are closed spatial coordinates or closed space-like geodesics. On the other hand, we usually assume that the time coordinate is not closed; i.e., we assume that the non-space-like geodesics (time-like geodesics and null geodesics) are not closed. This assumption has been called the chronology condition of spacetime23. This condition plays an important role for the concept of causality. Causality would break down and one could travel into 21 See Hawking, Ellis (1973). 22 Wigner (1972), p. 239. 23 Cf. Hawking, Ellis (1973), p. 189.

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one’s own past, if the chronology condition is not satisfied.24 The fact, that the time coordinate is distinguished from the spatial coordinates such that a particle´s world line can cross the t = constant line only in the positive direction of increasing t, is not determined by dynamical laws; because dynamical laws permit both directions, positive and negative. From this, it follows that (b), i.e. time flows in one direction, is compatible with dynamical laws, because dynamical laws allow both directions.

(c) Time is connected with directional processes In addition to the directional feature of time described in (b) which is not at all in conflict with dynamical laws, there are two further features of time which are connected with directional processes. (i) Assume a very large (long) but finite sequence of decimal places after 0, say the (finite) sequence of natural numbers, i.e. 0, 1 2 3 … 10 11 12 … 99 100 101 … etc. It can be proved that this sequence has a normal distribution. It will therefore be easily understandable that the probability of recurrence for the three numbers 1, 4, 2 (in this order) on decimal places will be not very low. It occurs in 142, in 1420, 1421 etc. In contradistinction to that the probability of the recurrence of an ordered sequence of 1010 numbers on decimal places as part of the above sequence will be very much lower. This has nothing to do with entropy or with the increasing of a certain physical magnitude. But it has to do with “direction” and asymmetry. (ii) Directional processes: First it should be clear that chronological time scales as they are used for time measurement do not define a direction of time even if they indicate that “time flows” in the sense of representing a sequence of partial ordering. Assuming events (states) S1, S2, … (∈ S), reference frames plus observer, RF, and chronometrical scales CS (mappings of durations onto real numbers, standard clock) the following postulates are basic for time T and time interval t: α) T is a function {S1, S2, RF, CS} → ℜ+ with T {S1, S1, RF, CS} = 0 β) For every state S1 relative to RF and CS, and for any value t ∈ ℜ+ there exists a second state S2 such that T (S1, S2, RF, CS) = t γ) Transitivity. For any triple of states (S1, S2, S3) relative to RF and CS it holds T (S1, S2, RF, CS) + T(S2, S3, RF, CS) = T (S1, S3, RF, CS) 24 See ch. 9 of Mittelstaedt and Weingartner (2005) for causality expressed by laws of nature.

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From these postulates it follows that for any two events S1S2 relative to RF and CS it holds that: T (S1, S2, RF, CS) = −T (S2, S1, RF, CS). This shows clearly that the above basic assumptions and postulates do not define a direction “in” time. They tell us only that the direction from S1 to S2 is the opposite of the direction from S2 to S1; but not which event is first in nature. On the other hand so-called directional processes (of nature) tell us unambiguously which event is first and which is second. Penrose25 lists seven such directional processes: (1) The decay of neutral K mesons in weak interactions. (2) The process of measurement in quantum mechanics, especially the socalled “collapse of the wave function”. (3) All processes in which entropy increases. (4) All processes of radiation. (5) All conscious mental processes. (6) The process of expansion of the universe. (7) The process of gravitational collapse ending in a black hole. Of these processes (1), (3), (4) and (6) are experimentally very well confirmed. (5) is very well confirmed by introspection and by the descriptions of the psychology of mental processes. The claim that (2) is a directional process is — at least to a considerable extent — a matter of interpretation of the quantum mechanical process of measurement. The time reversal of (7), leading to a white hole (no experimental evidence so far) is an open question such that (7) cannot be viewed as an unambiguous case of a directional process. Furthermore it is an open question whether processes (1), (4), (6) and perhaps (7) can be reduced ultimately to process (3).26

References Angelopoulos, A. et al. 1998 Physics Letters, B444, p. 43f. Aristotle 1985 (Phys) Physics, in: Barnes, J. (ed.) The Complete Works of Aristotle. 25 Penrose (1979). 26 Cf. Wheeler (1994). Concerning (7) the important question is whether the horizon area of the black hole can be proved to be proportional to measures of entropy which has been supported by Christodoulon, Beckenstein and Hawking.

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Boltzmann, L. 1897 „Zu Herrn Zermelos Abhandlung ‘Über die mechanische Erklärung irreversibler Vorgänge’, in: Wissenschaftliche Abhandlungen, vol. III, § 120, New York 1968. Davies, P. C. W. 1994 “Stirring up Trouble”, in: Halliwell, J. J. et al. (1994), pp. 119–130. Dirac, P. A. M. 1937 “The Reversal Operator in Quantum Mechanics”, Bulletin de l’Académie de Sciences de l’URSS 1937, pp. 569–575. Feynman, R. P. 1997 Six Not so Easy Pieces, Cambridge, Mass. Gell-Mann, M., Hartle, J. B. 1994 “Time Symmetry and Asymmetry in Quantum Mechanics and Quantum Cosmology”, in: Halliwell, J.J. et al. (1994), pp. 311–345. Halliwell, J. J. 1994 “Quantum Cosmology and Time Asymmetry”, in: Halliwell, J. J. et al. (1994), pp. 369–389. Halliwell, J. J., Perez-Mercader, J., Zurek, W. H. 1994 Physical Origins of Time Asymmetry, Cambridge. Hawking, S. W., Ellis, G. F. R. 1973 The Large Scale Structure of Space-Time, Cambridge. Kant, I. 1787 Kritik der reinen Vernunft. Kuhn, Th. 1978 Black-Body Theory and the Quantum Theory, 1894–1912, Oxford. Laplace, P. 1814 Essai philosophique sur les probabilités, Courcier, Paris, Engl. Translation: A Philosophical Essay on Probabilities, Dover, New York 1951. Mach, E. 1933 Die Mechanik in Ihrer Entwicklung. Leipzig, 9th edition. Mittelstaedt, P., Weingartner, P. 2005 Laws of Nature, Springer, Heidelberg. Newton, I. 1934 Mathematical Principles of Natural Philosophy, Cajori, F. (ed.), Berkeley 1962. Peirce, Ch. S. 1960 Collected Papers of Ch. S. Peirce, ed. by Ch. Hartshorne and P. Weiss, Harvard U. P., Cambridge Mass. Penrose, R. 1979 “Singularities and Time-Asymmetry”, in: Hawking, S. W., Israel, W. (eds.) General Relativity, Cambridge. Popper, K. R. 1965 “Of Clouds and Clocks”, in: Popper: Objective Knowledge. Oxford 1972, pp. 206–255. Prigogine, I. 1993 “Time, Dynamics and Chaos”, in: Holte, J. (ed.), Nobel Conference XXVI. Univ. Press of America. Schrödinger, E. 1944 What is Life?, Cambridge. Schwarzschild, B. 1999 Physics Today, Feb. 1999, pp. 19–20.

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Wheeler, J. A. 1994 “Time Today”, in: Halliwell, J. J. et al. 1994, pp. 1–29. Wigner, E. P. 1972 “On the Time Energy Uncertainty Relation“, in: Salam, A., Wigner, E. P. (ed.) Aspects of Quantum Theory, Cambridge. Wigner, E. P. 1932 “Über die Operation der Zeitumkehr in der Quantenmechanik”, Göttinger Nachrichten 31, pp. 546–459.

Time and Communication Kristóf Nyíri My main thesis in this paper, for which I will argue in section 3, is that with the mobile phone, time has become personalized. It is not just our perception of time that has changed, nor indeed merely our way of talking about time. What has changed is, in fact, the nature of time. Wittgensteinian received wisdom of course would not allow one to keep a straight face when mentioning the nature of time, or even when using the substantive “time” in earnest. I believe the received wisdom is wrong; an alternative philosophical strategy applicable to the problem of time is outlined in section 1.

1. Philosophical preliminaries It should be pointed out that though his therapeutic tone pervades the 1932– 35 lectures and dictations, and is dominant in the Philosophical Investigations (Kaspar and Schmidt 1992; Grundy 2005, 97; Reichenberger 2005), Wittgenstein himself did sometimes refer to time in a different key. In contrast to the position that puzzlement about the nature of time arises from a grammatical confusion (Wittgenstein 1979, 15; Wittgenstein 1958, 6; Wittgenstein 1953, §§ 89f.), he also made remarks suggesting that genuine issues might pertain to the phenomenon of time; that there might be room for exploration and insight here. According to one such remark, written in 1937 and still being experimented with in 1942/43 (in the text that was to become Part I of the Philosophical Investigations): “That times occur to us in coincidence with the clock; that we can estimate the time; is one of the reasons why what the clock measures, the time, is so important”.1 Or recall the remark, written in 1941: “could we talk about minutes and seconds, if we had no sense of time; if there were no clocks …; if there did not exist all the connexions that give our measures of time meaning and importance?”2 1 2

“Dass uns die Zeiten übereinstimmend mit der Uhr einfallen; dass wir die Zeit schätzen können; ist ein Grund, warum, was die Uhr misst, die Zeit, so wichtig ist”, Wittgenstein 2001, 517f. Cf. Wittgenstein 1978, 382. The sentence as printed there has “hours” instead of “minutes”; the same error occurs in the German editions as well. I am indebted to István Danka for alerting me to this lapse.

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On the other hand, when jotting down the remarks “This is the similarity of my treatment with relativity-theory, that it is so to speak a consideration about the clocks with which we compare events” (Wittgenstein 1978, 330) and “Einstein: how a magnitude is measured is what it is”3 Wittgenstein clearly saw himself as staying within the confines of grammar — compare his remark, written in 1937: “Du vergißt, was //glaube ich// Einstein //wie ich vermute// die Welt gelehrt hat: daß die Methode //Art & Weise// der Zeitmessung zur Grammatik der Zeit-Ausdrücke gehört” (Wittgenstein 2000: MS 119). Of course the similarity between Einstein’s approach and that of Wittgenstein soon breaks down. Where Wittgenstein warned against moving beyond the grammar of everyday language, Einstein sought to modify the everyday world-view embodied in our everyday grammar. As Wittgenstein said in 1935: “There is no trouble at all with primitive languages about concrete objects. … A substantive in language is used primarily for a physical body, and a verb for the movement of such a body. … We might say that it is the whole of philosophy to realize that there is no more difficulty about time than there is about this chair” (Wittgenstein 1979, 119). By contrast, Einstein certainly saw difficulties both with time and with primitive languages, and was intent on solving the former by clarifying the latter. As he wrote in 1934: “[t]he whole of science is nothing more than a refinement of everyday thinking” (cf. Miller 1984, 13). How can one avoid making time seem a “queer thing” (Wittgenstein 1958, 6), and still build up meaningful discourse about what time is? The philosophical strategy I believe to be the most promising here is to regard time as a kind of theoretical entity, in the specific sense Wilfrid Sellars gave to this term. The point where Sellars’ view of the nature of theories differs most significantly from that of, say, Carnap, Reichenbach, and Hempel, is his conviction that science is “continuous with common sense”. As he puts it: “the ways in which the scientist seeks to explain empirical phenomena are refinements of the ways in which plain men, however crudely and schematically, have attempted to understand their environment … since the dawn of intelligence” (Sellars 1963, 181–183). It is within the framework of everyday observational discourse that certain unobservable entities are first postulated, entities in terms of which certain properties of observable events become explainable. Now according to Sellars, time is just such a postu3

Written in 1929, and included in The Big Typescript, see Wittgenstein 2005: 208 and 488 for two different translations; the second is the correct one.

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lated entity, with “events in Time (or Space-Time) as metrical abstractions grounded in the reality of changing substances” (Sellars 1975, 282). There emerge “rules for coordinating statements concerning empirically ascertainable metrical relations between episodes pertaining to the things of everyday life and science, with statements locating these episodes, relatively to other episodes, in time, that is, with statements having the characteristic syntax of statements ‘about time’” (Sellars 1962, 551 f.). The advance of science, the physical theory of time, will tell us what time is (Sellars 1962, 593), but this advance has been underway all through the cultural evolution of humanity, from primitive thought through Plato, Aristotle, and Augustine, to modern and contemporary philosophy and physics.4 A great advantage of the specifically Sellarsian interpretation of time as a theoretical entity is that it allows for an amalgamation of social time, or time as a social construct, with astronomical time, or time as a construct of the physical sciences. The classic statement as to the originally social nature of time of course comes from Durkheim. It is clear, Durkheim wrote, that those indispensable fixed points with respect to which all things are temporally organized are the products of social life; that it is the periodicity of rites, feasts, public ceremonies, to which the division into days, weeks, months, years, etc. corresponds. Time, as opposed to duration, is time as lived by the group — social time; it is time, as Durkheim puts it, “tel qu’il est objectivement pensé par tous les hommes d’une même civilisation” (Durkheim 1912, Intr., sect. II). One and the same civilization — it should be stressed that this is a dynamic, rather than static, notion. Let me quote at some length the decisive sequel to Durkheim’s argument, by Sorokin and Merton (1937): The local time system varies in accordance with the differences in the extent, functions, and activities of different groups. With the spread of interaction between groups, a common or extended time system must be evolved to supersede or at least to augment the local time systems. … The final common basis was found in astronomical phenomena … Thus, the social function of time reckoning and designation as a necessary means of coordinating social activity was the very stimulus to astronomical time systems … 4

As Whitrow (1961: 58) puts it: “out of man’s primeval awareness of rhythm and periodicity there eventually emerged the abstract idea of world-wide uniform time”.

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Interaction between groups, as well as the coordination of social activities, essentially involve communication. And indeed there is an intrinsic connection between time and communication, whichever meaning of the latter term we focus on. “Communication” in its original, archaic sense means participation. Echoing Dewey (1916) and Heidegger (1927), this is the sense Carey singled out when describing the “ritual” view of communication. The ritual view, he wrote, “exploits the ancient identity and common roots of the terms ‘commonness’, ‘communion’, ‘community’, and ‘communication’. A ritual view of communication is directed not toward the extension of messages in space but toward the maintenance of society in time …” (Carey 1975). In another sense, communication of course means transportation — moving people and goods through space, in real time. And then there is communication in the sense of conveying information. It is the connection of time with communication in this latter sense that constitutes the topic of my paper. The connection can be seen in two broad perspectives. First, the communication of temporal information — communicating time. My argument in section 3 will be about communicating time in everyday life; at the present juncture, let me insert a brief reference to communicating time as a scientific issue. The reference of course is, once more, to Einstein, for whom the problem of synchronizing clocks at a distance was the starting point on the road leading to the special theory of relativity. Nor was this issue for Einstein, as has been brilliantly demonstrated by Galison (2003), an abstractly scientific one. The young patent office clerk in Bern had dutifully evaluated dozens of submissions having to do with the distant synchronization of clocks by electric means before hitting on the revolutionary thesis of his 1905 paper on electrodynamics. The second perspective is about the impact of communication technologies on our notions of time — and, with that, on the formation of the theoretical entity time itself. This is the perspective I shall pursue in section 2.

2. From cyclic time to timeless time According to a periodization germinating from McLuhan’s Toronto circle, the history of the technology of communication can be divided into the following main phases: 1. primary orality; 2. literacy; 3. the typographic phase (printing); 4. “secondary orality”, given rise to by electronic information processing and transfer. Elaborated, most notably, by Havelock (1963), Goody and Watt (1963) and Ong (1982), this periodization has for some

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time been rather widely accepted. I myself have adhered to it (cf. Nyíri 1991 and Nyíri 1992), until about the time I began working on my “The Picture Theory of Reason” (Nyíri 2001). Since then, I have realized that a more finegrained set of distinctions is called for. Currently, I would suggest something like the following series of divisions: 1. Mimetic communication, based on what has been referred to as “the emergence of the most basic level of human representation, the ability to mime, or re-enact, events” (Donald 1991, 16). Not only do we have good grounds to assume that language first emerged as a visual sign system, but clearly it today still retains a basic dimension of mimetic gestures. 2. The culture of primary orality, where words are exclusively spoken or heard, with the knowledge society possesses stored in easily recalled formulae, memorized through constant repetition of authoritative texts. 3. Pictorial communication, ranging from the earliest cave paintings through ancient pictographs, and through medieval and modern drawings, to photography, and on to twentieth-century iconic symbols (today, conspicuously, icons on digital displays). 4. Ideographs. 5. Syllabic and alphabetic writing. 6. Typography. 7. The age of secondarily oral communication, within which again several phases and dimensions must be distinguished: telegraphy, representing a step away from the silence of writing towards the world of sounds not because it involved clicks and clacks just as it did dots and dashes, but because it gave rise to an elliptic style reminiscent of spoken rather than written language; the telephone; the movie, both in its silent and sound film phases; radio broadcasting; television; and the various sound and video recording devices. 8. Computer-mediated communication, creating a kind of secondary literacy with e-mail (and its cousins instant messaging and mobile SMS), a return to writing in the age of secondary orality; creating, also, a network of users exchanging multimedia documents. What effect do these different modes of communication have on the evolving concept of time? Here the language of gestures provides a truly fertile initial medium. Gestures are movements, the meanings conveyed by them are created visibly in time. They necessarily create the experience both of “before” and “after”, as well as the experience of time consisting of extended intervals, the latter experience leading, say, to the Stoics’ idea of the “broad” present (Sorabji 1983, 25), or to James’ elaboration of the notion of “the specious present” ( James 1890, 608 f.), with this notion having interesting echoes in Wittgenstein’s middle period (Wittgenstein 1975, 98 and Wittgenstein 2005, 351). Miming, that is re-enacting, events must also generate a rudimentary consciousness of the difference between the present and

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the past — between what is in fact lived through, and what is only remembered. The experience of memory provides one of the main inspirations for the theoretical construct time. However, for this experience to become fully conscious, something like a verbal language must first emerge. In the case of autobiographical memory, it is with the development of linguistic skills that so-called childhood amnesia gradually diminishes, and first recollections arise (Draaisma 2004, 24 f.). Similarly with primitive group memory, which is embodied in myths and preserved through the basic information storage and retrieval activity of primarily oral societies: the recitation, that is repetition, of texts. To repeat is to re-live: time in the medium of primary orality is experienced as cyclic, rather than as linear. And it is of course a cyclic view of time that the daily movement of the sun, the changes of the moon, the seasons of the year, and the succession of generations in the animate world suggest. The idea of linear time is a culturally subordinate construct, one which did not become dominant prior to the age of the printing press. Jan Assmann provides a masterly summary (Assmann 1999, 27–38) of the simultaneous, but unequal, presence of the cyclic and the linear views in medieval Christianity (with the Church partaking in the sacred linear history leading to salvation, while events here in this world followed a cyclic pattern), in ancient Mesopotamia (with occasional attempts at retrospective political chronicles), and indeed in Egypt (where the construction of king lists represented rare and insignificant episodes within an overwhelmingly cyclic world view). The Egyptians did have a linear writing system just as the Mesopotamians did. However, the educated Egyptian was, also, immersed in a world of pictures, a world of images and ideographs (hieroglyphs). Now it is of course well-known that what these pictures depict conveys a notion of time recurring, or standing still. Also, the canonical style of Egyptian art, unchanging over thousands of years (Assmann 1992, 171–174), suggested and indeed upheld an idea of immutable time, where contact with the past meant repetition, not continuation.5 But the question we must here ask is whether it may not lie in the very nature of pictorial communication to give rise to a halting, as opposed to a sequential, view of temporality. And one 5

“Die Kanonisierung der Bildkunst … steht im Dienste der Wiederholbarkeit, nicht der Anschließbarkeit”, and similarly with texts: “Texte werden kopiert und variiert, aber sie werden nicht eigentlich interpretiert” (Assmann 1992, 177, 175). Yet here, too, one should pay heed to Assmann’s warning against an all too uniform view of ancient Egypt: “Man darf sich das Weltbild einer mehrtausendjährigen und vielschichtigen Kultur nicht zu monolithisch vorstellen” (Assmann 1975, 20).

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way to argue for this would be to point out that the handed-down is more powerful in pictorial than in textual form: a culture where the image dominates over text might well be more acquiescent, less prone to initiate changes in the pattern of events, than one in which text rules over the image. On the other hand, it should be stressed that even though, as Wittgenstein convincingly demonstrated (Wittgenstein 1953, § 22), isolated pictures are often ambiguous, a series of pictures, or a moving image, can very well be unequivocal, and can tell a story, i.e. recount events happening in time. Indeed one of the most interesting extended discussions by Wittgenstein on time (Wittgenstein 1958, 104–109), dealing with the question of “how a child might be trained in the practice of ‘narration of past events’”, begins with the introduction of a pictorial language involving two sequences of images running in parallel to each other. One sequence is the “sun series”, representing the passage of time during the day, the other the “life pictures”, showing the activities of a child. The two rows of pictures, when properly correlated, “tell the story of the child’s day”. Alternatively, the sun series can be replaced by writing a number against each life picture indicating the hour on the dial of a clock in the nursery. Interestingly, this pictorial training of the child does not seem to proceed beyond the boundaries of a single day. The sun completes its daily round, as does the hour hand of the clock; the picture series suggest a cyclic, rather than a linear, notion of time. Syllabic and alphabetic systems, with writing and reading proceeding from top to bottom, right to left, or left to right, in a definite direction (if we skip the boustrophedon, “as the ox ploughs”, early variants), obviously create a minimum experience of time being linear and having a direction.6 But the temporal world of manuscript cultures — think of Greece, think of the European Middle Ages — is still overwhelmingly cyclic. There are two broad reasons for this. First, there was a residual orality resulting from the phenomenon of reading aloud (typical before the advent of easily followable printed texts), with written lines, ultimately, still experienced as a fleeting succession of sounding syllables; secondly, text corruption was a common by-product of manual copying. The older a manuscript, the more reliable the text: there is decay, and a feeling that one should return to the beginnings. With the advent of the printing press, a radical change occurs. Every new edition produces identical, or indeed improved — corrected — texts. It was 6

While reading a linear text is a unidirectional process, looking at pictures involves to-and-fro scanning. This might be another cause for pictorial communication not engendering a linear notion of time.

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the fully developed age of the printed text, beginning with the seventeenth century, in which the idea of linear time became victorious. Newton’s vision of the constant linear flow of time, and of course Locke’s enthusiastic endorsement of that vision, would not have been imaginable without the suggestion of a “constant and regular succession of ideas” (Essay, II, xiv, 12) created by following the printed line. This was the age, too, in which the notion of progress, and modern historical consciousness, emerged. As Elizabeth Eisenstein wrote: “Before trying to account for an ‘idea’ of progress we might look more closely at the duplicating process that made possible not only a sequence of improved editions but also a continuous accumulation of fixed records. … the communications shift [precedes] … the beginning of a modern historical consciousness … by a century or more. The past could not be set at a fixed distance until a uniform spatial and temporal framework had been constructed” (Eisenstein 1979, vol. I, 124 and 301). Or to quote Sven Birkerts’ memorable formulation: “our sense of the past … is in some essential way represented by the book and the physical accumulation of books in library spaces. In the contemplation of the single volume, or mass of volumes, we form a picture of time past as a growing deposit of sediment; we capture a sense of its depth and dimensionality” (Birkerts 1994, 129). Historical consciousness, the ability, as J. H. Plumb puts it, “to see things as they were in their own time”, “the consciousness of a different past”, the “wish to understand the past in its own terms” (Plumb 1969, 82 and 118f.), did not, then, fully emerge before the seventeenth century. And after less than three hundred years, with the rise of telegraph news reporting, it had already begun to erode. Historical consciousness presupposes a definite point of view in time. Until the 1860s, the column reigned over the news even in the daily paper; there was a temporal perspective the newspaper conveyed. But then the daily paper became, to quote McLuhan, “a mosaic of unrelated scraps in a field unified by a dateline. Whatever else there is, there can be no point of view in a mosaic of simultaneous items” (McLuhan 1964, 249). In a much more tangible way, too, the experience of time was changed by the telegraph. Precise longitude determination and global mapmaking initially depended on the transportation of accurate timekeepers. After 1866, when the first transatlantic cable was successfully laid, long-distance synchronization of clocks became possible. By 1880, every inhabited continent was connected (Galison 2003, 132–144). Local times came to be elements within the overall framework of global time; there emerged the practice of almost real-time communication between people belonging to differ-

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ent time zones. The “mixing of tenses” bemoaned by Castells (1996, 433) began with telegraphy, the first medium which allowed a separation of the movement of information from the movement of people. The second such medium was telephony, with distant communication becoming actually realtime, and with the narrow broadband of telegraphy giving way to the much wider broadband of live human voice. In 1895 the Lumière brothers presented their cinematograph. With that, an extremely powerful new metaphor for the flow of time came into being (Draaisma 2004, 57ff.). Bergson made fundamental use of it; Wittgenstein, in the early 1930s, was infatuated with it (Reichenberger 2005, 255). In Creative Evolution there is an argument spanning some 40 pages (Bergson 1911, 304–345), in which the cinematograph simile (“the film of the cinematograph unrolls, bringing in turn the different photographs of the scene to continue each other”) is deployed to explain our inability to recognize real becoming behind a series of mental snapshots, to dissolve Zeno’s paradox of the flying arrow, to provide a context for the immutable eidos, to highlight both the parallels and the differences between modern and ancient science, and of course to plead once more for the Bergsonian notion of durée: “if time is not a kind of force, why … is not everything given at once, as on the film of the cinematograph?”. Wittgenstein seems to have read Bergson, and he, too, was impressed by the difference of the series of pictures existing, on the one hand, synchronously on celluloid, and on the other, creating a narrative in time on the screen: “If I compare the facts of immediate experience with the pictures on the screen and the facts of physics with pictures in the film strip, on the film strip there is the present picture and past and future pictures. But on the screen, there is only the present” (Wittgenstein, 1975, 83). Then there is, in the silent film, the effect of verbal language being subordinated to the secondary role of mere captions. The poet, playwright, and film critic Béla Balázs, in a book published in 1924 (Der sichtbare Mensch, “The visible man”), made the following observation: “In film … speaking is a play of facial gestures and immediately visual facial expression. They who see speaking, will learn things very different from those who hear the words”. Balázs expresses his belief that film will bring back “the happy times” when, in contrast to the times “since the spread of the printing press [when] the word came to be the main bridge between human beings”, “it was still allowed for pictures to have a ‘theme’, an ‘idea’, because ideas did not always first appear in concepts and words, so that painters would only subsequently provide illustrations for them with their pictures” (Nyíri 1999, 7f.). The pure

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pictoriality of the silent film was soon supplanted by the sound film, the first multimedia. But whether silent or sound, the experience of time given rise to by film was not the linear one suggested by written, especially printed, language. The more obvious influences of film on the experience of time are wellknown. The techniques of slow motion, fast motion, or running the film backwards create entirely new temporal impressions. Also, scenes alternate. With television, especially with cutting becoming ever faster, the breaking up of time as an ordered sequence continues.7 And with satellite channels, “TV’s electronic time zones are competing increasingly with … our internal biological clocks to determine our sense of time” (Ofield 1994, 593). What began with telegraphy, and continued with shortwave radio and longdistance telephony — the juxtaposing of different local times — went yet a step further with global television. The final step, of course, was the emergence of computer networks. Computers transformed our experience of time even before world-wide computer networks were built. In one of the founding analyses of the topic, Bolter argued that, for the computer programmer, time becomes finite, discrete, and — think of loops — cyclic (Bolter 1984, 100–123). But let us note that even for everyday users, certain time-related phenomena are changed. Word processing has a special significance here. The spoken word is flexible, elastic, but vanishes in the moment of speaking. Written language, and to an even greater extent, printed language, are enduring but rigid. A text that is stored in the computer, in contrast, is preserved, but changeable as well. The text called up from the memory of the computer is always simultaneous, lacking in all history. Age-old documents preserved in the computer carry no mark of temporality. Images called up from a CD-ROM or downloaded from the network might carry indications of their history; yet in their digitized form they belong to the here-and-now, with no difference whatsoever between original and copy. Clearly, this environment of timeless documents cannot remain without influence on our sense of time. Simple word-processing, global computer networks, and the world of digital multimedia documents all contribute, then, to the emergence of what Castells calls timeless time. As he puts it: “linear, irreversible, measurable, predictable time is being shattered in the network society … we are not just witnessing a relativization of time according to social contexts or alterna7

For a discussion from a complementary point of view, see Steininger 2005.

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tively the return to time reversibility … The transformation is more profound: it is the mixing of tenses to create a forever universe …, not cyclical but random” (Castells 1996, 433). However, as I attempt to show in the concluding section of this paper, the truly fundamental transformation in communications today — the triumphant progress of the mobile phone — does not further aggravate, but much rather alleviates the condition of timeless time.

3. Time and the mobile phone Back in 1934, Lewis Mumford noted that what is effected by “our closer time co-ordination and our instantaneous communication” is “broken time and broken attention” (Mumford 1963, 272). By contrast, I believe that the mobile phone gives rise to a new synthesis of what Mumford referred to as “mechanical time” and “organic time”.8 At the very beginning of Technics and Civilization Mumford gave a list of “the critical instruments of modern technology” (Mumford 1963, 4). The first two items on this list are the clock and the printing press. Now the two technical inventions whose significance is most plausibly paralleled by that of the mobile phone are the portable book and the portable clock. The portable hand-held book was an innovation, in 1501, of publisher Aldus Manutius. What this innovation enabled was communication, albeit unidirectional, with the absent author, anytime, anywhere; and access to information anytime, anywhere, as long as that information was contained in the books one carried around. The emergence of the portable clock, and the beginnings of the transition from the portable clock to the watch, took place over the course of the fifteenth century. The mechanical clock itself was invented in the thirteenth century. At first, it had no dial but it did strike the hours — it was in fact, as Landes puts it, an “automated bell” (Landes 2000, 81) — communicating time within the space of the monastery, or in the public space of the medieval town. The fourteenth century saw the spread of bell towers. Urban society increasingly depended on these, the “striking of the bells brought a new regularity into the life of the workman and the merchant” (Mumford 8

“[M]echanical time is strung out in a succession of mathematically isolated instants … [While] mechanical time can … be speeded up or run backward, like the hands of a clock or the images of a moving picture, organic time moves in only one direction — through the cycle of birth, growth, development, decay, and death” (Mumford 1963: 16).

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1963, 14). With the portable clock, public time could also be kept privately. By the nineteenth century, the regularity dictated by public time could no longer be experienced but as a tyranny of fixed schedules. As Georg Simmel wrote in his famous paper “Die Großstädte und das Geistesleben” in 1903, “The relationships and affairs of the typical metropolitan usually are so varied and complex that without the strictest punctuality in promises and services the whole structure would break down into an inextricable chaos. … If all clocks and watches in Berlin would suddenly go wrong in different ways”, Simmel continued, “all economic life and communication of the city would be disrupted for a long time. In addition, … long distances … make all waiting and broken appointments result in an ill-afforded waste of time. Thus, the technique of metropolitan life is unimaginable without the most punctual integration of all activities and mutual relations into a stable and impersonal time schedule”.9 By the last decades of the twentieth century, the rule of the clock became simply impractical in many domains of decentralized mass society, i.e. postmodern society. As Ling (2004, 62), in reference to Beniger (1986), puts it: “The demands for rapid and geographically dispersed coordination of small groups became more acute due to the rise of transportation systems and the differentiation of social functions.” It appears that in the postmodern world, the need for the possibility of frequent re-scheduling was there even before the mobile phone, the instrument par excellence for changing schedules while on the move, appeared on the scene.10 To a considerable degree, the mobile actually took over the functions of the clock. The co-ordination of social activity today relies, in no small measure, on mobile negotiation, rather than on keeping pre-defined schedules (Ling and Yttri 2002, 143f.). A different way of synchronizing activities has emerged: within the overall framework of fixed public time, windows of personalized time are opening up. In the constitution of the theoretical entity time, the building-block of Einstein’s situation-bound relative time there is now joined by the buildingblock of personalized time. 9

Simmel 1997, 177f. In the sentence “In addition …” I had to modify the translation “would make all waiting” to “make all waiting”. Simmel here is not continuing the speculation about what would happen if clocks went wrong, but is making a straightforward observation to the effect that since in the metropolis one has to travel longish distances to keep appointments, non-punctuality is all the more unpleasant. 10 “The mobile phone breaks the flow of information away from the scheduling necessary to ensure coordination of journeys” (Townsend 2000: 96).

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Literature Assmann, Jan 1975 Zeit und Ewigkeit im alten Ägypten: Ein Beitrag zur Geschichte der Ewigkeit, Heidelberg: Carl Winter Universitätsverlag. — 1992 Das kulturelle Gedächtnis: Schrift, Erinnerung und politische Identität in frühen Hochkulturen, München: C.H. Beck. — 1999 Ägypten: Eine Sinngeschichte, Frankfurt/M.: Fischer Taschenbuch Verlag. Beniger, James R. 1986 The Control Revolution: Technological and Economic Origins of the Information Society, Cambridge, MA: Harvard University Press. Bergson, Henri 1911 Creative Evolution, transl. by Arthur Mitchell, New York: Henry Holt. Birkerts, Sven 1994 The Gutenberg Elegies: The Fate of Reading in an Electronic Age, Boston: Faber and Faber. Bolter, J. David 1984 Turing’s Man: Western Culture in the Computer Age, Chapel Hill: University of North Carolina Press. Carey, James W. 1975 “A Cultural Approach to Communication”, Communication 2, 1–22. Castells, Manuel 1996 The Rise of the Network Society, Oxford: Blackwell. Dewey, John 1916 Democracy and Education: An Introduction to the Philosophy of Education, New York: Macmillan. Donald, Merlin 1991 Origins of the Modern Mind: Three Stages in the Evolution of Culture and Cognition, Cambridge, MA: Harvard University Press. Draaisma, Douwe 2004 Why Life Speeds Up As You Get Older: How Memory Shapes Our Past, Cambridge: Cambridge University Press. Durkheim, Emile 1912 Les formes élémentaires de la vie religieuse, Paris: Alcan. Eisenstein, Elizabeth 1979 The Printing Press as an Agent of Change: Communications and Cultural Transformations in Early-Modern Europe, Cambridge: Cambridge University Press. Galison, Peter 2003 Einstein’s Clocks, Poincaré’s Maps: Empires of Time, London: Hodder and Stoughton. Goody, Jack and Watt, Ian 1963 “The Consequences of Literacy”, Comparative Studies in Society and History 5, 304–345.

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Grundy, W. P. 2005 “Wittgenstein, Spengler and Time”, in: Friedrich Stadler and Michael Stöltzner (eds.), Time and History: Papers of the 28th International Wittgenstein Symposium, Kirchberg am Wechsel: ALWS, 96–98. Heidegger, Martin 1927 Sein und Zeit, Halle an der Saale: Niemeyer. Havelock, Eric 1963 Preface to Plato, Cambridge, MA: Harvard University Press. James, William 1890 The Principles of Psychology, New York: Henry Holt. Kaspar, Rudolf E. and Schmidt, Alfred 1992 “Wittgenstein über Zeit”, Zeitschrift für philosophische Forschung 46, 569–83. Landes, David S. 22000 Revolution in Time: Clocks and the Making of the Modern World, Cambridge, MA: Belknap Press. Ling, R. 2004 The Mobile Connection: The Cell Phone’s Impact on Society, Amsterdam: Elsevier. Ling, R. and Yttri, Birgitte 2002 “Hyper-Coordination via Mobile Phones in Norway”, in: James E. Katz and Mark Aakhus (eds.), Perpetual Contact: Mobile Communication, Private Talk, Public Performance, Cambridge: Cambridge University Press, 139–169. McLuhan, Marshall 1964 Understanding Media: The Extensions of Man, New York: McGraw-Hill. Miller, Arthur I. 1984 Imagery in Scientific Thought: Creating 20th-Century Physics, Boston: Birkhäuser. Mumford, Lewis 21963 Technics and Civilization, New York: Harcourt Brace & Company. Nyíri, Kristóf [ J. C.] 1991 “Historisches Bewußtsein im Informationszeitalter”, in: Dieter Mersch and Kristóf Nyíri (eds.), Computer, Kultur, Geschichte: Beiträge zur Philosophie des Informationszeitalters, Wien: Edition Passagen, 65–80. — 1992 Tradition and Individuality: Essays, Dordrecht: Kluwer. — 1999 “From Palágyi to Wittgenstein: Austro-Hungarian Philosophies of Language and Communication, in: Kristóf Nyíri and Peter Fleissner (eds.), Philosophy of Culture and the Politics of Electronic Networking, vol. 1, Innsbruck–Wien: Studien Verlag / Budapest: Áron Kiadó, 1–11. — 2001 “The Picture Theory of Reason”, in: Berit Brogaard and Barry Smith (eds.), Rationality and Irrationality, Wien: öbv-hpt, 242–266. Ofield, Jack 1994 Television”, in: Samuel L. Macey (ed.), Encyclopedia of Time, New York: Garland, 592–594. Ong, Walter J. 1982 Orality and Literacy: The Technologizing of the Word, London: Methuen.

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Plumb, J. H. 1969 The Death of the Past, London: Macmillan. Reichenberger, Andrea A. 2005 “‘Was ist Zeit?’ Wittgensteins Kritik an Augustinus kritisch betrachtet”, in: Friedrich Stadler and Michael Stöltzner (eds.), Time and History: Papers of the 28th International Wittgenstein Symposium, Kirchberg am Wechsel: ALWS, 253–255. Sellars, Wilfrid 1962 “Time and the World Order”, in: Herbert Feigl and Grover Maxwell (eds.), Minnesota Studies in the Philosophy of Science, vol. III, Minneapolis: University of Minnesota Press, 527–616. — 1963 “Empiricism and the Philosophy of Mind”, in: Wilfrid Sellars, Science, Perception and Reality, London: Routledge & Kegan Paul, 127–196. — 1975 “Autobiographical Reflections”, in: Hector-Neri Castañeda (ed.), Action, Knowledge and Reality: Critical Studies in Honor of Wilfrid Sellars, Indianapolis: Bobbs-Merrill, 277–93. Simmel, Georg 1997 “The Metropolis and Mental Life”, in: David Frisby and Mike Featherstone (eds.), Simmel on Culture: Selected Writings, London: SAGE, 174–185. Sorabji, Richard 1983 Time, Creation and the Continuum: Theories in Antiquity and the Early Middle Ages, Ithaca, NY: Cornell University Press. Sorokin, Pitirim A. and Merton, Robert K. 1937 “Social Time: A Methodological and Functional Analysis”, American Journal of Sociology 42, 615–29. Steininger, Christian 2005 “Time and Media Culture: Findings of Media and Communication Science”, in: Friedrich Stadler and Michael Stöltzner (eds.), Time and History: Papers of the 28th International Wittgenstein Symposium, Kirchberg am Wechsel: ALWS, 290–292. Townsend, A. M. 2000 “Life in the Real-Time City: Mobile Telephones and Urban Metabolism”, Journal of Urban Technology 7, 85–104. Whitrow, G. J. 1961 The Natural Philosophy of Time, London: Thomas Nelson. Wittgenstein, Ludwig 1953 Philosophical Investigations, Oxford: Basil Blackwell. — 1958 The Blue and Brown Books, Oxford: Basil Blackwell. — 1975 Philosophical Remarks, transl. by Raymond Hargreaves and Roger White, Chicago: The University of Chicago Press. — 1978 Remarks on the Foundations of Mathematics, transl. by G. E. M. Anscombe, Oxford: Basil Blackwell. — 1979 Wittgenstein’s Lectures: Cambridge, 1932–1935, ed. by Alice Am-

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brose, Oxford: Basil Blackwell. Wittgenstein, Ludwig 2000 Nachlass: The Bergen Electronic Edition, Oxford: Oxford University Press. — 2001 Philosophische Untersuchungen: Kritisch-genetische Edition, ed. by Joachim Schulte et al., Frankfurt am Main: Suhrkamp. — 2005 The Big Typescript: TS 213, ed. and transl. by C. Grant Luckhardt and Maximilian A. E. Aue, Malden, MA: Blackwell Publishing.

Perspektiven der Subjektivität: Das Verhältnis von Systemzeit und Eigenzeit in den perfektischen Tempusformen Richard Schrodt, Wien 1. Die Diskussion zu sprachwissenschaftlichen Themen über Tempus und Zeit ist angesichts der kaum zu bewältigenden Forschungsliteratur ein Wagnis. Die von Klein (1994, S. xii) gegebene Charakteristik der Forschungsliteratur „It is simply impossible to read, let alone to discuss, all possibly relevant literature. This has often been said before (even this has been said before).“ wird jede Tempustheorie überdauern. Angesichts dieser Problematik versuche ich, mich auf Grundsätzliches zu beschränken. Ich werde also nicht auf die zahlreichen verschiedenen Formalisierungen eingehen können, und auch bei den verschiedenen Beschreibungsmethoden beschränke ich mich auf das unerlässlich Notwendige. Mir geht es hier um ein Thema, das eher außerhalb der formallinguistischen Forschung steht und daher auch angesichts der intensiven Forschung nicht deutlich wird, nämlich um die Formen und Funktionen der Eigenzeitlichkeit in literarischen Texten. Ich werde mich dazu auf philologisch nachvollziehbare Argumente stützen und tempustheoretische Überlegungen so weit wie möglich auf ein Minimum beschränken.

2. Die Abbildung der zeitlichen Verhältnisse in den Sprachen scheint auf den ersten Blick unproblematisch zu sein: Der Strom der Zeit fließt von der Vergangenheit über die Gegenwart zur Zukunft, demnach müssten nur drei Tempora unterschieden werden. Da das zukünftige Geschehen meist einen modalen Wert hat (es ist erwartet, erwünscht, erhoff t oder befürchtet), kann es oft als Modusform angesehen werden. Ein universell gültiges Tempusschema müsste also mit zwei Tempora, Gegenwart und Vergangenheit, auskommen. Tatsächlich kann auch für manche europäischen Sprachen ein Zwei-Tempora-System angenommen werden, unter bestimmten methodischen Voraussetzungen auch für das Deutsche. Tatsache ist aber auch, dass für andere Sprachen eine größere Zahl an Tempuskategorien F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 317–335.

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angenommen wird. Es kommt nun darauf an, mit welchen Methoden eine solche Zahl ermittelt wird. Versucht man, Gebrauchskategorien zu erfassen, kommt man zu einer beinahe beliebig großen Zahl von Tempora, übrigens auch für das Deutsche. Führt man hingegen feste Kriterien ein wie etwa die Grammatikalisierung, so lässt sich diese Zahl deutlich reduzieren. Dabei muss natürlich erst bestimmt werden, was unter „Grammatikalisierung“ verstanden werden soll. Im Prinzip erlaubt der Zeitpfeil eine beliebige Unterteilung in kleinere Abschnitte, etwa „unmittelbare Gegenwart“ – „gerade vergangenes Geschehen“ – „unmittelbar bevorstehendes Geschehen“ und innerhalb der Vergangenheit etwa „länger – kürzer vergangenes Geschehen“. Diese Unterteilungen können einzelsprachlich auch durch feste Ausdruckskategorien bezeichnet sein. Eine (beinahe) beliebig feine Unterscheidung kann auch dadurch erreicht werden, dass man sich auf die Folge und das Ineinander von Zeitstufen bezieht (z.B. Harweg 1994). Sprachwissenschaftlich beschreibbare Komplexität entsteht also dadurch, dass auf der konzeptionellen Seite der Sprache verschiedene Schichtungen unterschieden werden können, während auf der Ebene der außersprachlichen Realität oft einfache und klar erkennbare Gegensätze vorhanden sind. Alle diese verschiedenen Beschreibungsmöglichkeiten beziehen sich auf grundlegende Tempuskategorien: Die Unterscheidung zwischen Gegenwärtigem und Vergangenem ist vom Bezeichneten her ebenso vorbestimmt wie etwa die Unterscheidung zwischen der Einzahl und der Mehrzahl (auch hier kann es auf der konzeptionellen Ebene verschiedene Formen der Pluralität geben). Man wird also eine entsprechende Kategorisierung als universalgrammatische Bezeichnungskonstante ansehen können.

3. Ein weithin anerkannter methodischer Ansatzpunkt der Tempuslinguistik ist die Tempuslogik von Reichenbach (1966 [1947]). Derzufolge bezeichnet das Tempus als Basiskategorie des Verbs die zeitliche Beziehung zwischen dem Sprechakt (S) und dem Zeitpunkt oder Zeitintervall des von der Aussage bezeichneten Sachverhalts oder Ereignisses (E). Aus diesen absoluten zeitlichen Relationen ergeben sich die Kategorien Präsens (E gleichzeitig S), Präteritum (E vor S) und Futur (E nach S). Einfachere Tempussysteme wie Nicht-Präteritum können mit E gleichzeitig S oder E nach S bzw. E vor S oder E gleichzeitig S formuliert werden. Das relative Tempus bezieht sich auf einen Referenzpunkt (R), welcher selbst nicht in einem Zeitverhältnis zum deiktischen Zentrum der Rede steht, sondern sich aus dem

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Kontext ergibt. Die Formulierungen entsprechen den absoluten Relationen mit dem Ersatz von S durch R. Das Plusquamperfekt ist demnach durch E vor R vor S wiederzugeben (Comrie 1985, 122ff.). Diese auf Reichenbachs Formulierung der universellen Zeitrelationen zurückgehende Beschreibung ist neuerdings in verschiedene Richtungen modifiziert worden (Überblick für das Deutsche bei Thieroff 1992 und d’Alquen 1997; intervallsemantische Fassung bei Klein 1994; mit Welke 2005 liegt nun eine umfassende und umsichtig argumentierende Darstellung vor). Das gilt v.a. für Reichenbachs Referenzpunkt, der ja mit dem Sprechzeitpunkt zusammenfallen kann und damit bei den einfachen Tempora überflüssig wird. Statt dessen sind auch Begriffe wie „Evaluationszeit“ und „Orientierungszeit“ in Gebrauch, die aber alle mehr oder weniger dasselbe meinen, nämlich einen Zeitpunkt (oder ein Zeitintervall), von dem aus das Ereignis betrachtet und zeitlich eingeordnet wird. Man wird auch damit rechnen müssen, dass es mehrere solcher Referenzzeiten in mehreren Schichtungen geben kann. Im Fall der indirekten Rede ist es weiters sinnvoll, einen dem Sprechzeitpunkt vergleichbaren zweiten Zeitpunkt anzunehmen, der nicht mit dem faktischen Äußerungszeitpunkt übereinstimmt, sondern ein zweites deiktisches Zentrum für die eingebettete Aussage bezeichnet. Dafür hat Thieroff (1992) den Terminus „Orientierungszeit“ vorgeschlagen. Da aber der Sprechzeitpunkt im Normalfall eben das erste deiktische Zentrum einer Aussage ist, könnte man auch hier von einer Orientierungszeit sprechen; dann gäbe es eben zwei verschiedene Orientierungszeiten. Ich werde hier die weitere Entwicklung der von Reichenbach ausgehenden Terminologie nicht verfolgen – es besteht sonst die Gefahr, sich im Gestrüpp verschiedener eben nur teilweise übereinstimmender Analysemethoden heillos zu verfangen. Diese Gefahr wird noch dadurch verstärkt, dass unter manchen Termini wie „Betrachtzeit“ ganz Verschiedenes, geradezu Gegensätzliches, verstanden werden kann. In meinem Beitrag werde ich mich einer an Reichenbach orientierten Terminologie bedienen, was mir schon dadurch gerechtfertigt erscheint, dass die Probleme im Reichenbach’schen Originalsystem (z.B. die Formalisierungen der futurischen Tempora) für mein Thema nicht bedeutend sind. Mir geht es hier nur um den Unterschied zwischen Perfekt und Präteritum, und für die Formulierung dieses Unterschied genügt es, für das Perfekt eine eigene Referenzzeit anzunehmen, die im Bereich des Sprechzeitpunkts liegt. Das Präteritum hingegen hat entweder keine Referenzzeit oder, wenn man aus methodischen Gründen von einer universalgrammatischen Referenzzeit ausgeht, diese Referenzzeit liegt im Bereich der Ereigniszeit.

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4. Mit dieser Formalisierung wird eine Funktion des Perfekts wiedergegeben, die traditionell etwa so formuliert wird, dass das perfektische Geschehen eine Auswirkung oder Nachwirkung in der (Sprecher-)Gegenwart habe. Es handelt sich um den typischen Fall des Resultatsperfekts: Ein Satz wie Ich habe das Buch gelesen soll bedeuten, dass das Buch als Gelesenes gesehen wird und dass mir sein Inhalt in irgendeiner Form präsent ist, während ein Satz wie Ich las das Buch auf das bloß vergangene Geschehen verweist und somit für die Nachwirkung dieser Handlung nichts aussagt. Solche Formulierungen sind aber vage und beziehen sich auf psychische Einstellungen und Vorgänge, die formalgrammatisch schwer nachvollziehbar sind. Die Problematik solcher Formulierungen zeigt sich besonders deutlich in der Geschichte der grammatischen Beschreibungen. Die derzeit neueste Auflage der Duden-Grammatik (2005, 514) drückt sich hier sehr vorsichtig aus: „Wenn nichts dagegen spricht, darf denn auch davon ausgegangen werden, dass das Geschehen aufgrund seiner Folgen oder der an ihm beteiligten Aktanten im Sprechzeitpunkt (noch) von Belang ist.“ An einer anderen Stelle (519) ist davon die Rede, dass „der Kontext das Geschehen oder dessen Folgen als relevant im Sprechzeitpunkt ausweist.“ Diese Formulierungen zeigen sehr gut die Problematik dieser Tempuskategorie im Deutschen. Folgende Konstellationen sollen für den Perfektgebrauch relevant sein: 1. die Eigenschaften des Geschehens selbst, 2. die Rolle der beteiligten Akteure, 3. der „Kontext“, also die sprachliche (und vermutlich auch außersprachliche) Umgebung des betreffenden Satzes, und schließlich 4. eine nicht näher bestimmte „Relevanz“, von der man nur vermuten kann, dass sie ein Merkmal der Aussageabsicht und damit der Sprecherintention ist. Es werden also nebeneinander Merkmale aus dem Bereich des Signifikats, der Textumgebung und der Sprecherhaltung berücksichtigt, wobei im Bereich des Signifikats nochmals zwischen Geschehensmerkmalen an sich und den beteiligten Personen (Akteuren) unterschieden wird. Besonders vage ist der Bezug auf den Kontext, denn hier kann ein ganzes Bündel von unterschiedlichen Merkmalen bis hin zu verschiedenen Textsorten gemeint sein. Die Problematik der Funktionsbeschreibung des deutschen Perfekts zeigt sich auch deutlich in der ersten Auflage der Duden-Grammatik (1959, 109), die methodisch der inhaltbezogenen Sprachwissenschaft nahe steht:

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Diese Formen bezeichnen meist Vergangenes, das sich noch irgendwie auf die Gegenwart des Sprechenden bezieht, ein Geschehen, das sich erst in der Gegenwart vollendet hat, Geschehnisse der jüngsten Vergangenheit. Der Sprecher wendet den Blick von der Gegenwart aus in die Vergangenheit zurück: „Es hat geschneit!“ ruft ein Junge, der früh am Fenster den in der Nacht gefallenen Schnee erblickt. Ebenso: Ich bin eben aus der Stadt gekommen. Jetzt habe ich den Schlüssel gefunden. Inge ist heute geprüft worden. Gestern ist ein Kind ertrunken.

Das gleiche Verhältnis ist auch dann gegeben, wenn sich der Sprecher so in die Vergangenheit zurückversetzt, daß ihm ein historisches Ereignis als gerade vollendet erscheint […]. Dies ist fast immer der Fall, wenn ein vergangenes Ereignis für sich betrachtet werden soll […]: Kolumbus hat Amerika entdeckt. Die Burg ist im 15. Jahrhundert erbaut worden. Dies gilt auch, wenn der Sprecher Allgemeingültiges im Perfekt ausdrückt […]: Ein Unglück ist bald geschehen.

Gegenüber der Neufassung treten zwei Momente stärker hervor: 1. die zeitliche Nähe zur Gegenwart und 2. eine besondere Sprecherhaltung, ein Rückblick oder eine Rückversetzung in die Vergangenheit, also Merkmale, die in der Neufassung zu den Punkten 1 und 4 gehören, wobei besonders das zweite Merkmal das recht vage Kriterium der „Relevanz“ mit etwas konkreteren Inhalten erfüllt. Immerhin wagt die Duden-Grammatik von 1959 eine genauere Beschreibung der psychischen Bedingungen für die Tempussetzung, ein deutlicher Vorteil der inhaltbezogenen Ansätze. Nimmt man alle diese Merkmale zusammen, so zeigt sich aber, dass die Merkmale auf der Ebene des temporalen Signifikats so undeutlich formuliert sind, dass sie wenig zur Erhellung des perfektischen Referenzbereiches beitragen. Etwas besser steht es mit den Merkmalen, die sich auf die Sprecherintention beziehen. Aber auch hier ist die alte Duden-Grammatik genauer, denn mit „Rückblick“ und „Rückversetzung“ sind zwei Sprecherhaltungen genannt, die im Grund psychologisch nachvollziehbare und damit klarer beschreibbare Kategorien sind. Dafür drängt sich geradezu der Terminus „Aspekt“ auf, doch davon soll noch später die Rede sein.

5. Wenn man beim Überbegriff „Relevanz“ bleibt, so ist damit gemeint, dass das Perfekt eigentlich eine modale Kategorie ist, gewissermaßen ein

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verstärkter Indikativ, mit dem die Aktualität des Geschehens deutlich hervorgehoben wird. Kann das deutsche Perfekt tatsächlich eine Moduskategorie sein? So sieht es jedenfalls Engel (1992). Zunächst zur morphologischen Ausgrenzung. Herkömmlich werden 4–6 Tempora unterschieden. Tempora sind Verbformen und haben mit der Zeit zu tun. Es gibt aber unterschiedliche Ausdrucksformen und Bedeutungsschwankungen. Geht man von sechs Tempora aus, so erscheinen sie in zwei Verbformen (Präsens, Präteritum) und vier verbalen Komplexen, davon drei zweigliedrig (Perfekt, Plusquamperfekt, Futur) und einer dreigliedrig (Futur II). Das sind aber natürlich verschiedenartige Konstruktionen. In den klassischen Sprachen (Latein, Altgriechisch) sind alle Tempora (reine) Verbformen. Es liegt daher nahe, hier eine Übernahme der Struktur der klassischen Sprachen auf das Deutsche zu vermuten. Zur semantischen Ausgrenzung: Herkömmlich wird angenommen, dass Informationen über Zeitverhältnisse bezeichnet werden. Aber bekanntlich ist das Präsens zeitindifferent, so wird es jedenfalls in vielen Grammatiken dargestellt. Das Futur hat oft (meist) „modale“ Bedeutung (Er wird krank sein), das Futur II ebenso (Sie wird es getan haben). Daher ist nach Engel der Zeitwert fraglich. Zeitliche Informationen werden auch durch andere Konstruktionen bezeichnet (ist am Bauen, will bauen, soll bauen, hofft zu bauen usw.). Daher liegt die Schlussfolgerung nahe: Zeitlichkeit ist mehr als die Tempora, und einige Tempora sind nicht oder nur eingeschränkt zeitlich zu verstehen. Der weitere Schluss: Tempora lassen sich nicht semantisch definieren. Schließlich kann man fragen, ob es überhaupt verbale Ausdrücke gibt, die sich semantisch ausgrenzen lassen. Nach Engel sind es die finiten Verben. Sie sagen etwas über die Realität des entsprechenden Sachverhalts aus, im Gegensatz zu den infiniten Verbformen. Dazu kommen noch zeitliche Bedeutungen bei bestimmten finiten Verbformen: Präteritum – Vergangenheit, Imperativ – Zukunft. Im Präsens wird ein Sachverhalt einem bestimmten Zeitraum oder Zeitpunkt zugeordnet, doch bleibt dieser Raum/Punkt grundsätzlich offen. Eine weitere semantische Kategorie zeigt sich am folgenden Beispiel: Alexander zerschnitt/zerschneidet den gordischen Knoten. Sind diese Sätze bedeutungsgleich? Tatsächlich bezeichnen beide Sätze den Sachverhalt als real. Normalerweise wird der erste Satz als ein Beispiel für das „historische Präsens“ angeführt: Ein vergangener Sachverhalt wird in die Gegenwart hereingeholt, wird gegenwärtig gemacht. Das ist aber nur möglich, wenn man das Präsens als „Gegenwart“ interpretiert; doch das wird bekanntlich bestritten. Hingegen kann man beim Satz im Präsens behaupten, dass der beschrie-

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bene Sachverhalt dem Sprecher näher steht als beim Satz im Präteritum. Das Präsens signalisiert also, dass ein Sachverhalt die Gesprächsbeteiligten unmittelbar angeht, für sie von Belang ist, und gehört damit einer modalen Dimension an. Dazu kommt noch das Merkmal des Gesprächspartnerbezugs, das ich aus Engels großer Grammatik (Engel 2004) ergänzt habe. In einer Tabelle zusammengefasst: Verbform

Realität

Zeit

Präsens

real

bestimmt, aber offen

+

belangreich

Präteritum

real

vergangen

0

nicht belangreich

real abgeschlossen

bestimmt, aber zeitlich nicht festgelegt

+

belangreich

Perfekt

Modali- Gesprächspartnertät bezug

6. An dieser auf den ersten Blick durchaus ansprechenden neuen Sicht des deutschen Verbalsystems fällt zunächst auf, dass Engel im Gegensatz zur grammatischen Tradition eine andere Auffassung von Modalität hat: Der Sachverhalt ist für die Gesprächspartner belangvoll. Die traditionelle Auffassung hingegen beruht letzten Endes auf dem Zusammenhang mit dem Status der Aussage als logisches Urteil (ein Erbe der logizistischen Grammatik). Diese Auffassung erlaubt eine klare inhaltliche Charakterisierung: Wenn ein Urteil ausgesprochen wird, dann steht der Indikativ; wenn kein Urteil (sondern Wunsch, Wille, Befehl, Möglichkeit, Unsicherheit usw.) ausgesprochen wird, dann kann der entsprechende Satz ein grammatisches Zeichen für die Modalität enthalten (Konjunktiv, Imperativ) oder diese Modalität ist lexikalisch bezeichnet (Modalpartikeln, Satzadverbien). D.h. in der traditionellen Sicht ist der verbale Modus eine grammatische Form, welche die durch den Sprecher angezeigte Geltung der Aussage bezeichnet. Für diesen Inhalt führt Engel aber eine eigene Kategorie der Realität ein, die vom Status der Verbmorphologie abhängt (finites/infinites Verb). Die traditionelle Bindung des Modus an die Urteilsqualität eines Satzes wird damit aufgegeben. Die verbalen Kategorien werden enger an die Verbgestalt gebunden; die satzzentrierte Fassung der semantischen Verbkategorien wird durch die verbzentrierte Fassung dieser Kategorien ersetzt.

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7. Das gibt an sich einen guten Sinn: Eine Grammatik als endliches Inventar von sprachlichen Zeichen verlangt auch eine Formgebundenheit der Beschreibung. Insofern ist es durchaus verständlich, wenn die zusammengesetzten Verbformen nicht zum Kernbereich der Grammatik gestellt werden. Weiters wird das Verb als strukturelles Zentrum des Satzes konsequent hervorgehoben, und das ist zweifellos eine sinnvolle Methode der grammatischen Beschreibung des Deutschen. Der spezifische Ansatz dieser Kategorisierung zeigt sich besonders deutlich am viel diskutierten Verhältnis von Perfekt und Präteritum. Nach Engel gibt es keine Übergänge zwischen diesen beiden „Tempora“ (auch im Original in Anführungszeichen) und daher auch weder Verwechslungsmöglichkeiten noch Zweifel über ihre Anwendung: „[E]in präteritaler Vorgang ist vergangen und ohne Belang für die Gesprächsbeteiligten, ein perfektischer Vorgang hingegen ist abgeschlossen und für die Gesprächsbeteiligten belangvoll.“ (Engel 1992, 62) Wenn auch kompetente Deutschsprecher „sich fragen, ob ‚abgeschlossen‘ und ‚vergangen‘ nicht zwei Namen für dieselbe Sache seien“, dann handelt es sich möglicherweise um eine Neutralisierung dieser Tempusopposition in bestimmten Nebensätzen wie Weißt du noch, wann sie hier eintraf/eingetroffen ist? Zum berühmten Schluss aus Goethes „Werther“ Handwerker trugen ihn. Kein Geistlicher hat ihn begleitet. erklärt Engel: „Demnach vermittelt der erste Satz, aus der Distanz gesehen, eine Hintergrundinformation. Der zweite aber bringt das Wesentliche, das den Leser anspringt, ihm keine Gelegenheit läßt, sich aus der Situation zu stehlen, sich unbeteiligt zurückzulehnen. Dies geht ihn unmittelbar an. Hier ist nichts austauschbar. […] Und im Grunde sind, sorgsam verwendet, Perfekt und Präteritum überhaupt nie austauschbar.“ (Engel 1992, 63) Man mag von Engels Tempustheorie halten, was man will: Aber nur sie erlaubt es, futurische Gebrauchsarten des Perfekts wie In einer halben Stunde habe ich den Brief geschrieben in ein argumentierbares Beschreibungssystem einzugliedern. Andernfalls müsste man die Orientierungszeit für diese Gebrauchsart nach die Sprecherzeit verschieben, was zwar formal durch die Koppelung der Orientierungszeit mit temporaldeiktischen Ausdrücken möglich ist, aber die Generalisierbarkeit der Tempusbeschreibung deutlich einschränkt.

8. Engels Tempustheorie hat sich nicht durchgesetzt. Sie ist aber für unser Thema von entscheidender Bedeutung, zeigt sie doch sehr deutlich die Ursachen für die Problematik der wissenschaftlichen Beschreibung des

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deutschen Tempus. In allen Modellen, die sich an Reichenbach orientieren und die ich hier mit der Bezeichnung „deiktisch“ benenne, ist das deutsche Perfekt (und universalgrammatisch jedes Perfekt) ein Vergangenheitstempus, denn das bezeichnete Ereignis liegt nun einmal vor der Referenzzeit bzw. vor der Orientierungszeit. Im „Abweichungsmodell“ von Engel (und in ähnlichen, nicht-deiktischen Modellen wie etwa bei Weinrich) ist das deutsche Perfekt ein „Gegenwarts“tempus, wobei über ein universalgrammatisches Fundament nichts gesagt wird. Es scheint so, als würden sich theoretisch die beiden grammatischen Komponenten des Perfekts, das Präsens des Hilfsverbs und das Präteritum des Partizips, verselbstständigen lassen. An diesem Punkt ist es nun unvermeidlich, den Terminus „Aspekt“ ins Spiel zu bringen. Aspektpaare, welcher Art auch immer, werden normalerweise mit den Termini „perfektiv – imperfektiv“ belegt. Ich gebrauche hier eine andere Terminologie: „Durativ“ ist der Aspekt, unter dem eine Handlung in ihrer Dauer gesehen wird; Anfang und Ende der Verbalhandlung kommen nicht in das Blickfeld. Demgegenüber ist „komplexiv“ der Aspekt, unter dem eine Verbalhandlung als unteilbare Ganzheit gesehen wird. Im Neuhochdeutschen (Nhd.) ist der Aspektunterschied nicht grammatikalisiert; er kann durch Fügungen wie etwa einen Pullover stricken (kursiv) und einen Pullover fertig stricken (komplexiv) verdeutlicht werden (Wortpaare wie etwas das bekannte jagen – erjagen geben hingegen einen Aktionsartunterschied wieder).

9. Aspektunterschiede können durch eine Vielzahl von grammatischen Formen kodiert sein. Im Nhd. stehen auch Formunterschiede wie an einem Brief schreiben / einen Brief schreiben und den Brief schreiben zur Verfügung: Im ersten Fall ist eine andauernde Handlung bezeichnet, die zwar Anfang und Ende hat, aber ohne dass diese Begrenzung kommunikativ wichtig wäre. Deshalb kann sich diese Phrase mit durativen Zeitadverbien wie stundenlang verbinden. Im letzten Fall ist das Ergebnis der Handlung, der geschriebene Brief, wichtig. Zeitadverbien wie heute beziehen sich auf die Gesamtheit der Verbalhandlung. In sie schreibt stundenlang den Brief (besser: sie schreibt den Brief stundenlang) überlagert das durative Zeitadverb den komplexiven Verbalhandlungskern mit dem Ergebnis einer kursiven Verbalphrase (daher steht auch das Zeitadverb in Ausdrucksstellung am rechten Satzrand). Objektiv liegt derselbe Sachverhalt vor, nur die Sichtweise ist eine Andere. Insofern kann man tatsächlich davon ausgehen, dass der As-

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pekt eine subjektive grammatische Kategorie ist. Es gibt aber grammatische Erscheinungen, die bestimmte Aspektformen begünstigen, sodass auf diese Weise eine Grammatikalisierung angebahnt sein kann. In nhd. Kausalsätzen z.B. ist in der Standardsprache das (aspektuelle) Perfekt üblicher als das Präteritum, weil die Verbalhandlung nicht in ihrem Verlauf und in ihrer Ausdehnung, sondern nur als punktuell gesehenes kausales Moment wichtig ist. Ein Kausalsatz im Präteritum ist aber keineswegs falsch: Gelegentlich ist gerade der Handlungsverlauf das kausal wichtige Moment, etwa in Sie war damals sicher zu Haus, weil in ihrer Wohnung die ganze Zeit das Licht brannte. Es können also auch in der Sprache Aspekte, besser gesagt: aspektuelle Formen, vorhanden sein, wenn dafür kein geschlossenes und systematisches Bezeichnungsinventar besteht. Die wissenschaftliche Kontroverse lässt sich damit auf einen einfachen gemeinsamen Nenner bringen: In deiktischen Tempussystemen ist das Perfekt ein Vergangenheitstempus; man braucht den Aspekt als Beschreibungskategorie nicht. Demzufolge hat das (deutsche) Perfekt keine aspektuelle Funktion. Das ist die gängige Meinung. In nicht-deiktischen Tempussystemen ist das Perfekt eine aspektuelle Kategorie (selbst wenn das manche Forscher wie Weinrich nicht wahrhaben wollen); man braucht das Tempus als Beschreibungskategorie nicht. Natürlich können Aspekt und Tempus kombiniert werden, weil es sich um prinzipiell voneinander unabhängige Kategorien handelt. Das deutsche Perfekt mag ein Tempus sein: Sobald man aber Ausdrücke wie subjektive Perspektive, Sicht von Außen, Totalität, Definitheit, Rückschau/Vorschau usw. gebraucht, kommt zum Tempus eine aspektuelle Qualität hinzu. Das Perfekt ist somit eine Art „subjektive Zeit“, oder genauer: Es kann zur Bezeichnung einer „subjektiven Zeit“ verwendet werden. Diese Ansicht wird auch in der neueren Forschung vertreten (z.B. Marschall und Valentin in Quintin/Najar/Genz 1997); diese Ansicht wird auch neuerdings von Welke (2005, 250) bestätigt: „Es geht um subjektive Eindrücke, um Interpretationen und Auslegungen […]“. Die Subjektivität des Perfekts, die ja schließlich auch ein textuelles Merkmal sein kann (dazu Confais 1995, Kap. 3), führt zu meinem eigentlichen Thema, der Eigenzeitlichkeit in der Sprache.

10. Ich führe den Begriff der Eigenzeitlichkeit mit einem Zitat aus dem Buch von Jakob von Uexküll und Georg Kriszat, Streifzüge durch die Umwelten von Tieren und Menschen, 1956, ein (30): „Die Zeit, die alles Geschehen umrahmt, scheint uns das allein objektiv Feststehende zu sein

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gegenüber dem bunten Wechsel ihres Inhaltes, und nun sehen wir, dass das Subjekt die Zeit seiner Umwelt beherrscht. Während wir bisher sagten: Ohne Zeit kann es kein lebendes Subjekt geben, werden wir jetzt sagen müssen: Ohne ein lebendes Subjekt kann es keine Zeit geben.“ Uexküll erläutert diese Eigenzeitlichkeit am Beispiel der Wahrnehmungsreize: Ein Moment als kleinster Zeitraum, innerhalb dessen alle Reize unabhängig von ihrer objektiven Zeitfolge als gleichzeitig empfunden werden, dauert beim Menschen eine 18tel-Sekunde, bei der Weinbergschnecke eine Viertelsekunde, bei der Pilgermuschel 13 Sekunden und bei der Zecke, die bis zu 18 Jahre ohne Nahrung in einem schlafähnlichen Zustand auf ihr Opfer warten kann, noch viel länger. Diese subjektive Zeit hängt bei den Lebewesen vom Rhythmus des Zentralnervensystems ab. Innerhalb der Impulse des Nervensystems steht die Zeit für das Lebewesen gewissermaßen still. Es ist klar, dass eine solche Vorstellung nur metaphorisch auf die sprachliche Gestaltung von Geschehnissen übernommen werden kann. Es gibt aber, so behaupte ich, grundsätzliche Übereinstimmungen von biologisch beschreibbaren Wahrnehmungsakten und sprachlichen Gestaltungsmöglichkeiten, die es erlauben, recht genau an den Begriff der Eigenzeitlichkeit bei Uexküll anzuknüpfen – abgesehen von den verschiedenen weiteren Verwendungsweisen dieses Begriffes, z.B. im Bereich der Sozialwissenschaften (vgl. etwa Nowotny 1993) oder in der Chemie und der Physik. Auch in der sprachlichen Gestaltung gibt es also so etwas wie einen erzählpraktischen Stillstand der Zeit, einen Zeitpunkt, an dem das erzählte Geschehen zu einer Art „Nullpunkt“ gerinnt und aus diesem Nullpunkt heraus eine neue Gestalt und damit auch eine neue Wirkungsweise gewinnt. Man muss nur Uexkülls „lebendes Subjekt“ durch das „sprechende/erzählende Subjekt“ ersetzen; allerdings stehen diesem „redenden Subjekt“, wie ich es hinfort zusammenfassend nennen will, grundsätzlich immer zwei Wahrnehmungsweisen = Erzählweisen zur Verfügung: die kontinuierliche, von Zeitpunkt zu Zeitpunkt (von Ereignis zu Ereignis) fortschreitende Erzählung und die momenthafte, das Geschehnis auf einen Punkt bringende „Totalfassung“. Diese „Totalfassung“ entspricht damit geradezu einem erzählpraktischen Wahrnehmungsakt: Hier sind nicht die einzelnen Ereignisse und ihre Folge wichtig, sondern das, was von diesen einzelnen Ereignissen als Eindruck und Wirkungsmoment auf das sprechende/erzählende Subjekt bleibt und was er/sie dem/der Leser/in als besonders hervorgehobene Aussage mitteilen will. Man kann das ein „resümierendes“ Perfekt nennen (vgl. dazu Vater 1997). Dem Rhythmus des Zentralnervensystems entspricht im Erzählakt

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der Rhythmus der Aussagegewichtung: Einzelne Begebenheiten verdichten sich zu einem Moment, das in seiner Bedeutung über diese Begebenheiten hinausreicht. Charakteristisch für die Sprache ist freilich, dass dieser Rhythmus rekursiv ist, d.h. er stellt sich auf verschiedenen Stufen der Komplexität in der gleichen Weise, aber in verschiedenen konkreten Formen, ein. Eines dieser Zeichen für den erzählpraktischen Nullpunkt ist das Perfekt.

11. Man kann diese Erzählpraxis nur an ganzen Texten sinnvoll erläutern, und das macht eine Demonstration schwierig. Ich werde mich daher hier auf zwei kurze Texte beschränken, auf Kürzestgeschichten, wie man sie auch genannt hat. Der erste Text stammt von Walter Jens und hat den Titel „Bericht über Hattington“. In dieser Geschichte in Form eines Erlebnisberichts wird erzählt, wie der Ausbruch eines Verbrechers mit dem Namen Hattington das Leben in einer Kleinstadt verändert. Das Grundtempus ist das Präteritum. Die Geschichte beginnt mit dem folgenden Absatz: Der Winter kam in diesem Jahr sehr früh; schon Mitte November hatten wir 15 Grad Kälte, und in der ersten Dezemberwoche schneite es sechs Tage lang hintereinander; am fünften, einem Mittwoch, brach Hattington aus. Er hatte offenbar damit gerechnet, daß der Schnee seine Spuren verschluckte – und diese Rechnung ging auf. Die Hunde verloren die Witterung, und die Gendarmen kehrten noch im Laufe der Nacht nach Colville zurück. Am Morgen darauf wurde unser Polizeiposten verstärkt, und Sergeant Smith bekam zwei neue Kollegen: man vermutete nämlich, daß Hattington versuchen würde, auf dem schnellsten Wege zu uns nach Knox zu gelangen; denn hier hatte man ihn, einen seit langem gesuchten Verbrecher, im Mai auf offener Straße verhaftet – wahrscheinlich auf eine Anzeige hin, die von der Kellnerin Hope und dem Tankstellenwart Madison kam, bei denen Hattington in Kreide stand. Die Annahme lag also nahe, daß der Zuchthäusler, um Rache zu nehmen, zuerst nach Knox kommen würde. Das Plusquamperfekt ist hier und in den folgenden Stellen der Ausdruck der Vorzeitigkeit, also ein deiktisches Tempus, das ein Ereignis bezeichnet, welches vor dem Geschehen der Haupthandlung liegt. Im nächsten Absatz werden die vergebliche Suche nach dem Verbrecher und das aufkommende allgemeine Misstrauen beschrieben, jemand hätte Hattington

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versteckt. Doch als bis über Weihnachten nichts geschah, kam das Leben in der Kleinstadt wieder in seine gewohnten Bahnen. Im folgenden Absatz wird berichtet, wie ein Mord und eine Misshandlung eines jungen Mädchens, für die Hattington verantwortlich gemacht wurde, das Leben in der Stadt aus dem Gleis warfen. Die Verdächtigungen kamen verstärkt wieder auf, es kam zu einer Hexenjagd und zu Randalen: Bald gab es kein Geheimnis mehr, das, von Schnüfflern entdeckt, nicht ans Tageslicht kam: Ehemänner, die einmal gefehlt hatten, sahen sich wie Verbrecher behandelt, harmlose Trinker wurden des Mordes verdächtigt; der Frauenverein ließ vor den Kino-Vorstellungen Zettel verteilen, auf denen sich die Bürger ermahnt sahen, den Umgang mit gewissen Leuten, wenn ihnen das Leben lieb sei, zu meiden. Auf der anderen Seite mehrten sich gerade in diesen Tagen unter den jungen Leuten Unordnung und Zuchtlosigkeit. Während die Älteren ihre Häuser nach Möglichkeit nur noch zur Arbeit oder zum Kirchgang verließen, versammelten sich die Jüngeren abends im Wirtshaus, tranken und johlten, pöbelten die Erwachsenen an und errichteten am Ende ein solches Schreckensregiment, daß wir ihrer nur mit Hilfe einer Art von Zivilpolizei, der Bürgerwehr, Herr werden konnten. Schließlich blieb kein anderer Ausweg, als die Rädelsführer kurzweg zu verhaften – und dabei kam dann heraus, daß auch die schlimmsten Radaubrüder sich eher aus Furcht, eines Tages Hattingtons Opfer zu werden, denn aus Übermut zusammenrotteten. Dieser Stelle folgt nun ein Satz im Perfekt: Das hat mir wieder einmal gezeigt, wie schnell die allgemeine Raserei im Schatten der Angst und des Schreckens gedeiht. Dieser Satz könnte auch im Präteritum stehen und wäre damit nicht auffällig. Das Perfekt bezeichnet hier aber nicht ein Geschehen, sondern den Eindruck des Geschehens auf den Sprecher/Erzähler. Dieser Eindruck wird noch durch das Prädikatsverb zeigen und die Ausdrücke der subjektiven Modalität mir wieder einmal bestätigt. Auch der folgende Objektsatz, der eine allgemein-menschliche und damit nicht auf die Ereigniszeit bezogene Verhaltensweise weist, verstärkt diesen Funktionsbereich des Perfekts. Man hat den Eindruck, diese Geschichte wurde eben deshalb erzählt, um genau diese Aussage zu belegen. Damit könnte sie auch schon zu Ende sein. Aber es wird (im Präteritum) weiter erzählt: Auch die Eltern standen im Radaumachen ihren Kindern nicht nach. Es folgt wieder ein Satz im Perfekt: Ich selbst habe Nächte erlebt, in denen man mich mehr als ein dutzend-

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mal anrief, um mich mit verstellter Stimme zum Boykott angeblich verdächtiger Bürger zu zwingen. Wieder wäre eine präteritale Fassung (Ich selbst erlebte Nächte, in denen …) nicht auffällig. Doch auch hier weist das Subjektspronomen in der 1. Person auf ein Begebnis, das sich aus der Folge der erzählten Ereignisse durch seine Sprecherbezogenheit heraushebt; nur handelt es sich hier um ein Begebnis, das in die Folge der berichteten Ereignisse eingebettet ist. Dieses Begebnis entzieht sich aber einer strikten Folgebeziehung: Es sind eben mehrere Nächte gemeint, die über die gesamte Ereigniszeit dieses Absatzes verstreut sind. Hier kann die Geschichte nicht zu Ende sein, sie geht weiter: Im darauf folgenden Absatz wird berichtet, dass am 17. März ein zweiter Mord geschah. Es kam zu hysterischen Ausschreitungen gegenüber grundlos Verdächtigten. Der nächste Absatz beginnt wieder mit einem Satz im Perfekt, das im nächsten Satz sofort ins Präteritum wechselt: Im April hat dann sogar Reverend Snyder, einer der letzten besonnenen Männer, kapituliert: von der Kanzel aus befahl er uns, den Mörder und seine Helfershelfer zu jagen. Von den perfektischen Sätzen könnte dieser Satz wohl am unauffälligsten ins Präteritum umgeformt werden. Hier zeigt sich tatsächlich das, was die neue Duden-Grammatik unter Merkmale des Geschehens und der beteiligten Akteure versteht: Das Geschehen erreicht seinen Höhepunkt, die Verfolgungsjagd wird durch einen bisher besonnenen Akteur abgesegnet. Ich verstehe dieses Perfekt daher nicht als Ausdruck einer Eigenzeit, sondern als textuell gebundenes Zeichen. Doch dann wird eine unerwartete Wende berichtet. Die Schneeschmelze begann. Die Sonne brachte alles an den Tag: am Karfreitag fand man Hattingtons Leiche, hundert Meter vom Zuchthaus entfernt. Weiter war er nicht gekommen bei seinem Ausbruchsversuch im Dezember. Der Schnee hatte die Spuren verschluckt, der Eissarg seinen Körper geschützt. Der letzte Absatz lautet: Von diesem Tage an begann es still zu werden, hier bei uns in Knox. Wer es irgend ermöglichen konnte, zog weg. Emily Sawdys und Madisons Mörder aber wurde niemals gefunden, das Vergehen an Helen Fletcher nicht gesühnt. Nur ich habe einen bestimmten Verdacht, doch ich schweige, und sonst weiß niemand, wer der Täter war. Eines aber ist sicher: es gibt nicht viele Leute in unserer Stadt, die frei sind von Schuld.

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Die Erzählung endet mit drei Sätzen im Präsens. Die ersten zwei beziehen sich auf Verhaltensweisen des Erzählers, der letzte bezeichnet eine Tatsache, die sich aus der Geschichte ergibt, und formuliert damit die poetische Botschaft dieses Textes. Die Geschichte kehrt so zum Erzähler zurück.

12. Das Perfekt als ein Moment der Subjektivität ist besonders charakteristisch für Erzählabschnitte, in denen ein Resümee gegeben oder eine allgemeine Folgerung gezogen wird. Solche Erzählabschnitte können Texte gliedern und sie zu einer Folge von verschiedenen Sinneinheiten strukturieren. Perfektformen kommen oft in einer Koda vor: Sie bezeichnet das Ende der Geschichte ausdrücklich oder enthält allgemeine Folgerungen und Beobachtungen, die sich aus der Geschichte ergeben (die „Moral“). Das, was in der Koda steht, tritt aus dem zeitlichen Rahmen der übrigen Erzählteile heraus und ist stärker mit dem Hier und Jetzt des Erzählenden verbunden. Derartige Textabschnitte können auch in manchen Textsorten am Anfang stehen, so z.B. in Agenturmeldungen von Tageszeitungen (Marschall 1995, Schecker/Padros/Jechle 1997). Ein besonders interessantes Beispiel für perfektische Ausdrücke am Anfang eines Textes ist die Erzählung „Seine k. und k. apostolische Majestät“ von Joseph Roth: Es war einmal ein Kaiser. Ein großer Teil meiner Kindheit und meiner Jugend vollzog sich in dem oft unbarmherzigen Glanz seiner Majestät, von der ich heute zu erzählen das Recht habe, weil ich mich damals gegen sie so heftig empörte. Von uns beiden, dem Kaiser und mir, habe ich recht behalten – was noch nicht heißen soll, daß ich recht hatte. Er liegt begraben in der Kapuzinergruft und unter den Ruinen seiner Krone und ich irre lebendig unter ihnen herum. Vor der Majestät seines Todes und seiner Tragik – nicht vor seiner eigenen – schweigt meine politische Überzeugung und nur die Erinnerung ist wach. Kein äußerer Anlaß hat sie geweckt. Vielleicht nur einer jener verborgenen, inneren und privaten, die manchmal einen Schriftsteller reden heißen, ohne daß er sich darum kümmerte, ob ihm jemand zuhört. Dieser Einleitungsteil gehört zum Hier und Jetzt des Erzählers. Der Autor hat heute das Recht vom dem zu erzählen, was ihm damals empörte. Das, was damals geschah, und das, was heute ist, erscheint auf schicksalshafte Weise verwoben, der Tod des Kaisers und das lebendige Herumirren des

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Erzählers, ein halt- und zielloses Herumirren, so als hätte der Tod der Majestät auch dem Erzähler den Boden entzogen. Es bleibt die Erinnerung. Sie drängt sich auf, sie braucht keinen Anlass, sie verschafft sich die Sprache des Dichters. Das, was folgt, ist die Rechtfertigung für das, was geblieben ist. Ich zitiere die nächsten beiden Absätze: Als er begraben wurde, stand ich, einer seiner vielen Soldaten der Wiener Garnison, in der neuen feldgrauen Uniform, in der wir ein paar Wochen später ins Feld gehen sollten, ein Glied in der langen Kette, welche die Straßen säumte. Der Erschütterung, die aus der Erkenntnis kam, daß ein historischer Tag eben verging, begegnete die zwiespältige Trauer über den Untergang eines Vaterlandes, das selbst zur Opposition seine Söhne erzogen hatte. Und während ich es noch verurteilte, begann ich schon, es zu beklagen. Und während ich die Nähe des Todes, dem mich noch der tote Kaiser entgegenschickte, erbittert maß, ergriff mich die Zeremonie, mit der die Majestät (und das war: Österreich-Ungarn) zu Grabe getragen wurde. Die Sinnlosigkeit seiner letzten Jahre erkannte ich klar, aber nicht zu leugnen war, daß eben diese Sinnlosigkeit ein Stück meiner Kindheit bedeutete. Die kalte Sonne der Habsburger erlosch, aber es war eine Sonne gewesen. An dem Abend, an dem wir in Doppelreihen in die Kaserne zurückmarschierten, in den Hauptstraßen noch Parademarsch, dachte ich an die Tage, an denen mich eine kindische Pietät in die körperliche Nähe des Kaisers geführt hatte, und ich beklagte zwar nicht den Verlust jener Pietät, aber den jener Tage. Und weil der Tod des Kaisers meiner Kindheit genauso wie dem Vaterland ein Ende gemacht hatte, betrauerte ich den Kaiser und das Vaterland wie meine Kindheit. Seit jenem Abend denke ich oft an die Sommermorgen, an denen ich um sechs Uhr früh nach Schönbrunn hinausfuhr, um den Kaiser nach Ischl abreisen zu sehen. Der Krieg, die Revolution und meine Gesinnung, die ihr recht gab, konnten die sommerlichen Morgen nicht entstellen und nicht vergessen machen. Ich glaube, daß ich jenen Morgen einen stark empfindlichen Sinn für die Zeremonie und die Repräsentation verdanke, die Fähigkeit zur Andacht vor der religiösen Manifestation und vor der Parade des neunten November auf dem Roten Platz im Kreml, vor jedem Augenblick der menschlichen Geschichte, dessen Schönheit seiner Größe entspricht, und vor jeder Tradition, die ja zumindest eine Vergangenheit beweist.

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Die Sinnlosigkeit der letzten Jahre des Kaisers war zugleich auch ein Stück der Kindheit des Erzählers: Die Sinnlosigkeit der Majestät hat dem Kind Sinn gegeben, und die kalte Sonne der Habsburger war immerhin eine Sonne. Die Trauer um den Kaiser ist zugleich auch eine Trauer um den Verlust der eigenen Kindheit. Alles das wird im Präteritum erzählt. Auch in den folgenden Textteilen, wo einige Szenen von der Abreise des Kaisers nach Ischl erzählt werden, findet sich kein Perfekt. Eine kleine Szene zum Schluss handelt von einer Frau, die ein Gnadengesuch in den Wagen des Kaisers wirft. Sie wird von Polizisten ergriffen und abgeführt. Der letzte Absatz lautet: Fort war der Wagen. Das gleichmäßige Getrappel der Pferde ging unter im Geschrei der Menge. Die Sonne war heiß und drückend geworden. Ein schwerer Sommertag brach an. Vom Turm schlug es acht. Der Himmel wurde tiefblau. Die Straßenbahnen klingelten. Die Geräusche der Welt erwachten. Auch hier keine Spur eines Perfekts. Die Ausfahrt des Kaisers hat keine Spur hinterlassen, nichts, das man sich merken, das man beachten müsste, nichts, das irgendwelche Folgen hätte. Die „Moral der Geschichte“ – es gibt sie nicht. Es bricht ganz einfach ein neuer Tag an mit dem Klingeln der Straßenbahnen und den Geräuschen der Welt. Die Erzählung verdämmert in der gleißenden Sonne des Tages. Sie geht einfach einem Ende zu.

13. Eigenzeitliche Textabschnitte sind sinnstiftende Formen sprachlicher Gestaltungen. Erzählen heißt vom Vergangenen berichten, und das Vergangene ist oft eine Verkleidung des Fiktiven. Vergangenes und Fiktives ist vom Erzählenden entfernt, beides durch einen „Erinnerungsschnitt“ vom Erzählerzeit/sprechpunkt entfernt. Aber alles Erzählte hat einen Grund, eine Ursache, erzählt zu werden, und es steuert oft auf diesen Grund und diese Ursache im Lauf der Erzählung immer wieder zu. Das sind die Momente, bei denen die literarische Eigenzeitlichkeit ins Spiel kommt. Diese Momente sind wohl nicht nur für die Literatursprache wichtig, sondern auch für das sprachliche Ausdrucksvermögen an sich: Vielleicht kann man auch die sprachgeschichtliche Tatsache, dass sich Perfektformen im Lauf der Sprachgeschichte immer wieder erneuern, damit verbinden. Heimito von Doderer (Die Wiederkehr der Drachen, 22) hat diese Zusammenschau der Zeitebenen so beschrieben:

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Es ist das Gesetz von der „Symbiose der Zeiten“, wie es A. P. Gütersloh nennt: daß nämlich nichts, was war, durch nichts, was inzwischen geschehen ist, sich abhalten läßt, zu sein. Anders, und sozusagen massiv, formuliert: Jede einmal ausgespielte Karte bleibt auf irgendeine Weise im Spiel. So in der Erdgeschichte, so im geschichtlichen, so in unserem persönlichen Leben: auch hier staut sich das Volk des Gewesenen in dichtem und buntem Gedränge hinter den Kulissen der jetzt eben gespielten Szene und in den Gängen zwischen jenen, bereit, hervorzubrechen und die Bühne zu überschwemmen, alle Handlung an sich zu reißen.

Quellen: Heimito von Doderer 1970 Die Wiederkehr der Drachen. Aufsätze/ Traktate/ Reden, München: Biederstein. Walter Jens 1963 „Bericht über Hattington“, in: Walter Jens, Herr Meister. Dialog über einen Roman, München: Piper, 32–38. Joseph Roth 1956 „Seine k. und k. apostolische Majestät“, in: Joseph Roth, Werke in drei Bänden, Köln-Berlin, Bd. 2, 328–333.

Literatur: Comrie, Bernard 1985 Tense, Cambridge: Cambridge University Press. Confais, Jean-Paul 1995 Temps – mode – aspect, Toulouse: Presses Universitaires du Mirail. d’Alquen, Richard 1997 Time, mood, and aspect in German tense, Frankfurt/ Main – Wien: Lang. Duden 1959 Grammatik der deutschen Gegenwartssprache, Mannheim: Bibliographisches Institut. — 2005 Die Grammatik, Mannheim: Bibliographisches Institut Brockhaus. Engel, Ulrich 1992 „Der Satz und seine Bausteine“, in: Vilmos Ágel / Regina Hessky (Hgg.), Offene Fragen – offene Antworten in der Sprachgermanistik, Tübingen: Niemeyer, 53–76. — 2004 Deutsche Grammatik, München: iudicium. Harweg, Roland 1994 Studien über Zeitstufen und ihre Aspektualität, Bochum: Brockmeyer.

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Klein, Wolfgang 1994 Time in language, London: Routledge. Marschall, Matthias 1995 Textfunktionen der deutschen Tempora, Genf: Slatkine. Nowotny, Helga 1993 Eigenzeit, Frankfurt/Main: Suhrkamp. Quintin, Hervé/Margarete Najar/Stephanie Genz (Hgg.) 1997 Temporale Bedeutungen – Temporale Relationen, Tübingen: Stauffenburg. Reichenbach, Hans 1966 [1947] Elements of Symbolic Logic, New York – London: The Free Press. Schecker, Michael/Elisenda Padros/Thomas Jechle 1997 „Textgliederung – und was sie leistet: Empirische Analysen zur Funktion der Vergangenheitstempora im alltäglichen Standarddeutsch“, in: Quintin/Najar/ Genz 1997, 167–177. Thieroff, Rolf 1992 Das finite Verb im Deutschen. Tübingen: Narr. Uexküll, Jakob von/Georg Kriszat 1956 Streifzüge durch die Umwelten von Tieren und Menschen, Hamburg: Rowohlt. Vater, Heinz 1997 „Temporale und textuelle Funktionen deutscher Tempora“, in: Quintin /Najar/Genz 1997, 23–40. Welke, Klaus 2005 Tempus im Deutschen, Berlin: de Gruyter.

Drei Pioniere der philosophisch-linguistischen Analyse von Zeit und Tempus: Mauthner, Jespersen, Reichenbach Elisabeth Leinfellner, Wien [A philosophical school in Tlön] reasons that the present is indefinite, that the future has no reality other than as a present hope, that the past has no reality other than as a present memory. Jorge Luis Borges, Tlön, Uqbar, Orbis Tertius

1. Einleitung: die semantische Rolle der Tempora Unsere alltägliche, ,psychologische‘, aber auch die abstrakte newtonsche oder mathematische Vorstellung von Zeit – die ,klassischen‘ Zeitvorstellungen – laufen darauf hinaus, dass es eine einfache lineare Abfolge von Vergangenheit-Gegenwart-Zukunft gibt. Analysiert man jedoch natürliche Sprachen, dann bietet sich ein anderes, komplexeres Bild: Würde man der einfachen, klassischen Vorstellung folgen, dann käme man in der Sprache mit drei Tempora aus: Präsens, Präteritum, Futurum. Ein Blick in die Grammatiken verschiedener Sprachen zeigt aber, dass es auch andere Tempora neben diesen dreien gibt, so im heutigen Deutsch das Plusquamperfekt und das Futurum exactum. Gewisse Tempora und auch temporale Ersatzformen lösen mehr oder minder glücklich das Problem, dass z.B. in einer Erzählung im Präteritum manchmal auch im zeitlichen Rückgriff eine Vergangenheit vor der Vergangenheit dargestellt werden muss, oder eine Zukunft in der Vergangenheit. Die Ursachen für eine nicht chronologische Darstellung sind vielfältig, z.B. stilistisch-textliche oder dass wir uns an Ereignisse nicht immer in der richtigen Reihenfolge erinnern. Weiters muss die Verwendung der Tempora in erzählenden Texten die Rekonstruktion des chronologischen Ablaufs von Ereignissen erlauben, z.B. aus praktischen Gründen. Dass Texte keineswegs immer chronologisch geordnet sind, hat in verschiedenen Disziplinen und oft im Gefolge des russischen Formalismus zu einer Unterscheidung zwischen zwei zeitlichen Achsen geführt, der chronologischen Achse der Abfolge der Ereignisse, und F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 337–361.

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der Achse der zeitlichen Ordnung, in der diese Ereignisse erzählt werden. Z.B. wenn die Vorgeschichte eines im Präteritums erzählten Ereignisses dargestellt wird, wird im Deutschen oft das Plusquamperfekt verwendet. Mauthner (1849–1923), Jespersen (1860–1943) und Reichenbach (1891– 1953) befassen sich ausführlich mit der Vielfalt der grammatischen Tempora, die von diesen Autoren dennoch als unzureichend kritisiert wird. Die chronologische Abfolge wird bei Mauthner diskutiert und bei Reichenbach als der Parameter des Ereignis-Zeitpunktes E, bzw. der Ereignis-Zeitpunkte Ei festgehalten. Der zweiten der genannten Achsen entsprechen bei Reichenbach und implizit auch bei Mauthner und eventuell Jespersen zwei andere Parameter, der Referenz-Zeitpunkt R, d.h. der Zeitpunkt, von dem aus ein Satz oder auch Text geäußert wird, und der Sprech-Zeitpunkt S. Eine Sprechzeit wird auch von K. Bühler in seiner Sprachtheorie (1934) eingeführt. Reichenbachs Analyse, 11 Seiten in Elements of Symbolic Logic (1947), war eine außerordentliche Leistung. Aber auch außerordentliche Leistungen entstehen gewöhnlich nicht aus einem Nichts. Die 11 Seiten sind der Endpunkt einer Entwicklung, die auch von Mauthner eingeleitet wurde: In einer sehr ausführlichen Rezension hat der bedeutende Sprachwissenschaftler Leo Spitzer 1919 besonders Mauthners Diskussion der Tempora hervorgehoben (1919, Sp. 208). Reichenbach selbst beruft sich auf den Linguisten Jespersen, dessen Analyse ein methodologisches Bindeglied zwischen der Mauthners und Reichenbachs ist. Wittgensteins (1889–1951) Sprachkritik an dem Wort „Zeit“ schließlich zeigt auffallende strukturelle Ähnlichkeiten mit der Mauthners, und Mach (1838–1916) war Mauthners großes Vorbild. Daher auch ein paar Bemerkungen vor allem zum Konzept der Zeit bei Mach und Wittgenstein. Das Problem der Tempora nimmt bei den drei „Pionieren“ noch viel speziellere Züge an: Es muss kognitiv, psychologisch oder sonstwie einen absoluten Bezugspunkt, einen Fixpunkt der temporalen Kennzeichnungen und der Tempora, allgemein: der zeitlichen Reihung geben, zumindest als Idealisierung. Dieser Fixpunkt ist bei Mauthner das sprechende und stets präsentische Ich, bei Reichenbach das Produkt des Ich, der Sprech-Zeitpunkt S, als der absolute, ,unbewegliche‘ oder permanente Bezugspunkt. Die Tempora können durch die Kombinatorik dreier Parameter, SprechZeitpunkt S, Ereignis-Zeitpunkt E und Referenz-Zeitpunkt R, auf einem Zeitpfeil charakterisiert werden, so explizit bei Reichenbach und im Ansatz bei Mauthner und auch Jespersen.

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Die zeitliche Reihung, die sich in der Reihung der Tempora als Koordinatensystem (in der ›langue‹) ausdrückt, sowie in der Reihung der temporalen Kennzeichnungen (in der ›parole‹), ist nach Mauthner auf einem Zeitpfeil beweglich. Das gilt auch für die Gegenwart, das Tempus Präsens und die Präsens-Kennzeichnung, obwohl Gegenwart, Präsens und PräsensKennzeichnung Nullpunkte oder Schnittpunkte sind. Was Gegenwart ist, wird zur Vergangenheit, was Zukunft ist, zur Gegenwart: Unser Gedächtnis behilft sich so, daß immer wieder das Vergangene zum Gegenwärtigen wird […] (B3, 247) Diese Verschiebung ist semantisch immer, aber keinesfalls immer syntaktisch-morphologisch, fassbar. Da sich der Nullpunkt nach Mauthner bewegt, muss sich auch das Koordinatensystem bewegen. Reichenbach löst dieses Problem, indem er für bestimmte Fälle vorsieht, dass es eine Reihung der Positionen des Referenz-Zeitpunkts R geben muss (siehe unten, § 5). Im Folgenden beschränken wir uns hauptsächlich auf Überlegungen zu Sätzen und Texten, die von zeitlich genau umrissenen Ereignissen handeln; sie gelten also nicht für Beschreibungen, nicht für Sätze, die habituelle Ereignisse darstellen, u.ä. Habituelle Ereignisse können im Deutschen z.B. durch Verben im Präteritum ausgedrückt werden. Wenn es in einem Heine-Gedicht heißt: (1) Es war ein schöner Page Blond war sein Haupt, leicht war sein Sinn; Er trug die seidne Schleppe der jungen Königin. (Heinrich Heine, Buch der Lieder) dann kann für „trug“ kein Ereignis-Zeitpunkt E festgestellt werden, da es sich um ein habituelles Ereignis handelt, das sich mehrmals innnerhalb eines zeitlichen Intervalls abspielt. Reichenbach, Mauthner und Jespersen haben sich auch mit solchen Fragen beschäftigt (vgl. § 6). Eine sich mit den Ereignissen auf dem Zeitpfeil bewegende Tempus-Kennzeichnung in der ›parole‹ verdeutlichen wir uns am besten an einem Beispiel: (2) (2a) Strickland’s injurious calm robbed Stroeve of the rest of his selfcontrol. (2b) Blind rage seized him […]. (W. Somerset Maugham, The Moon and Sixpence)

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Die temporalen Kennzeichnungen in (2a) und (2b) sind morphologisch in der ›langue‹ dieselben, „-ed“. Aber semantisch handeln (2a) und (2b) keineswegs von gleichzeitigen Ereignissen, sodass sich mit Mauthner auch „-ed“ mit den Ereignissen auf dem Zeitpfeil bewegt. Eine wichtige Frage ist: Wieviele und welche Tempora werden nach Mauthner, Jespersen und Reichenbach minimal benötigt, vorausgesetzt eine Sprache, die über ein am Verb orientiertes Tempussystem verfügt? Alle drei Autoren kommen hier zu sehr ähnlichen Ergebnissen (siehe § 9).

2. Sprachkritik am Begriff der Zeit Alle Begriffe sind, so Mauthner, Hypothesen, weiters metaphorisch und uneigentlich, daher trügerisch. Aber es gibt hier graduelle Unterschiede: Trügerisch, aber dennoch brauchbare Hypothesen sind solche, denen sensualistisch, d.h. in der adjektivischen Welt etwas entspricht. Das Wort „Zeit“ gehört morphologisch – erkenntnistheoretisch-sprachkritisch gesehen fälschlicherweise – zu den Substantiven. Zeit kann es nach Mauthner nur in der verbalen Welt geben; nur dort werden Veränderungen und Ereignisse, die aufeinander folgen, und an die man sich erinnert, metaphorisch abgebildet. Die verbale Welt ist die Welt, wo ein Mensch nach Heraklit nicht zweimal in denselben Fluss steigen kann – und nach Mauthner wäre es auch „nicht mehr derselbe Mensch, der zum zweiten Male hineinstiege“ (DBW 60f.). Während das Substantiv eine Art Gegenwart vermittelt, ist dies wegen der dynamischen Zweckgerichtetheit der Handlungsverben nach Mauthner unmöglich. Die Gegenwart und das Präsens als die Schnittpunkte oder Nullpunkte zwischen Vergangenheit und Zukunft und den entsprechenden Tempora seien sowieso nur kognitive und sprachliche Fiktionen. Die natürlichen Sprachen bilden also nach Mauthner die Welt nur sehr unvollkommen ab, und das gilt besonders auch für das Wort „Zeit“, die temporalen Kennzeichnungen des Verbs und die Temporal-Adverbien. Die Zeit ist nach Mauthner eine funktionale Bedingung auch der adjektivischen Welt (B1, 77; B3, 63f., 70; W3, 358, 475; DBW, 60ff., 156ff.; für eine Skala der relativen zeitlichen Stabilität in Relation zu den Wortarten siehe Frawley 1992, 65ff.; ebenso B3, 70f.). Ähnlich auch Wittgenstein über „Zeit“ als Substantiv: Die alles gleich machende Gewalt der Sprache die sich am krassesten im Wörterbuch zeigt & die es möglich macht, daß die Zeit personifiziert

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werden konnte, was nicht weniger merkwürdig ist als es wäre, wenn wir Gottheiten der logischen Constanten hätten. (VB, 46, 1931; vgl. DB, 74f.; BBB, 6)

3. Zeitkonzepte Mauthner trennt die objektive Zeit der klassischen Physik, weiters die objektive, weil messbare, Präsenzzeit der Psychologie und die objektive Zeit unserer Alltagsvorstellung, die alle klassisch am Raum abgeschätzt oder gemessen werden, von der subjektiven Zeit, der Zeit, die wir erleben, und deren Dauer relativ ist (ganz ähnlich Davies 1996, 283). Über die physikalische Zeit sagt Mauthner erstaunlicherweise etwas, das sich bei Davies wiederfindet: Mauthner denkt nämlich darüber nach, ob die physikalische Zeit, anders als bei Newton, wo sie eine abstrakte, mathematische Zeit ist, nicht doch als eine Art Kraft oder zumindest als eine funktionale Bedingung oder als eine Art Ursache aufgefasst werden sollte (W2, 319; W3, 475f.). Nach Davies: Space and time […] are not simply ‘there’ as an unchanging backdrop to nature; they are physical things, mutable and malleable, and, no less than matter, subject to physical law. (Davies 1996, 16) Wie Mach und Reichenbach, so lehnt auch Mauthner die Parallele von mathematisch-physikalischem Raum und mathematisch-physikalischer Zeit ab (vgl. etwa Reichenbach 1977 [1928], 130ff.; aber siehe unten, § 7, für Raum und Zeit in der natürlichen Sprache). Die subjektive, erlebte Zeit kann nicht auf eine psychologische oder auch vorgestellte, auf eine physikalische, physiologische oder gar auf eine inhaltslose mathematische Zeit zurückgeführt werden; daher kann sie weder gemessen noch in den Tempora ausgedrückt werden (vgl. Ploog 2000, 1162; W3, 441, 463; Mach 1905, passim; Mach 1986, 175; Einstein in Davies 1996, 269, siehe auch 266).

4. Das zeitlose Ich und das a-temporale Präsens Nach Mauthner hält das Ich, eine Funktion des Gedächtnisses, für uns, subjektiv, still, ist a-temporal, anders ausgedrückt: stets gegenwärtig. Es ist der Fixpunkt, auf den sich die vorgestellte Zeit, die temporalen Kennzeich-

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nungen und damit auch die Tempora beziehen. Wie bei Mach und Schlick, so ist auch nach Mauthner das zeitlose Ich dennoch eine Fiktion, wenn auch eine tief verankerte (Leinfellner 1995a, 74f.). Mauthner liegt hier auf einer Linie besonders mit Reichenbach. Nach Reichenbach scheint die Zeiterfahrung selbst mit der Erfahrung des eigenen Ich zusammenzuhängen: „Ich bin“ sei gleichbedeutend mit „Ich bin in einem ewigen Jetzt identisch mit mir selbst“ (1977 [1928], 131; ähnlich Schrödinger). Dieses Ich als Fixpunkt der zeitlichen Ordnung produziert den Sprech-Zeitpunkt S, der sich im abgeschlossenen, auch komplexen, Satz nach Reichenbach nicht verändert, noch verändert sich nach Mauthner sein kognitives Äquivalent, das idealisierte, stets präsentische Ich. Das Tempus Präsens wird von Mauthner ebenfalls als a-temporal angesehen, als bloßer Schnittpunkt oder Nullpunkt zwischen den Tempora der Vergangenheit und der Zukunft. Eingefügt in das Koordinatensystem der Tempora ist das Präsens mit diesem beweglich und zieht an dem präsentischen und dauerhaften Ich vorüber; es wandert mit jedem Satz auf dem Zeitpfeil (vgl. § 17 unten). Die Gegenwart ist, so Mauthner, ein „stets verlorener Besitz“ (B3, 68). Dem entspricht im Text eben der intern bewegliche Schnittpunkt oder Nullpunkt, ein ausdehnungsloses, unwirkliches ,interface‘ zwischen der grammatisch ausgedrückten Vergangenheit und Zukunft: das Tempus Präsens und die Präsens-Kennzeichnungen. Hier verstehen wir Mauthners Bemerkung, dass man eigentlich gar nicht, mit einem Null-Morphem als Präsens-Kennzeichnung, „es blitzt“ sagen könne, sondern nur im Präteritum „es blitzte“ (B3, 68f.). Wir verdeutlichen uns Mauthners Analyse an einer Anzeige für Volkswagen: (3) Er läuft und läuft und läuft. Wir haben dreimal dasselbe Verb im Präsens. Aber das System der Tempora als Koordinatensystem in der ›langue‹ und das temporale Kennzeichen (hier: ein Null-Morphem) in der ›parole‹ haben sich mit Mauthner bewegt: Jedes „läuft“ stellt eine neue Gegenwart dar (für Reichenbach siehe unten, § 12). Eine andere Auffassung des Tempus Präsens wäre, dass es unbestimmt ist. Von der subjektiv erlebten Gegenwart, also nicht vom Tempus Präsens, hat Mauthner dies explizit behauptet:

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Die Gegenwart läßt sich in Wahrheit nicht definieren, weil sie niemals existiert, weil wir sie erst dann empfinden, wenn wir zu ihr ein Stückchen Zukunft und ein Stückchen Vergangenheit mit hinzu rechnen […] (W3, 495f.) Ähnlich Wittgenstein: Das Gefühl ist nämlich, daß die Gegenwart in die Vergangenheit schwindet, ohne daß wir es hindern können. (PB, § 52, vgl. auch § 75) Diese subjektive Gegenwart ist nicht-linear. In (Ploog 2000) findet sich eine Abbildung, die diese Zitate und die Nicht-Linearität illustriert. Wir se-hen da, als Fläche dargestellt, eine Gegenwart, die ein Stück Vergangenheit gleichsam mitnimmt, und sich als flächige Pfeilform in die Zukunft erstreckt. Ganz ähnlich Jespersen über die Gegenwart und das Tempus Präsens (1965 [1924], 259).

5. Das Jetzt Für das Jetzt bei Mauthner müssen ähnliche Unterscheidungen getroffen werden wie für die Zeit: Erstens gibt es ein zeitlich ausgedehntes, messbares, psychologisches Jetzt, d.h. ein Jetzt, das gemessene, aber nicht erlebte Dauer hat, die Präsenzzeit. Eine Illustration: Es werden heute 24 Bilder/ Sekunde abgespielt, um die Scheinbewegung im Film zu erzeugen. Auch Wittgenstein hat das Abspielen des Films als Analogie zur Zeit erwogen, ebenso den Film als Illustration des Verhältnisses von Zeit und Gedächtnis (z.B. in PB, § 49–54; Bouwsma 1986, 13). Zweitens gibt es verschiedene Formen eines punktuellen Jetzt, die nach Mauthner a-temporal sind: die punktuelle, vom Gedächtnis gespeiste Vorstellung eines Jetzt und das punktuelle grammatische Jetzt, d.h. das Tempus Präsens als Tempus der Gegenwart.

6. Aspekt Jespersen hat den Aspekt kurz und prägnant in The Philosophy of Grammar (1965 [1924], 286-289) beschrieben, Reichenbach an Hand des ›progressive‹ und des ›imparfait‹ (1947, 290ff.). Im Temporalsystem ist ein wichtiges Moment der Aspekt. (In diesem

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Artikel wird kein Unterschied zwischen Aspekt und Aktionsart gemacht.) Darunter verstehen wir hier alle diejenigen semantischen und morphologischen Kennzeichnungen eines Verbs, die die innere Struktur der beschriebenen Ereignisse, Zustände etc. betreffen, am einfachsten an den ›extended tenses‹, wie dem englischen ›progressive‹ und im Französischen dem ›imparfait‹ im Unterschied zum ›passé simple‹ (›passé defini‹) festzumachen. Die erweiterten Tempora ›progressive‹ und ›imparfait‹ drücken relative zeitliche Dauer aus, nicht aber z.B. das ›passé simple‹. Im Deutschen kann der Aspekt gewöhnlich nicht eindeutig an den morphologischen Kennzeichen des Verbs erkannt werden (vgl. (1)). Die erweiterten Tempora der Verben erfordern eine Darstellung mit Zeit-Intervallen, was sowohl Jespersen als auch Reichenbach diagrammatisch festhalten ( Jespersen 1965 [1924], 278; Reichenbach 1947, 290ff.). Später hat man diesen Ansatz verbessert: Nunmehr werden alle Tempora, nicht nur die erweiterten, als Intervalle formuliert, sodass es keine Zeitpunkte mehr gibt, so etwa bei Couper-Kuhlen (1989). In den Beiträgen geht Mauthner von einer Diskussion des lessingschen Laokoon aus. Dort stellt Lessing fest, dass das Verb Handlungen ausdrückt, und dass daher ein Bild eine charakteristische Handlung als ,Momentaufnahme‘ darstellen müsse, welche Vergangenheit und Zukunft begreiflich mache. Dagegen wendet Mauthner sehr richtig ein, dass dies auf die modernen Stimmungsbilder und die moderne Stimmungspoesie nicht passe. Linguistisch ausgedrückt: Es muss neben dem Aspekt der Handlung zumindest auch den Aspekt des Zustands geben: In „Elissa geht schlafen“ wird eine Handlung ausgedrückt, in „Elissa schläft“ ein Zustand. Im Wörterbuch unterscheidet Mauthner implizit zwischen den Aspekten Handlung und Zustand und fügt eine weitere Kategorie hinzu, den Vorgang. Diese Unterscheidung zwischen den Aspekten Handlung, Zustand und Vorgang ist noch heute in der Linguistik grundlegend. Mauthner hat auch eine Art strukturalistischer, am Aspekt orientierter Komponenten-Analyse des Verbs vorgeschlagen, wie sie später, 1993, von Verkuyl genauer ausgeführt worden ist. Ebenso wie das Substantiv die pointillistischen adjektivischen Empfindungen oder die sie repräsentierenden Adjektive sprachlich zusammenfasst, so fasst nach Mauthner auch das Verb, zumindest das Handlungsverb, etwas zusammen, nämlich unzählige einzelne zweckgerichtete Teil-Handlungen oder auch die sie repräsentierenden Verben. Verben sind daher noch unwirklicher als die Substantive, die hypothetischen Zusammenfassungen von den allein wirklichen adjektivischen

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Empfindungen oder den sie repräsentierenden Adjektiven. Da wir nach Mauthner eigentlich nicht Handlungen wahrnehmen, sondern pointillistisch à la Mach nur aufeinander folgende Zustände, so müsste auch hier das ursprüngliche Verb letztlich entweder in intransitive Zustandsverben oder in Adjektive zerlegt werden. Tatsächlich weiß man heute, dass zumindest unsere Wahrnehmungen zeitlich ,gequantelt‘, also ruckartig, pointillistisch sind (Ploog 2000, 1163). Daher nimmt Mauthner an, dass intransitve Verben kognitiv den Adjektiven ähneln, wie in: (4) Der Himmel blaut. – Der Himmel ist blau. Eine interessante Parallele zu Mauthners Zerlegung der Verben in andere Verben findet sich in der KI bei Schubert 1976 und Cercone und Schubert 1974. Diese Autoren vertreten die These, dass die Analyse der Handlungsverben von mit Absicht verknüpften Handlungen – Mauthners „Zweck im Verbum“ – ausgehen sollte. Cercone und Schubert haben eine Analyse von Handlungsverben als Abfolgen von statischen Zuständen entwickelt.

7. Die Zeit als Raum Dreifach ist der Schritt der Zeit: Zögernd kommt die Zukunft hergezogen, Pfeilschnell ist das Jetzt entflogen, Ewig still steht die Vergangenheit. Friedrich Schiller, Spruch des Confucius

Ein zentrales Thema der Analyse temporaler Ausdrücke in der natürlichen Sprache ist auch: Räumliche Ausdrücke werden durch metaphorische Übertragung – „metaphorisch“ hier im gewöhnlichen Sinn verstanden – zeitlich. Der Grund dafür ist nach Mauthner der Vorrang des Sehens. Er schneidet damit eines der wichtigsten Themen der heutigen kognitiven Linguistik an. Zum Raum bemerkt Mauthner: Da jedes Individuum der stets wechselnde Mittelpunkt seines räumlichen Koordinatensystems ist, ist die Sprache des Raumes individuell. Diese Auffassung von situationsbedingten räumlichen Koordinaten ist, mutatis mutandis, eine gängige Auffassung der heutigen kognitiven Linguistik geworden. So führt Langacker den Begriff des Orientierungspunktes oder Marksteins (›landmark‹) ein. Der Markstein ist nach Langacker allerdings nicht das die Objekte im Raum betrachtende und sich bewegende

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Individuum, sondern ein stillstehendes Objekt, das den Bezugspunkt der Bewegung abgibt. Andererseits kann es eine Abfolge von Marksteinen geben (Langacker 1988, u.ö.). Kognitiv gesehen kann man aber Mauthner dennoch zustimmen: Auch der Markstein muss letztlich in Beziehung zum sich kognitiv orientierenden und sich bewegenden Individuum gesehen werden, soferne wir nicht eine Metrik in einem formalen Koordinatensystem, z.B. einer Landkarte, vorliegen haben; selbst dann muss der Standpunkt des Benützers der Karte diesem genau bekannt sein. Der im Raum unbewegliche Markstein der kognitiven Linguistik ist das räumliche Analogon des zeitlich stillstehenden, a-temporalen Ich bei Mauthner und des absoluten Sprech-Zeitpunkts S bei Reichenbach.

8. Die Repräsentation der linearen Zeit Eines der größten erkenntnistheoretischen oder kognitiven Probleme bei der Diskussion der Tempussysteme ist der Übergang von der klassischen Zeit unserer Alltags-Erfahrung, und insbesondere der vom Gedächtnis gespeisten, vorgestellten Zeit nach Mauthner, zum Tempussystem. Es ist ein attraktiver Gedanke, dass sich unsere klassischen Zeitvorstellungen semantisch-pragmatisch oder kognitiv direkt auf das grammatische System der Tempora abbilden ließen. Dem stehen nach Mauthner mindestens zwei fundamentale Hindernisse entgegen: (i) die semantischpragmatische oder kognitive Unbestimmtheit der Tempusformen; (ii) die Unvollständigkeit des Tempussystems. Ad (i). Die Tempora z.B. des Deutschen sind nicht ein-eindeutig an die verschiedenen zeitlichen Einheiten, die im Tempus-System repräsentiert werden sollen, geknüpft. So kann z.B. das Tempus des Präsens alles Mögliche ausdrücken: Es kann ein historisches Präsens sein, Zeitlosigkeit ausdrücken, die Zukunft, usw. Ein empirisches Beispiel für (i): In dem a-Satz von (5) handelt es sich um das historische Präsens mit der zeitlichen Funktion des Präteritums, in den (5b)–(5d)-Sätzen um das Futurum als Futurum der Vergangenheit: (5) (5a) In einer armseligen Backsteinhütte der Siedlung „Mein Peru“, Block A, Parzelle 3, bereitet die 18jährige Roxana Ayala, wie jeden Tag, für ihre vier Geschwister die Fischspeise zu. (5b) Roxana wird am 6. April, in der elften Woche der Cholera-Epidemie sterben. […] (5c) Vergeblich werden die Ärzte versuchen, durch künstliche Beat-

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mung die Patientin am Leben zu erhalten. (5d) Und danach wird ihre Mutter behaupten, (5e) am Abend zuvor habe ihre Tocher Reis mit Huhn gegessen. (Ruedi Leuthold, „Tänzchen mit dem Tod“) Die semantisch-pragmatische oder kognitive Unbestimmtheit der Tempora – hier des Präsens und Futurums – wird auch von Reichenbach festgestellt: Es werden, sagt er, die Tempora nicht immer so gebraucht, wie es seine noch zu erklärenden Schemata darstellen (1947, 292). Ad (ii). Das Tempussystem ist auch auf andere Weise ungenau. Erstens: Wir haben in keiner natürlichen Sprache, die Tempora des Verbs hat, so viele Tempora, als es zeitliche Bestimmungsstücke, z.B. historische Daten, Nanosekunden, Stunden usw. gibt, eine Sprachkritik, die wir in der heutigen kognitiven Semantik, so bei Frawley, wiederfinden (B2, 448f.; B3, 247; DBW, 69f.; Frawley 1992, 338). Diese Kritik ist zwar utopisch, aber zweitens: Das Tempussystem ist nach Mauthner, Jespersen und Reichenbach ungenau, weil zumindest im Deutschen und Englischen nicht einmal die minimal erforderlichen Tempora vorhanden sind – daher gibt es Ersatzformen. Manche Sprachen allerdings, wie z.B. das Vietnamesische, drücken Zeitbestimmungen nur adverbial, d.h. ohne Tempora aus. Im heutigen umgangssprachlichen Oberdeutschen kommt man durchaus mit zwei Tempora aus, dem Perfekt für alles Vergangene und dem Präsens für Präsens und Futurum. Um z.B. das Futurum auszudrücken, genügt das Präsens plus einem Temporal-Adverb: „Ich komme morgen“. Mauthner bemerkt, dass es kein eigenes Tempus gibt, das eine Zukunft in der Vergangenheit, also Reichenbachs ›posterior past‹, darstellt. Er bringt zwei schöne Beispiele aus dem Nibelungenlied: Im ersten Fall, (6c), wird mit einer Ersatzform, „mußten“ im Präteritum, gearbeitet, im zweiten, (7), mit einem Präteritum, das durch das Adverbial „seither“ modifiziert wird: (6) (6a) Kriemhild war sie geheißen, (6b) die war ein schönes Weib, (6c) Darum mußten noch viele Degen verlieren ihren Leib. (B3, 41) (7) Sie starben jämmerlich seither von zweier Frauen Neid. (B3, 41) Dazu kommt als (iii) noch die in der Einleitung festgestellte chronologische Nicht-Linearität vieler Texte.

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9. Wie viele Tempusformen benötigen wir? In den Beiträgen vertritt Mauthner daher Thesen, die zunächst, in unvollkommener Form, im 19. Jahrhundert von Madvig, Matzen, Kroman und Noreen diskutiert worden sind: Wieviele Tempusformen werden minimal gebraucht, um die als pragmatisch wichtig angesehenen zeitlichen Verhältnisse adäquat auszudrücken? Nach Mauthner würden wir für das Deutsche mindestens neun Tempusformen benötigen; es stehen aber nur sechs zur Verfügung. Zuerst die Grundlage dieser Überlegung: […] irgend eine Vergangenheit oder Zukunft wird gewissermaßen als ein Koordinatenursprung angenommen, auf welche sich wiederum eine andere Zeit als Vergangenheit oder Zukunft bezieht. (B3, 71f.) Daraus ergibt sich im Detail: Die Unbestimmtheit der verbalen Zeitformen scheint mir also recht mathematisch bewiesen zu sein. Unsere Stellung in der Zeit nötigt uns, mindestens 9 deutlich ausgeprägte verschiedene Zeitverhältnisse auszudrücken; wir aber besitzen nur 6 Verbalformen, mit deren Hilfe wir ungefähr sagen, was wir wollen. (B3, 42f., vgl. auch 39, 44, 247) Das sind dieselben Zahlen, zu denen auch Reichenbach in seinen Elements of Symbolic Logic kommt. Reichenbach beruft sich in vielem auf Jespersens The Philosophy of Grammar von 1924. Jespersen geht zunächst mit dem Latinisten Madvig auch von 9 Tempora aus, reduziert sie dann aber auf 7: Plusquamperfekt, Präteritum, Ersatzformen für die Zukunft in der Vergangenheit, Präsens, Futurum Exactum, Futurum, Ersatzformen für die Zukunft in der Zukunft (1965 [1924], 254ff.). Wie kommen Mauthner, Jespersen und Reichenbach auf die Zahl 9? Zunächst ist da die Erkenntnis, schon bei Madvig, dass die einfache Einteilung der Zeit in Gegenwart, Vergangenheit und Zukunft für die Grammatik nicht ausreicht, weil jede dieser drei Zeiteinheiten in sich wieder in Gegenwart, Vergangenheit und Zukunft gegliedert ist. Mauthner drückt das Problem auch so aus: Er betrachtet alle Tempora als relativ, d.h. relational, auch die sogennnten absoluten Tempora wie z.B. das Präteritum. Jedes Tempus bezieht sich immer auf eine Gegenwart als den Null- oder Schnittpunkt: auf die Gegenwart in der Vergangenheit, die Ge-

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genwart in der Gegenwart, und die Gegenwart in der Zukunft, in reichenbachscher Terminologie, und durchaus im Sinne Mauthners: auf ›simple past‹ (Präteritum) als das ,Präsens‘ für die zeitliche Vergangenheit, ›simple present‹ (Präsens) als das Präsens für die zeitliche Gegenwart und ›simple future‹ (Futurum) als das ,Präsens‘ für die zeitliche Zukunft (siehe (8)). Dementsprechend notiert Mauthner zum relativen oder relationalen Charakter aller Tempora, nicht nur der relativen Tempora der Schulgrammatik: Man hat die drei Zeiten der Vergangenheit, Gegenwart und Zukunft die absoluten Zeitverhältnisse genannt und sie so von den relativen Zeitverhältnissen, wie z.B. dem Plusquamperfektum, unterschieden. Natürlich sind diese Bezeichnungen nicht streng zu nehmen. Gegenwart und Zukunft beziehen sich immer auf die Gegenwart, sind immer relativ, und Plusquamperfektum, Futurum exactum usw. sind nur relativ in zweiter Potenz […] (B3, 71f.) Diese Konstruktion ist für die Analyse von Texten, insbesondere von erzählenden, sehr nützlich. Das reichenbachsche System der Tempora gründet sich auf diesen neun Zeiteinheiten und jeder dieser Einheiten entspricht eine spezifische Konstellation der drei zeitlichen Parameter E, R und S, manchmal auch mehr als eine (siehe (9)–(14)): (8) Die 9 Tempora nach Reichenbach: Reichenbach: Traditionelle Grammatik (verschiedene Bezeichnungen): anterior past Plusquamperfekt (Vorvergangenheit)* simple past Präteritum (passé simple = passé défini, Imperfekt, Mitvergangenheit; das ,Präsens‘ für die zeitliche Vergangenheit)* posterior past kein eigenes Tempus* anterior present Perfekt (Vergangenheit als Tempus) simple present Präsens (Gegenwart als Tempus)* posterior present Futurum (Futur I, Zukunft als Tempus; das Futurum der traditionellen Grammatik)* anterior future Futurum exactum (Futur II, Vorzukunft)* simple future Futurum (Zukunft als Tempus; das ,Präsens‘ für die zeitliche Zukunft) posterior future kein eigenes Tempus*

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Mit einem „*“ versehene Elemente finden sich auch auf der Tafel von Jespersen, der das Futurum nur einmal, das Perfekt gar nicht anführt (siehe unten). Reichenbach hat also Jespersens System verändert übernommen und durch eine Kombinatorik zeitlicher Parameter ergänzt. Aber schon Mauthner und Jepersen geben uns interessante Hinweise auf solche Parameter. Ich werde von Reichenbach ausgehen, ohne dass sich hier irgend ein direkter Einfluss Mauthners etwa über Jespersen nachweisen ließe. Jespersen hat allerdings den dritten Band von Mauthners Beiträge rezensiert (1914); auch hat er aus demselben Band zitiert (1965 [1924], 147). Ich stelle nun die drei reichenbachschen Parameter Sprech-Zeitpunkt S, Referenz-Zeitpunkt R und Ereignis-Zeitpunkt E genauer vor. Ihre Kombinatorik ist für Sätze gedacht. Reichenbach hat sich nur andeutungsweise mit Texten beschäftigt; er hat aber über die Analyse von komplexen Sätzen auch den Weg zur Analyse von Texten vorbereitet. (9)–(14) Parameter-Konstellationen für sechs Tempora nach Reichenbach (1947, 288ff.): (9) E – R – S (anterior past, Plusquamperfekt) (10) E,R – S (simple past, Präteritum, passé simple) (11) R – E – S; R – S,E; R – S – E (posterior past) (12) E – S,R (anterior present, Perfekt) (13) E,R,S (simple present, Präsens) (14) S,R – E (simple future, Futurum) Erklärung: „ – “ = zeitliches Auseinanderfallen von Parametern; der Beistrich „,“ = zeitliches Zusammenfallen von Parametern. Das Perfekt nimmt bei Mauthner, Jespersen und Reichenbach eine besondere Stellung ein. Reichenbach kennzeichnet das Perfekt mit (12) E – S,R (anterior present, Perfekt) d.h. dass Sprech-Zeitpunkt S und Referenz-Zeitpunkt R wie beim Präsens zeitlich zusammenfallen; nur der Ereignis-Zeitpunkt E liegt vor beiden. Er illustriert dies mit einer Passage aus Keats (1947, 289, vgl. 298): (15) Much have I traveled in the realms of gold, And many goodly states and kingdoms seen; Round many western island have I been […]

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Daher sei (15) kein Narrativ,sondern ein ich-betonter Bericht mit unmittelbarem Bezug auf die Gegenwart. Ganz ähnlich schon Mauthner: Nur das eigentliche Perfektum läßt sich nicht durch das Präsens ausdrücken, weil es eben ohnehin ein Präsens ist nebst einem adjektivisch gewordenen Verbum. (B3, 44) Aus demselben Grund lässt Jespersen das Perfekt auf seiner Tafel der Zeiteinheiten und Tempora von vornherein weg (1965 [1924], 269ff.; 1964 [1933], 237). Die Auffassung, dass das Perfekt u.a. auch dazu verwendet wird, um die Unmittelbarkeit eines Präsens zu erreichen, hat sich bis heute erhalten. Wir kommen also sowohl bei Mauthner als auch bei Reichenbach auf neun mögliche Tempora, denen im Standard-Deutschen nur sechs gegenüberstehen. Zwei dieser möglichen Tempora, ›posterior past‹ und ›posterior future‹, entspricht real kein eigenes Tempus. Das Futurum kommt zweimal vor, aber nur einmal ist es das klassische Futurum der Schulgrammatik.

10. Der Sprech-Zeitpunkt S Nach Reichenbach ist der Sprech-Zeitpunkt S derjenige Zeitpunkt, zu dem ein Satz als ›token‹ (Äußerung) hervorgebracht wird. Er ist ein Produkt des reichenbachschen stillstehenden, a-temporalen, stets präsentischen Ich und der absolute, abstrakte, idealisierte und unverrückbare Bezugspunkt, an dem sich die zwei anderen zeitlichen Parameter, der Referenz-Zeitpunkt R und der Ereignis-Zeitpunkt E, orientieren. Jeder Satz, bzw. erweitert auch jeder Text hat daher nach Reichenbach seinen unverrückbaren Bezugspunkt, den Sprech-Zeitpunkt S. In Anlehnung an die Physik könnte man metaphorisch von der ,Eigenzeit‘ des Satzes oder Textes sprechen. Der Sprech-Zeitpunkt S liegt idealisiert vom gerade Sprechenden her gesehen stets in der Gegenwart. Aber vom Leser her gesehen kann er genau so gut in der Vergangenheit liegen. Dies würde an der Kombinatorik der Parameter nichts ändern.

11. Mauthnersche Annäherung an den Sprech-Zeitpunkt S Mauthner führt in B3, 67 als erstes eine dem reichenbachschen SprechZeitpunkt S analoge Auffassung einer Sprech-Zeit ein, die am stets präsentischen Sprecher orientiert ist:

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Es bezieht sich das Zeitverhältnis immer auf das Subjekt, entweder auf das Subjekt des Satzes oder auf das den Satz aussprechende Subjekt. Dieses Subjekt vertritt die Gegenwart. Der Sprecher ist immer gegenwärtig, das grammatikalische Subjekt wird entweder als gegenwärtig gedacht oder mit der Gegenwart des Sprechers verglichen. (B3, 67). In diesem Zitat, B3, 67, wird, in reichenbachscher Terminologie, der Parameter Sprech-Zeitpunkt S von einem zweiten Parameter unterschieden, der entweder der Referenz-Zeitpunkt R oder der Ereignis-Zeitpunkt E sein könnte, eher letzteres. Im Prinzip wird nach Reichenbach S sowohl mit R als auch mit E verglichen. In bestimmten Fällen muss – entgegen Reichenbach – dennoch der Sprech-Zeitpunkt aufgehoben werden (siehe unten, § 12).

12. Probleme der empirischen Anwendung des SprechZeitpunktes S Analysieren wir folgendes Beispiel: (16) (16a) Bischof Samuel Wilberforce […] richtet [E1] an den Naturforscher Thomas Henry Huxley ironisch die Frage, ob er einen Affen lieber als seine Großmutter oder seinen Großvater haben wolle. (16b) Huxley […] erwidert [E2] mit gleicher Ironie […] (16c) Einige Zuhörer glauben [E3] gehört zu haben, Huxley wäre lieber ein Affe als ein Bischof. (16d) Es kommt [E4] zu einem Tumult. (Franz M. Wuketits, Darwin und der Darwinismus) Da das Präsens durch E,R,S, (13), gekennzeichnet ist, sodass S mit R und E gleichzeitig ist, müsste sich für (16a)–(16d) entweder der Sprech-Zeitpunkt S auf dem Zeitpfeil verschieben; anders ausgedrückt: er müsste, in Analogie zu R, in diesem Fall ›positional‹ gebraucht werden. Oder S müsste sich vervielfältigen. Derartige Lösungen des Problems sind bei Reichenbach nicht vorgesehen. Wie auch immer, S kann nicht gut mit allen Ei gleichzeitig sein, sondern nur mit einem. Oder wir verstehen S als ein Intervall, das mit allen Ei gleichzeitig ist, was aber nach Reichenbach ebenfalls nicht vorgesehen ist: Nur E und R können Intervalle sein (1947, 290f.). Von allen Sätzen von (16) kann nur einer der Bedingung für das Präsens, (13), genügen; in allen übrigen müssten sich die Ei von S distanzieren. Nehmen wir an, dieser eine Satz sei E4. Wir würden dann – hypothetisch – für alle

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übrigen Sätze E – R,S erhalten, das Perfekt (12). Fassen wir hingegen (16) von vornherein so auf, dass die Darstellung im historischen Präsens äquivalent der im Präteritum ist, dann würde für alle Sätze von (16) die Kennzeichnung für das Präteritum, (10), E,R – S, gelten. Es hat aber pragmatische oder semantische Gründe, warum das historische Präsens verwendet wird. Das historische Präsens übernimmt nämlich zwei Funktionen: die temporale des Präteriums, und eine pragmatische oder semantische, z.B. die, dem Leser etwas besonders eindringlich darzustellen, so als wäre es gerade aus dem Gedächtnis des Sprechers hervorgeholt worden. Dass man das historische Präsens nicht mit dem Präteritum gleichsetzen soll, sieht man an Texten, die beide Formen mischen: In dem Roman The Handmaid’s Tale von Margaret Atwood (1986) z.B. mischen sich Kapitel im Präteritum mit Kapiteln im historischen Präsens. Daher ist die absolute Gleichsetzung des historischen Präsens mit dem Präteritum keine besonders gute Lösung. Die Ungereimtheit kann hier nur beseitigt werden, indem man – entgegen Reichenbach – S tilgt. Nun ein anders ,gestricktes‘ Beispiel zum Thema „Sprech-Zeitpunkt“: Um ca. 1860 herum verfasste Jules Verne einen futuristischen Roman Paris im 20. Jahrhundert. Das Jahr 1860, die Gegenwart des Autors, sei der Sprech-Zeitpunkt S für alle Sätze des Romans. Von diesem Sprech-Zeitpunkt S her gesehen hätten die zeitlichen Parameter S, R und E für alle Sätze des Romans so kombiniert werden müssen, dass der Sprech-Zeitpunkt S und der mit S gleichzeitige Referenz-Zeitpunkt R vor den Ereignis-Zeitpunkten Ei liegen. Verne hätte also den Roman im Futurum, (14), S,R – E, schreiben müssen, während er in Wirklichkeit im ›passé simple‹, (10), E,R – S, geschrieben ist, wo S zeitlich nach den Ei und R liegt, also die Ereignisse als schon geschehen dargestellt werden. Hätte Verne seinen Roman im Jahre 2006 = Sprech-Zeitpunkt S als Rückblick auf das 20. Jahrhundert verfasst, würde die Kombinatorik von S, R und E genau so aussehen, wie sie in Vernes Roman tatsächlich aussieht. Dies ist eine paradoxe Situation, die nur beseitigt werden kann, indem man auch hier den Sprech-Zeitpunkt S tilgt. Ich verweise auf das Mauthner-Zitat B3, 67, § 11 oben. Eine einfache Lösung ist: Mit Reichenbach besteht der einzige Unterschied zwischen Präsens und Präteritum in der Position des Sprech-Zeitpunkts S auf dem Zeitpfeil. Das kann für Texte so umgedeutet werden, dass in fiktiven Narrativen S – und damit das Ich des Autors – im allgemeinen vernachlässigt werden kann (vgl. Maingueneau 2000, 57f.).

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13. Der Ereignis-Zeitpunkt E Der Ereignis-Zeitpunkt E ist nach Reichenbach der Zeitpunkt, zu dem sich etwas im Text ereignet. Da die Art, wie die Abfolge der verschiedenen Ei geschildert wird, nach ihm neben S auch zum jeweiligen Referenz-Zeitpunkt R relativ ist, wird dies unten behandelt (§ 15).

14. Die mauthnersche Annäherung an den EreignisZeitpunkt E Die mauthnersche Annäherung an den Ereignis-Zeitpunkt E findet sich wahrscheinlich im Zitat B3, 67, § 11 oben.

15. Der Referenz-Zeitpunkt R I tend to look at the world outside of Sweden as a literary phenomenon, something that exists in books and magazines. […] Paris is something which lives in the Goncourt brothers’ diaries, the most modern London is that of the early novels of Aldous Huxley. […] In my system, different times operate in different places. In Paris, for example, the mortar dust of the commune has hardly settled. What kind of time operates here? The now. (Lars Gustafsson, The Death of a Beekeeper)

Der Referenz-Zeitpunkt R ist nach Reichenbach der Zeitpunkt, von dem aus der Inhalt eines Satzes erzählt oder auch interpretiert wird. Reichenbach drückt sich hier nicht ganz klar aus: Einerseits spricht er vom „positional use of the reference point“, d.h. dem einen Referenz-Zeitpunkt R, der in gewissen Fällen verschiedene Positionen einnehmen kann; andererseits geht er so vor, also ob es entweder zeitlich verschiedene oder zeitlich inzidente, jedenfalls aber unterscheidbare Referenz-Zeitpunkte Ri gäbe, wie man auch an den Beispielen sehen kann (147, 293f.). Neuere Untersuchungen haben Reichenbachs Analyse des ReferenzZeitpunkts für bestimmte komplexe Sätze auch für Texte bestätigt: Es ist der Referenz-Zeitpunkt R, der die temporale Struktur eines erzählenden Textes kennzeichnet (vgl. Reichenbach 1947, 294f. für komplexe Sätze). Als erstes die Permanenz des Referenz-Zeitpunktes:

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(17) (17a) Elissa hatte den Brief schon aufgegeben, (17b) als Julian kam (17c) und die Neuigkeiten erzählte. In reichenbachscher Formulierung (1947, 293): (18) (17a′) E1 – R1 – S (17b′) R2,E2 – S (17c′) R3,E3 – S Nach Reichenbach fallen hier die drei Referenz-Zeitpunkte R1, R2, R3 für (17a), (17b) und (17c), zusammen, anders ausgedrückt: Es gibt einen Referenz-Zeitpunkt R, und drei Ereignis-Zeitpunkte E1, E2, E3. Reichenbach spricht hier von der Permanenz des Referenz-Zeitpunkts (1947, 294f.). Reichenbach erklärt Fälle wie (17) näher wie folgt: „als“ („when“ in seinem Beispiel) deutet darauf hin, dass (17a) temporal mit (17b–c) als Ganzes verglichen wird, und daher haben wir einen permanenten Referenz-Zeitpunkt R. Das gilt auch für ein Beispiel wie (19) (19a) Ich schickte ihn aus dem Zimmer, (19b) nahm meine Bücher, (19c) trug sie zum Bett. (Abraham Jehoschua, Das wachsende Schweigen des Dichters) Mit Reichenbach dargestellt (1947, 294f.): (20) (19a′) E1,R1 – S (19b′) E2,R2 – S (19c′) E3,R3 – S Die naheliegendste Vorstellung wäre hier, dass es sich in (19) um drei verschiedene Ei und drei verschiedene ,eigenzeitliche‘ Ri handelt. (19) ist aber ein Text, der von einem einzigen Referenz-Zeitpunkt vor dem SprechZeitpunkt S aus zu verstehen ist. Eine reichenbachsche Lösung ist also auch für (19), dass es zwar drei verschiedene Ei gibt, aber drei identische Ri, in anderen Worten: ein permanentes R. S wird wie immer für alle drei Sätze von (19) als permanent angesehen, sodass sich nur die zeitliche Distanz zwischen S einerseits und den Ei andererseits ändert, bezogen auf ein R. Diese Permanenz des Referenz-Zeitpunkts gilt aber nicht immer. Rei-

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chenbach entwickelt das Konzept des ›positional‹ Gebrauchs des ReferenzZeitpunktes R wie folgt: In (21) I met him yesterday. sind E und R gleichzeitig, R ist permanent. Wenn wir hingegen sagen: (22) Yesterday I had already met him. dann bezeichnet „yesterday“ den Referenz-Zeitpunkt R, aber E liegt (wegen des Plusquamperefekts) vor R. In (22) haben wir daher den ›positional‹ Gebrauch von R. Verständlich ist diese Analyse nur, wenn wir für (22) einen Kontext wie „I met him on October 14, the day before yesterday“ haben; siehe die Analyse von (23). Allgemein: Immer wenn wir keine identischen Referenz-Zeitpunkte Ri finden können, dann liegt der ›positional‹ Gebrauch von R vor, wie auch in einem anderen Beispiel von Reichenbach: (23) (23a) He was healthier (23b) when I saw him (23c) than he is now. Die reichenbachsche Analyse (1947, 295): (24) (23a′) R1,E1 – S (23b′) R2,E2 – S (23c′) S,R3,E3 (23a) und (23b) sind im selben Tempus, dem Präteritum; aber (23c) ist im Präsens. Hier kann kann wegen der Abfolge der Ei zwischen R1, R2 einerseits und R3 andererseits keine temporale Identität festgestellt werden, daher ist R in diesem Fall gewissermaßen ,beweglich‘, wird ›positional‹ gebraucht. Welche Art von Referenz-Zeitpunkt vorliegt, ist, wie man hier und schon bei (22) sieht, vom Kontext abhängig, so auch Reichenbach (1947, 288). Diese Darstellungen sind nicht grundsätzlich verschieden von denen bei Mauthner, aber natürlich um sehr Vieles genauer und auch eleganter. Anstelle der Beweglichkeit des temporalen Koordinatenystems bei Mauthner treten hier der ›positional‹ Gebrauch von R, bzw. seine Permanenz, in Verknüpfung mit der Abfolge von Ereignis-Zeitpunkten Ei.

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16. Mauthnersche und jespersensche Annäherung an den Referenz-Zeitpunkt R Weder Mauthner noch Jespersen haben explizit den Referenz-Zeitpunkt R eingeführt, obwohl Mauthner ihm sehr nahe kommt, näher vielleicht als Jespersen. Nach Reichenbach hätte Jespersen die drei Parameter für das Plusquamperfekt und das Futurum exactum angedeutet, aber nicht für die anderen Tempora (Reichenbach 1947, 290, Fn. 1; Jespersen 1965 [1924], 262f.). Es scheint mir aber, dass Jespersen nur die Abfolge von Ereignissen darstellt, und damit eher mit dem Ereignis-Zeitpunkt implizit auch den Referenz-Zeitpunkt; der Sprech-Zeitpunkt S fehlt. Mauthner und Jespersen diskutieren das Problem an Hand des grammatischen Jetzt, des Präsens, Mauthner auch kognitiv im Zusammenhang mit dem vom Gedächtnis hervorgerufenen, vorgestellten Jetzt. Jedenfalls, Jespersen nähert sich dem Referenz-Zeitpunkt R mit folgender Bemerkung: Die theoretische Gegenwart bewegt sich auf dem nach rechts gerichteten Zeitpfeil stets nach rechts weiter fort – und, so können wir hinzufügen: Um dies zu erkennen, brauchen wir einen Referenz-Zeitpunkt R: But what’s the present time? Theoretically it is a point, which has no duration, any more than a point in theoretic geometry has dimension. The present moment, “now,” is nothing but the ever-fleeting boundary between the past and the future […] (1965 [1924], 258; 1964 [1933], 237; vgl. Frawley 1992, 337f.) Die Gegenwart als „die stets flüchtige Grenze zwischen der Vergangenheit und der Zukunft“ ist das Gegenstück zu Mauthners „stets verlorenem Besitz“ (B3, 68). Viele Mauthner-Zitate treffen sich mit dem reichenbachschen Konzept des Referenz-Zeitpunktes R: Unter dem „Nullpunkt“ im folgenden Zitat müssen wir aber, entgegen der in diesem Artikel verwendeten Terminologie, das zeitlose, stets präsentische Ich als Fixpunkt verstehen: […] genau so wie der Schnittpunkt des Koordinatensystems für unsere Augen durch unser Gehirn geht, so ist der Nullpunkt [hier: Schnittpunkt oder Fixpunkt, EL] für die Erstreckung der Zeit immer unsere Gegenwart; der Nullpunkt bleibt bei uns, während wir in der Zeit weiterleben,

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wie das Koordinatensytem des Raumes sich mit uns bewegt […] (B1, 453, auch 77; B3, 129f.; W1, 555; vgl. auch PB, § 75). Am nächsten kommt vielleicht das folgende Zitat dem Konzept des Referenz-Zeitpunktes R: Nun aber können wir bei der Zeit wie beim Raum den Ausgang von einem Punkte nehmen, der vor oder hinter uns liegt. Messen wir von einem Punkte, der hinter uns liegt, so beziehen wir Vergangenheit und Zukunft auf diesen Punkt, so daß dessen relative Zukunft für unsere persönliche Gegenwart schon Vergangenheit ist. (B3, 40) „Persönliche Gegenwart“ ist eine Umschreibung für das stets präsentische Ich. Eine Analogie: Wenn wir von hier nach den USA reisen, dann verändern sich für uns die terrestrischen zeitlichen Koordinaten – aber das idealisierte Ich bleibt dasselbe.

17. Mauthners Analysen in reichenbachscher Form Was das Mauthner-Zitat B3, 40 (§ 16) beschreibt, nimmt das bei bestimmten Tempora vorkommende Auseinanderfallen von Referenz-Zeitpunkt R und Sprech-Zeitpunkt S bei Reichenbach vorweg. Nach Reichenbach ist das Auseinanderfallen von E, R und S die Konstellation für das Plusquamperfekt, (9), E – R – S, das folgerichtig bei Mauthner als Illustration von B3, 40 dient: (25) (25a) Nachdem das deutsche Volk Napoleon besiegt hatte, (25b) fügte es sich den alten Regierungen. (B3, 40) Wir vernachlässigen, dass auf den Nebensatz (25a) der Hauptsatz (25b) als Kontext folgt, und wir daher einen Fall wie (17) vor uns haben. Mauthner meint übrigens, dass in der Formulierung (26) (26a) Das deutsche Volk besiegte Napoleon und (26b) fügte sich dann den alten Regierungen. (B3, 40) (26a) den Sinn des Plusquamperfektums habe (vgl. Jespersen 1964 [1933),

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246f.). Heute sagen wir textlinguistisch entweder, dass (26) ikonisch die ordo naturalis wiedergibt (vgl. B3, 246f.), (25) aber nicht. Als nächstes eine reichenbachsche Analyse von Mauthners Beispiel (6c). Es handelt sich um eine ›posterior past‹, d.h. es soll eine Zukunft in der Vergangenheit ausgedrückt werden, wofür es kein eigenes Tempus gibt. Wir erhalten den ersten Fall aus (11): R – E – S. Der Erzähler beschreibt Ereignisse, die schon stattgefunden haben; daher befindet sich S zeitlich nach R und E. Der Referenz-Zeitpunkt R von (6c) ist die Vergangenheit, in der das Epos spielt. Der Ereignis-Zeitpunkt liegt aber in der Zukunft dieser Vergangenheit, also zeitlich nach R. Dasselbe würde für die Analyse von (7) gelten, wo das Tempus durch ein Adverbial ersetzt wird.

18. Ausblick Seit den drei „Pionieren“ Mauthner, Jespersen und Reichenbach hat sich die Analyse der Tempora in der natürlichen Sprache bedeutend weiter entwickelt. Erstaunlich viele der modernen linguistischen Analysen der Tempora sind direkt oder indirekt von Reichenbach beeinflusst. Ein Teil dieser Entwicklung wird in Leinfellner 1993 dargestellt; die ganze Entwicklung könnte man nur in einer Monographie beschreiben.

19. Ein Zitat anstelle einer Zusammenfassung Es gibt einen literarischen Autor, der sehr stark von Mauthner beeinflusst wurde, und das ist Borges. Einer seiner vielen Bemerkungen zum Thema „Zeit“ als ,Eigenzeit‘ (§ 10) und historische Zeit gibt diesem Artikel einen kognitiven Rahmen (für das Thema „historische Zeit“ bei Mauthner siehe den sehr ausführlichen Artikel „Geschichte“ in W1, 592ff.). Borges sagt da: In the first part of August, 1824, Captain Isidoro Suárez, at the head of a squadron of Peruvian hussars, decided the victory of Junín; in the first part of August, 1824, De Quincey published a diatribe against Wilhelm Meisters Lehrjahre; these events were not contemporary (they are now), since the two men died – one in the city of Montevideo, the other in Edinburgh – without knowing anything about each other … (Borges 1964, 222f.)

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Literatur Siglen: B1–B3, DBW, W1–W3 siehe unter Mauthner; BBB, DB, PB, VB siehe unter Wittgenstein. Adams, John K. 2000 „Narrative Theory and the Executable Text“, Journal of Literary Semantics 29, 171–181. Borges, Jorge Luis 1964 „A New Refutation of Time“, in: Borges, Jorge Luis, Labyrinths: Selected Stories and Other Writings, New York, NY: New Directions, 217–234. Bouwsma, O. K. 1986 Wittgenstein: Conversations 1949–1951, hg. von J. L. Craft und Ronald Hustwit. Indianapolis, IN: Hackett. Cercone, N. und Schubert, L. K. 1974 A Sketch of State-based Conceptual Representation (= TR74-19, University of Alberta), Edmonton: University of Alberta. Couper-Kuhlen, Elizabeth 1989 „Foregrounding and Temporal Relations in Narrative Discourse“, in: Schopf, Alfred (Hg.), Essays on Tensing in English 2, Tübingen: Niemeyer, 7–29. Davies, Paul 1995 About Time: Einstein’s Unfinished Revolution, New York: Simon & Schuster. Frawley, William 1992 Linguistic Semantics, Hillsdale, NJ: Erlbaum. Jespersen, Otto 1914 „Mauthner, Fritz, Beiträge zu einer Kritik der Sprache: Dritter Band […]“, Geisteswissenschaften Jg. 1, 915–916. — 1965 [1924] The Philosophy of Grammar, New York: Norton. — 1964 [1933] Essentials of English Grammar, University, AL: University of Alabama Press. Langacker, Ronald 1988 „A View of Linguistic Semantics“, in: RudzkaOstyn, Brygida (Hg.), Topics in Cognitive Linguistics, Amsterdam: Benjamins, 49–90. Leinfellner, Elisabeth 1993 „Reichenbachs Einfluß auf die Linguistik“, in: Haller, Rudolf und Stadler, Friedrich (Hg.), Wien – Berlin – Prag, Wien: Hölder-Pichler-Tempsky, 297–319. — 1995a „Die böse Sprache: Fritz Mauthner und das Problem der Sprachkritik und ihrer Rechtfertigung“, in: Leinfellner, Elisabeth und Schleichert, Hubert (Hg.), Fritz Mauthner: Das Werk eines kritischen Denkers, Wien: Böhlau, 57–82. — 1995b „Fritz Mauthner im historischen Kontext der empiristischen,

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analytischen und sprachkritischen Philosophie“, in: Leinfellner, Elisabeth und Schleichert, Hubert (Hg.), Fritz Mauthner: Das Werk eines kritischen Denkers, Wien: Böhlau, 145–163. Mach, Ernst 1905 Erkenntnis und Irrtum: Skizzen zur Psychologie der Forschung, Leipzig: Barth. — 1986 Auszüge aus den Notizbüchern 1971-1910, in: Haller, Rudolf und Stadler, Friedrich (Hg.), Ernst Mach: Werk und Wirkung, Wien: HölderPichler-Tempsky, 167–211. Maingueneau, Dominique 2000 Linguistische Grundbegriffe zur Analyse literarischer Texte, Tübingen: Narr. Mauthner, Fritz 1982 [31923] B1–B3 = Beiträge zu einer Kritik der Sprache 1–3, Frankfurt am Main/Wien: Ullstein. — 1996 [1923–1924] W1–W3 = Wörterbuch der Philosophie, Wien: Böhlau. — 1925 DBW = Die drei Bilder der Welt: Ein sprachkritischer Versuch, hg. von Monty Jacobs, Erlangen: Verlag der Akademie. Ploog, Detlev 2000 „Zeit und Zeitmaße im Gehirn“, Universitas 55, 1161– 1175. Reichenbach, Hans 1977 [1928] Philosophie der Raum-Zeit-Lehre, hg. von Andreas Kamlah und Maria Reichenbach, Braunschweig: Vieweg. — 1947 Elements of Symbolic Logic. New York: The Free Press. Schubert, L. K. 1976 „Extending the Expressive Power of Semantic Networks“, Artificial Intelligence 7, 163–198. Spitzer, Leo 1919 „Fritz Mauthner, Beiträge zu einer Kritik der Sprache“, Literaturblatt für Germanische und Romanische Philologie 49, Sp. 201–212. Wittgenstein, Ludwig 1965 [1958] BBB = The Blue and Brown Books, hg. von Rush Rhees, New York [etc.]: Harper & Row. — 1994 VB = Vermischte Bemerkungen: Eine Auswahl aus dem Nachlass, hg. von Georg Henrik von Wright, Frankfurt am Main: Suhrkamp. — 1999 DB = Denkbewegungen: Tagebücher 1930–1932, 1936–1937, hg. und komm. von Ilse Somavilla, Frankfurt am Main: Fischer Taschenbuch. — 1964 PB = Philosophische Bemerkungen, hg. von Rush Rhees, Frankfurt am Main: Suhrkamp. Für Vorschläge zur Verbesserung danke ich Werner Leinfellner und Michael Stöltzner.

Zeit, Performanz und die ontosemantische Struktur des Kunstwerks Constanze Peres, Dresden Die Relation von „Zeit“ und „Kunst“ eröffnet ein multiples Bezugsfeld. Es ist auf der einen Seite von der Struktur des Kunstwerks und den unterschiedlichen Symbolisierungsmodi der Künste markiert, auf der anderen Seite von der phänomenalen und theoretischen Perspektive auf „Zeit“. Das Thema soll in fünf Schritten entwickelt werden: I. Im ersten Schritt wird eine ontologische Struktur des Kunstwerks skizziert, die Temporalität als Konstituens enthält. II. Indem die Konstitution des Kunstwerks näher als Symbolisierung charakterisiert wird, ist die ontologische Struktur des Kunstwerks im zweiten Schritt genauer als ontosemantische Struktur zu bestimmen, die Performanz als Konstituens enthält. III. Im dritten kurzen Schritt ist zu klären, in welchem Sinne der Ausdruck „Zeit“ relevant für Kunst im hier diskutierten Theoriekontext ist. IV. Im vierten Schritt werden anhand von Beispielen künstlerisch symbolisierbare und symbolisierte Aspekte von Zeit(theorien) erläutert. V. Im fünften Schritt möchte ich ein aufsehenerregendes Kunstprojekt vorstellen, das eine vielfältige Bezugnahme auf temporale Aspekte exemplifiziert. Die Veröffentlichung von Vorträgen nach einem Symposion eröffnet die konstruktive Einarbeitung der in den Diskussionen aufgeworfenen Fragen, Einwände und Widerlegungen in die schriftliche Fassung. Innerhalb der fünf Schritte dieses Beitrages werden deshalb an entsprechender Stelle sachlich mögliche wie auch tatsächlich vorgebrachte Einwürfe dargelegt und ihre Erwiderungen im Rahmen der vorliegenden Konzeption zumindest skizziert.

I. Zeit und die ontologische Struktur des Kunstwerks 1. Die Struktur des Kunstwerks als eine ästhetisch-prozessuale 3-stellige Relation Bisher war schon mehrfach von „Struktur“ die Rede. Da der Begriff in der heutigen Philosophie (und Mathematik) in unterschiedlicher Weise verwenF. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 363–385.

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det wird, ist seine hier geltende Bedeutung festzulegen. Danach ist „Struktur“ im ontologischen Sinne der komplexe und intrinsisch in spezifischer Weise geordnete Zusammenhang von Entitäten bzw. einer bestimmten Entität. In diesem Sinne ist die Struktur der Entität „Kunstwerk“ näher als ästhetisch-prozessuale 3-stellige Relation zu kennzeichnen. Dabei enthält der Ausdruck „prozessual“ den temporalen Aspekt. Für „ästhetisch“ kann man einsetzen „ganzheitlich sinnlich-emotiv-intellektuell“, womit sowohl jemandes Zugangsweise zu etwas als auch die Zugänglichkeit von etwas näher charakterisiert wird. Entscheidend ist weiterhin die Tatsache, daß innerhalb dieser Relation jedes der drei Relata anders ontologisch bestimmt ist als außerhalb der 3-stelligen Relation: Innerhalb der Kunstwerks-Relation „Guernica“ ist Picasso eine andere – spezifisch künstlerische – Entität als in der persönlichen Paar-Relation mit Dora Maar. D.h. das Verhältnis jedes Relatums zur 3-stellig relationalen Gesamtentität ist kein äußerlichkompositionales, sondern ein intrinsisch-kontextuales Verhältnis. Dies ist so aufzufassen, daß keines der konstituierenden Relata ohne die anderen Relata existieren könnte oder positiv ausgedrückt, daß jedes Relatum nur im Kontext der ganzheitlichen relationalen Entität „Kunstwerk“ existiert und von daher zu bestimmen ist. Je nachdem, welchen Term man an die Subjektstelle des Satzes setzt, läßt sich die prozessuale 3-stellige Relation in ihrer einfachsten Form in drei Versionen ausdrücken, in einer produktionsästhetischen, einer werkästhetischen und einer rezeptionsästhetischen Version (vgl. Peres 2000, 27 ff.): V 1ae prod:

Jemand produziert – etwas als Kunstwerk X – für jemanden, der es als (dieses) Kunstwerk X1−n realisiert

V 2ae werk: Etwas als Kunstwerk X – wird von jemandem produziert – für jemanden, der es als (dieses) Kunstwerk X1−n realisiert V 3ae rez :

Jemand realisiert als (dieses) Kunstwerk X1−n – etwas (als Kunstwerk X) – das von jemandem als Kunstwerk X produziert wird.

2. Die drei Relata Die 3-stellige Relation ist, der Einfachheit halber in der Reihenfolge V 1 ae prod, näher zu erläutern: 1. Relatum: Die Instanz, die „etwas produziert“, kann Produzent oder Konstituent 1. Stufe genannt werden. Das sind z.B. Komponisten, Maler,

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Performer, Installationskünstler, Schriftsteller, Bildhauer usw. Diese Instanz produziert immer etwas für jemanden. Das ist nicht gleichbedeutend mit spezifischen Rezipienten, sondern meint lediglich Adressaten überhaupt. Es gibt immer mindestens einen Adressaten und das ist der Künstler, der mit und nach der Beendigung des Schaffensprozesses in Bezug auf die abgeschlossene Schöpfung nicht mehr als Künstler fungiert. Er hat dann sozusagen ‚die Seite gewechselt‘ und ist zum (häufig) ersten Rezipienten des Resultates des Schaffensprozesses geworden. 2. Relatum: Traditionell wurde in der Ästhetik das, was jemand produziert, als individuelles und singuläres „Kunstwerk“ bezeichnet. Das Problem ist bekanntlich, daß dieser Begriff eine ganze Reihe von Kunstwerken verfehlt. Zu diesem Typ des Kunstwerks gehören ein- oder mehrfach vollzogene Performances, die den Zuschauer als ko-agierende Instanz einbeziehen, ohne deren aktive Teilnahme die Performance nicht funktionieren würde. Aber auch die musikalische Partitur und der dramatische Text benötigen den Prozeß ihrer jeweiligen Aufführung für (oder mit) jemanden, um Musik- oder Theater-Kunstwerk zu sein. Deshalb bezeichne ich das „Etwas“, das der Künstler als Kunstwerk X produziert, als „Kunstwerk-Schema“, das Interpretation benötigt, um ein vollständiges Kunstwerk zu sein. Dieses Schema ist nicht abstrakt, wie der Ausdruck „Schema“ vielleicht suggerieren mag, sondern es ist singulär und durch seinen Urheber als unvollständig bestimmt, d. h. dazu bestimmt, durch seine verschiedenen Interpretationen individuell vervollständigt zu werden. Derart ist das singuläre und unvollständig bestimmte Kunstwerk-Schema offen für alle seine Realisationen. Entscheidend ist aber, daß auch die Kunstwerke, die durch sich selbst vollständig bestimmt zu sein scheinen – wie etwa Gemälde, Skulpturen, Romane oder Gedichte – ebenfalls singuläre und unvollständig bestimmte Kunstwerk-Schemata und offen für alle ihre Realisationen sind. Wenn es, sagen wir, 50 Interpretationen von Giorgones Gemälde „La tempesta“ oder von Thomas Bernhards Roman „Wittgensteins Neffe“ gibt, dann kann man nicht sagen, es gebe das eine Kunstwerk „La tempesta“ oder das eine Kunstwerk „Wittgensteins Neffe“. 50 Betrachter werden „La tempesta“ unterschiedlich sehen und erleben. 50 Leser werden sich z. B. den Neffen in der Klinik, seine Gestik, die Farben und Gerüche der beschriebenen Situationen oder die Stimmen der handelnden Personen unterschiedlich vorstellen, d. h. imaginativ ihre Romanwelt des Romans schaffen. In beiden Fällen haben wir 50 mehr oder weniger differierende Realisationen, d. h. indivi-

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duell realisierte Kunstwerke X1–50 und für „X“ setzen wir im ersten Fall „La Tempesta“ und im zweiten Fall „Wittgensteins Neffe“ ein. Die Offenheit für die Vielfalt der Realisationen impliziert nicht Beliebigkeit. Das singuläre und unvollständig bestimmte Kunstwerk-Schema verbietet die Möglichkeit irgendwelcher Interpretationsakte und verhindert ein „anything goes“. Immer sind es Realisationen dieses singulären Kunstwerk-Schemas. Man kann eben nicht „La Tempesta“ in der Betrachtung als Früchtestilleben und „Wittgensteins Neffe“ im lesenden Imaginieren als Göttermythos oder Comic realisieren. 3. Relatum: In jeder Kunstwerks-Relation gibt es eine Instanz, die das Kunstwerk-Schema als bestimmtes Kunstwerk erfährt und individuell vervollständigt. Diese Instanz ist ebenfalls mehr oder weniger produktiv, denn bereits beim Betrachten einer künstlerischen Photographie, beim Anhören einer Symphonie oder beim imaginierenden Lesen eines Romans sind selektierende, akzentuierende, löschende und ergänzende Prozesse im Spiel. Wir realisieren jeweils eine Welt-Version der, vielleicht nah beieinander liegenden, aber verschiedenen Realisierungsmöglichkeiten des gegebenen Kunstwerk-Schemas. Deshalb nenne ich die Instanz, die etwas als Kunstwerk X1−n realisiert, Konstituent 2. Stufe, KoKonstituent oder KonProduzent. In musikalischen, theatralen und ähnlichen Kunstwerken bedarf es zusätzlich einer medialen Instanz. Dirigenten, Performer, Musiker, Tänzer, Choreographen, Drucker, Schauspieler etc. sind die künstlerischen Konstituenten 2. Stufe, KonProduzenten oder KoKonstituenten des Kunstwerks X1−n. Innerhalb dieser Konstellation rückt der nicht-künstlerische Konstituent eine Stufe weiter. Er wird zum Konstituenten 3. Stufe (KonKoKonstituent oder KoKonProduzent) des Kunstwerks X1−n1−n. Im Überblick stellt sich die ontologische Struktur des Kunstwerks genauer so dar:

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3stellige → prozessuale Relation Bereich ↓ (einfach) kokonstituierende Künste

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Relatum 1

Relatum 2 Relatum 3

jemand produziert

etwas als für jemanden, der es als (dieses) KunstKunstwerk X1−n realisiert werkSchema X

der (künstlerische) Konstituent 1.Stufe produziert

Kunstder (nicht-künstlerische) Konstiwerktuent 2.Stufe kokonstituiert Schema X ein Kunstwerk X1−n

doppelt ko- der (künstkonstituielerische) rende Künste Konstituent 1.Stufe produziert

Kunstder (künstwerklerische) Schema X Konstituent 2.Stufe konproduziert das Kunstwerk X1−n

der (nicht-künstlerische) Konstituent 3.Stufe kokonstituiert das Kunstwerk X1−n1−n

Die hier entwickelte Struktur des Kunstwerks wirft eine Fülle von Problemen auf, von denen nur zwei als zentral herausgegriffen werden. Beide sind so komplex, daß sie einer eingehenden Diskussion bedürften. Die folgende knappe Darstellung kann deshalb nur die Richtung angeben, in der die Konzeption weiterzuentwickeln ist, aus der heraus die aufgeworfenen Fragen zureichend beantwortet werden können.

3. Kunstphilosophischer Einwand Eine wichtige Frage, die im Rahmen einer Produktionsästhetik an die hier vorgelegte Konzeption der dreistelligen Struktur des Kunstwerks herangetragen werden kann, lautet: Wo haben in dieser Struktur des Kunstwerks die spezifische Kreativität und die außergewöhnliche Leistung von Künstlern ihren Ort? Oder anders ausgedrückt: Gibt es einen Unterschied in den kreativen Prozessen der zwei bzw. drei Instanzen und wenn ja, wie ist er zu fassen? Die Antwort ist so zu skizzieren: Das genuin Kreative im Sinne des Erschaffens von Etwas vermindert sich von Stufe zu Stufe. Dabei wird der Gradunterschied zwischen dem künstlerischen Produzieren 1. Stufe und

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dem künstlerischen Konproduzieren 2. Stufe bereits groß sein, aber er wird relativ geringer ausfallen als zwischen diesen beiden Hervorbringungsprozessen und dem des nichtkünstlerischen KoKonstituenten. Neben diesen graduellen Unterscheidungen aber besteht der kategoriale Unterschied darin, daß die Realisation des jeweiligen Kunstwerks durch den bloßen kokonstituven Rezeptionsakt (3. Stufe) kein eigenständiges, künstlerisches Produkt für andere hervorbringt und setzt. Dem könnte entgegengehalten werden, daß Kunstwissenschaftler und -kritiker ihre theoretisch fundierten Interpretationen schriftlich niederlegen, öffentlich zugänglich machen und damit für andere hervorbringen. Das ist richtig, aber damit handelt es sich eben per definitionem nicht um künstlerische, sondern um kunstwissenschaftliche Produkte. Deren Rezeption kann allerdings hilfreich für das Erleben von Werken sein und derart in verarbeiteter Form (z. B. in einem durch das Wissen um die ikonographische Entschlüsselung veränderten Sehen eines Gemäldes) in den KoKonstitutionsprozess von Kunstwerken eingehen. Auch könnte man sich vorstellen, daß jemand, von der Lektüre von „Wittgensteins Neffe“ angeregt, anfängt, beispielsweise Comics zu zeichnen: In dem Moment aber, wo er den Roman nicht nur in der lesenden Imagination realisiert, sondern zeichnend in Comics transformiert, ist er bereits je nach Ausführung entweder zum künstlerischen Produzenten 2. Stufe dieses Romans (z. B. Illustrator) oder sogar zum künstlerischen Produzenten 1. Stufe eines neuen Kunstwerk-Schemas geworden.

4. Ontologischer Einwand In einigen Diskussionen mit Lorenz Bruno Puntel wurde die Frage nach der Identität des so konzipierten Kunstwerks vertieft. Ontologisch kann nämlich eingewandt werden: Wenn es der vorliegenden Kunstwerk-Struktur zufolge unzählige Realisationsmöglichkeiten eines Kunstwerk-Schemas wie der „Mona Lisa“ von Leonardo da Vinci oder der „Winterreise“ von Franz Schubert und bereits eine bestimmte Anzahl faktischer Realisationen gibt: Was ist oder worin besteht die Identität des Kunstwerks „Mona Lisa“ oder der „Winterreise“? Anders ausgedrückt: Wie steht es mit der Identität der Kunstwerke „Mona Lisa“ und „Winterreise“ durch alle ihre Realisationen und alle Zeiten hindurch und wie kann das Verhältnis der jeweiligen Realisationen untereinander gekennzeichnet werden? Dieser Einwand ist schwerwiegender und komplexer als der erste, da es sich um ein Subproblem der allgemeinen ontologischen Frage nach dem Sein bzw. der Seinsidentität von etwas handelt. Sie tritt in vielen Spiel-

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formen auf, von denen hier die Frage nach dem Verhältnis von Universale und Instantiierungen oder von type und tokens bzw. Typ und Vorkommnissen eine Lösungsrichtung vorzugeben scheint. Richard Wollheim hat diese Frage allerdings weitgehend in der Einschränkung auf die sogenannten nichtmateriellen Künste (z. B. Musik) behandelt (vgl. Wollheim 1982, 76 ff.). Innerhalb der vorliegenden Konzeption jedoch greifen diese Verhältnisbestimmungen nicht (am nächsten kommt vielleicht der Ansatz von Schmücker 22005a): Prima facie böte sich das Kunstwerk-Schema als Kandidat für dasjenige an, was an den Kunstwerken „Mona Lisa“ und „Winterreise“ durch alle ihre Realisationen und alle Zeiten hindurch der je identisch bleibende Orientierungspunkt bleibt. „Orientierungspunkt“ trifft zu, „identisch bleibend“ nicht: Konzipiert man nämlich die 3-stellige Relation des Kunstwerks ontologisch konsequent, so kann nicht eines der Relata – in diesem Fall das genuin erschaffene Kunstwerk-Schema – für sich beanspruchen, identisch zu sein und zu bleiben, während seine Realisationen je andere bzw. sich verändernde Entitäten sind. Als ein konstitutives Element der Gesamtrelation, d. h. als relationale Entität ist es produziert von einem Urheber, offen für alle seine Realisationen und in jeder Realisation anders konstitutiv. Deshalb muß gesagt werden: Das Kunstwerk-Schema und alle seine Realisationen durch alle historischen Zeiten „bis in alle Ewigkeit“ konstituieren zusammen die individuelle, komplexe und identische Totalität des Kunstwerks „Mona Lisa“ oder „Winterreise“. Oder noch anders: Das Kunstwerk, also die „Mona Lisa“ bzw. die „Winterreise“ ist die individuelle, komplexe und identische Totalität des erschaffenen Kunstwerk-Schemas und aller seiner mit-erschaffenen Realisationen1−n (2. und 3. Stufe) in der Zeit. Alle Realisationen1−n der „Mona Lisa“ oder der „Winterreise“ wiederum konstituieren unabhängig von ihrer jeweiligen Beschaffenheit und (valuativen) Qualität dadurch die Kunstwerkstotalität, daß sie Realisationen der „Mona Lisa“ oder der „Winterreise“ sind. Um darüber hinaus ihren internen Zusammenhang positiv zu beschreiben, könnte man mit Wittgenstein sagen: Die Realisationen eines Kunstwerk-Schemas bilden ein synchrones und diachrones dynamisches „Netz von Ähnlichkeiten“ (PU § 66) oder eine „Mona Lisa-“ bzw. „Winterreise-Familie“, innerhalb deren sich die Familienähnlichkeiten innerhalb einer Subfamilie und Generation, aber auch über die Generationen hinweg „übergreifen und kreuzen“ (PU § 67). Der Vorteil der Familienähnlichkeits-

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konstellation liegt darin, daß sie eine Bandbreite von ‚fast zum Verwechseln ähnlich‘ bis zu ‚fast schon nicht mehr ähnlich‘ umfaßt. Das sich unmittelbar anschließende Problem der Ähnlichkeitskriterien ist damit noch nicht gelöst, wäre aber eigens und auf einer anderen, nämlich der epistemologischen Ebene zu diskutieren (vgl. Puntel 2006, 3.2.2.4.1.3).

5. Temporale Phasen in der ontologischen Struktur des Kunstwerks Im Rahmen der dreistellig-relationalen prozessualen Konstitutions-Struktur des Kunstwerks gibt es auf diese Weise zwei bzw. drei temporale Phasen: TPh1

die temporale Phase 1, innerhalb deren der Konstituent 1. Stufe sein singuläres und unvollständig bestimmtes Kunstwerk-Schema X produziert, kurz: Konstitutions-Zeit;

TPh2

die temporale Phase 2, innerhalb deren der Konstituent 2. Stufe das Kunstwerk-Schema X als Kunstwerk-Schema X1−n realisiert, kurz: KoKonstitutions-Zeit;

TPh3

ggf. die temporale Phase 3, innerhalb deren der Konstituent 3. Stufe das Kunstwerk-Schema X als Kunstwerk-Schema X1−n1−n realisiert, kurz: KonKoKonstitutions-Zeit; im speziellen Falle des 1−n wiederholbaren Abspielens von Filmen, Musikwerken auf Tonträgern etc. muß die dritte temporale Phase der hörenden bzw. sehenden Realisierung eine zeitliche Subcharakterisierung erfahren: TPh31−n.

Einfach kokonstituierte Kunstwerke wie Gemälde, Skulpturen, Gedichte etc. benötigen nur zwei temporale Phasen: die Produktionszeit (TPh1) und die KoKonstituitions-Zeit (TPh2). Doppelt kokonstituierte Kunstwerke wie Tanz- oder Musikaufführungen können drei temporale Phasen für sich in Anspruch nehmen. In reinen performativen Improvisationen musikalischer, theatraler, körperkünstlerischer Art jedoch können alle drei Phasen zusammenfallen. Im Falle einer geplanten und angekündigten Performance fallen in der Regel die KoKonstitutions-Zeit des oder der künstlerisch Ausführenden (TPh2) und die KonKoKonstitutions-Zeit (TPh3) der nicht-künstlerisch Realisierenden zusammen; beide temporale Phasen können teilkongruent mit der Konstitutionszeit sein, wenn sich in und während der Performance neue produktive Elemente erst ergeben.

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6. Historische Zeit(phasen) Innerhalb der faktischen Werk-Genese können die temporalen Phasen als historische Zeiten oder Zeitphasen bestimmt werden. Ein künstlerischer Konstituent 1. Stufe kann eine oder mehrere historisch bestimmte Produktionsphasen benötigen, oder mehrere künstlerische Konstituenten (Autorenteams, z. B. das Ehepaar Christo) je eine oder zusammen eine Produktionsphase oder je mehrere, ggf. teilkongruente Produktionsphasen. Ein Kunstwerk-Schema kann z. B. im Falle einer Skulptur eine historisch abgeschlossene Zeit haben, im Falle mehrerer Editionen einer Partitur mehrere historische Zeiten. Im Falle der Konstituenten 1. und 2. Stufe wie auch der aufgeführten Kunstwerke 1−n und realisierten Kunstwerke 1−n1−n gibt es 1−n historische Realisierungszeiten. Insofern alle historischen und biographischen Zeiten erfüllt sind von qualitativen Faktoren wie sozialen Umständen, persönlichen Erfahrungen, Haltungen usw., könnte man sie auch mit Gadamer als „Horizonte“ bezeichnen. Ihm zufolge ist jede, wie ich es nenne, „Realisierung“ eines Kunstwerks eine „Horizontverschmelzung“ von Künstler und Rezipient und eine Seins-Weise des Kunstwerks (Gadamer 41975, 239 f., 356 f., 375).

II. Semantische Ebene: Ontosemantische Struktur des Kunstwerks In den letzten Jahren ist viel von der Performativität der Kunst die Rede. Eine kürzlich erschienene „Ästhetik des Performativen“ der Theaterwissenschaftlerin Erika Fischer-Lichte will – in Abgrenzung von hermeneutischen und zeichentheoretischen Ansätzen – als ‚neue Ästhetik‘ dem neuen Typ des Kunstwerks nach der „performativen Wende“ in den Sechziger und Siebziger Jahren des 20. Jahrhunderts gerecht werden. Dagegen möchte ich herausstellen, daß unter Voraussetzung einer geeigneten ontologischen Struktur gerade – und m. E. nur – mit einem symboltheoretischen Ansatz gezeigt werden kann, daß und wie Performanz allgemein ein zentrales Konstituens jedes Kunstwerks aller Zeiten ist. Fruchtbar ist hier der Rückgriff auf Austins Konzeption der „performativen Äußerungen“: danach implizieren performative Akte mindestens zwei Aspekte: erstens konstituieren sie etwas und zweitens sind sie selbstreferentiell und zwar im Sinne der Interrelationalität von Existenzweise und Darstellung (Austin 1965, 307 f., vgl. Fischer-Lichte 2004, 26 f., 32 ff.). Es ist allerdings zu klären, was das genau für das Kunstwerk heißt.

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1. Das Kunstwerk als Symbolisierung Kunstwerke beziehen sich immer auf etwas. Ihre ästhetisch-prozessuale 3-stellige Relation ist deshalb mit Nelson Goodman als Symbolisierung zu charakterisieren. Das Produzieren des Kunstwerk-Schemas ist ein Symbolisierungsprozeß. Das Kunstwerk-Schema ist eine vorliegende oder sich prozessual entfaltende Symbolisierung 1. Stufe. Seine Realisierungen 2. Stufe und 3. Stufe durch künstlerische oder nichtkünstlerische Interpreten sind prozessuale Symbolisierungen 2. Stufe und 3. Stufe. Nun beziehen sich auch Anzeigetafeln oder wissenschaftliche Studien auf etwas, ohne deshalb als Kunstwerke realisiert zu werden. Die ästhetische Symbolisierung der Kunstwerksrelation und ihrer Relata sind deshalb durch die Art der Symbolisierung zu kennzeichnen. Im folgenden soll ein für das Kunstwerk zentraler Symbolisierungsmodus skizziert werden, innerhalb dessen nicht nur die Temporalität, sondern auch die Performanz der Kunst zum Tragen kommt. Gestützt wird diese Konzeption von Goodmans Symboltheorie und besonders seiner Konzeption der exemplifikatorischen Bezugnahme (Goodman 1990, 93f.; 1987, 91ff.). Ich nenne diesen Symbolisierungsmodus selbstreferentiell-performative syntaktisch-semantische Verschränkung. Darin spielt die Verkörperung eine wesentliche Rolle. Statt Goodmans Terminus „exemplifikatorisch“ verwende ich deshalb die Ausdrücke „inkorporierend “, „inkorporativ“ oder „Inkorporation“, weil sie einerseits die Instantiierung und Singularität enthalten, die jeder Verkörperung innewohnen, andererseits den körperlichen und sinnlichemotiven Aspekt des ästhetischen Realisierens konnotieren. Ein künstlerisches Produkt kann mit zweifacher Vollzugsrichtung bezugnehmen und beides buchstäblich und/oder metaphorisch (vgl. Goodman 1990, 127ff.). In metaphorischer Symbolisierung wird entweder mit einer vertrauten Zeichengestalt auf eine neue Sphäre Bezug genommen oder in neuer syntaktische Weise auf eine vertraute Sphäre. Die beiden Vollzugsrichtungen unterscheiden zwei unterschiedliche Bezugnahmemodi, die im Kunstwerk zusammen oder getrennt auftreten können.

2. Denotierender Symbolisierungsmodus der Kunst Der 1. Bezugnahme-Modus ist die künstlerische Denotation. In ihr nimmt eine Symbolgestalt aus einem bestimmten syntaktischen Bereich hinweisend auf ein Objekt aus einer Sphäre Bezug. Alle sogenannten darstellenden bzw. mimetischen Kunstwerke denotieren etwas (für jemanden). Z. B.

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Goyas Gemälde „Saturn frißt seine Kinder“ denotiert buchstäblich den römischen Gott Saturn, der im Mythos bis auf Jupiter alle seine Kinder frißt, um sie als Machtkonkurrenten auszuschalten. Unter der Voraussetzung der (seit der Antike immer wieder vorgenommenen) Gleichsetzung von Saturn und Chronos (= Zeit) kann das Gemälde aber auch den grausamen Charakter der Zeit ausdrücken. Es denotiert dann metaphorisch die Zeit, die als sukzessive Abfolge von Zeiteinheiten verstanden, genau diese „ihre Nachkommen“, die Zeiteinheiten schluckt. Die denotativ-hinweisende Vollzugsrichtung kann vereinfacht so veranschaulicht werden: „X“

X

3. Inkorporierend-performierender Symbolisierungsmodus der Kunst Der 2. Bezugnahmemodus ist die künstlerische Inkorporation. In ihr kehrt sich die Symbolisierungsrichtung sozusagen um: Ein künstlerisches Produkt eines bestimmten syntaktischen Bereichs nimmt auf einen Gegenstand einer Sphäre Bezug, indem es einige charakteristische Konstituentia dieses Gegenstandes, die es selbst besitzt, beispielhaft an sich für andere zeigt und als bedeutsam vorführt oder eben performiert. Oder anders herum: Indem das Kunstwerk auf sich selbst, d. h. auf eigene syntaktische Konstituentia Bezug nimmt und sie performativ als bedeutsam zeigt, nimmt es auf Konstituentia anderer Gegenstände Bezug. Die inkorporative Bezugnahme ist mithin selbstreferentiell, performativ und syntaktisch-semantisch verschränkt. Ein Beispiel, das bereits einen Zusammenhang mit dem temporalen Aspekt herstellt: Die durch den künstlerischen Produzenten 1. Stufe Haydn komponierte und durch die künstlerischen Produzenten 2. Stufe (Orchester und Dirigent) erklingende Symphonie „Die Uhr“ inkorporiert buchstäblich in ihrer Klangabfolge das Geräusch tickender Uhren und ein (dadurch suggeriertes) gleichmäßig unterteiltes Fortschreiten der Zeit und zeigt dies performativ. Sie führt mit dieser inkorporierten Eigenschaft an ihrem Klangverlauf metaphorisch das Vergehen der Zeit vor. Dies wiederum kann von nicht-künstlerischen Konstituenten 3. Stufe metaphorisch als unerbittliches Verrinnen der eigenen Lebenszeit realisiert werden.

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Die inkorporativ-performative Vollzugsrichtung stellt sich verkürzt so dar: das Kunstwerk bezieht sich 1. auf charakteristische konstitutive Elemente von etwas, indem es sich 2. auf sich selbst, d. h. auf seine charakteristischen konstitutiven Elemente bezieht, und diese 3. an sich selbst als bedeutsam für andere performiert/vorführt. für andere

für andere

für andere „X“

X

für andere für andere

für andere

Die wohl lakonischste Charakterisierung dieses Symbolisierungsmodus – wenn auch ohne die eben ausgeführten Implikationen – findet sich bei Arthur C. Danto: „The work of art is about“ (Danto 1985, S. 20, vgl. 89, 111 f.). Hier ist also festzuhalten: Die ontologische 3-stellige prozessuale Struktur des Kunstwerks ist eine ontosemantische Struktur.

4. Kunstphilosophischer Einwand gegen die Semantizität der Kunst Als typischer und auch in der Diskussion vorgebrachter Einwand (Georg Franck) gegen das grundsätzliche „to be about“ des Kunstwerks wird immer wieder die Kunstgattung der Architektur angeführt: künstlerische Bauten bezögen sich auf nichts bzw. nicht auf etwas. Der Einwand trifft zu, wenn man Gebäude, was durchaus richtig ist, als Gebrauchsgegenstände einordnet: Insofern man sie bewohnt, in ihnen Güter lagert, sie als Arbeitsstätte benutzt etc., symbolisieren sie in keiner künstlerisch relevanten Weise; in dieser Hinsicht gelten sie nicht als Kunstwerke. Insofern jedoch Architekturkomplexe Merkmale an sich haben, die über Gebrauch und Funktionalität hinausweisen oder auf sie als bedeutsam verweisen, ist damit bereits ihre Semantizität gegeben: Der Grundriß eines normalen öffentlichen Gebäudes verweist auf nichts; der Grundriß der (zerstörten) Abteikirche von Cluny inkorporierte und führte buchstäblich mathematische Maßverhältnisse

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vor, die zugleich musikalische Schwingungsverhältnisse waren; der Grundriß sollte damit metaphorisch die himmlische Harmonie inkorporieren und performieren. Der Kubus eines einfachen Wohnhauses symbolisiert nichts; der preisgekrönte gläserne Bau des Dresdner Landtages inkorporiert und performiert buchstäblich Leichtigkeit und Helle und metaphorisch die (anzustrebende) politische Transparenz eines öffentlichen Parlaments. Ein hochaufragender Wasserturm ist nichts als ein Turm; die Türme des Wiener Stephansdomes wie auch die aufwärtsstrebenden Bündelpfeiler und hochgewölbten Spitzbögen jeder hochgotischen Kirche inkorporieren und performieren buchstäblich den Höhenzug, metaphorisch die Transzendenz eines Gotteshauses und die Transzendenzorientiertheit gotischer Religiosität.

5. Logischer Einwand gegen die 3-stellige ontosemantische Struktur des Kunstwerks Ein auf der logischen Ebene vorgetragener Einwurf (Nico Strobach) besagt, daß es sich unter Hinzufügung der semantischen Dimension bei der Struktur des Kunstwerks nicht mehr um eine 3-stellige, sondern um eine 6-stellige Relation handelt, sofern jedes der drei Relata auf etwas Bezug nimmt. Der Einwand ist unter der Voraussetzung richtig, daß man erstens Bezugnahme als zusätzliche Charakterisierung zur ontologischen Verfaßtheit des Kunstwerks ansieht und zweitens die drei Relata der 3-stelligen Kunstwerksrelation als gesonderte Komponenten auffaßt. Das wiederum kann man korrekt aus der logischen Perspektive tun, nur trifft es nicht die hier zugrundeliegende Ontologie und damit auch nicht die daraus abgeleitete Konzeption. Die zweite Voraussetzung verkennt, daß es sich bei jeder Kunstwerksrealisierung um eine „echte“ ontologische Relation handelt: es ist keine Beziehung zwischen für sich selbständigen Elementen, sondern von Elementen, die sich nur durch ihre Situierung in der Relation definieren, d. h. als Konstituentia in einer individuellen und komplexen ganzheitlich-prozessualen Entität. Diese komplexe Entität ist derart dreifach relational strukturiert, daß die Relata keine eigenständigen Entitäten sind und demzufolge nicht für sich Bezug nehmen können. Vielmehr sind sie in und bestimmen sie sich aus ihrer Einbindung in den Kontext der individuell-prozessualen Kunstwerk-Realisierung. Für die Widerlegung der ersten Voraussetzung müßte die hier zugrundeliegende Ontologie, die an so umfassende wie methodisch unterschiedliche Konzeptionen wie die von G. W. Leibniz und L. B. Puntel anknüpft, expliziert werden (vgl. z.B. Leibniz 1714 u.ö.; Puntel 2006). Da dies nicht möglich

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ist, sei hier nur so viel gesagt: Der Einwand geht, was durchaus üblich ist, davon aus, daß Sein und Bezugnehmen gänzlich verschiedenen Bereichen angehören. Wie Puntel in seiner struktural-systematischen Philosophie expliziert, sind gegenüber der traditionellen Dichotomie von Sein und Denken, Wirklichkeit und Symbolisierung, Ontologie und Semantik etc. die angeblich dichotomischen Elemente als die „zwei Seiten ein und derselben Medaille“ zu verstehen (Puntel 2006, 3.1.5 [2]). Begründet in dieser ausgeführten Theorie wird hier die Auffassung vertreten, daß die einfachen und basalen Seinseinheiten „Verhalte“ sind, zu deren Sein es wesentlich gehört, daß sie Bezug nehmen bzw. daß auf sie Bezug genommen wird. Die komplexe Entität des individuellen Kunstwerks, d. h. die jeweilige Kunstwerk-Realisierung kann demnach operativ unter ontologischer und unter semantischer Rücksicht erklärt werden, aber es handelt sich um die zwei Seiten einer ‚Sache’: die individuelle Kunstwerk-Realisierung ist/existiert als Ganzes, indem sie Bezug nimmt und nimmt Bezug, indem sie existiert. Insofern ist und bleibt innerhalb der vorliegenden ontosemantischen Konzeption jedes Kunstwerk seiner inneren Strukturierung nach eine 3-stellige Relation, deren Sein als komplexe Entität darin besteht, als Ganze (und mit bestimmten Merkmalen) Bezug zu nehmen.

III. „Zeit“ und temporale Aspekte in der künstlerischen Symbolisierung Um die künstlerische Symbolisierung von Zeit etwas ausführlicher an Beispielen erläutern zu können, ist zunächst zu klären, was es in diesem Zusammenhang heißt, von „Zeit“ zu sprechen und welche Aspekte von „Zeit“ relevant für das künstlerische Symbolisieren sind. Künstlerische Symbolisierungen der Kunst artikulieren weder eine theoretische Erkenntnis der Zeit, noch die bloße Alltagserfahrung, sondern eine ästhetische Erfahrung von Zeit. Das schließt nicht aus, daß alltägliche Zeiterfahrungen konstitutiv werden können. Aber es geht eben nicht um die triviale zeitliche Einteilung unseres Alltags, sondern um die persönliche oder existentielle Bedeutsamkeit, die beiläufige Alltagserfahrungen von Zeit ästhetisch gewinnen können. Es ist auch nicht ausgeschlossen, daß theoretische Zeitbegriffe und -modelle ästhetisch erfahren und künstlerisch artikuliert werden können. Dabei ist es unerheblich, ob das künstlerische Subjekt z. B. die spezielle Relativitätstheorie zunächst in ihren physikalischen Implikationen nachvollzogen und dann assoziativ-gestalterisch umgesetzt hat; es ist sogar unerheblich, inwie-

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weit und ob es diese Theorie überhaupt ,versteht‘. Entscheidend ist nur, daß in der künstlerischen Symbolisierung das ästhetisch erfahrene theoretische Zeitmodell künstlerisch transformiert wird. Im vorliegenden Zusammenhang ist es deshalb weder erforderlich noch erstrebenswert, einen eigenen zeittheoretischen Ansatz zu entwickeln oder sich argumentativ innerhalb eines bevorzugten Zeitmodells zu bewegen. Vielmehr interessiert hier „Zeit“ sozusagen extrinsisch aus einer bestimmten Perspektive. Es geht nicht um Zeit oder Zeittheorien als solche, sondern um künstlerisch symbolisierbare bzw. symbolisierte Aspekte von Zeit und Zeittheorien.

IV. Künstlerisch symbolisierbare Aspekte von Zeit(theorien) – Beispiele Für die künstlerische Umsetzung phänomenaler wie auch naturwissenschaftlicher und philosophischer Perspektiven auf „Zeit“ gibt es unabzählbar viele Beispiele. Zeit ist ein zentraler Bezugspunkt aller Künste von ihren Anfängen bis heute. Ich kann deshalb nur ausgewählte Aspekte herausgreifen und für einige von ihnen Beispiele anführen (vgl. Peres 2003 u. 2000, II.).

1. Sukzessivität Die Sukzessivität der Zeit kann mit McTaggarts B-Reihe der Zeit als Nacheinander von Entitätenzuständen, als Abfolge von Ereignissen, als interrelationale Zeit oder aus der 3. Person-Perspektive beschrieben werden, d. h. als ‚früher als – gleichzeitig mit – später als‘ (vgl. McTaggart 1993, bes. 68ff.). Dieser Aspekt, aber auch die Strukturierung oder Metrisierung in der Abfolge wird buchstäblich von serieller Musik, metaphorisch und buchstäblich von serieller Druckgraphik wie z. B. den Siebdruck-Serien Andy Warhols von Marilyn Monroe inkorporiert und vorgeführt. In beiden Beispielen ist das strukturierende Merkmal die metrisierende Wiederholung: Das liegt in der seriellen Musik auf der Hand; Warhols Serie aber inkorporiert die repetierende Abfolge metaphorisch, insofern das Nacheinander in die Syntax des Nebeneinander übertragen wird und buchstäblich, sofern das sehende Realisieren der Sukzessivität notwendig sukzessiv erfolgt.

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2. Bezugspunkt-relationale Abfolge Die bezugspunkt-relationale Abfolge von Zeit(en), oder mit McTaggart, die A-Reihe der Zeit (vgl. McTaggart 1993, bes. 68ff.) wird aus der 1. Personperspektive mit den Ausdrücken ‚vergangen – gegenwärtig – zukünftig‘ erfaßt: hier hinein fallen z. B. biographische Lebenszeiten bzw. -phasen. Sie können als in der Vergangenheit erlebte Ereignisse oder als gegenwärtig empfundene Abläufe einer oder mehrerer Personen und manchmal auch als deren Zukunftsprojektionen vom jeweiligen Erfahrungs(zeit)punkt aus z. B. im Film buchstäblich und metaphorisch inkorporiert, dargestellt und vorgeführt werden, etwa durch schwarz-weiße Rückblenden oder durch Überblendungen und Wiederholungen einer Szene aus verschiedenen Perspektiven.

3. Historische Zeit Personen, Situationen und Handlungen geschichtlicher Epochen gehören zum Grundbestand dessen, was darstellende Literatur, Malerei, Bildhauerei etc. buchstäblich denotieren und was Theater, Performances, Film buchstäblich inkorporieren und vorführen.

4. Lineare Zeit Die bisher aufgeführten Zeit-Aspekte suggerieren als zusätzliches Moment die Linearität, einige davon auch die Gerichtetheit der Zeit. Das unter 2. angeführte Beispiel des Genres Film kann aber auch gerade gegen diese klare lineare Abfolge von Ereignissen Filmszenen so ineinander- und übereinander blenden, daß für den Rezipienten ein schwer durchschaubares Kaleidoskop von Zeitphasen oder -fetzen entsteht, das sich der Rekonstruktion des Nacheinanders einer Erzählung entzieht. Auch der negative Bezug auf einen bestimmten Zeitaspekt ist eine Bezugnahme.

5. Netzartige Zeit Gegen die Linearität der Zeit stehen in der Folge beispielsweise des chemisch-physikalischen Modells von Ilya Prigogine irreversibel gerichtete, aber nonlineare und indeterministische bifurkative Netzwerke der Zeit (vgl. Prigogine 1995, bes. Kap. II) oder Ideen der Verräumlichung der Zeit. Indem sich z.B. in einer Komposition von Bernd Alois Zimmermann „von einem Punkt des klanglichen Einsetzens aus musikalische Bewegungen nach allen Seiten ausbreiten“, inkorporiert und performiert das aufgeführte Musikstück – das buchstäblich für den Hörer nacheinander abläuft –

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metaphorisch die, wie Zimmermann es nennt, „Kugelgestalt der Musik“ (vgl. Mehner 1997, 179f.).

6. Entschwindende – „verlorene“ – gesuchte – wiedergefundene Zeit als Zeiterfahrung Ein signifikantes literarisches Beispiel für die Symbolisierung entschwindender historischer Zeit (vgl. auch 3.) ist Kuno Raebers Roman „Sacco di Roma“ (1989). Er denotiert buchstäblich eine Reihe verschiedener historischer Ereignisse (wie den „sacco di roma“, also die Plünderung Roms durch deutsche Landsknechte von 1527). Diese werden aber nicht chronologisch geschildert, sondern als spiralförmige Geschichtsbewegung, in der die Zeit sich immer weiter dreht. Diese Chrono-Spirale des Entschwindens wird metaphorisch denotiert durch die zu Beginn beschriebene Bewegung des in einer Badewanne abfließenden Wassers, mit welcher der Roman beginnt: Das Wasser, kaum daß du den Pfropfen herausziehst, fließt aus der Wanne, langsam erst und dann schneller und schneller, den Abfluß hinunter, […] die Spiele des Schaums, seine Kapriolen im Abfluß, einmal dicht, einmal dünner, die Tempowechsel im Zurückweichen und in der Annäherung, immer dramatisch, Adagio, Crescendo, Presto, Molto Presto, Diminuendo und Presto wieder von vorn, das ist spannend zu sehen, […] wie sich eins aus dem andern ergibt, Streit und Versöhnung, Trennung, Umarmung, Widerstand und Ergebung, der Sacco di Roma, der Brand im Borgo wirbeln hinab […] Die Chrono-Spirale des Entschwindens in das verzweigte dunkle ‚Kanalsystem‘ der Zeit wird metaphorisch inkorporiert und vorgeführt durch die sichtbare und hörbare Sprache: der ganze Roman ist in einem einzigen Satz geschrieben, der keine Grenze, Pause, Zäsur vorgibt, sondern sich in rhythmischen Wiederholungen, Variationen, Paraphrasen kreiselnd durch die Lese- oder Hörzeit bewegt.

7. Strukturierte, gemessene, rhythmische, metrische Zeit In dem für diesen Zeitaspekt nächstliegenden Bereich der Musik kann nicht nur auf thematisch bezugnehmende Werke wie etwa Josef Haydns Symphonie „Die Uhr“ verwiesen werden (s.o. II.2.), sondern auf fast jedes Musikstück. Mozarts „Prager Symphonie“ z. B. hat kein anderes Sujet als

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eben das, was geschieht, wenn sie erklingt. Sie verkörpert buchstäblich eine in ihr bestimmt geregelte klanglich-rhythmische Ordnung eines Zeitablaufs mit spezifischen ästhetischen Qualitäten und führt sie vor. Auf andere Weise verkörpert Ravels (erklingender) Bolero performativ buchstäblich in ständiger leichter Variation zweier Themen einen sich in Geschwindigkeit, Lautstärke und Intensität steigernden Rhythmus, dessen Ausgangstempo dem normalen Herzrhythmus entspricht; er verkörpert performativ metaphorisch den sich bei Spannung beschleunigenden Herzrhythmus (was in der ersten Realisierung der Uraufführung bei einigen Hörern zu Herzproblemen geführt haben soll). Als Beispiel aus den Bildenden Künsten kann eine Installation der vorletzten Documenta gelten: dort konnte man durch einen Raum schreiten, in dem ein spiralförmiger Weg durch, von der Decke hängende, Maßbänder mit je unterschiedlichen Maßeinteilungen vorgegeben war. Das im Gehen erfahrene Kunstwerk verkörperte performativ buchstäblich räumliche Meßvorrichtungen als Lenkung unseres Gehprozesses; mit den je unterschiedlichen Maßeinteilungen denotierte es metaphorisch die Relativität der Zeitmessung; es verkörperte performativ metaphorisch die Vorstellung einer als Spirale verräumlichten offenen Zeit.

8. Die Zeit im Raum-Zeit-Kontinuum Die Zeit im Raum-Zeit-Kontinuum, wie es schon von Leibniz philosophisch begründet und schließlich in der speziellen und allgemeinen Relativitätstheorie Einsteins physikalisch-mathematisch expliziert wurde, hat als weitere Konnotationen das Problem der symmetrischen im Gegensatz zur asymmetrischen und der determinierten im Gegensatz zur indeterminierten Zeit. Die Künstler der klassischen Moderne reagierten nach 1905 und in den Folgejahren sehr deutlich auf das neue Weltbild und vor allem auf die Tatsache, daß durch die Relativitätstheorie die Spaltung der 3-dimensionalen Anschauungswelt von der mathematisch-physikalischen Begriffswelt vollzogen wurde: den gekrümmten Raum und die vierte Dimension des Raum-Zeit-Kontinuums kann man nicht empirisch erfassen oder imaginieren; beides kann man nur denken und berechnen. Das Auseinanderklaffen von „täuschender“ Erfahrungswirklichkeit und berechneter „wahrer“ Physikwirklichkeit wurde einerseits als Verunsicherung empfunden, andererseits als Befreiung von der mimetischen Norm, die sichtbare Wirklichkeit abzubilden. Das 1911 entstandene Gemälde „Der Lärm der Straße dringt ins Haus“

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des Futuristen Umberto Boccioni z. B. stellt eine Anschauungswelt dar, die wirkt, als sei sie aus den dreidimensionalen Fugen geraten: Innen- und Außenraum von Häusern und Straßenzügen klappen ineinander um; die Ansicht und das Sehen des Bildes scheinen in einen Geschwindigkeitswirbel gezogen zu werden. Das Bild denotiert metaphorisch das Phänomen der Geschwindigkeit und damit die Relation zeitlicher Abläufe zum Raum; und im Sehvorgang inkorporiert der Betrachter genau diese phänomenale Relation der zeitlichen Vorgänge zum Raum.

Umberto Boccioni „Der Lärm der Straße dringt ins Haus“, 1911 Einige der Ready-Mades von Duchamps wiederum existieren als in räumlich-zeitlicher Abfolge bewegt, d. h. sie inkorporieren diesen Aspekt buchstäblich.

9. Zeitdauer Verschiedene Arten der Zeitdauer sind künstlerisch relevante Zeiterfahrungen: Gegen die schnelle, kurz währende Zeit und Kurz-weil setzt z.B. der Regisseur Christoph Marthaler als Stilprinzip die lang währende Zeit und Lange-weile. Seine Inszenierungen wie etwa „Die Stunde Null oder die Kunst des Servierens“ (Hamburg 1995) denotieren sowie inkorporieren und performieren buchstäblich langsame, lang andauernde, sich rhythmisch wiederholende Handlungs- bzw. Geschehensprozesse.

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Thomas Manns Roman „Der Zauberberg“ stellt metaphorisch die unwirklich stillstehende Zeit dar, die Menschen in der ‚Watte-Welt‘ eines Sanatoriums als ihre wirkliche Zeit erleben.

10. Vergehen der Zeit und Vergänglichkeit Ein wesentlicher Aspekt ist die, existentiell an die Erfahrung der Sterblichkeit gebundene endliche, vergängliche Zeit. In der darstellenden Malerei und Literatur wimmelt es nur so von metaphorischen Denotationen der Vergänglichkeit (Vanitassymbolik), seien es die abgestorbenen Bäume und Ruinen in Caspar David Friedrichs Landschaften, die überreifen Früchte, umgestürzten Weingläser oder Salzgefäße in Chardins Stilleben oder der Sensenmann in Hugo von Hoffmannsthals „Jedermann“. Vergänglichkeit wird buchstäblich inkorporiert und der Erfahrung vorgeführt, wenn Paul Klee absichtsvoll für eine Reihe seiner Graphiken holzhaltige Papiere verwendete, die sich relativ schnell verändern, verfärben und schließlich zerfallen.

11. Unendliche Zeit und Ewigkeit Eine Fülle von Darstellungen unendlicher Zeit bzw. zeitloser oder überzeitlicher Ewigkeit findet sich z. B. in den religiösen Darstellungen und überlieferten Symbolen christlicher Kunst. Dieser Zeitaspekt kann, beispielsweise in Darstellungen der Auferstehung Christi oder der himmlischen Sphäre nach dem jüngsten Gericht prima facie nur metaphorisch denotiert oder inkorporiert werden. Kann man jedoch beim Betrachter den Glauben an die betreffenden religiösen Wahrheiten voraussetzen, so denotiert z. B. eine solche Darstellung die Ewigkeit nicht nur metaphorisch, sondern auch buchstäblich.

V. Multipler Zeit-Bezug: das Beispiel „Projekt Organ2 / ASLSP“ Eines der interessantesten Kunstwerke in diesem Zusammenhang ist das Projekt „Organ2 / ASLSP“ in der romanischen Burchardi-Kirche, Halberstadt (vgl. Göttert 2003; www.john-cage.halberstadt.de). Es thematisiert Zeit und symbolisiert eine ganze Reihe von Zeitaspekten auf vielfache Weise.

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1. Die Kunstwerk-Struktur Künstlerischer Produzent 1. Stufe ist John Cage. Das Kunstwerk-Schema ist sein 1985 konzipiertes und 1987 für Orgel komponiertes „Organ2 / ASLSP“ – „as slow as possible“. Künstlerische KonProduzenten 2. Stufe sind Organisten, Kunsttheoretiker, Orgelbauer, Theologen, Historiker, die eine spezielle Interpretation entwickeln: Deren Kern ist eine 639-jährige Aufführungszeit, von denen knapp 5 Jahre bereits umgesetzt wurden. Ihr Ausgangspunkt war die Frage: Was heißt bzw. wie langsam ist „as slow as possible“? Prinzipiell könnte das Stück unendlich lange aufgeführt werden. Faktisch bedeutet „so langsam wie möglich“ eine so lange Realisierungszeit wie die Blasebälge der Orgel halten. Ihr Anfang – und hier beginnt die interpretative Zeitkonstruktion – sollte im Jahr der Jahrtausendwende liegen. Da die erste Orgel der Geschichte in Halberstadt das erste Mal 1361 erklang, wurden die 639 Jahre vom ersten Orgelklang bis zum Jahr 2000 von dort aus symmetrisch in die Zukunft gespiegelt. Bis 2639 soll die Orgel zugleich immer weiter ausgebaut werden. Die Aufführung jedes der 9 Teile dauert (639 : 9 = 71) 71 Jahre . Dennoch war die Umrechnung der Partitur auf eine 639-jährige Zeitspanne äußerst kompliziert. Der definitive Beginn der Aufführung war am 5. September 2001, dem Geburtstag des verstorbenen John Cage. Bis heute wurden folgende temporale Phasen realisiert: 5. September 2001: Beginn mit einer anderthalbjährigen musikalischen Pause, in der kein Klang zu hören war, sondern allein die einströmende Luft in den ersten 6 fertiggestellten Orgelblasebälgen. 5. Februar 2003 erster Klangimpuls: der Akkord gis′, h′, gis″ erklingt: eine Gruppe von Schülern wird eingeladen, da sie in hohem Alter das Ende des ersten Klangkontinuums 70 Jahre später erleben könnten. Kleinste Veränderungen in dem über Jahre andauernden Ton sollen registriert und gemessen werden. 5. Juli 2004 zweiter Klangimpuls: Ergänzung durch ein e und ein e′. 5. Juli 2005 Pause (e, e′): nur gis′und h′ sind zu hören. Der nächste Klangimpuls a′, c″, fis″ ist für den 1. Januar 2006 festgesetzt. Das in der Realisierung befindliche Kunstwerk ist Organ2 / ASLSP Halberstadt1−x. Nicht-künstlerischer Kokonstituent 3. Stufe ist jeder, der während der 639

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Jahre die Kirche betritt und für eine mehr oder weniger lange Zeitspanne in das Klangkontinuum eintaucht, d. h. z. B. den gerade erklingenden Orgelintervall oder –akkord eine bestimmte Zeitspanne lang realisiert.

2. Symbolisierung von Zeit Das Gesamtprojekt „Organ2 / ASLSP“ inkorporiert und performiert: Buchstäblich eine relativ, d. h. maximal langsame klangliche Umsetzung, ferner die komplizierte neunteilige Strukturierung der Klang-Zeit, schließlich die zeitliche Begrenzung und Fragmentarizität des Hörens und Produzierens, d. h. die durch die Lebenszeit bedingte Unmöglichkeit, den vollständigen Realisierungszeitraum des Kunstwerks zu vollziehen; es inkorporiert und performiert metaphorisch die menschliche Vergänglichkeit. Indem 639 Jahre historische Orgel-Vergangenheit konzeptuell integriert und auf der Jahr-2000-Achse in die Realisierungs-Zukunft gespiegelt werden, inkorporiert und performiert ASLSP: Metaphorisch die bezugspunkt-bezogene A-Reihe der Zeit, eine gerichtet-symmetrische Auffassung von Zeit (um die Spiegelachse Jahr 2000) und weiterhin metaphorisch die „Entschleunigung“ der Zeit und „Entdekkung der Langsamkeit“ sowie buchstäblich die Antizipation der zukünftigen Vollendungszeit und daran anschließend metaphorisch das Vertrauen in die Kontinuität der Zukunftsentwicklung. Ferner inkorporiert und performiert ASLSP buchstäblich das den Menschen faktisch transzendierende Sein des aktuellen Kunstwerksprozesses wie auch die permanente Veränderlichkeit der Orgel und des andauernden Klangs und schließlich metaphorisch vielleicht die Vorstellung eines unendlichen, ewigen Klangs und das Gefühl, in die Ewigkeit gestellt zu sein.

Literatur Austin, John L. 1965 „Performative Äußerungen“, in: ders. Gesammelte philosophische Aufsätze, dt. u. hg. v. Joachim Schulte, Stuttgart: Reclam, 305– 327. Danto, Arthur C. 21993 Die Verklärung des Gewöhnlichen. Eine Philosophie der Kunst, dt. v. Max Looser, Frankfurt a.M: Suhrkamp. Fischer-Lichte, Erika 2004 Ästhetik des Performativen, Frankfurt a. M: Suhrkamp, 2004.

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Gadamer, Hans Georg 41975 Wahrheit und Methode. Grundzüge einer philosophischen Hermeneutik, Tübingen: Mohr Siebeck. Goodman, Nelson 1990 Weisen der Welterzeugung, dt. v. Max Looser, Frankfurt a.M: Suhrkamp. — 1987 Vom Denken und anderen Dingen, dt. v. Bernd Philippi, Frankfurt a.M: Suhrkamp. Göttert, Karl-Heinz 2003 Organ 2 /ASLSP, John-Cage-Orgel-Stiftung Halberstadt (Hg.), Halberstadt. Leibniz, Georg Wilhelm 1714 Monadologie (Die philosophischen Schriften, 7 Bde., hg. v. C. I. Gerhardt, Berlin 1875–90), Hildesheim/New York: Olms, 607–623. McTaggart, John Ellis 1993 „Die Irrealität der Zeit“, in: Walther Ch. Zimmerli und Mike Sandbothe (Hgg.) Klassiker der modernen Zeitphilosophie, Darmstadt: Wiss. Buchgesellschaft, S. 67–86. Mehner, Klaus 1997 „Das Antizipatorische in der Musik“, in: F. Gaede u. C. Peres (Hgg.) Antizipation in Kunst und Wissenschaft. Ein interdisziplinäres Erkenntnisproblem und seine Begründung bei Leibniz, Tübingen, 173–184. Peres, Constanze 2000 „Raumzeitliche Strukturgemeinsamkeiten bildnerischer und musikalischer Werke“, in: Tatjana Böhme u. Klaus Mehner (Hgg.), Zeit und Raum in Musik und Bildender Kunst, Köln/Weimar/ Wien: Böhlau, S. 9–30. — 2003 „Kandinsky, Leibniz und die RaumZeit“, Spektrum der Wissenschaft, Spezial 1 „Phänomen Zeit“, S. 84–89. Prigogine, Ilya 1998 Die Gesetze des Chaos, Frankfurt/Leipzig: Insel. Puntel, Lorenz Bruno 2006 Struktur und Sein. Ein Theorierahmen für eine systematische Philosophie, Tübingen: Mohr Siebeck (im Druck). Raeber, Kuno 1989 Sacco di Roma. Roman, Zürich: Amman, 1989. Wittgenstein, Ludwig 31975 Philosophische Untersuchungen (abgek. PU), Frankfurt am Main: Suhrkamp. Wollheim, Richard 1982 Objekte der Kunst, dt. v. M. Looser, Frankfurt a.M.: Suhrkamp. www.john-cage.halberstadt.de

Die Theorie der somatisch-neuronalen Entstehung von Werten, die a-chronologische Gedächtniszeit und die Verschränkung von Zeit und Bewerten Werner Leinfellner, Nebraska 1. Einleitung und Übersicht In diesem Artikel werden neuere Forschungsergebnisse der Neurowissenschaften, der Neurophilosophie, der Neuropsychologie und der Memetik, d.h. der Wissenschaft vom natürlichen und künstlichen Gedächtnis, zusammengefasst. Sie führen zu einer neuen, kognitiven und somatisch-neuronalen Theorie über die Entstehung oder Erschaffung und die Speicherung von individuellen und kollektiven Werten im Gedächtnis (Damasio 1994, 1999; Lane/Nadel 2002). Kollektive Werte sind solche, die in mindestens zwei individuellen Gedächtnissen vorhanden sind. Analog versteht man in der Demokratie unter kollektiven Werten solche, die von einer größeren Anzahl von Personen gemeinsam vertreten werden, z.B. von Personen in Parteien. „Kollektiv“ meint hier also keinesfalls „autoritär“. Die auf dem Gehirn beruhende somatisch-neuronale Erschaffung und Verarbeitung von Werten (value processing) läuft ständig und solange wir leben in einer a-chronologischen Eigenzeit, hier „Gedächtniszeit“ genannt, ab. Die Ergebnisse der neurowissenschaftlichen Forschung vereinen die somatisch-neuronalen, kognitiven Funktionen des menschlichen Gehirns bei der individuellen Entstehung von Werten mit der neuronalen Speicherung von individuellen und kollektiven Werten (Basar 1988b; Leinfellner 2006). Forschungen in diesem Gebiet beantworten die Frage: Woher kommen die Werte, und wie entstehen sie? Die Antwort beruht auf einer neurowissenschaftlichen, neurophysiologischen und psychologischen Aufklärung, wie Bewertungen und Werte somatisch-neuronal im menschlichen Gehirn in Zusammenarbeit mit den Sinnesorganen des menschlichen Körpers in einem ständigen Prozess erzeugt und gespeichert werden. Ohne diese Werte gibt es keine Entscheidungen, keine Handlungen und keine Lösungen von Konflikten (Damasio 1994, 1999; Leinfellner 2006; u.a.). Der Philosophie genügte ihr göttlicher F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 387–408.

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Ursprung oder ihre Schaffung durch Autoritäten wie Gesetzgeber und Regierende. Den Ursprung von kollektiven Werten erklärt man traditionell durch ihren mehrheitlichen Gebrauch in demokratischen Gesellschaften. Nach Damasio u.a. erschaff t der somatisch-neuronale Wertprozess während eines individuellen Lebens eine ungeheuer große Menge von Präferenz-Werten. Diese werden in unserem individuellen Gedächtnis in seiner Gedächtniszeit gespeichert; wir verfügen jederzeit über sie und können sie abrufen (Basar 1988b). Damasios Theorie der somatisch-neuronalen Wertschöpfung und der neuronalen Speicherung beantwortet eine die längste Zeit nur durch Vermutungen beantwortete Frage nach dem Ursprung der menschlichen Präferenz-Werte. Dies ist wichtig: Denn ohne stets neue Bewertungen, ohne ihre Verwendung und ihre empirische Realisierung geht in der menschlichen Erkenntnis und ihrer sozialen Verwirklichung überhaupt nichts. Wie dies für das menschliche Handeln und Entscheiden und für die Lösung von sozialen Konflikten aussieht, soll nur im Kontext von Demokratien und demokratischen Wohlfahrtsstaaten mit ihren Regeln dargestellt werden. Denn nur in Demokratien und Wohlfahrtsstaaten werden autoritäre Bewertungen und Meinungen abgelehnt, auch bei Konflikten zwischen egoistischen und altruistisch-kooperativen Werten. Unter „Meinungen“ verstehen wir hier komplexe Werte, Hierarchien von Werten, unabhängig davon, ob sie als wissenschaftlich erwiesen worden und/oder ob sie empirisch testbar sind (Leinfellner 2006). Die im Gedächtnis auftretende Eigenzeit als a-chronologische Gedächtniszeit unterscheidet sich von allen anderen Eigenzeiten: der Uhrenzeit der Naturwissenschaften, der kalendarischen Zeit, der Zeit des Alltags, der mathematischen, linearen Zeit Newtons und Leibniz’, der anschaulichen Zeit bei Niewentyts und Kant (Beth 1959, 69, § 18–19), der Zeit der Allgemeinen Relativitätstheorie und den 25 Arten von Eigenzeiten, die P. Davies aufzählt. Diese Gedächtniszeit hat aber für alle Sozialwissenschaften (Ökonomie, Soziologie, Politische Wissenschaften) die allergrößte Bedeutung, weiters für die Neurophilosophie, die Kulturwissenschaften, die Historiographie, die Theorie der Wert- und Nutzenmessung und die Memetik, und ebenso für den sozialen, kollektiven Gebrauch von Werten im Kontext von Demokratien und demokratischen Wohlfahrtsstaaten (Tulving 1983; Aungar 2000; Churchland 1989; Leinfellner 1988, 2006). Unsere individuelle und kollektive Geschichtszeit oder Geschichtlichkeit – nicht die Geschichte – beruht nicht auf der Alltagszeit, sondern auf dem individuellen und kollektiven Gedächtnis.

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Die Gedächtniszeit ähnelt aber sehr der evolutionären Zeit von Menschen in Demokratien, die von der Umgebung, von internen und externen Zufallsereignissen und von unseren politischen, sozialen und wirtschaftlichen Maßnahmen und Regeln statistisch-kausal abhängen. Man könnte auch sagen, sie ähnelt einem nicht-linearen und dynamischen Netzwerk. Das individuelle Gedächtnis speichert semantische Verschränkungen von primitiven Bewertungen mit zusammenhängenden Gedächtniseinheiten oder Erinnerungen, den Memen; letztere sind also die semantisch zu bewertenden Einheiten der Gedächtniszeit, sodass man auch sagen kann, ein Mem ist in der Gedächtniszeit „enthalten“, befindet sich „in“ der Gedächtniszeit, d.h. im Rahmen der Gedächtniszeit. Die Meme sind aber nicht so etwas wie „Jetzte“ auf einem Zeitpfeil oder wie die kalendarischen Daten der Alltagszeit. („Semantisch“ bezieht sich hier sowohl auf den Inhalt der Meme als auch auf ihre Bewertung, siehe unten.) Die Momente der klassischen mathematischen Zeit haben keinen empirischen Inhalt; sie können nur durch die abstrakt verstandenen Relationen Später und Gleichzeitig zwischen ihnen bestimmt werden. Die „Momente“ der Gedächtniszeit, besser „Einheiten“ oder „Meme“ genannt, sind also die wichtigsten semantischen Informationen. Aber nur die individuellen und die kollektiven Bewertungen im individuellen Gedächtnis geben den Memen oder zusammenhängenden Erinnerungen individuellen und sozialen Inhalt und unterscheiden sie dadurch semantisch. In den heutigen Neurowissenschaften, vor allem in Damasios bahnbrechenden neurophysiologischen und somatisch-neurologischen Forschungen, wurde zum ersten Mal die Frage nach dem Woher und dem Wie der Werte wissenschaftlich beantwortet. Ihre Vorstufen entstehen durch einen auf dem Gehirn beruhenden somatisch-neuronalen Bewertungsprozess (value processing) und werden dann neuronal gespeichert. Hauptsächlich handelt es sich um die Verschränkung der individuellen und der kollektiven Werte mit den Memen im individuellen Gedächtnis. „Verschränkt“ heißt hier nicht wie z.B. bei der Uhrenzeit, dass zwei Jetzte wie zwei Teilchen sich nach Russell überlappen können, wenn sie nur genügend zeitlich und räumlich nahe beieinander liegen (vgl. das Axiom der räumlichen Koinzidenz der Gleichzeitigkeit; Leinfellner und Leinfellner 1978, 111). Das Verschränkt-Sein von Memen und Werten hat Folgen: Die alte Bewertung eines geschichtlichen Ereignisses kann durch seine heutige Umoder Neubewertung beeinflusst werden. Der positive Wert der Schlacht bei Issos, 333, als Alexander der Große Europa vor dem Morgenland rettete,

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hat sich durch die heutige Bewertung geändert, so als ob eine zeitlose „Zeit“ zwischen beiden Bewertungen liegen würde. Obwohl beide Bewertungen kalendarisch und räumlich verschieden sind, wirken sie, wenn sie beide in einem individuellen Gedächtnis gespeichert sind, auf eine nicht kausale Art aufeinander ein. Zwischen beiden Bewertungen liegt also im Rahmen unserer a-chronologischen Gedächtniszeit eine Art Quasizeit oder – paradox ausgedrückt – „zeitlose Zeit“ (Barbour 1999, 217). Auch die Geschichtlichkeit illustriert diese „Zeitlosigkeit“. Napoleon wurde zuerst als Genie und Übermensch hoch bewertet, aber heute negativ als jemand, der die Sklaverei auf den Antillen wieder eingeführt hat und dessentwegen unzählige Menschen gestorben sind. Oder: Der Hochzeitstag vor zehn Jahren Kalenderzeit wird oft jahrelang positiv als der glücklichste Tag im Leben bewertet; aber wegen einer gegenwärtigen Scheidung wird er auf einmal negativ umbewertet. Plötzlich liegen beide Bewertungen, im kalendarischen Sinn zeitlos, im subjektiven Gedächtnis gespeichert beieinander. Wenn ich sage, „Im Frühjahr war ich glücklich“, dann kann ich im Herbst das Mem, die zusammenhängende Erinnerung an Ereignisse im Frühjahr, anders bewerten. Einstein sagte einmal bewertend, „Die glücklichste Idee meines Lebens war die Einführung der Lambda-Konstante“. Gerade diese Konstante bewertete Einstein aber nach zehn Jahren humorvoll als größte „Eselei“. Die meisten Menschen vergessen sehr leicht die kalendarischen Daten von Geburtstagen, Heiratstagen usw., und auch Personen-Namen. Ihr semantischer Wert, d.h. ihre Bewertung bleibt aber in unserem Gedächtnis gespeichert. Z.B. sagen wir: Das war die glücklichste Zeit, an die ich mich erinnern kann, aber leider erinnere ich mich nicht an die genaue (kalendarische) Zeit. Oder: Der Professor, den ich auf der Philosophie-Konferenz getroffen habe, hat mich beeindruckt; leider weiß ich nicht mehr, wie er heißt. Erst wenn zusammenhängende Erinnerungen, Meme mit ihrer Bewertung im individuellen Gedächtnis verschränkt werden, geben sie meinem Leben und der Gesellschaft einen Sinn und werden dadurch unterscheidbar (siehe Kap. 3). Die meisten Neurowissenschaften und die Neurophilosophie lehnen radikal die traditionellen Funktionen von „Geist“, „mind“ ab; sie ersetzen sie durch die Dynamik kognitiver Wert-Prozesse des neuronalen Gehirns. Diese neuen Wissenschaften erlauben es z.B., neue Hypothesen über die individuelle, somatisch-neuronale und auch über die kollektive Entstehung von Bewertungen, Werten und Meinungen aufzustellen und ihre Speicherung in individuellen Gedächtnissen (memories) mit einer individuellen, nicht-

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kalendarischen, a-chronologischen Gedächtniszeit neu zu deuten, kurz: Alles mit Allem zu vergleichen.

2. Damasios somatisch-neuronale Theorie der Entstehung von primitiven präferenziellen Bewertungen und Werten als nicht-bewusste Vorgänger bewusster wissenschaftlicher Werte Nach Damasios somatisch-neuronaler Theorie der Wertentstehung, die wir aus seiner umfassenderen kognitiven Theorie der Entstehung des Bewusstseins herausgreifen, beginnt der neuronale Wertprozess, ähnlich wie es auch von Basar und seinem Forschungsteam herausgearbeitet worden ist, mit der somatischen Apperzeption externer Wellenmuster, z.B. von externen Lichtwellenmustern beim Sehen (z.B. Bildern), Tonwellenmustern beim Hören (z.B. Melodien) usw. (Basar 1980, 1988b, 30ff; Leinfellner 1988). Die apperzipierten, von außen kommenden, externen Wellenmuster werden in den Sinnesorganen, den Außenposten des Gehirns, in interne, homomorphe verwandelt und auf afferenten Nervenbahnen elektrisch zum limbischen System des Gehirns, genauer: zur Amygdala weitergeleitet. Die Amygdala, das Hauptzentrum der internen, nicht-bewussten Wertentstehung, „aktiviert“ zwei Haupttypen von nicht bewussten oder halb bewussten primitiven (Proto-)Werten (Bewertungen; neuronale Vorstufen der späteren bewussten, sprachlich ausdrückbaren Präferenzwerte), die primären und sekundären Emotionen, die von den Gefühlen unterschieden werden. „Primitiv“ heißt hier einfach „intern“. Diese primitiven Werte oder Emotionen entstehen in verschiedenen, mit der Amygdala eng verbundenen Regionen des Großhirns, wie es das PET Imagining (Positronenemissionstomographie) zeigt (Damasio 1999, 61). Man darf nicht vergessen, dass auch diese primitiven positiven oder negativen Emotionen nur Zwischen- oder Vorstufen der Endprodukte, der bewussten, neuronal erzeugten Werte und ihres sprachlichen Ausdrucks sind. Bei einer Bedrohung unserer Existenz oder unserer Zukunft bestimmen Werte, die uns oft nicht bewusst sind, unsere Reaktionen und Handlungen innerhalb von 1/300 Sekunden! Wenn wir z.B. beim Vorübergehen an einem sehr baufälligen Haus Angst fühlen, kann uns dies als die Emotion der Angst, Furcht nur halb bewusst werden oder auch nicht bewusst bleiben; trotzdem werden wir automatisch handeln und meistens um ein

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sehr baufälliges Haus herumgehen. Dies hängt von unserer im individuellen Gedächtnis gespeicherten Eigenwertung des Risikos (risiko-freudig oder risiko-vermeidend) ab. Die primitiven positiven Werte und Entscheidungen erzeugen neuronal ein pragmatisches, emotionales Plus, ebenso wie die positiven Konsequenzen von schon gespeicherten Werten externer und interne Zufallsereignisse. Primitiv negativ sind sie, wenn sie ein Minus meiner Lebenserwartung verursachen, wie emotionale Bewertungen, die durch Angst, Körperschaden, Organdisfunktion, Leiden, Schmerz, Furcht oder durch die Konsequenzen negativer Zufallsereignisse hervorgerufen sind, oder auch durch ähnliche, bereits gespeicherte negative Werte (Erinnerungen, Meme). In bestimmten, mit der Amygdala verbundenen Regionen des rechten Großhirns entstehen primitive neuronale, positiv-hedonistische oder negative Erregungszustände, positiv oder negativ primitiv wertende Gefühle nach der traditionellen Bezeichnung. Gefühle sind die ersten primitiven neuronalen Gradierungen von noch ungeordneten internen primitiven Bewertungen, den primitiven Emotionen, die etwas später im neuronalen Prozess der Bewertung präferenziell und qualitativ geordnet werden können. Z.B. kann man nun den primitiven Wert eines Ereignisses als Mem, d.h. die entsprechende Emotion, im individuellen Gedächtnis dem primitiven Wert eines anderen gefühlsmäßig vorziehen. Es ist die statistische Ordnung des Vorziehens und des Gleichbewertens von Emotionen auf qualitativer Basis, die uns auch sprachlich bewusst werden kann. Die gefühlsmäßige Gradierung ist der erste Ansatz zu einer primitiven statistischen Skalierung von primitiven Werten, den Emotionen, durch die Bewertung der Emotionen durch Gefühle. All dies wird sofort im individuellen Gedächtnis gepeichert (Damasio 1994). Nach Basar u.a. können alle apperzipierten Wellenmuster, die im individuellen Gedächtnis neuronal gespeichert sind, über das Wernicke-Zentrum und das in der unteren Stirnhirnwindung der dominanten Hemisphäre gelegene Broca-Zentrum über das interne Nervensystem die 300 zum Sprechen verwendeten Muskeln aktivieren, um akustische Tonwellenmuster, nun in artikulierter Sprache, zu erzeugen. Dies dient uns zur externen Verständigung zwischen menschlichen Gehirnen, und auch dazu, die von „Damasio’schen“ Wertprozessen erzeugten präferenziellen Werte sprachlich sich selbst und anderen Menschen voll bewusst zu machen (Damasio 1999, 13). Es ist daher kein Wunder, dass die von der Amygdala ausgehende Weiterleitung der Emotionen zu den schläfenseitig liegenden Gehirnwindungen

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die neuronale und „positive“ Aktivierung von Lust, Glücksgefühl, Vergnügen buchstäblich als eine Art hormoneller Belohnung erzeugt, die also somatisch-neuronal verursacht wird. Populär spricht man hier vom internen neuronalen „hedonistischen highway“, auf dem (Proto-)Bewertungen intern entstehen (Howard 2000, 328). Kognitiv gesehen ist der somatisch-neuronale Wertprozess, der von primären und sekundären primitiv bewertenden Emotionen über Gefühle bis zu den sprachlich ausgedrückten Präferenzwerten und deren Speicherung in unserem Gedächtnis reicht, ein immerwährender, unglaublich schnell ablaufender Prozess der Wertentstehung und Wertverarbeitung. Dieser endet mit seiner Speicherung im individuellen Gedächtnis und wird dort im Rahmen der individuellen Gedächtniszeit omnipräsent gespeichert. Die Emotionen führen also durch ihre primitive Skalierung durch Gefühle letztlich zu präferenziellen, individuellen und kollektiven, oft sprachlich ausgedrückten, und im Gedächtnis gespeicherten Werten als Proto-Werten. Nach Damasio sind sie die Endprodukte des somatisch-neuronalen „value processing“. Präferenzen werden uns nicht nur sprachlich bewusst, sondern „präferenziell“ bezieht sich auch auf das handelnde, empirische Bewerten: Man zieht etwas empirisch vor, oder ist indifferent, und das geht natürlich nur, wenn die primitiven Werte vorher gespeichert worden sind. Zweifellos verfügen auch Tiere über präferenzielle primitive Bewertungen. Beim Menschen kommt ihr sprachlicher, ihnen bewusster Ausdruck in der vom Gehirn erzeugten Sprachfunktion hinzu. Während eines kontinuierlichen Bewertungsprozesses, z.B. wenn wir ohne uns etwas zu überlegen einem heran rasenden Auto ausweichen, oder wenn wir Freunde Feinden vorziehen, verwirklichen wir nur automatisch interne Präferenzwerte. Zusammen mit dem somatisch-neuronalen Wertprozess ereignet sich die momentane neuronale Speicherung „in a need for stability“ (Damasio 1999, 134). Jede somatisch-neuronale Analyse ist eine kognitive, neurowissenschaftliche Rekonstruktion, eine kognitive Theorie darüber, wie aus Empfindungen via Emotionen halb bewusste Gefühle und daraus primitive präferenzielle Bewertungen in unserem neuronalen Bewusstsein entstehen, wie sie neuronal gespeichert werden, und wie sie jederzeit wieder verwendet, d.h. abgerufen werden können (Damasio 1999, 37, 42).

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3. Das Verhältnis aller erzeugten, gespeicherten und bewusst gewordenen Werte zu den wissenschaftlichen und den empirisch verwendbaren Werten in Demokratien Wie zu erwarten: Für Damasio sind selbstverständlich nicht alle somatischneuronal produzierten Werte von vorne herein perfekte wissenschaftliche, sozial verträgliche und auch empirisch testbare Werte. Sondern im individuellen Gedächtnis existieren auch Werte, die nicht wissenschaftlich sind, und/oder solche, die nicht empirisch realisierbar sind, weiters auch künstlerische Werte, abstruse, postmoderne, negative Bewertungen, usf. Wir speichern also alle möglichen individuellen und kollektiven Bewertungen oder Werte und Meinungen in unserem Gedächtnis. Wegen der Redefreiheit hat dies in Demokratien die größten, bisher relativ wenig beachteten Konsequenzen. In Demokratien sind z.B. Kunst und Wissenschaft gesetzlich frei. Jedermann hat die Freiheit, seine individuellen ästhetischen, moralischen und politischen Bewertungen oder Werte, seine Meinungen zu äußern, zu vertreten und zu diskutieren. Ein Künstler kann die von ihm hoch bewerteten Bilder seines Gedächtnisses, seiner Vorstellung und Phantasie, d.h. seine bewerteten Meme (zusammenhängende Erinnerungen) malen oder in Stein herausmeißeln. Die in Demokratien herrschende Freiheit der Kunst und Wissenschaft garantiert auch, dass deren Werke von anderen Menschen bewertet werden können. Die Freiheit der Rede, die Freiheit zur Verwirklichung in der Kunst usw. erlauben, Werte, Meinungen unbeschränkt zu diskutieren, zu veröffentlichen und zu verbreiten. Kollektiv werden sie, wenn sie von Mehrheiten, z.B. in Demokratien, akzeptiert werden. Das in den Demokratien verankerte Mehrheitsprinzip hat auch negative Folgen. Es können sich auch bloß chaotische, phantastisch-postmoderne, nicht verwirklichbare und verrückte Werte und Meinungen verbreiten. Diese können die politische Mehrheit und damit unglücklicherweise auch die Mehrheit in Parlamenten erhalten, für Plato ein Grund, die Demokratie abzulehnen. Ein gutes Beispiel ist die vom südafrikanischen Präsidenten Mbeki vertretene Meinung über Aids. Derartige Vorgänge veranlassten den amerikanischen Präsidenten Abraham Lincoln zu der berühmten optimistischen Aussage über Demokratien: „You may fool all the people some of the time; you can even fool some of the people all the time; but you can’t fool all of the people all the time.“ Diese Grund„regel“ der Demokratie beruht ebenfalls auf dem Mehrheitsprinzip (Leinfellner 2006).

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Die somatisch-neuronale Erschaffung von allen möglichen Werten kann nicht ohne das individuelle Gedächtnis zustande kommen; sie spielt sich also im Rahmen einer Gedächtniszeit ab, die einzigartig ist, wie im nächsten Kapitel erörtert wird. Will man aus den Endprodukten der Erschaffung aller somatisch-neuronalen Werte, hier als die Werte aus einer Menge M gekennzeichnet, die wissenschaftlichen und empirisch anwendbaren aussondern, dann müssen wir sie aus der Menge M mittels eines Testverfahrens, das hier durch „RNNMS plus C1–C3“ charkterisiert wird, gewinnen (vgl. Kap. 6; Leinfellner 2006; „RNNMS“ ist ein Akronym und wird in Kap. 5 erklärt.). Fällt das RNNMS-Verfahren, oder fallen beide Teil-Verfahren, RNNMS und C1–C3, positiv aus, dann handelt es sich tatsächlich um wissenschaftliche und/oder empirisch anwendbare Bewertungen (Werte), die eine Teilmenge W aller somatisch-neuronal erschaffenen Werte M bilden, W ⊂ M. Genau so wie kognitive naturwissenschaftliche Theorien einen mathematischen Kern und eine empirische (experimentelle) Anwendung haben, so bildet sich in neuronal-somatischen Werttheorien ein mathematischer Kern, die RNNMS-Methode, und eine empirische Anwendung nach den C1–C3 Bedingungen. Nur für diejenigen Werte, Meinungen, die diesem Test RNNMS plus C1–C3 positiv genügen, gibt es eine wissenschaftliche, eine politisch-wirtschaftliche Kontrolle und kulturelle Anwendung, die empirische Messung von Werten, kurz: eine empirische, finite Interpretation des primitiven, somatisch-neuronalen Wertens, wie auch das quantitative Messen von Werten in modernen Nutzen- und Werttheorien und in sozialen und politischen, demokratischen Entscheidungstheorien wie der Spieltheorie. Allerdings, wie in Leinfellner 2006 bewiesen, sind Voraussagen mit wissenschaftlichen Werten nur statistische Erwartungen. Diese sind wiederum, in Demokratien und demokratischen Wohfahrtsstaaten, die wichtigsten Voraussagen, weil sie die Risiken und die künftige Wohlfahrt berechnen können. Nach Umfragen wünschen und erwarten 70% der Europäer, dass die individuelle und kollektive Wohlfahrt verbessert wird oder zumindest gleich bleibt.

4. Memetik und die individuelle und kollektive Gedächtniszeit Nach der gegenwärtigen „neuroscience“ beruht die neuronale dynamische Speicherung auf dem quantenphysikalischen Prinzip der Wellenüber-

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lagerung (Superposition) oder der Wellenresonanz (Basar 1980, 1988; Leinfellner 1985, 1988). Die Arbeiten von W. J. Freeman, E. R. John, W. J. Adey, H. Petsche und W. Leinfellner (alle in Basar 1988) berufen sich auf die neuronale elektromagnetische Wellenübertragung und -überlagerung zur Speicherung und zum Abruf von Gedächnisinhalten (Meme) aus dem Gedächtnis, worauf hier nicht weiter eingegangen wird. Wie gesagt, die somatisch-neuronale Entstehung von Bewertungen, ihr Abruf aus dem individuellen Gedächtnis und ihr sprachliches Bewusstwerden kommt in der unglaublich kurzen Zeit von 1/300stel Sekunde zustande. Das gilt auch für Bilder und Töne. Ein 84-jähriger oder 2.522.880.000 Sekunden alter, gesunder Mensch hätte so im Schnitt 7.568.640.000 mögliche Bewertungen seines Lebens hinter sich, und er hätte sie auch gespeichert. Die Memetik als Lehre von der menschlichen Gedächtnisspeicherung existiert so richtig erst seit 30 Jahren, seit R. Dawkin, E. Tulving u.a.; sie umfasst aber auch die Memetik der Speicherung in Computern. Die Memetik als die Wissenschaft von der menschlichen neuronalen Gedächtnisspeicherung beruht auf Erinnerungseinheiten, den Memen, wie schon beschrieben. Im individuellen Gedächtnis sind Meme hauptsächlich mit Bewertungen verschränkt, die nur in einer bestimmten Form der Zeit existieren, der nicht-kalendarischen, a-chronologischen Gedächtniszeit. Es können bestimmte Meme oder zusammenhängende Erinnerungen z.B. in Bildern dargestellt und, vor allem, semantisch-sprachlich, z.B. in der Dichtkunst beschrieben werden. Hier sind wir nur an den Bewertungen der Erinnerungen interessiert – leere Werte gibt es nicht. Ihre funktionale, gedächtnisspezifische Ähnlichkeit mit Genen ist offensichtlich, schon allein deshalb, weil sie dynamisch sind, sich replizieren, und weil sie mutieren können. Meme oder zusammenhängende Gedächtnisinhalte benötigen aber keine Reduktion auf ihre genetische Basis, obwohl beide dieselbe evolutionäre Dynamik aufweisen. Unser neuronales Gehirn besitzt, nach Harsanyi, die Fähigkeit, beides, die individuellen und die kollektiven Bewertungen und Präferenzen, zusammen finit allgegenwärtig (quasi-allgegenwärtig) zu speichern und die Bewertungen zu vergleichen (Leinfellner 1985). Die nichtkalendarische, a-chronologische Gedächtniszeit, mit der Bewertungen im individuellen und kollektiven Gedächtnis omnipräsent gespeichert werden, ist nach der heutigen Auffassung nicht nur eine der zahlreichen Arten von Zeit, sondern auch die fundamentalste kognitive Zeit, die Zeit in den Sozialwissenschaften, die Gedächtniszeit. Sie hat folgende charakteristische Eigenschaften:

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1. Mit Bewertungen verschränkte, zusammenhängende Erinnerungen, die bewerteten Meme im Rahmen der Gedächtniszeit, sind quasi-allgegenwärtig, omnipräsent und existieren als finite Einheiten; sie haben eine eigene Quasi-Ubiquität mit allen anderen bewerteten Memen. Im Gegensatz zur Alltagszeit und zur Anschauungszeit bedeuten „finit“, „quasi-allgegenwärtig“ und „omnipräsent“, dass die individuelle und kollektive, a-chronologische Gedächtniszeit nach oben finit offen ist, und dass wir in ihr Alles mit Allem vergleichen, dass wir vorziehen, gleich bewerten und umwerten können. In unserem Gedächtnis kann ein bewertetes Mem gespeichert sein, dessen eine Bewertung Jahrtausende zurückliegt, und wir können die alte Bewertung dieses Mems mit seiner heutigen Bewertung vergleichen und dadurch verändern. Aber es muss, solange wir uns im Rahmen der Gedächtniszeit befinden, nicht dasselbe Mem zu verschiedenen Zeiten sein, das wir vergleichen: In der Mythe gibt es das bewertete Mem des Apfels vom Baum der Erkenntnis, und dieses kann heute mit einem anderen bewerteten Mem, dem der Sünde und dem Fluch der Arbeit, verglichen und dadurch verändert werden. Ein anderes, ähnliches Beispiel: Der altägyptische Gerechtigkeitsbegriff (Mem) des Ma’at als die sozialgerechte Verteilung des nationalen Reichtums kann mit dem Begriff (Mem) unserer sozialen Gerechtigkeit und mit dem der Menschenrechte in Demokratien verglichen und als (nahezu) identisch interpretiert werden (Assmann 1995, 55). Im Alltagsleben und in der normalen Wert- und Nutzentheorie können wir, anders als im Gedächtnis, nur relevante Ereignisse, Objekte usw., die vom selben Typ sind, zur gleichen Kalenderzeit miteinander vergleichen, z.B. Äpfel mit Äpfeln zu einer bestimmten kalendarischen Zeit, aber nicht Äpfel mit Schreibmaschinen. 2. Die Gedächtniszeit ist evolutionär-dynamisch: Alle ihre bewerteten Meme können sich jederzeit mit anderen bewerteten Memen über Jahrtausende hinweg verknüpfen, verschränken und ohne lokal-kausale „Berührung“ aufeinander einwirken (vgl. die Einleitung zu „Netzwerk“). Normale, alltägliche Einwirkungen, die in der Alltagszeit oder in der mathematischen Zeit traditioneller Naturwissenschaften ablaufen, brauchen eine kausale Nahwirkung. Im Rahmen der Gedächtniszeit ergeben für den Menschen angenehme, hedonistische oder unangenehme somatisch-neuronale Bewertungen die so wichtigen Sinngebungen unseres individuellen und kollektiven Lebens, eben weil wir in der Gedächtniszeit Alles mit Allem vergleichen können. Nur in einer individuellen, nicht-kalendarischen Gedächtniszeit,

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die omnipräsent ist, kann also Alles mit Allem wieder und wieder von uns verglichen werden. In der Nutzentheorie und den traditionellen Wertlehren andererseits können nur relevante Ereignisse desselben Typs miteinander verglichen werden, Gewicht mit Gewicht, Länge mit Länge, ein Kulturfakt mit einem Kulturfakt derselben Art, ein Artefakt mit einem anderen Artefakt desselben Typs. 3. Nach Harsanyi (der den Nobelpreis für seine Nutzentheorie erhalten hat) sind individuelle und kollektive Werte omnipräsent in unserer Gedächtniszeit vorhanden. Der Mensch kennt seine eigenen egoistischen Werte; aber in Demokratien muss er sie mit den kollektiv-sozialen, z.B. juristischen Werten vergleichen. Ausnahmen sind „Wolfskinder“ wie Caspar Hauser, die ohne Sprache und ohne Gesellschaft kein Gedächtnis für kollektive Werte besitzen. Die meisten Individuen in Demokratien verfügen im Rahmen ihrer Gedächtniszeit omnipräsent sowohl über die individuellen als auch die kollektiven Bewertungen. So wird die Welt zu meiner Welt. Sie können im individuellen Gedächtnis weiters individuelle, extrem egoistische Werte von extrem altruistischen unterscheiden, die extrem autoritären von den extrem kooperativen. Z.B. sind shareholder values extrem egoistische Werte, stakeholder values berücksichtigen auch die Gesellschaft, die Umwelt usw. Diese Unterscheidungen sollte man in Demokratien in Schulen erwerben, erwirbt sie aber oft erst nach einem langen Leben in Demokratien. 4. Die individuelle Gedächtniszeit ist nicht-linear: Wir kennen nicht das kalendarische Datum unserer allerersten Erinnerung. Dazu kommt ihre unbestimmte oder verschwommene Onmnipräsenz. Die Alltagszeit hingegen beruht auf der deterministischen Kausalität, wie sie auch in den traditionellen Wissenschaften vertreten wird. Die Gedächtniszeit ist für die somatisch-neuronale Verarbeitung von allen Werten im Sinne Damasios wichtiger als die kalendarische Zeit, die nichts, nämlich keine Werte, produzieren kann.

5. Konsequenzen für die Sozialwissenschaften: Antizipationen als Voraussagewerte F.P. Ramsey stellte sich schon früh und als erster die Frage, wie man bewusste Präferenzen, die er einfach als empirisch gegeben ansah, wissenschaftlich definieren und von nicht-wissenschaftlichen Werten oder Bewertungen trennen könnte (Ramsey 1921). Nach fünfzig Jahren Erfor-

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schung dieses Problems durch Ramsey (R) selbst, Nash (N), NeumannMorgenstern (NM), Suppes (S) und Leinfellner stellte sich heraus, dass die RNNMS-Kriterien plus den C1–C3 Kriterien die einzigen Bedingungen sind, mit denen es gelingt, wissenschaftliche und/oder empirische Werte W in Demokratien zu definieren. Die Konsequenz davon ist, dass es neben den wissenschaftlichen und empirisch möglichen Bewertungen und Werten W eine Unmenge N von – bis jetzt! – nicht-wissenschaftlichen und empirisch nicht vertretbaren Werten im individuellen und kollektiven Gedächtnis geben muss: unsinnige, phantastische, widersprüchliche, irrationale, sinnlose und absurde. Letzere existieren ebenso gespeichert neben den Werten W in unserem Gedächtnis. Solche unsinnigen etc. Werte machen einen nicht unerheblichen Teil der heutigen postmodernen, „anything goes“ Werte und Wertregeln nach P. Feyerabend aus. Es ist erstaunlich, wie viele Werte der Menge N angehören, die alle im Gedächtnis existieren und munter gebraucht werden. Die abgetrennte Menge N von nicht-wissenschaftlichen und nicht empirisch nachprüfbaren Werten, wobei N ⊂ M, wird ebenfalls im Gedächtnis neuronal gespeichert. N ist sozusagen ein Überschuss. In Feyerabends und in der postmodernistischen Philosophie ist zwischen wissenschaftlichen und nicht wissenschaftlichen, z.B. literarischen Werten kein Unterschied, nach der Devise „anything goes“, wozu unsere freie demokratische Gesellschaft beiträgt (Bullock und Trombley 1999, 673ff.). Wegen der Freiheit der Rede und der Meinungen würden, nach dem Postmodernismus, bald alle Unterschiede zwischen den wissenschaftlichen und/oder empirisch anwendbaren Werten W und den nicht-wissenschaftlichen Werten N verschwinden, und, als ein bedrohlicher Nebeneffekt, z.B. auch die zwischen den terroristischen Feinden der Demokratie und ihren Verteidigern nach Popper. Wenn es also in der Menge M aller möglichen, von der somatischneuronalen Wertproduktion erzeugten Werte, plötzlich einen allzu großen Überschuss, eine allzu große Submenge N von nicht-wissenschaftlichen Werten, N ⊂ M, gibt, die in verschiedenen Gedächtnissen gespeichert ist, dann entsteht ein soziales und politisches Problem mit den Werten, oder sogar ein Chaos von Werten. Nur die Werte W befähigen Menschen und Regierungen in Demokratien, sich zu entscheiden und soziale Konflikte sozial optimal, z.B. durch Risikominimierung zu lösen, d.h. statistisch zu berechnen und empirisch zu verwirklichen. So ist die Spieltheorie heute in der Lage, alle mikroökonomischen Voraussagen mit Risiko-Erwartungen auch zu berechnen und soziale Konflikte zu lösen.

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Zukünftige Werte und Voraussagen bestimmen in Demokratien künftige Gewinne oder Verluste an Wohlfahrt unter Unsicherheit und Risiko (Leinfellner 1985). Das beste Beispiel ist heute die Mikro-Ökonomie. Die Methode RNNMS plus C1–C3 kann z.B erfolgreich benützt werden, um die von der somatisch-neuronalen Wertschöpfung erzeugten und aus dem individuellen Gedächtnis abgerufenen Werte aus der Menge M zu prüfen: ob sie wissenschaftlich sind, ob sie empirisch angewandt werden können, und, im speziellen Fall, ob sie die demokratische Wohlfahrt stabil halten, erhöhen oder vermindern. Dadurch vermeiden wir das Regiment chaotischer Werte, Bewertungen und Meinungen (Leinfellner 2006, 139f.). Mit „Antizipationen“ bezeichnen wir nach Nash (2002, 39) gespeicherte, halb bewusste und bewusste präferenzielle Werte und deren Vorstufen, bewusste oder halb bewusste Vorgefühle, Ahnungen, Vorausahnungen, Einschätzungen, utopische Werte, Erwartungen, Vorwegnahmen, Meinungen, ästhetische Werte, normative Werte, wissenschaftliche und/oder empirisch anwendbar Werte, aber nur dann, wenn sich diese Werte usw. auf die Zukunft beziehen. Leinfellner 2006 beschäftigt sich mit der Frage, wie man die RNNMS-Kriterien plus C1–C3 auf solche Antizipationen anwenden kann, um festzustellen, ob sie wissenschaftlich und/oder empirisch anwendbar sind und demokratisch akzeptiert werden können. Antizipationen als Voraussagewerte oder künftig erwartete Werte sind stets wahrscheinliche Werte, in der Form [βA1, (1 − β)A2] durch bedingte Wahrscheinlichkeiten aneinander gekoppelt, wobei die Ai Alternativen sind. Wir folgen hier Savages These: Schon lange bevor der Mensch mit Zahlen rechnen konnte, konnte er die Wahrscheinlichkeiten von Risiken einschätzen. Die Wahrscheinlichkeiten β können verschieden gesehen werden, entweder als gespeicherte vergangene, ex-post Verteilungen (Frequenzen) von Werten, Meinungen z.B. in Demokratien, oder als bedingte Wahrscheinlichkeiten, die künftige Veränderungen von Verteilungen anzeigen, z.B. von t1 nach t2. Wie schon erwähnt, werden die nicht-wissenschaftlichen und empirisch nicht vertretbaren Werte N im Alltag oft anstelle wissenschaftlicher gebraucht und missbraucht, in Demokratien wegen der Rede- und Diskussionsfreiheit auch öffentlich, z.B. bei sozialen und politischen Entscheidungen. Dies kann früher oder später zu Katastrophen führen, wie wir dies auch heute in Demokratien erleben.

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Einerseits werden solche Werte N fälschlich oder demagogisch in der Politik wie wissenschaftliche, empirisch verwirklichbare, und sogar als normative Werte angewandt, z.B. in Wahlversprechen. Ihre fälschliche Verwendung hat aber auch viel zum Zusammenbruch deterministischer Vorausagen in der Ökonomie und in den Sozialwissenschaften beigetragen. Andererseits enthält N im individuellen Gedächtnis auch Werte, die früher oder später, wenn sie die Bedingungen RNNMS erfüllen, verbessert, zu wissenschaftlichen umgeformt und daher umgewertet werden können, d.h. solche die potentiell wissenschaftlich sind. Daher oben die Formulierung „bis jetzt“. Dasselbe gilt für C1–C3. Diese Umwertung ist eine gängige evolutionäre Methode des Entdeckens von neuen Werten, neben der Innovation, dem Erfinden, die durch Ausnützung positiver Zufälle im Verein mit Bayes’schem Lernen neue Werte schafft (Leinfellner 2000). Eine Beispiel für eine Umwertung, allerdings nicht innerhalb von N, sondern innerhalb von W, ist der Banach-Raum. Bis 1910 gab es für ihn keinerlei empirische Interpretation. Aber 1910 wurde er von Hilbert als empirischer Raum der Quantenphysik „entdeckt“. Das „entanglement“ ist auch ein Beispiel für eine Umwertung innerhalb von W, hier der Quanthentheorie. Die Verschränkung, das „entanglement“ in der Quantenphysik galt zunächst als eine kuriose physikalische Eigenschaft von Mikro-Teilchen. Einstein bezeichnete sie als „spukhaft“. Zeilinger wies nach, dass das „entanglement“ wissenschaftlich möglich ist und empirisch existiert (Zeilinger 2005). Ein letztes Beispiel: Schillers Forderung nach Gedankenfreiheit im „Don Carlos“ wurde erst später gesetzlich als ein höchster demokratischer Wert entdeckt und zur garantierten Freiheit der Rede und Diskussion in Demokratien umgeformt. Durch das wissenschaftliche Testen mit der Methode RNNMS plus C1C3 wird zumindestens in Demokratien die Ausbildung von quantitativen Werten, von Wertmessungen und von Prognosen möglich und dynamisch. Die wissenschaftlichen und/oder empirisch verwendbaren Werte W sind die wichtigsten menschlichen Bewertungen, die in den Prognosen über das künftige Wohl und Wehe, d.h. über die Risiken unserer individuellen und kollektiven Wohlfahrt in Demokratien und demokratischen Wohlfahrtsstaaten verwendet werden können. Sie erlauben zwar keine traditionell deterministischen Vorausagen, sondern nur statistisch-evolutionäre über das erwartete durchschnittliche Auf und Ab von Wirtschaftswachstum, Arbeitslosigkeit, von internen und externen Zufällen in den dynamisch-

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evolutionären modernen Demokratien. Da diese multifeedback Systeme sind, in deren evolutionären Verlauf der Mensch und seine demokratischen Regierungen selbst eingreifen können, treten sie an die Stelle deterministischer Voraussagen. Solche nicht deterministischen Voraussagen haben aber den Vorteil, dass sie auch unter Unsicherheit und Risiko funktionieren und berechnet werden können.

6. Die RNNMS-Methode schließt auch die somatischneuronale Erklärung der Entstehung von Werten mit ein Die RNNMS-Kriterien sind nicht nur eine auch psychologische, wahrscheinlichkeitstheoretische Rekonstruktion des bewussten wissenschaftlichen Bewertens, gegeben Antizipationen von präferenziellen Bewertungen, sondern sie schließen auch die Damasio’sche somatisch-neuronale Wertentstehung in unserer Gedächtniszeit ein (Leinfellner 2006). Wenn die Antizipation A1 der Antizipation A2 vorgezogen wird, dann werden zwei bewertete Alternativen A1 und A2 im individuellen Gedächtnis statistisch mit einer frequentistischen Wahrscheinlichkeit β (Gebrauchswert) bewertend zusammengefasst, ausgedrückt als: [βA1, (1 − β)A2]. Solche Rekonstruktionen sind Mischungen von Antizipationen, deren Wahrscheinlichkeiten die Verteilung, die Addition und Multiplikation, sowie die Wertordnung und Skalierung von Präferenzen ermöglichen und festlegen, z.B ob etwas, ein Kulturfakt, ein Artefakt, ein Technifakt, „überwiegend, mehrheitlich“ pragmatisch nützlich oder pragmatisch gut im Vergleich zu einem anderen sein wird. Trivial heißt dies: Kein Wert kommt allein. Alle Werte in Antizipationen sind komplementär. Auch ein guter Mensch kann nicht zu 100% gut sein, sondern ist z.B. zu 95% = β gut, z.B. altruistisch und hie und da zu 5% = 1 − β schlecht, z.B. egoistisch. In christlichen Religionen kann er z.B. 5% lässliche Sünden begehen, die ihm vergeben werden, wenn er bereut. β ist dann 0,95 und 1 − β 0,05. So werden schlecht bewertete Handlungen dann geduldet, wenn sie z.B. nur einmal unter hundert als gut bewerteten Handlungen vorkommen (Leinfellner 1985). Die Rekonstruktion des Antizipierens als Savage’sche Vorform des bewussten Bewertens fängt also immer mit Paaren von Antizipationen βA1 und (1 − β)A2 an. In nicht quantifizierter Form sind diese Antizipationen schon im neuronalen Gedächtnis gespeichert, und sie werden durch Wahr-

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scheinlichkeiten β und (1 − β) komplementär zusammengehalten. Sie bilden ein einheitliches Paar, zunächst ein Tupel, dann aber auch Tripel, Quadrupel etc., wobei β auch durch Zufallsereignisse zustande kommen kann. Der Gesamtwert V ist mit V = [v(βA1) + v(1 − β)A2] schon im Gedächtnis abgeschätzt gegeben. Z.B: Ein Ehepaar beschließt, im Schnitt einmal pro Monat ins Theater und zweimal ins Kino zu gehen. Dies wird für beide eine primitive, erwartete und kollektive Bewertung. Statistisch drückt also die größere empirische Häufigkeit aus, dass die Antizipation A1 „besser“, d.h. ihr geschätzter oder gefühlter Wert größer oder besser als der von A2 ist, vice versa. Dies gilt auch von den Vorstufen des Wertens im somatisch-neuronalen Prozess der Verarbeitung von Werten, z.B. dass im Durchschnitt A1 „besser“ („>“) als A2 ist, etc. Der βWert der Antizipation, d.h. der gradierende antizipierte Schätzwert (die „futures“) von Ereignissen, Dingen, Kulturfakten Mentifakten, Soziofakten, Artefakten wird so erhalten. Der Gebrauch von Zahlen a, b, c für die quantitative Wertskalierung ist nur eine historisch späte, mathematischstatistische, quantitative Repräsentation davon, dass somatisch-neuronale primitive wahrscheinliche Werte als Vorstufen einer Wertskalierung fähig sind. Wenn wir eine Münze aufwerfen, um zwischen A1 und A2 zu entscheiden und so eine 50% : 50% Ausbeute erhalten, dann wissen wir, dass wir eine ½ A1- und eine ½ A2-Erwartung haben, eine statistisch-stochastische Gleichwertigkeit der individuellen Bewertung, die wir dem Aufwerfen von Münzen überlassen haben. Damit wird der Zufall zum Bewerter. Dies ist aber keine deterministische Voraussage, ob Kopf oder Adler als nächstes aufscheinen wird. Wenn wir 75% : 25% Ausbeute haben, dann zeigt dies ex post, welche Alternative im Schnitt, z.B. bis jetzt, vorzuziehen sei. Eine Antizipation „ist“ die künftige, in Wahrscheinlichkeiten ausgedrückte Bewertung durch ein Individuum, mit einer ex ante Erwartung, einer künftigen, möglichen Verteilung von β. Die Antizipation ist statistisch und liegt in zukünftigen Bereichen, erst recht, wenn sie von unvorhersagbaren Zufallsereignissen abhängt. Sie sind nicht deterministische, sondern irreduzibel-statistische Voraussagen, die wegen der Unberechenbarkeit von Zufällen nicht näher bestimmt oder deterministisch vorausgesagt werden können. Auch so einfache Voraussagen wie „Morgen werde ich Frühstück essen“ gehören hieher. Wenn die Wahrscheinlichkeit β ist, wobei 0 < β ≤ 1 gilt, und wenn A1 und A2 zwei Antizipationen sind, dann gibt es immer eine Antizipation, die einen Wahrscheinlichkeitswert hat, den wir mit [βA1 + (1 − β)A2] in der Rekonstruktion repräsentieren können.

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Das RNNMS-Verfahren (Kriterien) plus den empirischen C1 und C2 Kriterien umfasst Bedingungen, die auf alle gegebenen Antizipationen und Präferenzwerte angewendet werden können. Man kann dann prüfen, ob gegebene Präferenzwerte wissenschaftliche Bewertungen sind, und ob sie in allen Entscheidungsprozessen z.B. der Nutzen-, Spiel- und Entscheidungstheorie empirisch angewandt werden können. Zusammengefasst: Wie das neuronale menschliche Bewerten die Zukunft mit ihrer Unsicherheit und ihrem Risiko bewältigt und optimal meistert, kann kognitiv nur mit Hilfe von stochastischen Wahrscheinlichkeiten und der finiten Wahrscheinlichkeitsrechnung rekonstruiert werden. Die RNNMS-Methoden der Rekonstruktion sind nicht logisch-deduktiv, sondern beruhen auf Ersetzung der Werte durch finite Wahrscheinlichkeiten (Leinfellner 2006). Die rekonstruktiven Kriterien 1–6 sind: 1. Wenn Wahrscheinlichkeiten zwischen 0 und 1, 0 < β ≤ 1, zwei alternative, im Gedächtnis gespeicherte Antizipationen A1 und A2 charakterisieren, und beide mit β und 1 − β komplementär voneinander abhängig sind, dann heißt dies, dass ihre erwarteten Werte nicht klassische Kolmogoroff ’sche Wahrscheinlichkeiten sein können; sie können aber eingeschätzt und berechnet werden. Daher gibt es weiters immer eine andere Antizipation, die wir als eine wertmäßige Alternative, [(βA1) + (1 − β)A2], darstellen können, wenn folgende weitere Kriterien gelten: 2. Wenn ein Individuum die Antizipationen im Gedächtnis bewertet, dann kann es sich frei entscheiden, welche Antizipation es einer anderen primitiven Antizipation vorzieht, oder ob es beide gleich bewertet. 3. Nach all dem Gesagten muss der somatisch-neuronale Bewertungsprozess, die Bewertungsskalierung oder die Wertordnung vieler aneinander gereihter Antizipationen, die im Gedächtnis und im Rahmen der a-chronologischen Gedächtniszeit gespeichert sind, transitiv sein, um als Wert im Sinn der Wissenschaft angesehen zu werden. Die RNNMS-Kriterien fordern: Wenn A1 besser als A2 ist und A2 besser als A3, dann ist auch A3 besser als A1. Auf diese Weise wird in Demokratien festgestellt, was sozial optimal ist. Durch die Anwendung der RNNMS-Methode wird getestet, ob multiple Antizipationen transitiv und demokratisch akzeptierbar sind (C3). Die Transitivität bezieht sich auch auf die statistische Transitivität von Verteilungen. 4. Wenn A1, A2 und A3 wie in der zweiten Annahme oben im individuellen Gedächtnis gegeben sind, dann gibt es eine Wahrscheinlichkeits-

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Kombination von A1 und A3, die als Ganzes gleich bewertet, eingeschätzt wird wie A2. Dann gilt weiter: 5. Ist β gegeben, wobei 0 < β ≤ 1, und sind A1 and A2 präferenziell oder emotional gleich, dann sind βA1 + (1 − β)A3 und βA2 + (1 − β)A3 primitiv gleich oder präferenziell wertgleich. Man kann dann die Antizipationen A1 und A2, da sie wertgleich sind, gegenseitig austauschen. Die Antizipationen A1 und A2 sind dann emotional, primitiv und in der Rekonstruktion mathematisch gleich, d.h. A1 kann für A2 wertmäßig substituieren. A1 wird so in jeder Wertordnung, die die Regel 1–5 erfüllt, wertgleich mit A2 sein. Diese Annahmen (Kriterien) genügen bereits, um eine präferenzielle Wertordnungfunktion oder statistisch-mathematische „utility function“ für den Bewertenden aufzustellen. Diese Wertfunktion ist nicht eindeutig; sie steht einfach für den neuronalen Wertoperator, der hier das somatischneuronale Gehirn ist. Die RNNMS-Rekonstruktion ermöglicht, dass man heute reelle Zahlen als numerische Bewertungen oder als Werte v(a), v(b), v(c) benützen kann, vorausgesetzt a > 0. Sie ordnet mathematisch jeder Antizipation einen Zahlenwert (wahrscheinlichen Wert) zu. Wenn u solch eine Wertfunktion des bewertenden neuronalen Operators oder der Operator ist, dann ist es auch au + b, vorausgesetzt a > 0. Wenn Großbuchstaben Antizipationen bedeuten, und kleine Buchstaben reelle Zahlen für ihren Wert, dann müssen Wertfunktionen oder menschliche Wertzuordungen u die folgenden formalen Eigenschaften im Alltag, in jeder Wohlfahrtsökonomie, in demokratischen Wohlfahrtsstaaten, in der Mikroökonomie, in der Theorie des erwarteten Nutzens haben, sodass: 6. Wenn u in u(A1) > u(A2) ist, dann drückt dies zahlenmäßig und wissenschaftlich aus, dass A1 einen größeren Präferenzwert als A2 für den individuellen Wertoperator oder für das Individuum (Menschen) als Operator hat. Wenn nun 0 < β ≤ 1 dann ist u[βA1 + (1 − β)A2] = βu(A1) + (1 − β)u(A2), was genau die primitive Linearität einer statistischen Wertfunktion ausdrückt, deren Werte Wahrscheinlichkeitsverteilungen von Werten sind (Nash 2002, 38ff.; Leinfellner 2006, 139f.). Das bisherige Resultat ist, dass auch wissenschaftliche Voraussagen nur Antizipationen sein können, nur statistische, stochastische Erwartungen von individuellen und kollektiven zukünftigen Chancen und Risiken für einzelne und viele Menschen. Dies gilt für die Menge W von Werten. In der Praxis kommen alle Werte und Antizipationen vielleicht nur in Mischungen von positiven und nicht-positiven, erwarteten Werten [βA1 + (1 − β)A2] vor. Es bleibt uns nur übrig, die positiven so schnell als möglich zu verwirklichen.

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Werte, die außerhalb dieser wissenschaftlichen und/oder empirisch verwirklichbaren Wertinseln W liegen, wie die bisher noch nicht überprüften Werte N, z.B. postmoderne oder phantastische Werte, können aber jederzeit, sollten sich inzwischen Veränderungen, Zufallsereignisse ereignet haben, die den individuellen und kollektiven Wohlfahrtsstandard verbessert oder verschlechtert haben, neuerlich wissenschaftlich überprüft werden; besonders können sie entsprechend den Kriterien C1–C3, die Leinfellner 2000 und 2006 vorgeschlagen hat, auf ihre empirische Anwendbarkeit im Vorhinein überprüft werden. Das lässt uns die Krise der Voraussagbarkeit z.B. in der gegenwärtigen Ökonomie, in den Politischen Wissenschaften und in der Soziologie in einem neuem Licht sehen. Erstens: Alle Voraussagen in den Sozialwissenschaften können nur als statistische, sich stets ändernde Erwartungen individueller und/oder kollektiver Verbesserungen oder Verschlechterungen der individuellen und kollektiven Wohlfahrt berechnet werden, die ja die Hauptkriterien demokratischer Wohlfahrtsstaaten sind. Daraus folgt zweitens, dass z.B. Wirtschaftsprognosen in demokratischen Wohlfahrtsstaaten nur kurzfristig, ceteris paribus, solange keine einschneidenden Veränderungen oder Zufallsereignisse auftreten, gemacht werden können. Empirische Verwirklichungen nach C1–C3 können durch Bayes’sche Lernprozesse ergänzt und verbessert werden, die in Demokratien dazu führen, dass die individuelle und die kollektive Wohlfahrt verbessert oder zumindest nicht verschlechtert wird. Zum Schluss eine Kurzfassung von C1–C3: C1: Präferenzen, die omnipräsent im individuellen Gedächtnis gegeben sind, müssen zumindest potenziell verwirklichbar sein. C2: Präferenzen müssen zumindest transitiv geordnet werden können, um wissenschaftlich zu sein. C3: Präferenzen müssen mit demokratischen Regeln vereinbar sein.

Literatur Assmann, J. 1995 (2. Auflage) Ma’at: Gerechtigkeit und Unsterblichkeit im Alten Ägypten, München: Hanser. Aungar, R. 2000 (Hg.) Darwinizing Culture: The Status of Memetics as a Science, Oxford: Oxford University Press. Barbour, J. 1999 The End of Time: The Next Revolution in Physics, Oxford: Oxford University Press.

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Basar, E. 1980 EEG-Brain Dynamics: Relation Between EEG and Brain Evoked Potentials, I und II, Amsterdam: Elsevier. — 1988a (Hg.) Dynamics of Sensory and Cognitive Processing of the Brain, Berlin: Springer. — 1988b „EEG-Dynamics and Evoked Potentials in Sensory and Cognitive Processing by the Brain“, in: Basar 1988a, 30–55. Beth, E. 1959 The Foundation of Mathematics: Studies in Logic and the Foundations of Mathematics, Amsterdam: North Holland. Bullock, A. und Trombley, St. (Hg.) 1999 The Norton Dictionary of Modern Thought, New York: Norton. Churchland, P. S. 1989 Neurophilosophy, Cambridge: MIT Press. Cuccurullo, L. / Mariano, E. (Hg.) 2005 Contesti e validità: del discorso scientifico, Rom: Armando. Damasio, A. 1994 Descartes’ Error: Emotions, Reason, and the Human Brain, New York: Putnam. — 1999 The Feeling of What Happens: Body and Emotions in the Making of Consciousness, New York: Harcourt, Brace & Co. Davies, P. 1995 About Time: Einstein’s Unfinished Revolution, New York: Simon and Schuster. Dennett, D. 1991 Consciousness Explained, Boston: Little, Brown & Company. Galavotti, M. C. 2006 (Hg.), Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle, Dordrecht: Springer. Götschl, J. 2005 „Self-organization: Epistemological and Methodological Aspects of the Unity of Reality“, in: Cuccurullo / Mariani 2005, 107–134. Harsanyi, J. C. 1976 Essays on Ethics, Social Behavior, and Scientific Explanation, Dordrecht: Reidel. Howard, P. J. 2002 The Owner’s Manual of the Brain, Atlanta: Bard Press. Kotulak, R. 1997 Inside the Brain: Evolutionary Discoveries of How the Brain Works, Kansas City: Andrews McMeel. Lane, R. D. und Nadel, L. 2002 (Hg.) Cognitive Neuroscience of Emotion, Oxford: Oxford University Press. LeDoux, J. 2002 „Cognitive-Emotional Interactions: Listen to the Brain“, in: Lane und Nadel 2002, 129–155. Leinfellner E. 2006 „Drei Pioniere der philosophisch-linguistischen Analyse von Zeit und Tempus: Mauthner, Jespersen, Reichenbach“, in diesem Band.

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Leinfellner E. und Leinfellner W. 1978 Ontologie, Systemtheorie und Semantik, Berlin: Duncker und Humblot. Leinfellner, W. 1985 „A Reconstruction of Schlick’s Psychosocial Ethics“, Synthese 64, 317–349. — 1988 „The Brain-Wave Model as a Protosemantic Model“, in: Basar 1988a, 349–354. — 2000 „The Role of Creativity and Randomizers in Societal Human Conflict and Problem Solving“, La Nuova Critica II, Nuova Serie, 5–29. — 2006 „The Somatic Neuronal Foundation of Human Preferential Evaluation from Ramsey to Damasio“, in: Galavotti 2006, 139–153. Nash, J. 2002 The Essential John Nash, hg. von H. W. Kuhn und S. Nasar, Princeton: Princeton University Press. Ramsey, F. P. 1931 The Foundations of Mathematics and Other Logical Essays, hg. von R. B. Braithwaite, London: Kegan Paul. Savage, L. J. 1954 Foundations of Statistics, New York: Wiley. Tulving, E. 1983 Elements of Episodic Memory, Oxford: Clarendon. Zeilinger, A. 2005 Einsteins Spuk, München: Bertelsmann.

Für wertvolle Diskussionen danke ich J. Barbour (South Newington, UK), H. Götschl (Universität Graz, A), E. Leinfellner (Universität Wien, A), J. Schank (University of California, Davis, USA), F. Stadler (Universität Wien, A), P. Suppes (Stanford University, USA) und M. Stöltzner (Universität Wuppertal, D).

Wittgenstein und Sraffa Zeitproduktion durch Zeit Peter Weibel, Karlsruhe In seinem berühmten Tractatus Logico-Philosophicus (1921) präsentierte Ludwig Wittgenstein seine Bildtheorie der Sprache: „Der Satz ist ein Bild der Wirklichkeit“ (Tractatus Logico-Philosophicus, 4.01). Sprache, wenn sie adäquat benutzt wird, bildet die Realität ab. Daher ist es die Herausforderung der Philosophie, die Sprache zu entwirren, um Probleme zu klären. Die kritische Betrachtung der Sprache ist das Ziel des Philosophen. Zwischen Sprache und Realität besteht eine Verbindung, eine gemeinsame logische Form. Alles was ist kann in einer formalen Sprache ausgedrückt und beschrieben werden. Was nicht gesagt werden kann, darüber sollte man schweigen. Während seiner Zeit in Cambridge modifizierte Wittgenstein seine Bildtheorie radikal. Daher wird auch gerne von Wittgenstein I und Wittgenstein II gesprochen. Kurz gesagt kann man von der Bildtheorie der Sprache (Wittgenstein I) und der Theorie des Sprachspiels (Wittgenstein II) sprechen, wie am besten in Wittgensteins Philosophischen Untersuchungen (Philosophical Investigations) beschrieben. Doch wie ist Wittgensteins radikale Abkehr von seiner zuerst aufgestellten Theorie der Sprache zu erklären? In seiner Theorie des Sprachspiels wird Bedeutung nicht abgeleitet von der logischen Form wie in der Bildtheorie der Sprache, sondern es wird davon ausgegangen, dass die Bedeutung eines Satzes allein durch seinen Gebrauch hergestellt wird. Der Gebrauch gibt den Worten eine Bedeutung, nicht ihre logische Form. Im Vorwort zu den Philosophischen Untersuchungen gibt Wittgenstein einen Hinweis darauf, wie er zu dieser neuen Sicht der Dinge gelangte: „Mehr noch als dieser – stets kraftvollen und sicheren Kritik [von Frank Ramsey] verdanke ich derjenigen, die ein Lehrer dieser Universität, Herr P. Sraffa, durch viele Jahre unablessig [sic!] an meinen Gedanken geübt hat.“ Wer war dieser Piero Sraffa, der so nachhaltigen Einfluss auf Wittgenstein hatte? Sraffa war ein enger Freund des italienischen revolutionären Marxisten Antonio Gramsci. Während Gramsci, von Mussolini inhaftiert, im Gefängnis saß, versorgte Sraffa ihn mit Büchern, Schreibutensilien etc. und blieb in engem Kontakt mit ihm. Unter anderem von Gramscis Philosophie der Tat beeinflusst, schrieb er einen kritischen Artikel im Manchester F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 409–417.

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Guardian, der Mussolini verärgerte und Sraffa gefährdete. Aufgrund dessen lud John Maynard Keynes Sraffa ein, an das King’s College nach Cambridge zu kommen. Sraffa wurde dort Bibliothekar und später Fellow am Trinity College. Schnell wurde er ein wichtiges Mitglied der akademischen Welt – er gehörte zu der so genannten „Cafeteria Gruppe“, der sich auch Frank Ramsey und Ludwig Wittgenstein anschlossen und die das von J.M. Keynes 1921 aufgestellten Traktat über Wahrscheinlichkeitstheorie (Treatise on Probability) untersuchten. Zusammen mit Keynes versuchte Sraffa die Theorien von Friedrich Hayek zu widerlegen (Sraffa, 1932). Ab 1931 begann er zudem, die Schriften von David Ricardo (Nationalökonom, 1772–1823) zu sammeln und zu editieren, die 1953 publiziert wurden. Seine Einführung in das Buch ist bemerkenswert und gab der Neoricardianischen Schule und der Kapitalkontroverse (Cambridge-Capital-Controversy) einen großen Aufschwung. Doch vor allem Sraffas Buch Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory (Warenproduktion mittels Waren), das er in den 1920er Jahren begonnen hatte und das 1960 publiziert wurde, bestätigt Ricardos Theorie für die Moderne. Sraffas Einfluss auf Wittgenstein kann offensichtlich als Übertragungsfunktion betrachtet werden. Sraffas Ansichten stammen von einer Philosophie der Aktion, die Gramsci prägte und die auch Wittgenstein beeindruckte. Indem Sraffa mit Gramsci und später mit Wittgenstein diskutierte, waren es Gramscis Ideen, die, durch Sraffa gefiltert, Wittgensteins Sicht der Welt prägten. Die Wendung von der Bildtheorie der Sprache hin zur Theorie des Sprachspiels geht also auf Sraffa und Gramsci zurück. Wittgenstein war generell in politischen und ideologischen Dingen sehr konservativ gewesen, doch in seiner späten Theorie der Sprache kam er zu einer sehr demokratischen, um nicht zu sagen marxistischen Sicht der Dinge. Sraffa selbst ging über Gramscis Theorie hinaus und verfasste seine eigene ökonomische Theorie. Man kann drei Entwicklungsstufen erkennen: 1. Ricardo definierte als Ware von allem die Arbeit. Arbeit wiederum definiert den Wert und Preis einer Ware. 2. Marx definierte den Wert und Preis einer Ware ähnlich wie Ricardo, jedoch etwas präziser, da er eine neue Kategorie einführte: die Zeit – und zwar im Sinne der Arbeitszeit. Diese neue Kategorie ermöglichte es ihm, seine Theorie über den Mehrwert aufzustellen. 3. Sraffa vernachlässigte die Kategorie Arbeit als Quelle für den Wert einer Ware. Er schlug eine Input-Output-Analyse der Waren vor, die

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dann den Wert der Waren bestimmen sollte. Die Frage ist also, wie viele Waren braucht man, um eine andere Ware produzieren zu können. Wie viele Waren muss man produzieren, um eine andere Ware konsumieren zu können. Es gibt zwei Möglichkeiten, wie Mehrwert (den der Kapitalist auf dem Rücken der Arbeiter austrägt) produziert werden kann. Der Arbeitstag, also die Arbeitszeit, kann verlängert werden, was Marx als absoluten Mehrwert definierte, oder technische Innovationen können die Arbeitszeit für einen Arbeitsprozess verkürzen – ein relativer Mehrwert. In jedem Fall hat man eine Gleichung, in der die Zeit eine Rolle spielt. Entweder die Arbeitszeit ist länger, damit mehr Güter produziert werden können, während der Lohn gleich bleibt; so entsteht ein Mehrwert für den Kapitalisten, der die Produktionsmittel besitzt. Oder die Arbeitszeit wird aufgrund von technischen Innovationen kürzer und die Löhne werden aufgrund der kürzeren Arbeitszeiten gesenkt, ohne dass die Produktion weniger wird; so entsteht der Mehrwert für den Kapitalisten. Man kann also sagen, Arbeit ist produktiv, wenn sie relativen Mehrwert produziert. Genauer, die Arbeitszeit ist die Quelle für den Mehrwert. Das Zeitverhältnis ist die Quelle für den Mehrwert. Diese Betrachtung verändert unsere Einstellung bezüglich des Konsums. In der klassischen Wirtschaftstheorie wurde der Konsum als unproduktiv betrachtet, da kein Mehrwert erzeugt wurde – im Gegenteil, der Konsument „verlor“ dadurch sogar Geld. Wenn man die Produktivität mit dem Mehrwert verknüpft, erscheint der Konsum, da hier kein Mehrwert produziert wird, als unproduktiv. Verknüpft man aber die Produktivität nicht mit dem Mehrwert, sondern sieht Produktion und Konsumption als separate Kategorien, kann man folgendermaßen abstrahieren: es gibt eine Produktionszeit und eine Zeit des Konsums. In beiden Fällen haben wir es mit bestimmten Zeitverhältnissen zu tun. Wenn man nicht genug Produktionszeit produziert (äquivalent zum Geld, das man bekommt), hat man nicht genug Zeit für den Konsum (äquivalent zum Geld, das man ausgeben kann). Daher muss man zu Marx Ausspruch „Geld als Wertmaß ist notwendige Erscheinungsform des immanenten Wertmaßes der Ware, der Arbeitszeit“ (Marx 1962, S. 109) hinzufügen: die Maßeinheit für Wert ist Zeit, und zwar Arbeits- und Konsumptionszeit. Arbeit und Konsum sind Vektoren der Zeit. Aber der Wert der Zeit ist variabel. Für den ungelernten Arbeiter oder Experten auf seinem Gebiet hat Zeit einen

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anderen Wert, was sich in einem unterschiedlichen Preis niederschlägt. Wie werden also Werte zu Preisen? Das berühmte Transformationsproblem der Ökonomie wird durch den Faktor Zeit gelöst. Zeit kann als Geld, in Zahlen und Ziffern ausgedrückt werden. Ricardo definierte Arbeit als Wert und Zeit als dessen Maßeinheit. Der Preis ist die Zeitform des Wertes, und Geld ist die Wertform der Zeit. Daher können Werte in Preise transformiert werden bzw. Wertformen in allgemeine Geldware. Zeit als Maßeinheit der Arbeit wird zum alleinigen Wertmaß. Der homo economicus ist ein Mensch der Zeit. Auch der Raum wird dem homo economicus zu Geld, weil der Raum ja nur Statthalter der Zeit ist. In einem Roman von Raymond Chandler sagt der Wirt zu seinem Gast, der zu lange an einem Tisch sitzt, zu lange einen Raum okkupiert und zu wenig konsumiert: „Man, this is money space“. Nach „time is money“ nun auch „space is money“, denn auch der Raum verfällt schon seit langem der Wertform des Geldes. Die verkaufte Zeit steht über dem Eingang zur Industriegesellschaft. Chronometrie, gemessene Zeit, ist die Voraussetzung für eine Marktökonomie, in der Geld zur höchsten Ware wird. Wenn also Taylor und Gilbreth als Betriebsingenieure Zeitstudien betrieben, dann in der richtigen Einschätzung der Lage, dass Zeit das edelste Metall ist, nämlich Gold, das eigentliche invariante Wertmaß. Die Zeit- und Bewegungsstudien um das Ende des 19. Jahrhunderts begleiten eine Industrialisierung der Zeit, sie beweisen den Beginn einer Herrschaft der Zeit. Zeitstudien waren Sondierungen des Kapitals, der abstrakten Wertform Geld, sie bereiteten den Boden für den subtilen Kapitalismus der postindustriellen Gesellschaft. Der durch den Tauschwert eingeleitete Abstraktionsprozess, der sukzessiv Gegenstände in Waren und Waren in Zeichen verwandelte, dieser Prozess der Auflösung des Objekts, hat sich im postindustriellen Kapitalismus, der ein Feudalismus der Zeit ist, so zugespitzt, dass das Geld selbst verschwindet, weil es noch zu sehr seine Herkunft vom Objekt, von der Ware verrät. Im Zuge der Temporalisierung des Raumes durch die beschleunigte Maschine wurde auch die Arbeit temporalisiert. Nicht nur die Bewegung wurde vom Raum unabhängig, sondern verabsolutiert und verabschiedet haben sich auch der Wert von der Arbeit und der Preis vom Wert. Das führte zu einer Verselbständigung der Preise und der Geldware, zur Abstraktion der Geldzirkulation, die fast irreal und immateriell wurde. Aktienmarkt, Börsengeschäfte, Wertpapiere, Termin- und Geldwechselgeschäfte, wo Geld mit Geld getauscht wird und der Profit aus der bloßen

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Zeit-Differenz (!) und nicht mehr aus irgendeiner Form produktiver Arbeit entsteht, belegen diese beschleunigte Abstraktion der Arbeit, welche die Arbeit selbst tilgt. Der reine Geldmarkt löscht das Geld – wie im Raumzeitalter der Raum verschwindet. Das Auftauchen der Scheckkarte, die das Geld in dem Maße verdrängt, dass man in den USA sofort der Kriminalität verdächtigt wird, wenn man mit Geld statt mit einem Scheck bezahlt, bezeugt das Verlöschen des Geldes als Objekt, als Ware zweiten Grades, belegt die Verabsolutierung des Geldes zur Immaterialität. Nach der Arbeit und nach der Zeit hat sich auch das Geld so abstrahiert, dass es nicht nur als Objekt zweiten Grades verschwindet, sondern in einer paradoxen Drehung sogar zum Gradmesser von Armut wird. Wer wirklich reich ist, hat kein Geld mehr, zumindest in der Tasche, sondern eine Scheckkarte. In besseren US-Hotels bekommt man mit Bargeld kein Zimmer, sondern nur mit Scheckkarte. Wer Bargeld hat, ist der Ärmste. Wer Bargeld hat, ist im Zeitalter der Scheckkarte, des temporalisierten Geldes, in Wirklichkeit bar des Geldes. Scheck und Kredit zeigen den Kapitalismus als Geschäft mit der Zeit, als Ergebnis der fortschreitenden Abstraktionen und Verselbständigungsprozesse von Arbeit, Wert, Zeit, Geld als die Parameter und Stützen unserer Ökonomie. Dieser vom Tauschwert verursachte Abstraktionsprozess hat die Werttheorie zur Preistheorie verwandelt. Dieser Prozess hat die Arbeit bis zur Verdrängung abstrahiert, so dass im „ökonomischen Kalkül“ (Charles Bettelheim) eine Verschiebung von der Warenproduktion durch Arbeit zur „Warenproduktion mittels Waren“ (nach der gleichnamigen Publikation von Piero Sraffa, 1960) stattfindet. Die Ökonomie der Preise hat sich in der Tat durch die kapitalistische Produktionsweise und Zirkulation des Kapitals so sehr von der Realität der Arbeit entfernt, dass Sraffa zu einem strukturalen mathematischen Modell greifen musste, um die Marktgesetzlichkeiten erklären zu können. Auch die Ökonomie ereignet sich im mathematischen virtuellen Raum. Unter dem Diktat der Zeit und der Chronometrie wird auch die Ökonomie zur Ökometrie. Die letzte Stufe der Abstraktion der Arbeit ist die gegenwärtige Zeitproduktion mittels Zeit. Die Ökonomie lebt davon, dass die Arbeitenden ihre (Lebens)Zeit investieren und verkaufen (in Form von Arbeitszeit), um Geld zu verdienen, damit sie ihre Freizeit (in Form von Waren, Erholung und Subsistenzmitteln) kaufen können. Zeit ist es, was investiert, gekauft und verkauft wird. Daher sind die so genannten Zeit- und Termingeschäfte die lukrativsten. Die höchste, abstrakte Form des Kapitalismus ist die Monetarisierung der Zeit. Das ist die logische Konsequenz jener Gleichung,

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die Arbeitszeit und Wertzeit, Zeitwert und Arbeitswert gleichsetzte. Auch der Marxismus hat sich diesem fatalen Horizont nicht entzogen. Die Inflationen und Krisen des Kapitalismus entstehen aus Zeitimplosionen im ökonomischen Kalkül. Benjamin Coriat beschreibt in L’atelier et le chronomètre (1979) den Beginn des Zeitkapitalismus im Fordismus. Im Zeitalter der Arbeitsökonomie gab es die Angst vor der Überproduktion von Waren. Eine Krise der Überproduktion entstünde, würde nicht gleichzeitig eine Massenkonsumption von Waren eintreten, indem diese billig genug sind, dass die Arbeiter sie kaufen können. Deswegen hat Henry Ford bekanntlich gesagt, unsere Arbeiter sollten auch unsere Kunden sein. In einer Art Selbststeuerung des Marktes waren die Arbeiter in großem Maße auch Konsumenten der Waren, die sie produzierten. Auf dieser Basis der Massenproduktion und Massenkonsumption wurde intensiv Kapital akkumuliert, entstanden die großen Vermögen Amerikas. Diese Selbstregulation geriet aber in eine Krise der Überproduktion, als der wichtigste Markt für solche Massenprodukte, nämlich der Haushalt, gesättigt war. Der zweite Grund war die beschleunigte Produktivität, nämlich die Ersetzung von Menschen durch Maschinen im automatisierten Produktionsprozess, sie hat das Problem der Überproduktion drastisch verschärft. Die durch die Automation arbeitslosen Arbeiter verfügten einerseits über eine Überfülle von Zeit, andererseits über keine Kaufkraft, da es sich um eine Überfülle von Stehzeit handelte. Eine Schere begann sich zu öffnen: die Produktion von Massenwaren beschleunigte sich, aber gleichzeitig reduzierte sich die Masse der Arbeiter, welche die Massengüter kaufen konnten. Ein und dieselbe Quelle, nämlich Maschinen, war sowohl für das Ansteigen der Produktivität wie das Absinken der Kaufkraft verantwortlich. Je mehr durch Maschinen produziert wurde und je mehr Arbeiter durch maschinelle automatisierte Produktion „freigesetzt“ wurden, umso mehr Waren gab es, aber ebenso umso weniger Arbeiter, die genug Geld verdienten, um die von den Maschinen produzierten Güter kaufen und konsumieren zu können. Der Widerspruch ist am besten in monetärer Zeit formulierbar. Die Arbeiter waren die Konsumenten der Zeit, die sie produzierten. Als jedoch die Automation immer mehr Zeit produzierte, hatten die Arbeiter immer weniger Zeit, diese zu konsumieren. Eine Überproduktion nicht nur von Konsumgütern, sondern von Zeit war die Folge. Das Problem des Güterkonsums verdeckt nur das Problem des Zeitkonsums, die Überproduktion von Zeit. Die ökonomische Gleichung Arbeit-Wert-Ware-Zeit-Geld wurde durch den Faktor Zeit, durch die maschinelle Beschleunigung der Zeit, aus dem

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Gleichgewicht gebracht. Das ist das eigentliche Problem heute hinter dem so genannten demographischen Problem, dem Verhältnis von Rentnern und Arbeitenden. Die Automation hat zuviel Zeit erzeugt, eine Überproduktion von Waren und von Zeit. So hat ein komplexes äquilibristisches Spiel von Arbeitszeit, freigesetzter Zeit, von Freizeit und freier Zeit, von Produktions- und Konsumptionszeit begonnen. Das neue Ziel des Kapitalismus ist also, ein Maximum an Zeit (Konsum) mit einem Minimum an Zeit (Produktion) zu erzeugen, um der Überproduktion (von Waren und Zeit) zu entgehen. Die Theorie des „menschlichen Kapitals“, die jedes Individuum zum Produzenten (oder Künstler) macht, ist von Gary Becker (Human Capital: A Theoretical and Empirical Approach, 1964) bis zu Joseph Beuys (Kreativität alias menschliche Produktivität als Kapital) die neokonservative Konzeption des homo economicus. Wenn ein Individuum (theoretisch verkürzt) zum Produzenten von Waren wird, wird deren Wert gemessen als Maß der Zeit, den das Individuum damit verbringt, sie zu konsumieren, also als konsumierte Zeit. In einer Konsum orientierten Gesellschaft, deren Vektor der Abbau der Überproduktion ist, ist nicht mehr – wie bei Ricardo – die Arbeit die einzig wirkliche Reichtumsquelle der Nation, sondern der Konsum, zumindest in den neokonservativen Köpfen. Ein perpetuum mobile der Zeit ist der uneinlösbare Kern der neokonservativen ökonomischen Theorie. Das Gespenst der Zeit-Ökonomie heißt Überproduktion von Zeit. Denn um Zeit zu gewinnen, müssen die Individuen Geld verdienen, da ja Zeit Geld und Geld Zeit ist. Um Geld zu verdienen, müssen die Individuen einen Teil ihrer Zeit der Lohnarbeit widmen. Damit diese Zeit der Lohnarbeit nicht zu lang wird, um Zeit in der Produktion von Waren (Konsumzeit) zu gewinnen, muss das Individuum Maschinen bauen, welche den Produktionsprozess beschleunigen, d.h. Wissen erwerben. Das wiederum kostet Zeit; Maschinen sind also gespeicherte und gesparte Zeit. Aber um diese Zeit zu speichern, um mit den Maschinen die Zeit der Lohnarbeit zu verkürzen, um Zeit zu sparen, durch Maschinen, welche Güter schneller produzieren, muss das Individuum vorher diese Zeit hineinstecken, investieren, also die Zeit der Konsumation verzögern und verkürzen. So entsteht in Wirklichkeit ein neuer Asketismus, ein ungeheurer Zeitdruck. Das Individuum wird nämlich, um ein hohes Einkommen zu erreichen, das ihm scheinbar mehr Güter und Konsumzeit verspricht, den Großteil seiner Zeit dem Geldverdienen widmen, was ihm den Kauf jener Maschinen ermögli-

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chen soll, die Zeit sparen, um mehr Geld zu verdienen. Aber wenn die Zeit des Individuums nichts wert ist, sein Lohn ohnehin zu gering, wird es immer mehr Zeit haben, um immer weniger Güter zu konsumieren. Verlust an Zeit für die einen und Verlust an Waren für die anderen sind das Ergebnis der Schere der Zeit: Die einen, deren Zeit teuer ist, weil Experten, werden immer mehr arbeiten müssen, die anderen, die ungelernten Arbeiter, werden immer weniger arbeiten können. Das ist das wahre Gesicht der Zeitakkumulation statt der Kapitalakkumulation. „Kapitalakkumulation wird für ein Individuum nichts anderes sein als die Vermehrung der Waren, die ihm die Mühen ersparen, sich die Zeit zu nehmen, sie zu konsumieren.“ (Alliez/ Feher, 1986) Geld kristallisiert Zeit als Produktionszeit in Form von Bezahlung. Verdientes Geld ist dann gesparte Zeit. In Form von Kapital speichert Geld die Zeit der Warenkonsumation. Das Kapital wird zum Geber, Versorger der Zeit. Die absolute Chronokratie/Herrschaft der Zeit hat also begonnen, ein neuer Feudalismus der Zeit, wo es Zeitsklaven und Zeitherren gibt, aneinandergekettet im gemeinsamen Wunsch, die Zeit profitabel zu machen. Wie es im Feudalismus des 11. Jahrhunderts eine Unterernährung der Armen und ein Überfressen der Reichen gab, so im neuen Feudalismus immer mehr Konsumationszeit (die ja gespeichertes gespartes Geld ist) für die Reichen und immer weniger Konsumationszeit für die Armen (was äquivalent ist mit „freier Zeit“, frei gesetzter Zeit). Wer also keine Zeit hat im Sinne von Zeit als gespartes Geld, als Kaufkraft, muss sich Zeit kaufen. Aber wie, wenn er kein Geld hat, sich Zeit zu kaufen und sein Wunsch nach Zeit ja gerade daher rührt, dass er kein Geld hat? Indem er eben seine Zeit verkauft. Aber wie, wo wir doch davon ausgegangen sind, dass er keine Zeit hat? Im Zeitalter der Raumpolitik konnte man auf Grund und Boden ein Lehen aufnehmen. Im Zeitalter der Chronokratie kann man auf seine Lebenszeit ein Lehen aufnehmen. Das Individuum, dessen gegenwärtige Zeit nicht ausreicht, verkauft seine künftige Zeit, es nimmt ein Lehen auf seine künftige, erst in Zukunft sich kapitalisierende Zeit; es nimmt ein Dar-Lehen, einen Kredit auf. Es tauscht mit Zeit. Es kauft sich Konsumationszeit, damit es sich jetzt schon die Güter kaufen kann, für die es noch nicht genügend Geld (Zeit) produziert hat und die es jetzt konsumieren will, indem es künftige Zeit verkauft. Dafür muss dieser Zeitknecht natürlich Zinsen zahlen, Zeitzinsen. Er muss nämlich zumeist ein Drittel mehr an Zeit in Zukunft investieren, produzieren, bezahlen, was die Ware jetzt an Zeit gekostet hat. Das Individuum verliert also Zeit, Lebenszeit bei diesem Zeittausch in

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Form eines Darlehens. Du leihst dir Zeit von der Bank, du nimmst ein Lehen auf deine Lebenszeit bei der Bank auf – die sich wiederum gegen dein frühzeitiges Ableben, dein Ableben vor der Zeit, versichert, was auch du wiederum bezahlen musst –, damit es dir ermöglicht wird, jetzt schon Produkte anzukaufen, die du dir noch nicht leisten kannst, weil du dafür noch nicht genug gearbeitet hast, Geld verdient hast, Zeit produziert hast. Leih- und Lehn-Zeit sind gewissermaßen simulierte Zeit, Falschzeit. Der eigentliche Lohn der Arbeit heute ist Lehn-Zeit. Aus der Tatsache, dass heute fast alle, Privatpersonen wie Betriebe, von Krediten, Darlehen leben, erkennen wir, in welchem Maße wir schon in einer Chronokratie leben. Nicht mehr die Gegenwart genügt in dieser Wirtschaft, um die notwendige Zeit zu produzieren; alle brauchen mehr Zeit als sie haben. Daher wird auf Kosten künftiger Lebenszeit, auch künftiger Generationen, Zeit für den gegenwärtigen Verbrauch gekauft und geborgt. Alle machen Schulden auf ihre Lebenszeit und die Zeit ihrer Nachfahren. Surrogat-Zeit beherrscht die Ökonomie. Schulden auf Zeit bedeutet verschuldete Zeit als neues inflationäres Virus der Weltökonomie, die eine Zeitökonomie ist.

Literatur Eric Alliez, Michael Feher 1986 „The Luster of Capital“, in: Zone 1/2, New York, 352. Becker, Gary 1964 Human Capital: A Theoretical and Empirical Approach, New York: Nat. Bureau of Economic Research. Coriat, Benjamin 1979 L’atelier et le chronomètre: essai sur le taylorisme, le fordisme et la production de masse, Paris: Bourgois. Marx, Karl 1962 Das Kapital, Band 1, Berlin: Dietz. Sraffa, Piero 1932 „Dr. Hayek on Money and Capital“, in: Economic Journal XLII, 42–53. — 1960 Production of commodities by means of commodities: prelude to a critique of economic theory, Cambridge: Cambridge Univ. Press. Wittgenstein, Ludwig 2003 Tractatus Logico-Philosophicus, Frankfurt/M.: Suhrkamp. — 1971 Philosophische Untersuchungen, Frankfurt/M.: Suhrkamp.

A Mini-Guide to Logic in Action Johan van Benthem, Amsterdam & Stanford 1. The dynamic turn Classical logic is about propositions which we can know or believe, and unchanging inferential relationships between them. But inference is first and foremost an activity, for which propositions are merely the input, and the result. In recent years, there has been a growing awareness that various activities of reasoning, evaluation, belief revision, or communication, are themselves typical themes for logical investigation, and that their dynamic structure can be studied explicitly by logical means.1 For instance, it seems strange to study only the statics of what it means to ‘know’ a proposition, when knowledge usually results from basic actions of learning that we perform all the time, such as asking a question and getting an answer. Indeed, asking questions and giving answers are just as much logical core activities as drawing conclusions! This line can be extended: the natural dynamic counterpart of static epistemic logic is the theory of arbitrary individual or social learning mechanisms. Similar trajectories from static to dynamic arise when we look at inference in such stages, first as a zero-agent mathematical relationship between static propositions, then as a one-agent activity of drawing conclusions, and finally as a many-agent interactive process of argumentation. This broadening of perspective, sometimes called the ‘Dynamic Turn’, started around 1980 with work on interpretation procedures for natural language, as well as belief revision in artificial intelligence. But how should logic incorporate actions as first-class citizens into its scope? Plausible formal frameworks to this effect come from the philosophy of action, temporal logic, and systems for analyzing programs in computer science, such as dynamic logic. Moreover, further influences have come from process theories in computer science, as well as game theory, and this contact between disciplines is 1

The same ‘static’/‘dynamic’ distinction makes sense when we extend our notion of classical logic, e.g. by including definitions and expressive power of languages. Expressive power has to do with activities of evaluation of statements, making distinctions between given situations, and so on.

F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 419–440.

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still continuing. This paper sketches one trajectory of ‘dynamification’, reflecting my personal interests. A much broader survey is given in (van Benthem 1996).

2. From epistemic logic to communication 2.1 Questions and answers A typical illustration of the Dynamic Turn arises in epistemic logic. Let us move beyond the usual concerns that most of us were raised with, such as ‘Is knowledge true justified belief ?’, or ‘Which modal axioms should we choose for the epistemic operator K?’. Instead, consider the most basic episode of communication. I ask you a simple YES/NO question “Is Amsterdam at the same latitude as Peking?”, and you answer me truly. By the way, the actual answer is “No.” 2 Now much more information flows in this simple question-answer episode than meets the eye. Under normal circumstances, my question is only felicitous when certain preconditions are satisfied. First, I indicate to you that I do not know the answer. But there is more. The fact that I am asking you indicates that I think it is at least possible that you know the answer.3 Now to the effects of the answer. By telling me, you make me learn the relevant fact P. But more is true afterwards. You know that I know, I know that you know that I know, and so on to any depth of iteration. We achieve what is called common knowledge in the philosophical and logical literature. These are the so-called postconditions of a truthful answer.4 Incidentally, most preconditions and postconditions noted here involve knowledge about other people’s knowledge. This may seem a somewhat redundant social side-effect of communication. But in reality, such iter2 3 4

At least, according to the little globe standing on my desk as I write this. These are normal cooperative questions. Neither condition holds when a teacher asks a didactical question to students in class — or in games, where questions may serve to mislead an opponent. ‘Preconditions’ and ‘postconditions’ are standard notions from program analysis in computer science.

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ated knowledge levels are often crucial to effective physical action. Suppose that I know that you have stolen my watch and are now wearing it, but also I know that you do not know that I know it. Then I will try to quickly grab it back. But if I think that you may know that I know you have it (note that this involves three iterations!), I must retrieve my stolen watch in some more sophisticated manner. Thus both communication and physical action involve careful handling of knowledge assertions of various shades.

2.2 Epistemic logic The preconditions and postconditions of the preceding episode can be written in standard epistemic logic, which is an excellent formalism for displaying knowledge about facts and about other people’s information. For instance, the question indicated that the following was true, with K for the epistemic modal operator “knows that” and for the dual modality “holds it possible that”: ¬KI P & ¬KI ¬P (KyouP ∨ Kyou¬P) Moreover, after the answer has been given, the following are true: KyouP, KI P, KyouKI P, etc. all the way to common knowledge C{you, I }P. More precisely, these epistemic formulas refer to the usual semantic models M = (W, {∼j }j , V ) for epistemic logic, consisting of a set W of possible worlds (the ways the actual world might be), accessibility relations ∼j for each agent j, and a valuation function V giving each proposition letter a truth value in each world. A formula Kj P is then true at a world s if P is true in all worlds t with s ∼j t. The much stronger formula CG P is true if P holds at all worlds that are reachable from s by any finite chain s ∼j t ∼j k … where the relations may be for arbitrary agents. For convenience, one often assumes that the ∼j are equivalence relations, making the logic a poly-modal S5 system in a lan-

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guage with a common knowledge operator. But similar ideas will work for much weaker logics, modelling agents’ belief instead of knowledge.

2.3 Dynamics: changing information states But there is more to be done. An explicit account of what happens in a question-answer episode does not just record statements that are true before and after. It will also model the change of information state directly, in terms of transitions between states in some information space: old state

update action

new state

To make this precise, we need to ‘dynamify’ traditional epistemic logic. First, for the successive information states in a conversation, we can take epistemic models (M, s) as above with a designated actual world s for the real state of affairs. These models describe ‘snapshots’ of the current information available to the agents. Normally, we keep one such model M fixed, and evaluate formulas φ as to their truth or falsity in some world. But now, we must look at sequences of such models, because speech acts of assertion change them according to some update rule. E.g., in a simple question/answer scenario, the initial model might be as follows, indicating that the Questioner (Q) does not know whether P, but the Answerer (A) does:

P

Q

¬P

The black dot stands for the actual world. (In this particular model, by the rules of epistemic logic, Q even knows that A knows the answer — though this is not strictly required for asking a genuine question.) Next, A’s answer triggers an update of this information model, eliminating the option not-P, to yield the one-point diagram

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P At this stage, P has become common knowledge between Q and A. The general dynamics here is as follows. Public announcement φ! of a true proposition φ eliminates all those worlds from the current model which fail to satisfy φ: from (M, s)

to (M|φ, s)

φ ¬φ

With larger epistemic models, world elimination acquires striking effects.

2.4 Games Card games are nice examples, with non-trivial information flow even in simple cases. Let three players 1, 2, 3 draw a card from ‘red’, ‘white’, ‘blue’, with an actual distribution rwb. Each sees only his own card. The epistemic model is this rwb

rbw

1 3

2 bwr

2 wbr

3 3

1 brw

2

1 wrb

The diagram says the following. Though they are in rwb, no player knows this. As they ponder their group situation, they must take into account all 6 worlds. Now 1 says truly: “I do not have the blue card”. What do players know about the cards after this event? Solving this in

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words can be complicated, but here is the correct update, removing the two worlds starting with b: rwb

1

rbw 2 wbr

3 1 wrb

This shows at once that 2 knows the distribution, 3 knows that 2 knows, and 1 knows only that 2 or 3 knows. But, e.g., it is not common knowledge that 2 knows! For, 1 thinks it possible that 2 has the blue card, in which case the first assertion would not have helped her. The diagram also predicts the effects of further assertions. E.g., if 3 now were to say truly “I still don’t know”, only the left-most worlds would remain, and 2 would find out the correct distribution.

2.5 More general update Models like this clarify, e.g., the famous Muddy Children puzzle and other scenarios, as shown in (Fagin, Halpern, Moses & Vardi 1995). A simple exposition of the ideas and resulting general questions is found in (van Benthem 2002). Such scenarios have been the starting point for a whole line of research on update mechanisms for more sophisticated forms of communication, including hiding, forgetting, or cheating. These may mix public and private information (as happens with security protocols on the Internet), where agents may even become systematically misinformed. The best current system is product update for states with actions: see (Baltag-MossSolecki 1998, van Ditmarsch 2000).5

3. Epistemic process logics 3.1 Dynamic logic The preceding dynamification still has no explicit calculus for defining update actions and reasoning about them. A truly two-level static-dynamic 5

In general product update, epistemic models may also grow in size, as a conversation or a game proceeds, and there may be no straightforward descent to common knowledge of the actual world!

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system implementing the Dynamic Turn imports an idea from computer science, viz. the coexistence of propositions and action expressions in socalled dynamic logics. These can describe conditions true in states resulting from performing actions: [a]φ

φ holds after every successful execution of action a.

In the same vein, we can now state epistemic effects of communication, such as [A!]Kj φ after a true public announcement of A, j knows that φ. This combined language mixes modalities from dynamic logic with epistemic modalities. Their order records the interaction of preconditions and postconditions. For instance, here is a simple statement, that may seem obvious: [A!]CG A public announcement leads to common knowledge. We will see later how plausible this is as a general logical law of communication. As another illustration, here is a valid principle in the obvious semantics relating knowledge achieved after a public announcement to what agents know beforehand: [A!]Kj φ ↔ (A → Kj [A!]φ) This says that knowledge of φ afterwards corresponds to knowledge of a suitably relativised version of φ beforehand. This is just one law for reasoning about communication in a complete system of dynamic-epistemic logic for public announcement, which is known to be axiomatisable and decidable. This seems the simplest logical calculus of communication.6 More sophisticated systems exist for more complex product updates. Thus, 6

This calculus is a basic epistemic logic plus simple reduction axioms decomposing postconditions recursively. But there are subtleties, as the reduction axiom for common knowledge after an announcement requires enriching the static base language with an operator CG (A, φ) of conditional common knowledge within the set of worlds satisfying A (van Benthem, van Eijck & Kooi 2005). Thus, a dynamic superstructure may also suggest modifications of its static base structure.

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dynamic-epistemic logic promises a more systematic logical taxonomy and understanding of general communication.

3.2 Analyzing speech acts All this takes a new look at old issues in philosophy. Consider complete epistemic specifications in speech act theory. Say, what do we learn from a public announcement of φ? The above ‘learning principle’ suggested it always produces common knowledge of φ. But this is false! E.g., if A had answered φ “You don’t know it, but P”, this would have been true, the same update would have occurred, but the assertion φ would become false by the very update, because Q now knows that P! Philosophers will recognize Moore’s Paradox here, now as an issue in dynamic epistemic logic.7 Thus, update logics in the Dynamic Turn take up old issues with new techniques. Indeed, even a simple formula like [A!]Kj φ encodes ideas from linguistic speech acts, philosophical epistemology, and program logics in computer science.

3.3 Program structure But the analogy between communicative actions and programs goes still further. Computer programs are typically constructed from basic actions hardwired in a computer using software constructions, such as composition conditional choice guarded iteration

S;T, IF P THEN S ELSE T, WHILE P DO S.

Especially, the latter structure is typical for computation, where we may not be able to tell beforehand how often the computer has to repeat some instruction. But this analogy persists for communication. A public announcement is a basic instruction, which modifies an information state in a way that is hardwired into our social conventions, or even our brains. But on top of that, there is ‘communicative software’. We can give people more complex instructions like 7

The technical question which forms of epistemic assertion do produce common knowledge when announced is still open. A connection with the ‘Fitch Paradox’ is explored in (van Benthem 2003c).

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“First ask how she is doing, and then state your request”, or “If the teacher asks A, then say B, else say C”. And even iterations occur. Thus we can think of conversation as a sort of imperative programming, where the ‘machines’ are the social settings that we influence. A nice concrete example of iteration occurs in the following well-known puzzle: Muddy Children: After playing outside, two of three children have mud on their foreheads. They all see the others, but not themselves, so they do not know their own status. Now their Father comes and says: “At least one of you is dirty”. He then asks: “Does anyone know if he is dirty?” The children answer truthfully. As this question–answer episode repeats, what will happen eventually? Nobody knows in the first round. But upon seeing this, the muddy children will both know in the second round, as each of them can argue as follows: “If I were clean, the one dirty child I see would have seen only clean children around her, and so she would have known that she was dirty at once. But she did not. So I must be dirty, too!” This reasoning is symmetric for both muddy children — so both know in the second round. The third child knows it is clean one round later, after they announced that. The puzzle is easily generalized to other numbers of clean and dirty children. It involves an iteration “keep stating your ignorance until you know”, which may be repeated any number of times, depending on the composition of the group. To analyze this puzzle completely, we need a dynamic-epistemic logic which allows for complex actions π in assertions [π]φ. Axioms for such constructions are known from computer science, such as the program reduction law [S;T]φ ↔ [S][T]φ.

3.4 General logic of communication There is much more to logic of communication. (Van Benthem 2002) explores new sorts of issues, such as “Tell All”:

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How to describe the best possible outcome that can be achieved by a group of agents that are out to inform each other optimally? Issues of group communication and collective knowledge are taken further in (Roelofsen 2005). A very rich source is (Baltag-Moss-Solecki 2003). E.g., it contains the result that the dynamic-epistemic logic of public update with Kleene iterations of assertions added is undecidable. Thus, the background logic of puzzles like Muddy Children8 is rich enough to encode significant mathematical problems! This is one of many ‘complexity thresholds’ in the spectrum of human communicative activities.

4. Revising beliefs and expectations 4.1 From update to revision Information update is just one cognitive activity that we engage in. Another key source for the Dynamic Turn is the theory of belief revision (Gärdenfors & Rott 1995), which highlights the interplay of three processes: (a) information update adding certain propositions, (b) information contraction leaving out certain propositions, (c) belief revision changing prior beliefs to accommodate new ones. Belief revision theory proposes representations of information states plus an account of the revision process via basic postulates, and optional ones reflecting more conservative or more radical policies for changing one’s beliefs. Moreover, there is not just transformation of propositional information. One can also change agents’ plausibility orderings between worlds, or their preferences, or indeed any parameter in logical semantics that admits of meaningful variation over time.9 It is still an open issue how to best combine these ideas with epistemic update logics as proposed above. One way of doing this works by dynamifying conditional logic, the study of implications A ⇒ B interpreted as saying that

8 9

Or those late-night alcoholic conversations where we tend to repeat ourselves. Even the language itself, encoding the conceptual framework, may be subject to explicit revision.

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B is true in all most preferred or most plausible worlds satisfying A. Dynamic actions then involve changes in plausibility orderings, in addition to just removing worlds or uncertainty links. Some relevant publications are (Veltman 1996, Aucher 2003, van Benthem & Liu 2005). Eventually, something like this must be done, even when modelling very simple scenarios in understanding conversation, as we shall see with the case of games below.

4.2 Learning theory Evidently, people have various strategies for revising theories, or just their ordinary opinions. Belief revision theory is not out-and-out dynamics yet, as those processes themselves are not manipulated as first-class citizens in the calculus. An example of the latter move is the modern theory of learning mechanisms, merging ideas from the philosophy of science, mathematical topology, and computer science. (Hendricks 2002) makes an extensive plea for the broad epistemological relevance of this move. Update, revision, and learning form a coherent family of issues, going upward in complexity and range from short-term to long-term cognitive behaviour. (Van Benthem 2003a, 2005) discuss the whole picture in some more detail, including connections with contemporary epistemology.

5. Goals, strategies, and games 5.1 The broader setting of communication Public announcements are building blocks for arguments or conversations. But in those larger settings, we do not just ask what people are telling us, but also why. What are my partners trying to achieve, and for that matter, what are my own goals in choosing what to say or ask? E.g., consider the following scenario. A has to choose an action, and then the turn passes to E, who can choose an ending from x, y or from z, u. The first to know where the story ends wins a prize. Imagine that players have made up their mind what to do in each case. A E

E x

y z

u

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Now suppose A asks E: “What are you going to play on the left?” This is a genuine question, as A does not know, and A even knows that E knows the answer. But there seems to be more information in this question than just these preconditions from the earlier epistemic update logics. For, why would A ask this? It only seems to make sense to know this if he is going to play ‘left’. But the latter information would tell E exactly what is going to happen, since she already knows her own move, and so she can win the prize even before answering the question. So, is it justified for E to conclude this? It depends on what sort of conversational partner she takes A to be: rational, stupid, etc. Moreover, pay-offs matter. Suppose that announcing the wrong solution makes the prize go the other player. Then A might have just asked the question in order to fool E into making a wrong announcement. The considerations in this simple example all point toward strategic interaction in rounds of conversation, and planning for various future contingencies. A good paradigm extending update logics for this broader purpose is found in game theory. Games are a model for a group of agents trying to achieve certain goals through interaction. They involve two new notions compared with what we had before: agents’ preferences among possible outcome states, and their longer-term strategies providing successive responses to the others’ actions over time. In particular, strategies take us from the micro-level to a description of longer-term behaviour.

5.2 Conversation games Consider two people who are not equally informed. I do not know if we are in Holland (P) or not (¬P), and if the year is 2004 (T ) or not (¬T ). You know that I do not know the place, but think that I might know about the time. But I do know whether we are together for a good reason (R), whereas you don’t. In fact, we are in Holland in 2004, and indeed for a good reason. Here is a concrete epistemic model for this situation, with the black dot indicating the actual world:

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P, T, R

¬P, ¬T, R

1 2

P, T, ¬R

2 1

¬P, T, ¬R

Now we want to discover the true situation — and the one who finds out first wins. I can ask you a question first, and it needs to be genuine: in particular, I do not know its answer. Then you can ask, and so on. At each stage, someone who knows the precise facts can announce this, and wins. (There might be a draw if both announce simultaneously). Now I can clearly ask better or worse questions. Suppose I ask you about the time. Then you learn that I do not know if T holds, which eliminates the two bottommost worlds. But then you know the facts (as we are really in the black world with P, T, R, and there are no uncertainty lines from there left for you), and so you win at once. Therefore, I should rather ask about the place (P). This gives away no information which you don’t already have, because it is compatible with all four worlds. But your positive answer eliminates the two right-most worlds, after which I know the facts and you still do not know about R. This choice between better and worse questions (or things to say in general) is the beginning of a game dynamics of conversation, where players must select questions so as to profit most while leaving their opponents in the dark as much as possible. Whether this can be done depends not just on the epistemic model, but also on the schedule of questions and answers. (Clearly, you could win the above game if you could start.) But matters of timing, too, are very much a feature of real games.10

5.3 Game theory and logic Game theory studies sets of strategies that reflect optimal long-term behaviour for players, according to Nash equilibria or other plausible notions of game solution, where players do not gain by deviating from their strategy given what the others have chosen. These notions apply to concrete games 10 There is much more to the issue of asking best questions in a conversational setting, and real conversation games might easily involve more probabilistic considerations; cf. (van Rooy 2003).

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of any sort (economics, war, amusement), but also to generic games for social activities of language use or logical reasoning. Much of the mathematics of the field is about finding equilibria and their properties, for players having more or less information at their disposal. There are many techniques for this, from leaf-to-root analysis of game trees to much more complex results (Osborne & Rubinstein 1994). Despite obvious differences in scope and aims, game theory and logic also have natural connections. (Van Benthem 1999–2003) presents a panorama of games inside logic that model semantic evaluation, argumentation and other activities. This idea of logic games may be extended to uses of games as a model for interactive computation. The result is a merge of logical calculi for programs and logical calculi for defining games and studying their computational properties (Parikh 1985, Abramsky 1998).

5.4 Game logics The other side of the contact between logic and game theory are logical investigations of deliberation, decision and action by players. For general games, this involves an abstraction step as compared with the earlier update logics. We have a complete game tree of all possible moves, with players’ turns indicated at the nodes, and we wish to analyze which particular sequence(s) of actions will be taken by agents who can reflect on their strategies. For a simple example, consider the following three game trees, with respective values for A, E indicated at the end: A E

E 1,0

0,1 0,1 1,0 (a)

A

A E

E

1,0 ½,1 0,1 1,0 (b)

E

E 1,0

0,1 0,1 1,0 (c)

Each of these games is a model for a modal logic of its basic actions — in this case, ‘left’, ‘right’. Game structure and strategies may then be formulated in standard terms. E.g., out of her 4 possible strategies (maps from turns to moves), the best strategy for E in the first game (a) is to do the opposite of what A has done: “if he has gone left, go right — if he has gone right, go left”

(#)

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This strategy is a simple program that can be studied in a standard dynamic logic (van Benthem 2001b). Interpreting the value ‘1’ as ‘winning’, we see that this is a winning strategy for player E: by following it, she wins no matter what A plays. Most logic games go no further than this notion. But in the middle game (b), with finer preferences among outcomes, better predictions can be made. Again E will play strategy (# ) at her two turns, assuming she is rational. But given that, A will choose left, as it will give him ½, as opposed to the 0 on the right. This predicts the unique ‘subgame-perfect’ Nash equilibrium of this game, which lets E play her winning strategy, while A plays ‘left’. In logical terms, an argument like this involves expressions for values of nodes, perhaps even a full-fledged preference logic (cf. van Benthem, van Otterloo & Roy 2005 for a powerful modal approach). Finally, the game (c) introduces a new feature, viz. imperfect information. At her turn, E does not know what move was played by A, as indicated by dotted line between the two nodes in the middle. Imperfect information arises in many games, e.g. because of restricted powers of observation — as in card games. Such games are models for a joint dynamic-epistemic language with basic actions a, b, … corresponding to the moves, and epistemic operators Kj standing for players’ knowledge. This language can express special information patterns in games, such as the fact that player E does not know which move will make her win: ¬KE winE ∧ ¬KE winE It can also express general laws describing special types of agent. (Van Benthem 2001a) has typical illustrations of this interface between logical and game-theoretic notions. For instance, players j with perfect recall of the past history of the game will have an ignorance pattern satisfying this knowledge-action interchange axiom: Kj [a] φ → [a]Kj φ This is like an earlier axiom for actions of public announcement, which basically related [A!]Kj φ to Kj [A!]φ.11 This assumed perfect recall for all agents involved. By contrast, players with some finite bounded memory will only 11 The axiom for update logic has an equivalence between the two operator orders for [ ] and K. The extra implication reflects a further condition that players never lose ignorance ‘spontaneously’.

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remember the past up to some fixed ‘window’, and their behaviour will satisfy different logical laws.12

5.5 Information update in games In this game setting, the earlier update logic still makes sense. Intuitively, players move through a game tree, as moves are played. That is, at every stage, they learn more about events that took place, while their horizon of possible future developments decreases. First, consider the former. Imperfect information games as described here encode structural uncertainty about the game, which gets modified systematically by observing moves. For this purpose, one can use the earlier epistemic update mechanisms, as a means of explaining how the dotted uncertainty lines arose in the above pictures. One starts at the root, perhaps with some initial epistemic model M. In general, players have only partial powers of observation for moves as the game unfolds. This may be encoded in an epistemic action model A consisting of concrete events, with uncertainty relations between them indicated as for worlds. E.g., I may observe that you are drawing a card, but for all I know you are either drawing the Queen of Hearts or the King of Spades. Both actions will occur in A, but there will be an uncertainty line for me between them. Now, successive layers of the game tree arise by computing successive update products in the sense of Section 2: M,

M x A,

(M x A) x A,

etc.

Given this special update mechanism, their pattern of dotted lines for the complete game tree will satisfy special requirements (one of them is the above perfect recall), which can be determined precisely. The full story is in (van Benthem 2001a).

12 For further topics at the interface of logic and game theory: cf. (van Benthem 1999–2003, 2003b), on powers of players, structural notions of game equivalence, operations constructing new games and their algebra, and analysis of game-theoretic equilibrium concepts in fixed-point logics. Many other interesting strands are found in (Stalnaker 1999, Pauly 2001).

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5.6 Managing expectations But information update by observed events is only half of the story of reasoning in games. Even when they know the whole game structure perfectly well, including all past moves, players still play a game with expectations about their own future behaviour and that of others — and that anticipation is also the essence of all human activity. Stable predictions of this sort are indeed the point of the game-theoretic notion of a strategic equilibrium. But expectations can really be of any sort. Perhaps, you suspect that I have a one-bit memory, remembering only the last move that was played, so that my behaviour only depends on what you did just before. Now, as moves are played, some of those expectations may be refuted. Say, E was expecting A to start by playing ‘left’ in game tree (b), but instead, A plays ‘right’. In this case, expectations about the other player need to be revised, and we enter the area of belief revision, as briefly considered in Section 4. A proper account of the two sorts of mechanism combined: information update and expectation management, seems just around the corner in current logical studies of games.13 A new website with information on research in the area of logic, games and computation is http://www.illc.uva.nl/lgc.

6. Temporal evolution We started with the logic of single steps in communication, and the corresponding updates of information states for groups of agents. Then we moved to longer-term behaviour in games, where players want to achieve goals through finite sequences of actions, responding to what others do. This requires stronger logics, including reasoning about strategies. But eventually, communication and games lie embedded in an even larger temporal setting of human practices over time. We briefly consider some aspects of this more general perspective here. 13 Abstract games and update by observing moves still relate to concrete conversation in many ways. Suppose that players have already chosen their strategies in a game tree, but the art is now to find out where the game will end. The player to know this first gets a prize. This is again an imperfect information game where information can be revealed through statements and questions. In particular, just failure to claim the prize, implying ignorance of where the game will end, can convey useful information, as it may rule out certain moves. See (van Benthem 2004) for details.

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6.1 Finite versus infinite Games seem finite terminating activities, like proofs or talks. But computer science also studies useful infinite processes, like the running of an operating system allowing many special-purpose programs to perform finite tasks. The same dichotomy occurs with cognitive processes in the Dynamic Turn. Some activities are meant to terminate, others provide the operating system for these. Examples of the latter are logical proof systems, or Grice’s wellknown maxims in running conversation. Likewise, game theory also studies infinite games and players’ behaviour in them, such as repeated Prisoner’s Dilemma in social co-operation.

6.2 Temporal logic To study these phenomena, the above logical systems need to be embedded in a temporal system, allowing for discussion of epistemic multi-agent protocols over time, and other long-run notions. E.g., a protocol may encode general regularities relevant to communication, like my knowing that you speak the truth only half of the time. The usual picture here is the familiar tree of forking paths: h t h′ This temporal universe, with epistemic structure added, seems the right stage for putting together single update steps, finite game-like activities, and relevant infinite processes running in the background.14,15 (Pacuit 2005) 14 Cf. the computer run model of (Fagin et al. 1995), the infinite games of (Abramsky 1996), the protocol model for messages in (Parikh & Ramanujam 2003), the universe for learning mechanisms in (Kelly 1996), or the philosophical theory of deliberation and action in (Belnap et al. 2001). 15 Uncertainty between finite sequences of actions in these models naturally generalizes earlier notions from dynamic epistemic logic. E.g., in the Tree setting, epistemic product update says that two sequences X, Y are indistinguishable if they are of equal length, and all their matching members Xi, Yi are indistinguishable. By

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establishes connections between the different temporal-epistemic frameworks available to-day — while (van Benthem & Pacuit 2005) extend this to dynamic-epistemic logic.

6.3 Dynamic logic and dynamical systems In modern game theory, this infinite setting leads to evolutionary models, where social behaviour is analyzed in terms of equilibrium features of infinite dynamical systems, often with a state-transition function of some biologically-inspired sort (cf. Osborne & Rubinstein 1994, Hofbauer & Sigmund 2002). This is a very different mathematical style of thinking about long-term behaviour (van Benthem 2003a, Sadzik 2005), where stable structures emerge as statistical properties of populations. There is an interesting challenge how to interface this with the logical approach of this paper.

7. From analysis to synthesis The final relevant aspect of the Dynamic Turn that we wish to high-light lies on a different dimension. Most of logic is about analyzing and understanding given behaviour, of language users or reasoners. But of equal interest is the undeniable fact that logical investigations also create new ways of expressing ourselves, reasoning, and computation. Well-known examples in computer science are formal specification languages, or logic programs. But the same move from analysis to design makes sense in general cognition. For instance, any working voting procedure is a designed piece of ‘social software’ (Parikh 2002), where we create a new pattern of behaviour for beneficial purposes. Analyzing these may be hard by itself16, but designing better ones is even more of a challenge! And the same is true for the stream of new games that appear in this world, and which are assimilated into our repertoire of human activities.17 The systems of this paper can also be used in this more ‘activist’ mode, as a way of designing behaviour, and changing contrast, systems based on finite automata for their memory will only require indistinguishability up to some fixed finite set of preceding positions. 16 A nice example is the impenetrable selection procedure for the Doges adopted in Venice in 1268, with its vast array of safeguards against family influence and patronage, including many stages of voting plus drawing by lots. Norwich‘s History of Venice (Vintage Books, New York, 1989) says it “must surely rank among the most complicated ever instituted by a civilized state”. 17 Cf. also the study of ‚mechanism design‘ in modern game theory.

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the world. An example from the original update logics is the ‘Moscow Puzzle’ (van Ditmarsch 2002): “A gets 1 card, B and C 3 cards each. What should B, C tell each other, in A ’s hearing so that they find out the distribution, while A does not?” Going beyond such puzzles, one might even think about creating new games, and other practices, using dynamic logics as a means of suggesting possibilities, and as a way of keeping our thinking straight about the intended effects.

8. Conclusion This paper has sketched a broad view of logic in a setting of communication, computation, and cognition. This merges the traditional analysis of reasoning and definition with that of revising beliefs, planning actions, playing games, and their embedding in longer-term patterns of social behaviour. We gave some examples of how this might be done — but admittedly, most of this is still wishful thinking, rather than solid experience. But then, experience does tell us that wishes may come true.18

References Abramsky, S. 1998 “From Computation to Interaction, towards a science of information”, BCS/IEE Turing Lecture. Aucher, G. 2003 A Joint System of Update Logic and Belief Revision, Master of Logic Thesis, ILLC University of Amsterdam. Baltag, A., Moss L. & Solecki, S. 1998 “The Logic of Public Announcements, Common Knowledge and Private Suspicions’”, Proceedings TARK 1998, 43–56, Morgan Kaufmann Publishers, Los Altos. Updated version, 2003, Department of Cognitive Science, Indiana University, Bloomington, and Department of Computing, Oxford University. Belnap, N., Perloff, M. & Xu, M. 2001 Facing the Future, Oxford University Press, Oxford. 18 Acknowledgment. This is an update of an earlier paper that appeared in a special issue of Philosophical Researches, Beijing 2003. I thank the editors for their permission to reuse it.

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Benthem, J. van 1996 Exploring Logical Dynamics, CSLI Publications, Stanford. — 1999–2003 Logic in Games, electronic lecture notes, ILLC Amsterdam & philosophy Stanford (occasional paper versions). — 2001a “Dynamic Epistemic Logic”, Bulletin of Economic Research 53:4, 219–248 (Proceedings LOFT-4, Torino). — 2001b “Extensive Games as Process Models”, Journal of Logic, Language and Information 11, 289–313. — 2001c “Logics for Information Update” Proceedings TARK VIII (Siena 2001), Morgan Kaufmann Publishers, Los Altos, 51–88. — 2002 “One is a Lonely Number: on the logic of communication”, Tech Report PP-2002-27, ILLC Amsterdam. To appear in: P. Koepke et al. (eds.), Colloquium Logicum, Muenster 2001, AMS Publications, Providence. — 2003a “Logic and the Dynamics of Information”, in: L. Floridi (ed.), Minds and Machines 13:4, 503–519. — 2003b “Rational Dynamics and Epistemic Logic in Games”, in: S. Vannucci (ed.), Logic, Game Theory and Social Choice III, University of Siena, department of political economy, 19–23. To appear in The International Journal of Game Theory. — 2003c “What One may Come to Know”, Tech Report PP-2003-22, ILLC Amsterdam. Appeared in Analysis 64 (282), 2004, 95–105. — 2004 “Representing Arbitrary Games through Conversation”, manuscript, ILLC Amsterdam. — 2005 “Epistemic Logic and Epistemology: the state of their affairs”, ILLC Tech Report, Amsterdam. To appear in Philosophical Studies. Benthem, J. van, Eijck J. van & Kooi, B. 2005 “Logics for Communication and Change”, in: R. van der Meyden (ed.), Proc’s TARK 10, Singapore, 253–261. Benthem, J. van & Liu, F. 2005 “Dynamic Logics of Preference Upgrade”, Workshop on Theories of Belief Revision, ESSLLI Summer School, Edinburgh. To appear in Journal of Applied Non-Classical Logics. Benthem, J. van, Otterloo, S. van & Roy, O. 2005 “Preference Logic, Conditionals, and Solution Concepts in Games”, ILLC Amsterdam. To appear in H. Lagerström, ed., Festschrift for Krister Segerberg, Uppsala University. Benthem, J. van & Pacuit, E. 2005 “Dynamic-Temporal Logics of Protocols and Process Structure”, working paper, ILLC Amsterdam. To appear in Proceedings in AiML, College Publications, London.

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Ditmarsch, H. van 2000 Knowledge Games, dissertation DS-2000-06, ILLC Amsterdam and University of Groningen. — 2002 “Keeping Secrets with Public Communication”, Department of Computer Science, University of Otago. Fagin, R., Halpern, J., Moses, Y. & Vardi, M. 1995 Reasoning about Knowledge, The MIT Press, Cambridge (Mass.). Gärdenfors, P. & Rott, H. 1995 “Belief Revision”, in: D. M. Gabbay, C. J. Hogge & J. A. Robinson (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming 4, Oxford University Press, Oxford 1995. Hendricks, V. 2002 “Active Agents”, PHILOG Newsletter, Roskilde. In: J. van Benthem & R. van Rooy (eds.), special issue on Information Theories, Journal of Logic, Language and Information 12:4, 469–495. Hofbauer, J. & Sigmund, K. 2002 Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge. Osborne, M. & Rubinstein, A. 1994 A Course in Game Theory, The MIT Press, Cambridge (Mass.). Pacuit, E. 2005 “Comparing Temporal Epistemic Logics”, CUNY Graduate Center New York & ILLC Amsterdam. Parikh, R. 1985 “The Logic of Games and its Applications”, Annals of Discrete Mathematics 24, 111–140. — 2002 “Social Software”, Synthese 132, 187–211. Parikh, R. & Ramanujam, R. 2003 “A Knowledge Based Semantics of Messages”, CUNY New York & Chennai, India. In: J. van Benthem & R. van Rooy (eds.), special issue on Information Theories, Journal of Logic, Language and Information 12:4, 453–467. Pauly, M. 2001 Logic for Social Software, dissertation DS-2001-10, Institute for Logic, Language and Computation, University of Amsterdam. Roelofsen, F. 2005 Dynamic Logics for Distributed Knowledge, Master of Logic thesis, ILLC Amsterdam. van Rooy, R. 2003 “Quality and Quantity of Information Exchange”. In: J. van Benthem & R. van Rooy (eds.), special issue on Information Theories, Journal of Logic, Language and Information 12:4, 423–451. Sadzik, T. 2005 “Exploring the Iterated Update Universe”, Graduate School of Business, Stanford & ILLC Amsterdam. Stalnaker, R. 1999 “Extensive and Strategic Form: Games and Models for Games”, Research in Economics 53:2, 93–291. Veltman, F. 1996 “Defaults in Update Semantics”, Journal of Philosophical Logic 25, 221–261.

On the Problem of Defining the Present in Special Relativity: A Challenge for Tense Logic Thomas Müller, Bonn*

1. Introduction According to our commonsense view, time can be divided up into the past, the present, and the future. The present (“the now”), separating the past from the future, plays a special role in this picture, and our commonsense view accordingly affords the present a distinguished metaphysical status: presentism, which is arguably the commonsense metaphysics of time, holds that what exists at any given time is exactly that which is present at that time. At any rate, the present plays a central role in language: In English and in other Indo-European languages, all finite verb forms carry tense, an indexical temporal determination relative to the present. Tense logic, developed by Prior starting in the 1950ies, maps the temporal determinations of natural language onto a formal, modal-logical calculus. The project of tense logic is connected with two claims. The first is expressive adequacy: the tense-logical calculus claims to represent adequately the temporal distinctions present in natural language. The second is metaphysical adequacy: tense logic is often connected with the metaphysical doctrine of presentism mentioned above. This was certainly Prior’s own view when he wrote that “the present simply is the real considered in relation to two particular species of unreality, namely the past and the future” (Prior 1970, 245). Special relativity (SR), the physical theory formulated by Einstein in 1905, has often been viewed as a challenge to our commonsense views of both space and time. Minkowski, in giving his famous geometrical interpretation of space-time (1908), proclaimed that the distinction between space and time must fall, and many philosophers have followed suit. For example, Quine in Word and Object remarks that “Einstein’s relativity principle […] *

Thanks to the audience in Kirchberg for fruitful discussion, and to Cord Friebe for detailed comments on a previous draft.

F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 441–457.

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leaves no reasonable alternative to treating time as spacelike” (Quine 1960, 172). Apart from arguably urging us to revise some of our most basic assumptions about space and time, special relativity has also been claimed to show that the tense-logical project must founder. Massey expresses the point succinctly when he writes that tense logic is “ill-advised because grounded in bad physics” (Massey 1969, 31), arguing that tense logic is bound to a prerelativistic, Newtonian, empirically refuted conception of time. The central thrust of Massey’s challenge is that given SR, a notion of the present such as is required by tense logic cannot even be defined. Prior for one certainly took this challenge quite seriously, and some of his latest papers (e.g., Prior 1968 and 1970) deal with that issue, which is still the subject of an intense debate.1 In this paper we wish to give a novel answer to the relativistic challenge by showing the feasibility of the tense-logical project with respect to both claims mentioned above. To this end, it will be important to separate clearly the two aspects of the relativistic challenge referring to these two claims: one should distinguish (1) the question, belonging to the philosophy of language, of whether a tense-logical language can work in the context of special relativity, from (2) the metaphysical question of whether presentism, claiming a metaphysically distinguished status of the present as that which alone is real, stands any chance in the face of the empirical success of special relativity. These two questions have often been identified. Separating them will allow us to answer the relativistic challenge in two steps. In what follows, we will first show how a tense-logical language can work in special relativity, building upon previous work of our own (Müller 2002, 2004). We will then tackle the more difficult, metaphysical aspect of the relativistic challenge by showing how an indeterministic conception of ontological (causal) determination based on Belnap’s theory of branching space-times (Belnap 1992) offers a fruitful interpretation of presentism in the context of special relativity.

1

Recent works on the problem of the present in SR include Mellor (1998), Müller (2002), and Rakić (1997).

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A

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B

e

*

Figure 1: Two observers, A and B, and a distant flash, e.

2. Expressive adequacy of tense logic Both the linguistic and the metaphysical aspect of the relativistic challenge can be illustrated by Figure 1, which gives a space-time diagram of two observers, A and B, observing a distant flash. The solid vertical line indicates the t-axis of the diagram, which coincides with A’s world line. The world line of B, who is moving relative to A, is indicated by the dashed vertical line, which is at an angle to A’s world line. Observers A and B coincide (meet) at the diagram’s origin. Special relativity gives a clear verdict as to which events are simultaneous for any observer at any given point on her world line. With respect to the origin, the simultaneity hypersurface of observer A coincides with the solid horizontal line, i.e., the diagram’s x-axis. For B, the simultaneity hypersurface at the origin is represented by the dashed horizontal line, which is at an angle to that of A. The linguistic challenge now comes about as follows: The distant flash, e, is present for A, while it is future for B. Thus A can say truly, “The light is flashing now”, while B can say truly, “The light isn’t flashing now, but it will be flashing”.

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A and B thus say contradictory things — and still, both are right. This is an odd situation, and it is not rendered any more acceptable by a standard tense-logical formulation, which would look like this (taking φ to stand for the present-tense “The light is flashing now” and using the future operator F for “it will be the case that”): Observer A: φ Observer B: ¬φ ∧ Fφ These two sentences together yield the contradiction φ ∧ ¬φ. Surely, a logical calculus that allows one to deduce a contradiction from true descriptions of a perfectly reasonable situation must be ill-conceived? This verdict is strengthened when one observes that the depicted situation, giving rise to the contradiction, is inherently relativistic: In Newtonian space-time, the simultaneity hypersurfaces for A and B at the origin necessarily coincide, blocking the contradiction. This looks like a vindication of Massey’s complaint that tense logic is “illadvised because grounded in bad physics”. However, there is a simple and, moreover, tense-logically natural answer to the purported difficulty. Tense logic takes the perspectival nature of assertions seriously and thus provides the natural resources to cope with the problem of the depicted situation. In tense logic, like in any branch of modal logic, sentences are evaluated locally, with respect to a so-called index of evaluation, and that index can be shifted via modal operators in accord with the standard Kripke semantics for modal languages. For example, in linear tense logic, the index consists of a point in time, and sentences are evaluated in such a way that the present refers to the moment of evaluation. A tense operator like “it will be the case that” has the effect of shifting the index of evaluation (in that case, to the future). In this way, tenses and other temporally indexical expressions are handled easily. The question of how the index of evaluation in a modal language looks like must be answered by considering the relevant indexicals. E.g., a language that allows spatial reference via the expressions “to my left” or “to my right” (such as English) needs to represent the spatial orientation of the speaker as part of the index of evaluation. Systematic alterations of that index allow us to resolve indexical references made by others. Thus, if you say, facing me, “There is a stone 1 m to my right”, I know that you are referring to a place that is 1 m to the left from where you stand. (Note that this

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implies chaining two indexical references together: One reference to your position, the other to the left, relative to that position.) In order to make tense logic work in the context of special relativity, we need to specify the index of evaluation appropriately. As it turns out, all that is needed is to include the speaker’s frame of reference in the index of evaluation, and to introduce modal operators that shift that index. To be fair, one has to concede that English does not have any corresponding indexicals — we do not distinguish grammatically between your frame of reference and mine. However, that is explained very easily: In our everyday situations (including the physicists’ laboratory!), there is no need for such a distinction. It simply does not happen that people pass each other with speeds anywhere near a significant fraction of the speed of light. When one looks at philosophical discussions that try to depict such situations, one invariable encounters scenarios like people riding their bikes at a speed of 0.9 c. These rather inadequate examples2 just show — to sound a Wittgensteinian note — that relativistic encounters are not part of our life form. Certainly a life form has a history, and it can change. We can (can we?) imagine a human life form that lives in outer space and that has to deal with situations in which people do pass each other at really high speeds frequently. We can rest assured that that life form will have developed their language correspondingly.3 In the rest of this section, we will sketch one possible approach to such relativistic talk.4 The basic idea of a relativistic tense logical language, or “logic of points of view”, is that (1) the index of evaluation includes an inertial frame and (2) there are available modal operators to shift that index. We will present 2

3

4

For simple physical reasons, one cannot ride a bike or anything else at any significant fraction of the speed of light on the surface of the earth, since one would invariably start to escape the earth’s gravitational field. Frictional effects would cause anybody approaching very high speeds in the earth’s atmosphere to burn to ashes anyway. Technology certainly can drive changes in the way we talk, and perhaps even in our conceptual scheme. E.g., medical advances may already have altered our conception of illness. To make an empirically unfounded guess: today, it seems that it is possible to classify a person as terminally ill even though that person does not show any symptoms of illness, while such classification was not possible a few hundred years ago. That approach was inspired by some remarks in Prior’s (1968) paper on “Tense logic and the logic of earlier and later”, where he alludes to a “logic of points of view”. Some of Prior’s ideas have been developed in Müller (2002, written in German; cf. also the English article Müller 2004).

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the simplest framework here, i.e., a propositional modal language, where the propositions refer to what is true or false at space-time points. The basic formal fact behind the logic of points of view is that the transformations between the inertial frames of special relativity form a group, viz., the (proper, orthochronic)5 Poincaré group P. An element of that group is described by 10 real parameters: 4 for a spatio-temporal translation, 3 for rotation, and 3 for a Lorentz boost. A frame of reference can be described by an element of the Poincaré group, interpreted as a transformation relative to some arbitrary, fixed frame. The group structure secures that there is a transformation between any two frames, each such transformation has an inverse, and composition of transformations is associative. Formally, we take an index of evaluation to be a frame f, and we associate with each Lorentz transformation l ∈ P from the (proper, orthochronic) Poincaré group a modal operator l . For the identity element of the group, I, the corresponding modal operator, I , will result in no change of the index of evaluation. The semantic clause for the modal operators builds upon the notion of a model M, specifying for each atomic proposition and for each space-time point whether the proposition is true there. The clause for the atomic propositions is, accordingly: (atomic) M, f B φ if and only if at the origin of f, φ is true according to M. The clauses for the propositional connectives are standard. The clause for the modal operators reads as follows: (modal) M, f B l φ if and only if M, f ′ B φ, where f ′ is the frame of reference f transformed by l. The group structure of P then secures the following formal facts about the modal operators: • For each l there is some l –1 such that both l l –1 and l –1 l act as I . I.e., each modal operator can be “cancelled” by another modal operator. This generalises the fact that in tense logic, e.g., “it will be the case a day hence that it has been the case a day before” acts like “at present”. 5

We thus disregard spatial or temporal mirror images.

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• Any two modal operators can be combined to form a single operator: l l ′ acts like l ′D l , where “D” denotes group multiplication in P. • The combination of modal operators is associative. So far, the language sketched does not address the problem of the present at all. However, it is only a small step towards meeting the relativistic challenge. Among the modal operators, there is a (4-parameter) subgroup describing pure spatio-temporal translations by a space-time vector Δ s . Thus, the language has a full set of metric spatio-temporal determinations. Relative to any inertial frame, these determinations allow for a unique decomposition into a spatial and a temporal component — one can read the spatio-temporal vector Δ s as a combination (Δ x ,Δt ). The language thus has the resources to interpret “presently” as “presently here or somewhere else”, i.e., as a purely spatial translation. What about the mentioned problem of A and B saying contradictory things? A very general fact about communication with indexicals shows that (a) there is no inconsistency and even (b) how A can use B’s statement in a meaningful way. (a) By assumption, A’s and B’s frames of reference, fA and fB , are different. In terms of the semantics sketched above, the fact that both A and B are right means that there is a model M such that M, fA B φ and M, fB B ¬φ, which is absolutely unproblematic. (A basic tense-logical analogue would be obtained by considering, e.g., “It is raining now” and “It isn’t raining now”, uttered at different times.) The relativistic challenge has dissolved. (b) Furthermore, our framework allows us to understand how B can make good sense of what A says. Quite generally, in indexical communication we have to presuppose that the hearer “knows where (at which index) the speaker is” (think of the example about left and right given above). In our case, this means that B knows the transformation lBA from her frame of reference to A ’s frame. If A then says “at a spatial distance Δx , φ” ( l φ), B can prefix this sentence by the transformation lBA , pulling it back to her frame. The resulting indexical sentence, lBA l φ, evaluates like a sentence l ′ φ, where l′ is no longer a purely spatial translation. Thus in the example, B can infer that A’s sentence, “the light is flashing now” means for her, “the light will be flashing”. The language described above has many shortcomings, to be sure. It does not address the difficult question of how to interpret persisting objects in special relativity, nor have causal operators been introduced. For our purpose, →

→

→

→

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however, the important issue is just that the language is feasible, and that it meets the relativistic challenge easily. There is no problem, really, in talking about the present in a relativistic setting.

3. Metaphysical adequacy of tense logic: Presentism in the context of special relativity Before going on to discuss the metaphysical aspect of the relativistic challenge, we would like to express a doubt about the relevance of the whole discussion. In taking special relativity to raise a metaphysical question, that theory is afforded a metaphysical status, which presupposes a realistic attitude towards special relativity. Whatever one thinks about scientific realism, a minimal condition that any realistically interpreted theory must meet is empirical adequacy. Special relativity, however, is not empirically adequate in an unqualified sense. Of course, the theory is empirically highly successful within its domain of application — but that domain does not encompass all there is. Metaphysics, however, is about everything. Gravitation, the force that keeps our feet on the ground, lies outside the scope of special relativity, and the same holds true of many other phenomena. Thus, the metaphysical impact of special relativity can at best be limited, and the following discussion should be read in that light. If the solution offered here seems artificial, one should not forget that the problem is an artificial one to start with. Given that proviso, we wholeheartedly agree that the language part met the easier of the two challenges put forward against the tense-logical project. You talk about your present, I talk about mine, and we can still reach agreement. That is nice, but none too spectacular. What about ontology? Can the relativising strategy of the previous section be adapted to meet the metaphysical challenge? The prospects don’t look good — the notion of a relativised ontology will probably appear incoherent. Are there other options? In what follows, we will first try to phrase the metaphysical challenge in a way that allows for a formal treatment. We will then sketch the known options of solving that formal task. Finally and most importantly, we will suggest a novel approach that we claim successfully answers the metaphysical challenge. 3.1 From metaphysical intuition to a formal question It has been said above that tense logic is often connected with the metaphysical view of presentism, according to which that which is real at any

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moment, is what is present at that moment. The task thus is to show a way in which presentism may be sustained against the background of special relativity. The basic problem here is to say what “the present” is at any moment. Taking “moment” to be “space-time point” (the only reasonable option), the problem depicted in Figure 1 becomes pressing: relative to the origin, there is no such thing as “the present”— there is A’s present, B’s present, and so on. In the last section, these relativised notions of the present were enough to secure communication between A and B — but can a frame-relativised notion of “the real” be appropriate? Hardly anybody thinks so. Gödel for one strongly expressed his view that a relativised ontology is a contradiction in terms: “existence by its nature is something absolute” (Gödel 1949, 258n). Rather than opting for a maverick position, we will accept that aspect of the metaphysical challenge: Whatever the question is, it can’t have a frame-relative answer. That is: If A’s and B’s positions coincide,6 then A and B have to give the same answer when asked what is real, i.e., ontologically present with them. Formally, we are thus after a one-place notion of “the present”, given a space-time point, or, equivalently, a two-place relation of “being ontologically present with” between space-time points, which is frame-independent. Furthermore, it will be uncontroversial to claim that (i) each event is ontologically present with itself (reflexivity) and (ii) if e is present with f, then f is also present with e (symmetry). Many philosophers have argued that the relevant notion must be transitive, too: If e is present with f, and f is present with g, then e is present with g as well.7 We will accept this intuition. Thus, the sought-for notion of presentness must be an equivalence relation that 6

7

Another one of these tellingly inadequate ways of talking. Coinciding with the position of another observer at a high relative speed means annihilation for both, of course. This is more than just a witticism, since it points to the fact that the notion of “spatiotemporal coincidence” operative in special relativity refers to an extended region of space-time, and it may be a pragmatic matter how big that region is. Einstein conceded that point when even in his original definition of simultaneity (Einstein 1905) he referred to events “in the immediate neighbourhood” (“unmittelbare Umgebung”) of the observer.— While our own solution will be to argue for a notion of the present as a spatio-temporally extended region, we will not exploit the pragmatic aspects here mentioned. By this we only strengthen our position, allowing less resources for meeting the challenge. Thereby we do not wish to claim that the pragmatic resources available are useless. Cf. e.g., van Benthem (1983), Stein (1991), Clifton and Hogarth (1995), and Rakić (1997).

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is not based on frame-relative notions. What are the candidates? How can one define the present in special relativity? 3.2 Available options At first sight, the prospects look none too good. An argument made rigorous by van Benthem (1983, 25f.; cf. also Stein 1991 and Clifton and Hogarth 1995) shows what the trouble is: Any relation definable on the basis of special relativity alone must be invariant under all of the theory’s automorphisms, including the full Poincaré group and contractions. Now these automorphisms can be exploited systematically to show that, given an equivalence relation R and just two points x ≠ y such that xRy, we already get x′Ry′ for any x′ and y′. Thus, van Benthem’s theorem states that the only equivalence relations definable on the basis of special relativity are trivial, viz., the identity relation and the universal relation. Taking “present with” to mean identity would lead to the verdict that each event is present only with itself, while according to the universal relation, any two events would be present with one another. The first option thus leads into straightforward “solipsism of the present moment”, while the second leads to a notion of “the present” devoid of any discriminatory significance. Special relativity alone, e.g., in the form of Robb’s (1914) axiomatisation,8 does not provide the resources for a satisfactory definition of “the present”, i.e., of the sought-for relation of “ontologically present with”. Contrapositively, this means that any satisfactory definition of the present on the basis of special relativity will have to avail itself of an extension of that theory. Which extensions are reasonable? In the literature, Rakić’s dissertation (Rakić 1997) is the most advanced study of possible extensions so far. Rakić first makes the requirements on a satisfactory definition of “the present” more stringent. Basically, she is after a notion of the present as a space-like hypersurface, which may seem reasonable enough.9 She then points out that the mere fact that such a hypersurface cannot be defined does not preclude adding such a hypersurface. One is thus led to the question, not whether “the present” can be defined, but whether adding “the present” to the basic structure of special relativity has any unwelcome ef8 9

Cf. Mundy (1986) for a perspicuous presentation of an equivalent, much simpler system. Our own solution will deviate from Rakić’s already at this point.

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fects. It turns out not to: Rakić proves that a satisfactory notion of “the present” may be added conservatively to Robb’s axiomatisation of the SR causal relation.10 So far so good. One may be satisfied with this result for a number of reasons. E.g., one may welcome the technical result because one believes that there is a physical explanation of a preferred frame after all, even though one outside special relativity. General relativity in many models provides the necessary resources in terms of a definable notion of “cosmic time”. Given a cosmic time for any event, two events can be defined to be present with one another if and only if they happen at the same cosmic time. “The present” is then just the set of all point events that happen at the same cosmic time as the origin. This is a perfectly good equivalence relation with a clear, satisfactory interpretation. Whatever the ultimate justification for preferred hypersurfaces, and no matter how nice the technical result mentioned above is — one needs to concede that by going that way, one is leaving orthodox special relativity. We are sympathetic to those who would hold that this may be a good thing, but if we are looking for a sustainable notion of presentism within special relativity, we need to search for other options. 3.3 A novel approach based on branching space-times What are candidates for a less “intrusive” addition to the framework of SR? Add we must, but we may be able to add something less controversial than preferred hypersurfaces. There seem to be two good options: Adding indeterminism, and adding persisting objects. We will not explore the option of introducing objects, since that would presuppose a lengthy discussion of persistence in SR.11 Thus, the option we wish to explore here is whether indeterminism can offer the additional resources needed for a satisfactory definition of the present in special relativity. Indeterminism is the thesis that there is more than just one possible future. That thesis needs to be spelled out carefully in the context of SR, as SR itself is one of the very few deterministic theories of physics (cf. Earman (1986), esp. Chap. 4, for discussion). There is, however, a well worked10 This means that all new theorems of the extended theory include the new presentness relation, i.e., the set of theorems that can be formulated in the old theory is left unchanged. Cf. Rakić (1997, 50ff.) for formal details. 11 Cf. Friebe (2005) for a promising approach, which starts by clarifying the concepts of perdurantism and endurantism with respect to SR.

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out formal theory of indeterminism in the context of special relativity, viz., the theory of branching space-times developed by Nuel Belnap (1992). That theory generalises the well-known theory of branching time by allowing for histories (complete courses of events) to be not just linearly ordered sets, but space-times with a relativistic causal ordering.12 Branching space-times will form the background for what follows. Rather than giving a full presentation of the formalism (for which cf., e.g., Belnap (1992, 2002, 2005)), we will only highlight informally those aspects of the theory that are immediately relevant in the present context.13 In employing an indeterministic theory for giving a formulation of presentism compatible with SR, we first have to specify what presentism means in an indeterministic universe. In view of our aim of establishing presentism, we are looking for a single relation between space-time points that allows for two interpretations, viz., as “present with” and as “real relative to”. We will argue that the sought-for relation is the relation of being “ontologically codetermined with”, and we will specify ontological determination in terms of indeterministic causation: Two events are ontologically co-determined if and only if they have the same (indeterministic) causes. Our interpretation of indeterministic causation is indebted to Belnap’s new theory of causation (Belnap 2005), which uses the formal framework of branching space-times to determine the causae causantes, or originating causes, of indeterministic events.14 Belnap’s theory singles out so-called basic transitions, consisting of a point event and one of its immediate possible fu12 The branching time framework was developed by Prior in the 1950ies and 1960ies in his attempt of giving a semantics for the future-tense operator of tense logic. For a good overview of the so-called Prior-Thomason semantics, cf. Belnap et al. (2001), Chap. 6–8. 13 Rakić in her dissertation (1997) already provided a first attempt at employing branching space-times for a clarification of the problem of the present in the face of special relativity. However, her approach does not use the full strength of the theory. Rather, she first gives an interpretation of “the present” in terms of preferred hypersurfaces, as outlined in the previous section, and then extends that reading to the branching framework. Our approach will proceed differently, using the branching aspect directly. 14 As Belnap argues convincingly, the relata of the causal relation are in general not just events, but transitions, consisting of an initial I (“first this”) and an outcome O (“and then that”); an event may be viewed as a transition with an empty initial. For our purposes, it will be sufficient to consider events as that which is caused; this corresponds to the first stage of Belnap’s analysis. Cf. Belnap (2005) for details.

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tures, as the basic building blocks of the causal structure of our indeterministic world. For any event e, the set of its causae causantes, CC(e), is defined to be the set of those indeterministic basic transitions in the past of e that were responsible for bringing about e instead of one of its alternatives.15 We will follow Belnap in accepting a claim that may be rather controversial: Deterministic events (those occurring in every history) are not caused in any non-trivial sense — they happen anyway. Only indeterministic events have non-trivial causes, i.e., a non-empty set of causae causantes. For a deterministic event e, CC(e) = Ø, so in some sense, “nothing happens”. Following this line of thought, we can make good sense of a notion of objective change that is also tied to indeterministic events: There is change from event e to event f if and only if, given that e and f can occur within a single possible course of events (i.e., in some history), e and f do not occur in exactly the same histories, i.e., CC(e) ≠ CC( f ). Change conceived of in this way is not a language-relative thing, but something rooted in the objective, indeterministic structure of our world. We do not need to trouble ourselves with trying to find out the most basic predicates with which to describe the world in order to capture change (a gruesome task, as every good man knows) — we just need to ask whether something else could have happened. If yes, we have change; if no, we don’t. It is now just a small step to arrive at our indeterministic conception of presentism — we just need to accept a variant of the thesis that “time involves change”, so that the present is that during which there is no change. The present of e can then be determined as that region of space-time in which there is no objective change relative to e. Formally, we define: e PRES f

if and only if

CC(e) = CC( f ).

The present conceived of in this way, as the region that is ontologically codetermined with, and thus real in relation to, the origin, by having the same causae causantes, can have various geometrical shapes. Figure 2 illustrates two possibilities:

15 For the formal definition of CC as well as for illustrations, cf. Belnap (2005).

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e e

A

B

Figure 2: A: The present of e is equal to the shaded, extended region, including the solid lines, but excluding the dashed lines. Dots mark indeterministic events. B: The present of e as a space-like hypersurface. If the indeterministic events acting as initials of basic transitions form a discrete set, which appears like a natural assumption, then the present will be a region of space-time that has both a space-like and a time-like extension (cf. Figure 2.A, which includes three indeterministic events). In the extreme case in which indeterministic events are distributed densely, we can recover the preferred hyperplanes mentioned above (cf. Figure 2.B).16 At first, this consequence of our definition may appear weird: How could something be present which is in the causal future? Again, separating the two aspects of the relativistic challenge pays off. The linguistic, frame-relative notion of the present indeed needs to single out a space-like hypersurface. Ontologically, however, if nothing happens, time is just a coordinate. An ontological notion of time by assumption only comes into play once there is objective change. With respect to the situation of Figure 1, the question of whether the distant flash is ontologically present with the origin can now be answered. The answer depends on the distribution of indeterministic events in the causal 16 The situation is actually more complicated than that. If the densely distributed indeterministic events act independently, the situation can collapse to what above was called “solipsism of the present moment”: the present of an event can consist of just that one event. A preferred hyperplane is obtained if the indeterministic events act in a correlated fashion, exhibiting what Belnap has called “EPR-like funny business”. Cf. Belnap (1992) for a discussion of “branching along a hyperplane”, and his (2002, 2003) for a discussion of the notion of “funny business”.

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past of the flash and of the origin. The two are present with one another if CC(flash) = CC(origin), but not otherwise — and this holds for both observers, A and B. Apart from now fully answering the relativistic challenge, we claim that our approach also allows us to understand what is good about the “block universe” view of Minkowski space-time, according to which everything is there at once, so that the only available reading of “the present” is the whole “block” of Minkowski space-time itself. This intuition does not only mesh well with the formal result reported in the previous section, according to which the universal relation is, apart from solipsism, the only definable candidate notion for “present with”— the intuition is also reproduced in our theory! If our world is completely deterministic, so that a branching model contains only one branch, i.e., a single Minkowski space-time, then all events have the same set of causae causantes, viz., the empty set. Thus, no change, no time, and everything is ontologically present with everything else. Only indeterminism can save us from the block universe view. To summarise our results: • In special relativity, with or without indeterminism, a tense-logical language is not only formally unproblematic, but also pragmatically useful, as speaker and hearer can resolve indexical references in communication. • Ontologically, bare Minkowski space-time may be viewed as a “block universe” with the trivial notion of presentness as the universal relation. However, given indeterminism, a non-trivial and positively illuminating notion of the present can be defined. • Thus, far from foundering on the special theory of relativity, the tense logical project should be seen as a fruitful challenge to that theory. Tense logical considerations urge us to extend the theory of relativity in a way that allows us to recapture both the scientific results on which that theory is based, and our deeply held commonsense notions of space and time, central to our own conception of ourselves as real agents. Indeterminism provides the crucial ingredient that solves the problem of defining the present in special relativity.

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References Belnap, N. 1992 “Branching space-time”. Synthese 92: 385–434. — 2002 “EPR-like “funny business” in the theory of branching spacetimes”. In T. Placek and J. Butterfield (eds.), Non-locality and Modality, Dordrecht: Kluwer 2002, 293–315. — 2003 “No-common-cause EPR-like funny business in branching spacetimes”. Philosophical Studies 114:199–221. — 2005 “A theory of causation: Causae causantes (originating causes) as Inus conditions in branching space-times”. British Journal for the Philosophy of Science 56: 221–253. Belnap, N., M. Perloff, and M. Xu 2001 Facing the Future. Oxford: Oxford University Press. van Benthem, J. 1983 The Logic of Time. Dordrecht: Reidel. Clifton, R. and M. Hogarth 1995 “The definability of objective becoming in Minkowski space-time”. Synthese 103: 355–387. Earman, J. 1986 A Primer on Determinism. Dordrecht: Reidel. Einstein, A. 1905 “Zur Elektrodynamik bewegter Körper“ [“On the electrodynamics of moving bodies”]. Annalen der Physik 17: 891–921. Translation in Einstein et al. (1923), 35–65. Einstein, A., H. A. Lorentz, H. Weyl, and H. Minkowski, eds. 1923 The Principle of Relativity. London: Methuen. Friebe, C. 2005 “Time and existence in special relativity”. In F. Stadler and M. Stöltzner (eds.), Time and History. Papers of the 28th International Wittgenstein Symposium, Kirchberg am Wechsel: Austrian Ludwig Wittgenstein Society, 81–82. Gödel, K. 1949 “Some observations about the relationship between relativity theory and Kantian philosophy”. In Collected Works, Volume 3, ed. by S. Feferman et al., Oxford: Oxford University Press 1995, 230–260. Massey, G. 1969 “Tense logic! Why bother?” Noûs 3: 17–32. Mellor, D. H. 1998 Real Time II. London: Routledge. Minkowski, H. 1908 “Raum und Zeit” [“Space and time”]. Address delivered at at the 80th Assembly of German Natural Scientists and Physicians, Cologne, 21 September 1908. Translation in Einstein et al. (1923), 72–91. Müller, T. 2002 Arthur Priors Zeitlogik. Paderborn: Mentis. — 2004 “The language of special relativity”. In R. Bluhm and C. Nimtz

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(eds.), Philosophy — Science — Scientific Philosophy. Proceedings of GAP.5, Paderborn: Mentis, 1–9. Mundy, B. 1986 “Optical axiomatization of Minkowski space-time geometry”. Philosophy of Science 53:1–30. Prior, A. 1968 “Tense logic and the logic of earlier and later”. In Papers on Time and Tense, Oxford: Oxford University Press, 116–134. — 1970 “The notion of the present”. Studium Generale 23: 245–248. Quine, W. V. 1960 Word and Object. Cambridge, MA: MIT Press. Rakić, N. 1997 Common Sense Time and Special Relativity, Dissertation, ILLC Dissertation Series 97-02, University of Amsterdam. Main results also in her article: Past, present, future, and special relativity. British Journal for the Philosophy of Science 48: 257–280 (1997). Robb, A. 1914 A Theory of Space and Time. Cambridge: Cambridge University Press. Stein, H. 1991 “On relativity theory and the openness of the future”. Philosophy of Science 58: 5–23.

A Deontic Logic with Temporal Qualification Eduard F. Karavaev, St. Petersburg There are more than enough (syntactic, semantic, pragmatic) difficulties in constructing deontic logic. We suppose that at least some of them can be overcome by means of temporal qualification of the relation of deontic alternativeness and by modification of the so-called “standard” models. We use a temporal structure that is discrete, branching “to the left” and infinite in both directions. The system of tense-logic based on this structure has been developed thanks to works by J. P. Burgess. The essence of our development consists in providing the system with facilities for comparison of times of events being found on different “branches” of a set of possible courses of events.

I There are more than enough difficulties in constructing deontic logic. The characters and the types of these difficulties — syntactic, semantic, pragmatic — are various ones. From them many relevant questions arise. For example, how we ought to qualify the status of variables? That is, whether they are thought to represent categories of human actions or to express sentences that represent states of affairs? If the latter is the case, what is it to be such a state of affairs? We suppose that all obligations are conditional ones, though the conditions are expressed in various linguistic forms and often some obligations appear to be unconditional ones. Let us consider an example: “Do not p forget to switch off your mobile telephones during the presentation ; however if p you did and q during the presentation a bell rang r you ought to apologize ”. In symbolic form we have got: O¬p & ((p & q) → Or). At first sight it might seem that only the second obligation Or is conditional and the first one is unconditional. But rather soon1 we find that the first obligation is supplied with a condition —“during the presentation”. 1

As Wittgenstein writes (in “Diaries”) a decomposed sentence speaks more than a non-decomposed one.

F. Stadler, M. Stöltzner (eds.), Time and History. Zeit und Geschichte. © ontos verlag, Frankfurt · Lancaster · Paris · New Brunswick, 2006, 459–467.

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However, it seems that the technical means for the expression of conditional obligations are rather more complicated than the ones for unconditional obligations. So, instead of a formula O (B/A) where A is a condition for B to be obligatory we may write: (A → OB) & (¬A → ¬OB). The second member of the conjunction allows us to account for the relevancy A with respect to B. Otherwise we could fall into a fallacy and would formalize O (B/A) as (A → OB). Then O (B/A) would be true whenever A is false and/or OB is true. When constructing systems of deontic logic we use some principles which appear to be quite plausible we often encounter various paradoxical conclusions. So, for example, if we combine a collection principle, CP: OA & OB → O (A & B) with the Kantian principle (“ought” implies “can”), KP: OA → ◊B we may obtain2: (OA & OB) → (¬◊(A & B) → ◊(OB & B)): 1. (OA & OB) — a premise 2. ¬◊(A & B) — a premise 3. OA & OB → O(A & B) — CP 4. O(A & B) — 1, 3, modus ponens 5. O(A & B) → ◊(A & B) — KP [6.] ◊(A & B) — 4, 5, modus ponens So, the lines 2 and 6 yield a contradiction. Or let us combine two other principles that also appear to be quite plausible ones. According to the first of them, if we are obligated to provide a state of affairs A and it is the case that some other state of affairs B would be incompatible with A then we are obligated to act in such a way that there would no be B: OA & (B → ¬A) → O ¬B. According to the second principle: O¬B ¬OB. Then the following formula is proved: (OA & OB) → (¬◊(A & B) → (OB & ¬OB)): 1. OA & OB — a premise ¬ 2. ◊(A & B) — a premise ¬ 3. ◊(A & B) → ¬(A & B) — a modal principle 4. ¬(A & B) — 2, 3, modus ponens 2

Here we follow a principle formulated by G. Vico: “verum quod factum”, i.e. “it is possible to understand only what you have made yourselves”.

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5. ¬(A & B) 6. B → ¬A 7. OA 8. OA & (B → ¬A) → O¬B 9. OA & (B → ¬A) 10. O¬B 11. O¬B → ¬OB 12. ¬OB 13. OB [14.] OB & ¬OB

461

— 4, the removal of necessity — 5, propositional calculus — 1, the removal of conjunction — the first principle — 7, 6, the introduction of conjunction — 9, 8, modus ponens — the second principle — 10, 11, modus ponens — 1, the removal of conjunction — 13, 12, the introduction of conjunction

So, the line 14 is an inconsistent formula.

II It seems quite natural, after G. H. von Wright (1996), to accept the following definitions. Definition 1: A state of affairs that can be produced or destroyed, prevented from coming about or from vanishing (if it is just there) is called a doable state. Definition 2: A state of affairs is called doable in pragmatical sense if its obtaining or not-obtaining in given conditions can be a result of human action. Definition 3: (1) By a genuine norm is called such an obliging norm, O-norm or a permitting norm, P-norm the content of which is a doable state in the pragmatical sense. (2) A norm the content of which is a necessary or impossible state of affairs is called non-genuine, or spurious. Definition 4: (1) A set of O-norms is deontically consistent if the conjunction of their contents expressed a doable state of affairs is an obtainable formula. (2) Every set of P-norms is deontically consistent. As it is known, in semantical investigations of deontic logic a so-called “standard model” appears as the following:

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Eduard F. Karavaev

μ = 〈W, R, V 〉 in which W is a (non-empty) set of possible worlds, R is a binary relation of deontic alternativeness, and V is a mapping from the set of propositional letters Var = ⎨p0, p1, p2, …⎬ to subsets of the set W. So a proposition pi is true in a possible world α if and only if α is included into a subset wi of W where wi is a product of the mapping. Commonly the evaluation is extended to the use of deontic operators. For example: V(OA) = ⎨α ∈ W: ∀β ∈ W (αRβ ⇒ β∈ V(A))⎬. Let us note, in standard models there is no expression of dependence of norms on time.

III We suppose that at least some of the difficulties in constructing deontic logic can be overcome by means of temporal qualification of the relation of deontic alternativeness and by modification of the so-called “standard” models. We use a temporal structure that is discrete, branching “to the right” and infinite in both directions. The system of tense-logic based on this structure has been developed thanks to works by J. P. Burgess (1979; 1980). The essence of our development consists in providing the system with means for the comparison of times of events being found on different “branches” of a set of possible courses of events. The basic relation of precedence is defined as a degree of an elementary relation