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Space. A History

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Any SZ2 i 2 _ OXFORDPHILOSOPHICAL CONCEPTS Christia Mercer, Columbia University



Efficient Causation Edited by Tad Schmaltz




Edited by Justin E. H. Smith

Edited by Eric Schliesser

Edited by Remy Debes

The Faculties Edited by Dominik Perler



Edited by Dmitri Nikulin Moral Motivation

Edited by Iakovos Vasiliou Eternity Edited by Yitzhak Melamed

SelfKnowledge Edited by Ursula Renz

Edited by G. Fay Edwards andPeter Adamson Pleasure


. ay

Edited by AndrewJaniak

Edited by Lisa Shapiro Health Edited by Peter Adamson Evil Edited by Andrew Chignell Persons Edited by Antonia LoLordo


The Self Edited by Patricia Kitcher


Modality Edited by Yitzhak Melamed

Human Edited by Karolina Hubner

The World-Soul Edited by James Wilberding

Love Edited by Ryan Hanley

Edited byJulia Jorati




Oxford University Press is a departmentofthe University of Oxford.


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and education the University’s objective of excellence in research,scholarship, of Oxford University by publishing worldwide, Oxford is a registered trade mark Press in the UK andcertain other countries.

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Introduction 1



CIP data is on file at the Library of Congr ISBN 978-0-19-991412-8 (pbk.) ISBN 978-0-19-991410-4 (hbk.)

1. Space in Ancient Tithes: From the Beginning to Aristotle 11

1337986 42


da Paperback printed by Marquis, Cana dStates ofAmerica Unite , ry,Inc. Binde nal Natio eport Bridg Hardback printed by

Kul. 43593

Reflection: Body Space/City Space: Veiling as an Embodied Spatial Practice 52 BANU GOKARIKSEL

2. Imagine a Place: Geometrical and Physical Space in Proclus 63


Reflection: Ants in Space 98 NICOLE E, HELLER

3. Concepts ofSpace in the Fourteenth Century: Works of Nicole Oresme ~



KARD C2 RSUeaPLedd 2 emagg Leyes

andSelected Earlier Work for Comparison 104 EDITH DUDLEY SYLLA



Reflection: Space, Vision, and Faith: Linear Perspective in Renaissance Artand Architecture 176 MARI YOKO HARA

Series Editor's Foreword

4. Geometry and Visual Space from Antiquity to the Early Moderns 184 GARY HATFIELD

Reflection: Space for Thought 223

JENNIFER GROH 5. Space in the Seventeenth Century 230 ANDREW JANIAK the Cosmic Space 270 Reflection: Chemical Laboratory and


6. Space in Kantian Idealism 280 MICHAEL FRIEDMAN Reflection: Non-Euclidean Geometry JEREMY GRAY

alize key concepts in the history of philosophy, to render that history 306

e 312

e on Spac Reflection: A Mathematical Sculptor’s Perspectiv GEORGE HART BIBLIOGRAPHY 319


Oxford Philosophical Concepts (OPC)offers an innovative approach to philosophy’s past andits relation to other disciplines. As a series, it is unique in exploring the transformations of central philosophical concepts from their ancient sources to their modern use. OPC hasseveral goals: to makeit easier for historians to contextuaccessible to a wide audience, and to enliven contemporary discussions by displaying the rich and varied sources of philosophical concepts still in use today. The meansto these goals are simple enough: eminent scholars come together to rethink a central concept in philosophy’s past. The pointofthis rethinking is not to offer a broad overview but to identify problems the concept wasoriginally supposed to solve and investigate how approachesto themshifted over time, sometimes

radically. Recent scholarship has madeevidentthe benefits of reexamining the

standard narratives about western philosophy. OPC’s editors look beyond the canon and explore their concepts over a wide philosophical

landscape. Each volumetraces a notion fromits inception as a solution to specific problems through its historical transformationsto its modernuse,all the while acknowledgingits historical context. Each OPC volumeis a history of its concept in thatit tells a story about changing solutions to its well-defined problem. Many editors have



foundit appropriate to include long-ignored writings drawn from the

Islamic and Jewish traditions and the philosophical contributions of women. Volumes also explore ideas drawn from Buddhist, Chinese,

Indian, and other philosophical cultures when doing so adds‘an especially helpful new perspective. By combining scholarly innovation with focused and astute analysis, OPC encourages a deeper understanding

Abbreviations and References

ofour philosophical past and present.

One of the most innovative features of Oxford Philosophical Concepts is its recognition that philosophy bears a rich relation to art, music, literature, religion, science, and other cultural practices.

Theseries speaks tothe need for informed interdisciplinary exchanges. Its editors assume that the most difficult and profound philosoph-

ical ideas can be made comprehensible to a large audience and that materials not strictly philosophical often beara significant relevance to

Ancient Works DK = Fragmente der Vorsokratiker. Edited by H. Diels and W. Kranz. Berlin, 1951.

These philosophy. To this end, each OPC volume includes Reflections. are short stand-alone essays written by specialists in art, music, litera-

t ture, theology, science, or cultural studies that reflect on the concep 2

from their own disciplinary perspectives. The goalof these essays is to

enliven, enrich, and exemplify the volume’s concept and reconsider the boundary between philosophical and extraphilosophical materials. OPC’s Reflectionsdisplay the benefits ofusing philosophical concepts >



References to worksby Aristotle use so-called Bekker numbers, which correspondto the pagination and line numbers of Bekker’s 1830 edition ofthe text. These will be found in the marginsofanyreliable edition or translation.

and distinctions in areas that are notstrictly philosophical and encourage philosophers to move beyond the borders oftheir discipline as presently conceived.

The volumes of OPC arrive at an auspicious moment. Many

philosophers are keen to invigorate the discipline. OPC aimsto pro-

voke philosophical imaginations by uncoveringthebrilliant twists and unforeseen turns ofphilosophy’s past. Christia Mercer Gustave M.Berne Professor ofPhilosophy Columbia University in the City ofNew York


AT = Ocuvres de Descartes, Edited by Charles Adam and Paul Tannery. Paris: Vrin, 1996.

H = Treatise ofMan. Edited and translated by Thomas Steele Hall. Cambrdige, MA: Harvard University Press, 1972. Thetitle ofthis work is moreusually translated as Treatise on Man, and that formis used herein. O = Discourse on Method, Optics, Geometry and Meteorology. Translated by Paul J. Olscamp. New York: Bobbs-Merrill, 1965. The title of Descartes’s Dioptrique is translated herein as Dioptrics (as op-

posed to Optics).




A/B: A = correspondsto thefirst-edition pagination ofKant,Critique of Pure Reason (1781); B = corresponds to the second-edition pagination ofthe Critique (1787). and References to all other texts in Kant’s corpus are to the volume Schriften gesammelte Kants page numbers in the Akademie edition, (Berlin, 1902—-).



rs, which References to works by Plato use so-called Stephanus numbe s ofa famous Renaissance correspondto the pagination and lin e number th e late sixteenth century. That edition ofPlato’s texts that appeared in

e text. Anyreliable edition use d theletters a—e to split up sections ofth the margins. edition or translati on will include these numbers in

MICHAEL FRIEDMANis SuppesProfessorof Philosophy of Science at Stanford

University and works on the history and philosophy ofscience fromthesixteenth through the twentieth centuries. The author or editor of eleven books and numerousarticles, Friedmanis a leading figure in philosophyofscience, in the study of Kant’s thought, andin the’ interpretation of logical positivism and its aftermath. His books include Foundations ofSpace-Time Theories (Princeton, 1983), Kant and the Exact Sciences (Harvard, 1992), and Kant’s Construction of

Nature (Cambridge, 2012). BANU GOKARIKSEL is Professor of Geographyand the Royster Distinguished

Professor for Graduate Education at the University of North Carolina, Chapel Hill. The author of numerousarticles concerning women and society in the Islamic world, Gékariksel’s research has been funded by the National Science Foundation andhas been covered widely in the international press, including by NPRandLe Monde.

JEREMY GRAYis Emeritus Professor of Mathematics at the Open Universityin England. A leading figure in the history of modern mathematics, he is the author ofnumerouspapers and many books, including Henri Poincaré: AScientific Biography (Princeton,2012), The Hilbert Challenge (Oxford, 2000), and Ideas of Space: Euclidean, Non-Euclidean and Relativistic (Oxford, 1989). JENNIFER GROH is a professor of psychology and neuroscience and of neurobiology at Duke University. Groh’s lab is funded by the National Institutes of Health and the National Science Foundation. She is the author of numerous




scientific papers andofMaking Space: How the (Harvard, 2014).

Brain Knows Where Things Are

Entightenmentsin Revolutionary Europe (Pittsburgh, 2016).

ersity. Heis now a

eana oarou. ivers eities lay at un ispla ently on disp ematical sculptor, with sculptures curr Msalti or of many papers and of United States. He is also the auth ering (Springet, 1995). Analysis: Algebras and Systems,for Science andEngine ral . Incll.eceu ara.te in a : is the Adam Seybert Professor GARY HATFIELD ‘ ey . The author or € itor hy at the University of Pennsylvania Phil nis oht in figure vrxe g leadin ld is a ny ern science and philosophy, Hatfie s on mod bookpeek

my i a focus on the history of nana and philosophyof science, with oBy: pom Kant 8 eption, and the philosophy of psycho

theories of perc toaphyef Theories of Spatial E erecpior J The Natural and the Normative: © eption and Cognition: Essays in Helmholtz (MIT, 1990) and Perc Psychology (Oxford, 2009). ene at

Anthropoc andthe curator of the NICOLE HELLERis a museu tn fellow . Heller holds a PhD in

Historyin I ittsburgh the Carnegie Museumof Natural

gle fellow

iversi rsity and waspreviou. slya Goo biological sciences from Stanford Unive for science communication.

s Chair of philosophy at Duke ne ANDREW JANIAK is Professor and i. .

or or editor of four where he is also a Bass fellow. Janiak is the auth

uding Newton as Philosop le about Isaac Newton and modernphilosophy, incl ac et

also the (Cambridge, 2008) and Newton (Wiley-Blackwell, 2015). Heis

She is also the author of Affinity, That Elusive Dream: A Genealogy of the Chemical Revolution (MIT Press, 2003) and The Imagined Empire: Balloon

owin art history at Colu. mbia MARI YOKO H ARAis a Mello n Postdoctoral Fell sheis curnee on Renaissance art history, University. The author of several papers itec -arch -ar inter Peruzzii andtheart ofpainter rently writing a book about Baldassare in sixteenth-centuryItaly. years at many e an d computer scii ence fore GEORGE HART taught engineering ee

Univ Columbia University and Stony Brook



ote teaching an of Project Vox,a digital humanities endeavor designed to prom n. rn wome research of neglected philosophical work by early mode

MI GYUNG KIM is a professor of history at North Carolina State University: A leading historian of chemistry, she is the author of many papers concerning the history of chemistry and the place ofscience in the French enlightenment.

MARIJE MARTIN is the C. J. de Vogel Professor of Ancient and Patristic

Philosophy at Vrije Universiteit Amsterdam. The author of numerous papers on ancient mathematics, science, and philosophy, Martijnis also the author of -Proclus on Nature (Brill, 2010).

BARBARA M. SATTLERis a seniorlecturer in philosophy at the University of St. Andrews in Scotland, having previously taught at Yale University and the University of Illinois at Urbana-Champaign, She is the author of numerous articles concerning science and philosophy in antiquity, including papers on Aristotle, Plato, Parmenides, and Zeno.Sheis also the author of The Concept of

Motionin Ancient Greek Thought:foundationsin logic, method and mathematics (Cambridge, 2020). EDITH DUDLEY SYLLAis Professor Emerita of History at North CarolinaState

University, where she taught history ofscience for manyyears. The author ofnumerousarticles and the editor of several books,Sylla is one of the world’s leading historians focusing on medieval science. She has written ona wide rangeoftopics stretching fromthe thirteenth to the eighteenth century and has discussed the

thought of many prominenthistoricalfigures, including Jacob Bernoulli, John Buridan, William of Ockham, and Nicole Oresme. .

Introduction AndrewJaniak

Space is ubiquitous. So are spatial concepts. Scholars in architecture, art history, mathematics, cosmology, ecology, neuroscience, sculpture, chemistry, and geography employ concepts of space and articulate concepts with spatial components. It would be hopeless tolist them all, and equally fruitless to search for patterns among them,or

for their common node. One needsa specific focal point. In ourcase, the history of philosophy—andthe waysin which philosophersin different eras have pondered space—is our focus. We will also consider some of the myriad intersections between philosophical discussions of space and treatments in other disciplines and enterprises. Some of these intersections are obvious: philosophers and scientists in the nineteenth century were deeply influenced by and played important roles in articulating the new non-Euclidean geometry developed by mathematicians like Bolyai and Lobachevsky. The intertwining of AndrewJaniak, IntroductionIn: Space. Edited by: AndrewJaniak, Oxford University Press (2020). © Oxford University Press, DOI: 10.1093/080/9780199914104.003.0001



sophical treatments the history of geometry with the history of philo are less obof space is intimate and obvious. Other intersections by the development early modern philosophers influ enced

vious: Were

gence of microscopyaf-

of Renaissance perspective? Howdid the emer ill emphasize some of fect philosophical concep tions of space? Wew on philosophers’ viewsin varthese connections through theess ays or “Reflections,interspersed ious eras and throughour short essays,

throughout the volume. historically, Since our volume is organized

harmfully anachronistic presupposition

ourfirst task is to avoid a

in approaching space. We do

in this book—from early antiqnot presuppose that each era broached ut what philosophers who thought abo uity to modernity—involved a single concept of assume that there is we d o Nor e. spac call now we be concepts with e. There may very well space whose history we can trac t SPACE


ugh time, that can be tr aced thro an unambiguous history several guiding have joind y adopted we ead, Inst m.! the of s is not one o phers in previous era loso Did philos : ing low fol the g udin incl ions, e quest have a conceptof spac

Did they think about something called space? atial sigut concepts with sp at all? Or were they primarily thinking abo place?

city, ora bject, a person, a nificance, such as the concept of an 0 topic ofphil-

s a significant Just as important, did they regard space @ due to ate? If they did, was that osophicalreflection, analysis, and deb matheor ones arising from theology, reasons internal to philosophy although our ofMatics, natural science, or else where? In this sense,

e the alternative, ficialtitle is Space: A History, we mi ght also embrac it took centuries

pace The Emergence ofa Concept. Unsurprisingly,

or our modern SPACEto emerge, or perhaps for a single concept to me Tracing that emergence is our joint task in this volume.

confusion, arises fom nnPorential anachronism, and attendant ce on historical perspective for philosophy, coupled


i the 1 Followin g convention, i word“spece?

both space itselfand the

ve from hereafter denoted SPACE, thereby distinguishing it


with the failure to apply the same wise rule to other disciplines. For instance, we may insist that philosophersin thefirst half of the seven teenth centurytreated space differently fromthose in the secondhalf, and then forget to treat accompanying developmentsin, say, ex eri mental technology with the same historicalcare. Perhaps for obvi reasons, it is especially important for our approach to recogniz histor ically shifting conceptions of geometry during the cpochsunder sandy in this volume. To a contemporary reader, geometry is important in

this context becauseit just is the science of space. Concomicand it may seem obvious that the approach outlined in Euclid’s Elements vas the first systematic attempt to understand the nature of space. But i earlier historical periods, philosophers and mathematica “idt 0 necessarily regard Euclid as providing a doctrine or creatment f ne thing called space or as a description of what we would c th hs"sical space (as Jeremy Gray discusses in his Reflection), That may und odd to modernears, but caution in this area seems to be warif redb the ancient and medieval evidence. Indeed, some influentialseh | . ship suggests that in previous periodsin history, Euclid was som: re es

regarded as discussing notspace (or SPACE) butthe constructionof gures and their various properties. A fully historical approach, then will provide not only a historically precise conception ofphil hy but an equally precise, historically rich conception of hs vat : disciplines “heses with which philosophers engaged in dial ialogue ve throughout The Oxford Philosophical Conceptsseries includes a novel element mentioned earlier: it enriches the chapters written on diff


in philosophical history with short essays called “Refl . ‘ong? hat tackle related developments in myriad other fields TheRefle .

in this volume fall into two broad categories.First “ome ofchemare easily paired with closely related philosophical essays. Because Gary nae’ chapter concernsphilosophical theories ofvision and optics fom¢ e Middle Ages to modernity,it is nicely paired both with Mari

Hara’s essay on theuseofperspective in Renaissance painting and with




ndingof visualspaceiy Jennifer Groh’s contribution on the understa thy historian Hara documents contemporary neuroscience. The art influentiall, ian Renaissance artists famousl y and

ways in which Ital y and optics to introduce linea employed their knowledge ofgeometr architecture, thereby bridging thy perspective into their painting and nce. Groh’s essay, written from thy expected gap between art and scie ws the ways in which populations of perspective of a neuroscientist, sho y ofsome res rors the perceived topograph mir at th map a m for s ron neu ng Kim’s Reflection

Gyu such as sp aces, which discusses figures on early modern laboratory Janiak’s chaptet 0 an of science Mi gion in the world. The histori

paire d with Andrew Boyle and Newton, is easily that theearly modern tions ofspace. Kim shows

early modern concep

air pump—

ed equipment—such as Boyle's chemical laboratory contain o served i on of emptyspace, bur als designe dto operationalize the not case, two Reflections inIn another to represent an ideal social order. zes emy Gray con cisely characteri Jer r. the ano e on h wit tersect nicely cians came ' x story of how mathemati ple com and fascinating the a set of possible as merely one among Euclidean geometry


lines than is ed 4 different notion of olv inv h ic wh of me so geometries, ion of tulate. The veritable explos contained in the famous parallel pos pis story—fromthe develo philosophical idea s that accompaniedth g ofa priori knowledge—is ment of conventionalis im to the rethinkin lptor George Hart Reflection by the m athematical scu

well known. The temporary artistry by brings Gray’s discussion into dialogue wi th con features both showing how onecan sculpt objects to express intriguing of Euclidean and of non-Euclidean spaces. The remarkable images ©

a wide Hart’s sculptures in this volume will assist readers in imagining array of spaces.

The Reflections in the second broad category are notpaired one-to-

One with our philosophical chapters or with other Reflections. They cbproach space From the perspective of other disciplines—especially thas hiloce re y and ecology—thatraise intriguing questions, ones phers may have overlooked. These Reflections help to


enrich our overall treatment ofspace andits concepts over the centuries The Reflection by the cultural geographer Banu Gokariksel discusses public spaces in Istanbul from a sociopolitical perspective, indicatin

(e.g.) that a particular place within a globalcity can be gendered in vat ious ways that are renderedlegible primarily through detailed scholarship on the group. Philosophers and mathematicians have often noted that a space mightbe occupied or empty, curved orflat; it expandstheir imaginations to show howa space mightbe gendered. The Reflection by the biologist Nicole Heller, which concerns the methods used in ecology to infer evolutionary facts from the patterns of organisms in space, serves to challenge philosophical assumptions about what counts as an object or a body occupying space. When the object in question is an ant colony—it is the entity falling underselection pressure rather than the individual organism—we encounterintri vin questions about the sense in whichit behaves bothlike a single object occupying somelarge space andlike a series of autonomous oneities each moving through space independently of the others. Clearly, ant colonies pose tough questionsfor the metaphysically minded. | | When wejointly conceived of this volume’s contents,it wasinitiall tempting to include only Reflections that would pair neatly with out philosophicalessays. But weresisted the temptation.Just as the first set of Reflections may enrich the way in which philosophers think about their own history, the secondsetof Reflections may prod philoso hers to think about space from a novel perspective. Afterall shilosopher may notordinarily think that a space can be gendered and that loca tion can help to constitute one’s gender identity. So we include these disciplinarily varied and highly creative Reflections to avoid the attractive, but ultimately unwise, plan of neatly matching the contents of each chapter with a specific and clearly related Reflection. It might make sense to approach certain concepts, especially concepts “hot ate primarily or solely philosophical in character, in that way-_one thinks

of consequentialism or idealism—butphilosophers do nothold exclusive, or even primary, sway over the conceptofspace. They never have.



derable depth Indeed, philosophers have thought about space in consi

geographers, biologists over the centuries, but so have architects and

ctions do not comand geometers, artists and physicists. OurRefle ry approaches to space, prehensively represent these various disciplina a provide an entrée into which maybe an impossib le task, but they do

about space a philosowider world, enriching the set of p ossible ideas

pher might have in her scholarly repertoire. tre atment of SPACE ends Even a causal reader will notice that our work of Immanuel Kant. We have in the high Enlightenment with the up to the present. This choice is chosen not to bring the discussion compatriots simply not intendedto signify that philosophers and their sigIn a way, it is meant to stopped thinking about space a fter Kant. more opposite! Space became nify, or to admit, something like the very a s is evident already in the important after Kant, not less. First ofall, ction of the Critique of “Transcendental Aesthetic,” the first main se the di scussion of space Pure Reason (first edition, 1781), Kant placed of the phical revolution. His tre atment

at the very heart of his philoso it is a pure intuit ion rather representation of space—contending that concept (Begriff) than an empirical represent ation of any kind, or a to undergird rranscendental ideof any kind—is already understood on space andits alism in somesignificant ways. S econd, this emphasis ing representation was prescient: philoso phical developments (follow science Kant’s work) conspired with parallel develop ments in natural topic of and mathematics to ensure that space would remai na major s. As consideration throughout the nineteenth and twenti eth centurie

saw therise Gray chronicles in his Reflection, the nineteenth century

ment involving of non-Euclidean geometry, a revolutionary develop the work of Bolyai, Lobachevsky, and Gauss. This striking change in

mathematics was perhaps overshadowedby the even more impressive

beginning of the new century: by 1905 Einstein’s first revolutionary

ideas, containedin thespecial theoryofrelativity, taught the worldto speak of “space-time” rather than space andtime separately. A mere decade later, Einstein’s second burst of revolutionary ideas, expressed



in his general theory, meantthat space would forever be thoughtof as exhibiting non-Euclidean features (such as variable curvature) based on the distribution of mass-energy. These developments were not merely revolutionary in the sense that they overturned centuries of assumptions in mathematics, philosophy, and science—they went so far as to trespass the boundsof whathad been assumed to be possible at all. Ir’s shocking enough to hear that Euclid did not have the final word on geometry after roughly two thousandyears; it’s another thing entirely to learn that the real world is non-Euclidean and that space itself interacts causally with material objects on a vast, cosmological scale. These are ideas that Isaac Newton would not have dreamed of onhis wildest day. Given all the excitement concerning space in the nineteenth and twentieth centuries, the reader may wonder whythis volume endsjust before these developments took shape.Intriguingly, these developments are not merely exciting per se; they are intimately connected—one

mightsay, internally relaed—to the development ofanalytic philosophyitself. As is well known,analytic philosophy emergedin theearly twentieth century in part through the influence of logicalpositivism on the pragmatism-infused thought ofAmericanslike W. V. O. Quine and many others. Andforits part, logical positivism wascentrally concerned with the revolutionary developments in mathematics and sci-

ence during its own heyday, developments in which SPACEplayed a leading role. It is no coincidence, for instance, that Moritz Schlick, a founding memberof the Vienna Circle, published one of the very first books concerned with the general theory of relativity (in 1917), with Hans Reichenbach publishinga distinctive treatment of relativity

just three years later.” Similarly, Rudolf Carnap’s most importantearly 2 Moritz Schlick’s Raum und Zeit in der gegenwiirtigen Physik (Springer, 1917) went into several editions; by 1920 it was already onits third edition and was translated into English that same year

by Henry Brose of Christ Church, Oxford,as Space and Time in Contemporary Physics (reprinted was certainly tracking “contemporat f ses" the edition Schlick’stoaccount 1963). in to Doverhad by of 1920 be changed reflect the remarkable confirmation of Einsreln' eneral theory of relativicy che year before! See also Hans Reichenbach, Relativirdtstheorie und Evkennonis prior




of philosophy, physics,

publication, which involved the intersection m(Space) and appeared just 4 and mathematics, was entitled Der Rau of the 19208, Reichenbach few years after Schlick’s work.3 By the end work on space and time. Since had already contributed another major central to science and mathematics the discussion ofspace and time was central to the newly emerging phiduring the Vienna Circle, they were censince philosophy of science was losophy of that time. In tandem, philosophy in the English-speaking tral to the development of analytic

world in the previous century, and

space and time, in turn, have always

of science, the lirerature been central topics within philosophy

analytic philosophy on space and time is



thar recent twentieth century mean These facts about the lion’s share of the

have received the developments concerning space no The interested reader will have attention in the past few decades. just accounts of space and time from trouble finding philosophical century OF more. However,scholarly about any moment in the past antiquity, the medieval period, treatments of space and of SPACE in period are much less common. the Renaissance, and the early modern asa kind ofsupplement to the existing In this way, our volume can serve



1960 as The Theory of ated by Mari a Reichenbach in (Berlin: Springer, 1920), which was transl s). Pres a rsity 0 f Californi Relativity and a priori Knowledge (Berkeley: Unive ien Erginzungshefte 56 3 Rudolf Carnap, Der Raum: Ein Beitrag zur Wissenschafislebre, Kant-Stud (Berlin: Reuther and Reichard, 1922). Gruyter, 1928), translated in 4 Hans Reichenbach, Philosophie der Raum-Zeit-Lehre (Berlin: De (NewYork: Dover). an 1957 by Maria Reichenbach andJ. Freund as The Philosophy ofSpace d Time Vienna Circle (see, the of ¢ member by In addition to the translations of important publications g philosophy dealing with space and e.g» notes 2 and 4 above), major works in English-speakin Grinbaum, Philosophical

time published over the past fifty years include the following: Adolf Stein, “Newtonian SpaceProblems of Space and Time (New York: Knopf, 1963); Howard to the Philosophy Introduction 1 Fraassen, van Bas 174-2100; (1967): io Quarterly Texas time,” Spacetime of Time and Space (New York: Random House, 1970); Larry Sklar, Space, Time and (Berkeley: University of California Press, 1974); Jill Van Buroker, Space and Incongruence (Dordrecht: Reidel, 1981); Michael Friedman, Foundations of Space-time Theories (Princeton,

NIPaireeton University Press, 1983); John Earman, World Enough and Spacetime (Cambridge, \: Ml Press, 1989); Robert DiSalle, Uneerstanding Spacetime (Cambridge, UK: Cambridge University Press, 2006). Theliterature is simply too vast to doit justice here.


philosophical scholarship.¢ The monumental importance of the twentieth century for thinking about space cannot be denied. But perhaps we can supplement our knowledge ofthat time by producing a volume in which wegive pride ofplace to figures whoseideas about space are less well known. Hence in this volume,in lieu of discussing Riemann and Einstein, Poincaré and Minkowski, we emphasize the likes of

Aristotle and Proclus, Hobbes and Kant, Ibn al-Haytham and Leibniz. One happy consequenceofour approachis that our volumeexhibits

a kind of intellectual unity, Forall of the figures in this volume, geometry was Euclidean, space and time were separate things, and both geometrical space and the physical space in which welive could be fundamentally understood through the kindsofreasoning already codified in antiquity. Naturally, the figures discussed in this volume engagedin

vociferous disputes on a widerangeoftopics: the finitude ofthe world, space’s basic relation to matter, the possibility of the vacuum, God's

relation to’space and the world, and so on. Butall of these disputes occurred within certain basic intellectual confines. Obviously, Euclid

and Aristotle lived in a fundamentally different world than Kant, but in this respect their intellectual horizons were remarkably and profoundly aligned. For this reason, the Reflections by Gray and Hart, which concern the emergence of non-Euclidean geometry in history and the possibility of its representation through mathematically sophisticated sculpture, serve to forma kind ofintellectual boundaryfor our volume. Since they discuss the developments in mathematics that would help to usher in a new world, the world that we now occupy,

6 Penthe very bese works, ones with the most sophisticated historical accounts of the development of physics and concomitant ideas about space and time, will tend to give pride of place to the nineteenth and twentieth centuries, even while acknowledging the Significance of earlier ideas a Burs Although they occupy decidedly distinct philosophical perspectives, Cushing and orretti oth illustrate this broad pointin their works from the very late twentieth century. See James Cushing, Philosophical Concepts in Physics (Cambridge, UK: Cambridge University Press

Lminneniretne The Philosophy ofPhysics (Cambridge, UK: Cambridge University Pesss 19 9), posh ond

egin with antiquity (Aristotle, Prolemy,etc.) bur move quickly on to



storthey are profoundly important. But from theperspective ofthehi

ical figures thar appear throughout our volume, these developments ,Lying were notfar offon the horizon—they were beyondthe horizon

they occupied an beyondthe limits of what was considered possible, imagination. This intellectual space exceeding the limits of the human like to live in that world, volumetries to capture a bit of wh at it was whichis now long passed.



Barbara Sattler


In the roughly four hundred years fromthe earliest Greek texts to Aristotle, many of the most basic questions about space were raised for the first time in western thought and answeredin a greatvariety of

ways. These are questions such as Whatis the task ofspace—isit to answer the question where some bodyis situated and whereit is moving, oris it rather to delimit one thing from another and thus to be a condition for plurality? Does it have an internal structure? If so, whatis its internal structure like? Is spaceitself some kind ofa bodily entity, or is ic nonbodily,as, for example, a vacuum? And what do the answers

to these questionstell us aboutits ontological status? Is it of the same

status as the bodily things? Is it something more fundamental, given that everything seems to need to be somewhere in order to exist? Or Barbara Sattler, Space in Ancient Times In: Space. Edited by: Andrew Janiak, Oxford University Press (2020), © Oxford University Press. DOI: 10,.1093/080/9780199914104.003,0002




lly dependent is itjust a feature of bodily things and as such ontologica on bodies?

dupon,I will While most of these questions will at least be touche concentrate here on one impor tant stream in the development of spaofspace tial thinking during the early ancient period: the establishment

rtheless be combined asa magnitude independent oftime that can neve philosophical with time. If we look at prephilosophical or the earliest wesee accounts ofspace,for ex ample, at Hesiod’s and Anaximander’s,

notclearly that time and space (and sometimes al so matter) are often this might be distinguished: something is described as far off, where

d Anaximander’s meant either temporally or spatially or both. An ich everything develops, apeiron seemsto be the basic ground out ofwh materially, temporally,



limit of an object’s surrounding is often understood as a mere notion of a vessel. As suchit seems not only to be too narrowto be ofany u for understanding motion butalso to be merely an account of lace,

not of space. I will show, however, that Aristotle in fact orovides us with an understandingof topos that prepares the notion of a general frame of reference, which allowsfor locating things in the Wo ida well as for an account of motion. What enables Aristotle to com bine space with time in his account of motion and to reply to Zeno’

paradoxes of motion is his understanding of both cme and s veea continua.


Butbefore we jumpinto the unfoldingofthisstory, let us start with a few general methodological considerations, _

clear distincas well as spatially. This lack of a

e an tion between time and spaceleadsto problems when wetryto giv anding of the relaaccount of motion, as motion requires an U nderst s Leucippus and tionship between time and space. With the atomist

a cl ear notion of Democritus we will see a first attempt to develop are completely silent on Space in contrast to matter. But the atromists the notion of time. And simply omitting one of the two magnitudes, of time or space, when accounting for mot ion leads to another kind

problem, as Zeno’s paradoxeswill show.

h Plato's The next steps in this: development can be see n wit

conTimaeus: drawing on atomistic as well as Pythagore an ideas cerningspace, it posits time and space as two completely indepen dent

to makethe unimagnitudes. Timeis created and introduced in order which some receptacle, verse more intelligible, while space—Plato's scholars have also understood as exhibiting features of matter—is unented and an essential part of what Plato's “creator” god starts out vanand hes to putinto order. In fact, time and space are so different hal it is unclear how they could be combined atall for the purposes of

8iving an accountof motion. Awe task, central for any natural philosophy,is finally tackled in

tistotle’s Physics. Aristotle’s conceptionoftoposas thefirst immobile

11. MethodologicalPrelude: Problems and Possible Criteria Jor Space e period dealt with in this chapter raises at least two serious problems for an investigation of space: 1. Several of the thinkers we will consider do not have an explicit account, notion, or conceptofspace; explicit discussion af

how to understand spaceonly starts with Plato and Aristotle I hope to showthatit is nevertheless fruitful to look at the

pre-Platonic thinkers deal with what we consider to be s wal ideas or notions, But in order to understand how far im ‘ict accountsofspace can be foundin these thinkers, we swelves will have to think aboutpossible criteria for space that all to identify such spatial notions, , “ones 2. Some of the most importantspatial terms in Greek in this

period—chéra, topos, diastéma, and kenon—do not ne match one to one with ourspatial vocabulary.

Letus look at both points in somewhat more detail

il _





is orientable,bounded, orinan internal structure and, ifso, whetherit homogencous or not, isotropic finite, or its opposite. Is it everywhere

importantfor the ancients. At least within the Pythagorean tradition this distinction does not seemto be crucial, as one and the samespatial notion, the void, is used to separate mathematical entities as well as sensible things.’ So we shouldleave it open as a possibility that our distinction between physical and geometrical space may not be a distinction the ancients would draw. It might seem that they are compelled to make such a distinction given that the prevailing notionsof the universe as the most compre_ hensive physical space assume it to be finite and bounded, whereas (Euclidean) mathematical spaceis infinite. However, while this sounds

are the consequences of ular dimension? Is it metrizable?2 And what

cussion ofit in preclassical and classical ancient times; spatial notions

. LI. POSSIBLE CRITERIA FOR SPACE things, on how What we understand by space depends, among other al bodies in it is related to bodies: Is it itself a body like other physic le, a con the world (presumably just somewhat bigger), as, for examp If the lattes, tainer? Oris it genuinely different fromordinary bodies? is spacejust the relation of bodies or it s own entity? whetherit has ore, we will have to look at the ques tion of Furtherm

Does space possess a particor anisotropic, continuous or discrete?!

ple, if we assume assuming space to have a certain structure? For exam it seems that a linear motionwill our space to befinite and bounded,

have to stop at a certain point.

dingof space, however, The most important point for an understan to be: Is spacethat seemsto be the question ofwhat we take it s function

ough which somein which something can besituated, that in and thr thing from anthing can move, or that which separates or delimits one d on we take the main task of space to be will also depen

other? What

space. As physical whether we think ofit as physical or mathematical

t ofthe objects ofour space, it should explain at least one decisive aspec ateness experience: either the possibility of their mo tions ortheir separ on the other or (at least in part) their shape. Asa mathematical space, hand, oneofits main virtues should be thatit allows for all possible

like a screamingtensionto us, there does not seem to be an explicit disare just used by mathematiciansas wellas by natural philosophers. And while wefind an explicit discussion ofphysical space in Aristotle, we do not have any evidence ofa discussion of space by the mathematicians during these times.*

1.2. A BriefLook at Someofthe Main Spatial Terms 1.2.1, TO KENON

s normally translated as “empty” or “void” the nominalization of this adjective, to kenon,as “the void.” In commonlanguageonly the adjective is used, while the substantive expressionis tied to philosophical contexts. In its first philosophically interesting use, with Melissus










mathematical constructions in such a way thatfor whatever construc-

tion we perform, we will never run out ofspace. However, while we take the distinction between mathematical and

3 DK 58 B30. 4 AsMax Jammer, Concepts of Space: The History of Theories of Space in Physics (Chelmsford,

physical spaceto bevital, we will have to investigate whetherit is at all

' : ourien nor 3) points out, also the anisotropyof space with Aristotle and the inhomogeneity “ space wid h P ato seemto make physical space incompatible with the geomerticalspace used in

2 Some of these features we are more used to from mathematical spaces (¢.g., orientabilicy), some ftom physical ones,


—____—_1 By “isotropy” » we normally understand that something exhibits equal properties in all directions, ; ve Se while “homogeneity” means that something is uniform throughout.

reek mathematics (ifwe can take Euclid's Elements as evidence). Aristotle does of course distinguish between physical and mathematical things, but since mathematica are abstractions from physical things, mathematical space is not a separate space over and above the physical one, And for him mathematical spaceis in fact not infinite bur just as big as we need, , so hedo es. not fface a discrepancy betweena finite physical world and an infinite mathematical space. 4 For the discussion ofta kenon,, chéra, yaand topos | am a especially ind b i ‘

ofSpace in Greck Thought (Leiden: Brill, 1994), chapter 2 vindebted so Keimpe Algras Concepts




Eleyjc and the atomists, to kenonis a physical interpretationofthe

with the multifaceted notion of nonbeing, whatis not. In accordance rent ways by the atomig., use of nonbeing, to kenon gets used in diffe

his refutation of and Aristotle takes up these different notions in potential config. the void in the Physics, where he also points out :



between them.® ' ither to t can refere In general, to kenon means “emptiness”; as suchi thing orpart of (1) emptyextensi on orspace or to (2) a specific empty

a thing(like an empty vessel). 7 Butit can also refer

to (3) space orplace ‘

as such.8



on, which can be Occy. Chora is a two- or three-dimensional extensi “ground ae pied. The basic meaning of chéra is “land,” “region, can also mean “stretch,” “fe 4 it is applied to a smaller extension it

or should be. Zopos is or “place”—it points out the place where oneis a. But while chéra already appears largely used synonymously with chér ? in Homer,topos cannot be found before Aeschylus.

ike e to tr anslate 3 a and fopos, peoplelik ship i of chér i Asforthe relation e is no one-to-one chéra as “space” and fopos as “place. ” LJowever, ther and space; the adequate match between sopos and place and chéra n bothare used tion depends very much on the contex t!0 Whe



space against the notion of 6 For example, when he uses the notion of a void as an occupier of Text, ed, W. D. Ross Revised A Physics: Aristocle’s Aristotle, ff 2174 in space the void asitself (Oxford: Clarendon Press, 1936), g Inthefirst senseit is independent of any possible thing chat may bein it; in the secondsense, by


together, topos may denote a part of chéra. But in contrast to chéra, topos can also be used to denoterelative location orpositionin relation to a surrounding. And topos is often understoodto befully occupied place, while chéra as only partly occupied. Topos can also denote the underlying extension not of individual things but of the whole universe andis thus used forindicating what we wouldcall“space.” With the Hellenistic schools, the Epicureans and the early Stoics, chéra, topos, and kenon becometechnical terms. Topos refers to the

space thatis occupied by a body and kenoto the space that is not occupied by a body. For the Epicureans chéra indicates the space a body is moving through," while for the Stoics chérais an interval partly occupied by a body andpartly unoccupied.!? But the time we are looking at, which is before the early Stoics and Epicureans, does not use these

termsin a fixed, technical way. 1.2.3. DIASTEMA

Diastémabasically means“distance.” It can refer to distancein general or to a specific distance, as, for example, the distance and hence interval between notes in music or the distance between the center of

a circle and any point on its circumference, hence the radius. These distances need not bespatial; for example, the diastéma between two numbers such as 1 and 2 is what we would understandas the interval between the two numbers.” Butit is prominently usedalso for spatial extension, suchas the spatial extension between bodies, and also covers what we wouldcall spatial dimensions: length, breadth, and depth.

contrast, it is dependent on the thing that is empty (¢.g.,0n the vessel).

Cf,for example,Aristotle's report on such a usage in Physics, 214814. 9 For chéracf.for example, Homer, iad XXIIL 521, or Odyssey VIII, $733 for topos, Aeschylus, 7he Persians, 769. 10 In Timaeus 19a, for example, Plato talks about the ehéra of a person ina class society dependent on his ability; we would probably translate it as the “place” (rather than “space”) thar the less

deserving should change with the more deserving, For the relation between chdna and topos see especially Algra, Space, 33~38; also Benjamin Motison, On Location: Aristotle's Concept ofPlace (Oxford: Oxford University Press, 2002), 23, 121-32.

11 In this way the Middle Ages also employed the distinction betweenplace, which refers to location, and space, whichis employed in contexts of motion. 12 Cf, for example, Sextus Empiricus, Adversus mathematicos lo.2—4, in Sexti Emptrict Opera

(Teubner: Leipzig, 1914-61); Richard Bett, ed. and trans., Sextus Empiricus: Against the Logicians (Cambridge, UK: Cambridge University Press, 2005). 13, See,for example, Aristotle, Physics 202418, 14 See, for example, Aristotle, Phystes, 20944.




by the Greeks up to Looking at some of the main spatial terms used are ofimportance: the Aristotle, we see that the following spatial ideas nce; chdra and topos refer to term diastéma expresses the idea of a dista

the notion ofacertain extended area, but also

toa specific point or sec-

motion needs to be something thatallowsfor

and separation.

m kenos conveys the idea that there tion within such an area; and the ter


be something we would characterize as spatial. Also Aristotle in his Physics understands it as a first notion, or proto-notion, of space,

since it shows that we need something where all other things can then comeinto being. This fits also with the next thing that comes into being, Gaia, “broad-breasted earth.” Thus westart out with the spatial dimensions of depth (chasm) and breadth (broad-breasted). Presumably,ifearth has breadth,shealso has length, so that we haveall three spatial dimensions(or, as is more commonin ancient times,all


21. Hesiod

ant image of spacein GreekliteraIn Hesiod we find thefirst signific h several passages that do ture: chasm. But weare also co nfronted wit time. Right fromthevery notclearly distinguish between space and and genealogy of the beginning of Hesiod’s Theogony—his cosmogony gods—temporalandspatial notions are closely intertwined:

then broad-breasted In truth, first of all Chasm came to be, and

possess Earth, the ever immovable seat ofall the immortals who of the depth snowy Olympus’s peak and murky ‘Tartarus in the broad-breasted earth, and Eros.... From ChasmErebos and black Night came to be; and then Aether them after and Day came forth from Night, who conceived and bore Sky, equal mingling in love with Erebos. Earthfirst of all bore starry to herself, to cover her on every side, so that

she would be the ever im-

translation) movable seat for the blessed gods. (lines 116-27; Most’s

six spatial extensions,since each of our three dimensions has a to and fro).!6 In addition,earth also possesses the height of Mount Olympus, which makes her “the ever immovable seat ofall the immortals”; she provides /ocation for the gods. Thusalso the gods are given a space, snowy Olympus, before they comeinto being, Earth is the mainspatial reference point:it is from earth that we can say ‘Tartarus is below and,later on, that Ouranosis above. However, while earth is seen as something clearly limited,!” in contrast to the indeterminate Tartarus, we are not given any clear shape and size of Gaia;'8all we hearis that Gaiais encircled by Oceanus.

While it seems we get only a spatial setup of the universe in the beginning,it is actually fromthe veryfirst spatial notion, chasm, that we get the first temporal notion: from chasm night came into being. And

fromnight and darkness (Erebos)” day is generated, so somethingelse that we would characterize as temporal. There is no indication in the

Hesiod’s term indicates a gap or opening; cf. also Geoffrey Stephen Kirk, John Earle Raven, and Malcolm Schofield, eds. The Presocratic Philosophers: A Critical History with a Selection of Texts (Cambridge, UK: Cambridge University Press, 1983), 37 (hereafter KRS), who point outthat the

term “chaos” comesfrom the rootcha, which means“gape?“gap? or “yawn?

First of all Chasm cameto be. A chasm (Greek, tos) usually is some

5 Hesiod gap within a (or between two) spatially extended things.! does not determine the chasm in any way further, but ic seems to of the Theogony 15 The Greek term og is often eranslaced as “chaos.” However,in his translation Mostrightly points our that chis misleadingly suggests a jumble of disordered matter, By contrast,

16 Cf, for example, Aristotle's Physics, which talks about three dimensions (Physics 209a4-6)orsix extensions (Physics 2o8b12~14). The dimension of depth has the up and the down,etc.

17 Sheis limited above by Ouranos, and for her below, we are told in lines 621-22 that Obriareus, Cottos, and Gyges have to dwell under the earth, at the edge or limits of the earth, By contrast, Pontos(sea) is explicitly called boundless (apeivon) in line 678,

18 Gaia is on the one handtreated as a person, onthe other hand as a place, in which,for example, Zeus can be hidden as in a container(lines 479~83), 19 Erebosis also seen as aplace; see line 669 and Homer, Odyssey X, 528 and XII, 81.




in text that now the dimension weare lookingat is changing;rather,

o be fromanthe same genealogical way as onespati al notion comest temporal no-~ a so , earth) other(e.g., Ouranos from Gaia, heaven from l chasm. tion like night comes to be from the spatia

is also clear fromm That time and space are not strictly distinguished

Theogony 721-25 we read: other passages in Hesiod. For example, in

For a brazen anvil For so far is it fr om earth to murky Tartarus.

ld reach s heave n nine nights and days wou falling down from en anvil falling from the earth upon the tenth: and aga in, a braz Tartarus upon the tenth. earth nine nights and days would reach

and night spreads in triple line Round it runs a fence of bronze,

above grow the roots of the all about it like a neck-circlet, while

translation earth and unfruitful sea. (Hugh G. Eyelyn-White’s with alterations)

ce—“round it runs Tartarus is described as being encircled by a fen r something sp atial. But a fence of bronze”—-which seems normalfo line all about it like a nee . then we hear that “night spreadsin triple oral unit, night, circlet.”” Thus something thatis usually seen as a temp is here treated as somethingspatial.

ween heaven and Furthermore, the way in which spatial distance.s bet

d is in terms of earth and between earth and Tartarus are determine sky to earth time, namely in terms of anvil days: ten anvil days from is the and ten more from there to Tartarus. An anvil day presumably

distance an anvil will fall in a day. You might think that “day” here is a spatial unit, since we are talking about a day in the sense of how much space an anvil covers in a day.

Thus a day seems to be a way to determine a unit ofspace. After all,


awfully long.?? Of course, nobody knew howfar an anvil would fall in a day. (How would they have found out?) So thisis notreally an easier way to determine a certain spatial extension. And even if we treat it as straightforwardly a unit of space, it wouldstill originally be some temporal unit that was then turned into spatial one. Summing up, we can say that there are three main spatial notionsin Hesiod:

1. With chasm wegetthe idea of a where in which things can comeinto being. Butit isa mere openingthatis in no way further determined and thus does notprovide any further spatial orientation. 2. With Tartarus and Ouranusweare given a basic below and above and can determine a motion’s up and down. 3. Withearth all three dimensionsare fully unfolded.Ie is extension thatis in the foregroundand thepossibility of location, not so much actually tracing downa specific spot where somethingis situated, Inspite of these spatial notions, we saw that Hesiod does not strictly distinguish between spatial and temporal notions, as becomes especially clear when weare dealing with limits. That a clear-cut distinction between time and space might not be something regarded as a matter

of fact or even desirable we also see in Anaximander. 22. Anaximander and the Early Cosmologists

Cosmology can beseen asa, if not the, starting pointof philosophy. Thales, the first philosopher, allegedly predicted a solar eclipse, and

this is before the time of having an Ur-meterin Paris, with the help of which atleastall scientific measurements are done. So perhaps this isa

way for Hesiod to indicate for people in Boeotia, as well as in Athens, in Asia Minor, and on the Peloponnese howlongthis distanceis: it is

20 It is not uncommon alsoinlater times to use the time a normal journey would take in orderto indicate a distance; cf, for example, Herodotus I, 5, where he indicates the part above Lake Moeris

that has been gained from theriver as “upto a three-daysail.”




an an important part of what we know aboutthe Joni

thinkers is how

naturally not only they thought the universe was se t up. Cosmology

al arrangementof the includes some thinking about space, the spati phenomen a together: the world, butit also brings spatial and temporal

s in space is us ed to locomotion of heavenly bodies and thus a proces time. determine temporal units and to calculate a cle ar center of the uniof idea Many cosmologies work with the

ed with a special value, verse, usually the earth. This center may be load ;”it is usually defined not as we see, for example, with the Pythagoreans dy in But alrea

lute center. in relation to somethingelse bur as an abso mists—we seemto encounter Anaximander—and later on also in the ato ca se we can talk abouta center the idea ofinfinitely many worlds;?2 in this universe as such. only relative to our world, not ofthe the of the position of th e earth in Thales not only gives an account —

is whyit is not “falling down’ universe—it is Hoating on water, that important question for many but he also seems to have raised an there is one basic entity, prinphilosophers to come, namely whether in

plurality of phenomena ciple, or element that can help explain the thatit is

is such a principle and our world. Thales thinks that there

ts. water, which can then turn into the other elemen mony (Physics 204b22-29)s To judge from Aristotle’s testi this basic question of Thales Anaximander seems to have reacted to water as this basic prinby criticizing Thales’s attempt to establish

since it is one speciple: water cannot be a suitable first principle


ofit. Instead, we need somethingthatitself is not one of the four elements and thus cangiverise to all four equally. Anaximander’s first principle, the apeiron—the Greek termliterally means “unlimited,” “infinite,” or “indecerminate’—can guarantee this by being

itself indeterminate and unlimited. By being indeterminate and unlimited the apeiron can be the source for the abundantarray of phenomenaand the incessant process of their coming into being andpassing away,It allows for unlimited processes in what we today would see as three different ways, as a material, spatial, and tem-

poral ground ofthings:”4 “Anaximander... said that the source and elementofexisting things wasa certain natureof the apeiron, from which comeinto being the heavens and the world in them. This is eternal and unageing, andit also surroundsall the worlds” (DK 12 A u, Bz Hippolytus Ref. I, 6, 1-2, myitalics). 1. The apeiron is that from which (ex hés) the heavens and the worlds come into being. Thus it seems to be some matterlike stuff out ofwhich everything can comeinto being.The generation of everything outofit is possible,sinceit itself is not determined andlimited in the way someparticular piece of

matter or material thingis. 2. The apeironis “eternal and unageing”** and thus temporally infinite. The infinite durationis a necessary condition for the

unceasing processes of the phenomenal world.

rise also to the other cific element and thus does not qualify to give for example,it elements. If water itself is seen to be cold and wet, stuff, can come out is unclear how its opposite, the dry and warm

23 ve get areflection ofthis thoughtlater on, in Plato's account ofthe receptacleinthe Timaeus: there all elements comeinto being in the receptacle, and the receptacle is preparedforthis by bei pitself yeep: free of any ofthe features the elements possess, occupied by earth 21 However, since the center in the universe is the most valuable place, it cannot be

for the Pythagoreans, but has to be occupied by whatitself is most valuable: byfire.

22 Though thereis some dispute among scholars whether Anaximander did indeed assume infinitely many worlds and whether,ifhe didso, these infinitely many worldsare coexistent or successive; cf.

12.4 f ,

24 SeeDK 12 Ag-i.

25 KRStranslate arché evenas “material principle,” 26 For apeiron used in a temporal sense, se ¢ also fra

it is


the ie is eeein ve ible,” and Simplicius apeiron where in KRS),that 113claimed rang (ft. Phys,there amortal ... and ind estructible,” : is “immortal

we hear that infinitely many worlds cometo be and pass awayad infinitum (ep’apeiron)




and thusis the most 3. The apeiron surroundsall the worlds

embracingspace.” It-allowsall motions and place in it.?8

changesto take

has spatial, temporal, as well ay We see that Anaximander’s apeiron fdoes notspecify whether material connotations. Anax imander himsel

spatial principle. Indeedit he takes it to be a material, tempora 1, or plur.ality of phenomena and seems the apeiron can explain the rich |

g features ofall three. their generation exactly by possessin notions to be found with theearly There are of course other spatial the Pythagorean notion of rheee Presocratics. There is, for example, d as a mit between numbers”? an as a separator: void is understoo of space, but

unphysical notion thus as what we would think of as an gly, there as delimiting bodies. Accordin

it is subsequently also seen c difference between mathematics does not séem to be a systemati oreans. And there is Parmenidess and physical space for the Pythag plete: giving an account of the connple powerful analogy of a sphere for ivi ever, the next accountof a spatia homogeneity of whattruly is. How ant i ortan s’s ’ most imp ides the one by Parmenide centrate on is i con l wil I ion not : student, Zeno.


2.3. Zenos Paradoxes ofTopos and Motion While with Hesiod and Anaximander wesawthat time and space are not clearly distinguished, with Zeno wefind the consistency ofthe notion of motion andofthe central spatial notion ofzopos put into question. In his paradoxoftopos Zenoposes the problem of whether place does indeed exist, His paradoxes of motion point out some possible problems with - the internal structure of motion andspace as well as problemsthatarise if we give an accountof motionsolely in terms of the space a motion covers withouttaking into accountthetime this motiontakes. 2.3.1, ZENO’S PARADOX OF TOPOS We find the earliest explicit formulation of Zeno’s topos paradox in Aristotle’s Physics 209a23~25:°° “Zeno’s difficulty demands somie explanation: forif everything thatexists has place,it is obvious that also

place will have a place, and this will go on ad infinitum,” This paradox claims thatif everything that exists is in a place and placeitself exists, then place will be in a place, ad infinitum. Or, in somewhat moredetail: 1. Whateverexists is in something. 2. Whateveris in somethingis in a place. 3. If place is something thatexists,it has to be in something and

fern apeiron is stressed by KRS,since they think chatthe 27 ‘Thespatial characteristic ofthe apeiron ve si ese enmse t KR er, Howev r, mande ty before Anaxi is used only as indicating spatial infini erat s ited in extent and duration. And Aristotle un that one understanding is to take it as unlim tands ; ¢

s 20 sar6fE. and 204b22fF he unders Anaximander's apeiron also as material when in Physic are opposed to each other.Also Chat es which ts elemen the apeiron as something beyondorbesides (New York: Columbia University Press, i” Kahn, Anaximander and the Origins of Greek Cosmology re « . eidea ofspace with che “the material which fillsit. 1960), 233 understandsche apetron as combiningth interest in cosmology 28 That Anaximanderis interested in spatial notions can be seen not only in his around buralso in the fact thar heis the first thinker reporte dito conceptualize a map of the area ‘oO


the Mediterranean Sea in the Greek world; see DK 12 Al. not need anything to We usually tend to think that numbers are naturally discrete, so chey would

“separate” them, However, with Philolaos, who is the first Pythagorean of whomwehave trustworthy fragments, discreteness seems to be groundedbythe void. SeeC. A, Huffinann, Philolius of Croton: Pythagorean and Presocratic. A Commentary on the Fragments and Testimonia with Interpretive Essays (Cambridge, UK: Cambridge University Press, 1993). Void is scen as granting separation and location (DK 58 B 30) but notyet as a basis for motion.

thus in place. 4. But then place would be in place, which again would have to

be in a place ad infinitum. Andthe implicit conclusion to be drawn fromthis is that place does notexist," because if there were an infinite series, 30 For a moredetailed discussion of this paradox and another paradox of Zeno connecting topes and motion (DK 29 B4), see my paper “Space and Place in Zeno” more details, such as that nothing is in itself and that there are no circular chains in yet For filling 3t of places, see Morison, On Place, who devotes his chapter 3 to this paradox. See also Aristotle,

Aristotle's Physics, Books II and IV, ed. Edward Hussey (Oxford: Oxford University Press, 1983) , 1193 Jonathan Barnes, Zhe Presocratic Philosophers (London: Routledge,1982), 256-58.

Peay BEB EO) liam): \ Png cy . Wald EI OVEIPLED, i «




er to the question we would notbe able to give a genuine answ where somethingis.*?

o’s Timatres, cit vari ation ofthis paradoxin Plat

We can find an impli aplace everything that exists must be in whenit is claimed in 52b that in

s notdo so cannot exist But

and occupy somespace, while what doe

e claim ina dreamlikestate, since wha the Timaeus we are told this is a n is not in anything.” This variatio truly is—for Plato, the Forms— ng efirst premise, that everythi in the Timaeus attacks the truth ofth by Aristotle will block the conclusion there is has to be in something.

to be false; in his Physics he pos demonstrating the second premise t ways. So while be iz something in eight differen

out that things can itis of location is a prominent sense, being in something in the sense in some place can thus be in something by no meanstheonly one, and in : x does raise the question what other sense. But Zeno’s parado

and Aristotle ill see the atomists, Plato, existence place has, and wew

analyzing this problem further.



If something moves overa certain spatial distance, for instance, a

runner wants to cover a certain finite race course, hefirst has to cover

half of this distance. Of the secondhalf, the runner again has to cover the first half, and then again thefirst half ofthestill remaining distance, etc. So he will have to pass an infinite numberofspatial pieces

before reachingthe end ofthefinite distance, and hewill have to pass these infinitely manyspatial pieces in a finite time.I will not deal with the first point here, how finite spatial extension can contain infinitely manyparts; instead I will briefly talk about the second point, that the runner seemingly has to cover an infinite numberofspatial pieces in afinite time. With Zeno’s paradoxes weare thus facing the problem that of the two magnitudes whichare necessary to determine movement, namely time and space, the latter is thought to be infinite and the formerfinite. . Aristotle—oursource for this paradox—reacts to this problem by pointing out thatif the spatial magnitude covered in a motionis continuous andinfinite, so (and in the very same way)is the time taken, andvice versa. Whatleads to the problem under consideration hereis



we t. in : we get While the paradox ofplace raises problems ises 2 ox ch , r parad of place or space as beingitself a body, Zenos runne ; out u tne ints i Ir points i an extension. problems ifwe think ofspace simply as an extended spatial magnitude problematic part-whole relation that get into if i n our acseems to display, and it shows the problems we time com: count ofmotion we concentrat¢ only on space andleave out ‘ : 34 pletely. The basic setup of the paradoxis as follows. i Iit into

thus the fact that Zeno does nottake time'into accountsufficiently.For, although timeis explicitly mentioned in the paradox whenasking for the possibility of covering the infinitely manyparts in afinite time,it is

ignored in the process of division. Otherwise, if Zeno had considered time as well as space and as a magnitudelike space,?> he would have put the division roughly like this: First the runner has tocover half the racing course in half the time. Of the remaininghalf-course, he then has to cover anotherhalf in half of the half time, and so on ad infi-

nitum. IfZenohadreally considered thecorrelation of time andspace,

32. Cf. Morison, On Place, 92-95.

participace in 33, One may object thar the Forms could be said ¢ o be in the individual chings which seemto sugmight Plato e middl the in chem; at least the language of the “presence” of the Forms ndent gest that (as, for example, the Phaedo). However, weare also told thar the Formsare indepe other, ofthis relationship. So while usually they will be present in some individualsensible thing or their existenceis in no way dependenton these sensible things, Andthesense of “in” in which the Formsare in a sensible thing is noc the local one employed in the second premise.

_ 34 See Aristotle, Physics 233a21~26, 239b11-14, and 263a4—11.

35 The problemis not only to take time into account at all—Zeno might have said that whatis

movingisfirst here, ater there, and before that at another place withoutthis changing the paradox at all—but to take ir into accountas something which canbe divided in exactly the same way as space can and whichhasto be divided wheneverthe distance of a movementis divided. However, this presupposes a basic similarity in the structure oftime and space, which we find only in book 6 ofAristotle's Physics.




ng offofthe

to a marki each marking offof a spatial part would have led

corresponding temporalpart (see Figure 1.1). 2








between timy

.** In the following wewil and space-—though only by neglecting time

ween matter andspace with see the beginning of a clear distin ction bet the atomists. ofthe Void 2.4. The Atomists and the Notion onsthat can allowfor terms, we can frame an account of locomotion. Putting it in abstract for x to move, problem motion raises as follows: In order

The notion ofa void is one of thefirst spat ial noti

the basic Buty cannot there has to be somey in which or into which x can move. move in of be the very same kind of th ing as x, for then x could not kind in the same into y without there being two things of the same lemthattheintrospot, which would lead to collocation. It is this prob seems to react to. duction of the void into the philosophical discussion Being, as whatis full. In fragment 7 Melissus?” understands whatis,

what is empty, Whatis full can move only into whatis not full, into 36 Fora more thorough discussion ofthis par:adox, sce myarticle “Aristotle's Measurement Di lemma,” Oxford Studies in Ancient Philosophy 52 (2017): 2$7-30h and especially my unpublished book cal manuscript “Natural Philosophy in Ancient Greece: Logical, Methodological, and Mathemati Foundationsfor the Theory of Motion,” chapter3. 37 ‘Thereis a dispute amongscholars who introduced the notionofa void first into the philosophical discussion ofthe time, the atomists or Melissus (leaving out potentially earlier claims aboutthe void by the Pythagoreans because of unclear sources).


Melissus claims that there is no void and that Being, whatisfull,is ac-

distinction betweena spatial and a material notion: whatis, Being, are

of time andspace. Figure 11 Zeno’s runner paradox—division

inction With Zeno, we see the beginning ofa clear dist

the void.°8 Thus void seems to be a necessary condition for motion. cordingly unmoved and unlimited. The atomists take up thestrict division between what is and what is not from the Eleatics in the physical interpretation it is given by Melissus. But they understand Nonbeing qua void as a basic concept on a par with Being and thusturn the ontological distinction between whatis and whatis not into what today we would understand as a clear

Hy 1


the atoms; what is not, Nonbeing, is the void. The atomsare eternal andindivisible bodies, and all the phenomena we experience in the world come to be out of an arrangementof these atoms. Being that out ofwhich all the phenomena cometobe, the atoms can be seen as predecessors ofa notion of matter. When the atomsget attached to each other, these phenomena cometo beiv the void. The void can thus be seen as a basic notion of space that is a condition for what we experience as “generation” and for the locomotionsof the atoms which underlie this “generation.” The atomistic distinction between atoms and the void also allows for avoiding Zeno’s paradox oftopos. We saw that this paradox claims that if everything that exists is in a place andplaceitself exists, then place will be in a place, ad infinitum. The atomistic position allows us to block this regress by understanding void in such a way thatititself

does not exist in the samesense in which the bodily things exist that are in a place. While void in some sense exists, it is not a Being, not

an atom.


With the void of Leucippus and Democritus, we get not only some

spatial notion but something that can be seen atleast as a basis for a full notion of space. The void is, on the one hand, introducedas separating whatis, atoms, It thus guarantees their plurality. On the

38 Here we also seem toget a prefiguration ofthe distinction between space and matter,




atoms can move andthus allows other hand,it is that #2 which the fall it allows for

in which the atomsare, for motion.3? And by being that gis in it, everything can be placed location in the sense that everythin pletely without internal structure, in it. But since the void is com a bodyis ition to define the particular pos it itself does not help us in determine its particularplace is

can occupying; the only way we relation to other bodies.

mere extene, the void seemsto be Asthat in which the atoms mov atoms; the void force or power upon the

sion, without exerting any

re seems

the ions of the bodies. And thus has no influence on the mot show any the at omists do not ly, ing ord acc ed; olv inv ion ict to be nofr we would think tion of motion—as ua in nt co the for t oun acc to d nee They explain 5 Newtonian universe. it to be the case in a frictionles accounted n. Such changes are io mot of ns ctio dire the changes onlyin the void: the ms and th eir motionsin ato the of up set c basi the by for bumpinto each extreme rapidity. and atoms move continuo usly with ed.” s for th e atomsinvolv path new to ead | then ions collis other; these ctions between duct of intera Changes ofdirection are thus the pro different atoms. to be an mistic void seems ato the in est manif space of idea The homogeneous, and iso early and rough predecessor to an infinice, obliged to mists, the Epicure ans, felt tropic space.4! But the later ato iticism s that had i space 42 in order to answer.a cl itICci i otropic ass : ume an anis tely ” Ariscocle and Simplicius repea 29: the atoms “cravel in the void. 39 Cf, eg., DK 68 A3z, line in the void; Simplicius even explte move s atom the that m clai sts’ atomi detestimony ofthe

provi cius’s commentary on De Cuelo 294.33. itly identifies rapos with the void. Cf. Simpli : Leuctjppus and Democritus 40 See Ps-Plutatch, Epitome L4, in C. C. W. Taylor, The Atomists ty of Toronto Press, a Commentary (Toronto: Universi Fragments: A Text and Translation with the atoms can also stay together after a collision, 1999), testimonium 79a.If their shape allows, until some other collision separates them. with the whirls when a world comes into 41 Anyfurther structure, any anisotropy comes in only existence.

opic space” which seems to be introduced with the atomistic notion of the voi d, “seems to have been foo abstract even for the theoretically minded atomists,” he nevertheless points our how via

42 While Jammer, Concepts, 11-13 thinks thar “the idea of a continuous hom geneousand isotr

picurus and Lucretius we get to exactly such an idea, choughthen spaceis takento be anisotropic,


been prominently raised by Aristotle in his Metaphysics. Aristotle there complains that the atomists had not explained why the atoms move in thefirst place rather than stand still.49 Epicurus answers this by claiming that the atoms naturally move downward due to their weight.44 While Epicurus understandsspacestill as homogeneous,it is not isotropic any longer; it has the same structure everywhere and is not curved, but it does not display the same characteristics in all directions because the downward direction is distinguished vis-d-vis the other directions: it is the preferred direction for the motions of the atoms, This anisotropy of space, however, raises the problem that nowthereis no reason left why atomsshould bumpagainst each other rather thanjustall fall downin straightlines, The introduction of the swerve is meantto ensure that the atomsstill bump into each other and the phenomenaluniverse remains in motion.45 It seems that Leucippus’s and Democritus’s notion of the void is

indeeda very fruitful, evenif notfully developed, notion ofspace.46 Andit allows the atomist to distinguish clearly the void qua the empty

from the full qua atoms, and thus to pave the way for distinguishing matter and space.In the next section wewill see that Plato prepares the groundfordistinguishing time and space not simply by neglecting time, as we saw Zenodo,butby giving both, time and space, their own


“ . 43 “The questionof the origin and nature of motionion in in things thi they [the atomi omis blithely as the others” (Metaphysics 985b, lines 19-20),




ts] foo ignored ustas

44 ve have conflicting reports for the Presocratic atomists’ assumption of weight: Aetius seemst . foe : t ‘ ‘ : ° c sim that itis only Epicurus who introduces weight asa characteristic ofthe atoms, while Aristotle an his commentarors ascribe weight to Presocratic atoms, But in anycase, the pre-Epicurean + ' > ° . atomists do not seemto employ weight in order to accountfor the motions ofthe home, 45 Cf Lucretius, De rerumnatura 2.225~50,

46 For a discussion of why Aristotle does not consider the Presocratic atomists to be predecessorsfor his investigation of space and David Sedley’s claim that we should not understand their void as a

notionof space, see my unpublished book manuseript “Ancient Notions ofSpace,’ chapter3 .





3.1 Plato’s Timaeus

the late Plato7 of phe In order to fully understand our se nsible world, ical pringjple Timaeus claims, we need to introduce a th ird metaphys :

in addition to Being and Becoming( 52a) known from earlier Platonic

1, That in whicha sensible thing as an image of a Form appears can neither be the imageitself, nor theintelligible paradigm. It has to be something over and above the Formand the thing that has comeinto being,a third kind: the receptacle. The Formguarantees that a sensible thingis the particular thing it is, for example,fire, becauseit is an image of the Form offire.

Butthe receptacleis responsible for the thing to be a sensible

Academy texts, what hecalls the receptacle. From the time of Plato's

thing,sinceit is once the image of the Formappearsin the receptacle thatit is the sensible thing we perceive,

ofboth, fn understood as space or as matter or as showing features has beenthe prefeyred the twentieth and twenty-first century space

is actually understood option, even if it is usually notclarified what

as some king of by space. Nevertheless, understanding the receptacle Plato cot 52a in space seems a secure interpretation given that time can be ine ced calls the receptacle chdra. While a notion of Within the metaphysical framework we are used to from the mile

Plato, a notion of space, the receptacle, requires a developmentof the : Whole metaphysical basis. But why is the receptacle so fundamental for the structure ° the

World? According to Plato’s Timaeus, the sensible things are copies of need somethingjy images of intelligible Forms. But as images they

more, the which they can appear, andthis is the receptacle.*” Further the explain to helps cle World is moving and changing and the recepta n es. es Finally, in spice of all the process taking initiatio of these process Place, there is also some stability within chis sensible world, which allows Us to understand the world, and the receptacleis also involved

this task, Let me briefly explain how the receptacle is meant to fulfill e

ch of these three tasks.



The Timaeus calls the sensible things images (eikdn, 5202), appearances (phantasma, 52¢3), or aes (mimémata, socs) of the Forms. And in s2cq Plato makes it clear that the images and PPSatances need somethingiz which to appear.


le shouly pe onward, there have been debates whether the receptac

According to the Timaeus, the cosmosis set up by a divine

craftsman: in the beginningthere are traces of the four elements,fire, air, water and earth, that appearin the receptacle and movein a disorderly way; the demiurge bestows order ontothese traces, shapes them into geometrically structured elements andall there is into the dodecahedron ofthe worldbody, and formsthe well-ordered cosmos in such a waythatit

itself is an image of the eternal paradigm. Thereceptacle fulfills crucial functions both before and after this “creation” of the cosmos. Before the creation,itis

centrally involved in the initial motion of the elementaltraces. Theinitial motion in the physical world is caused by the uneven powersofthe traces in the receptacle: because oftheir unevennessthe traces movethereceptacle and are movedbyit in turn; thus the receptacleis itself moved and causes motion

(52e-53a). Furthermore, while the inirial motion of the traces and the receptacleis chaotic,it leads, nevertheless, to some sorting.

This is why Plato comparesit to the motions of a winnowing basket: shaking the traces, we get the heavy and dense parts on oneside and the light and rare ones on anotherside. The immediate effectis a first spatial order:fire traces come to be

here, water traces there.





into eachother,that in 3. While the elements constantly change stable. In this way the which they appear, the receptacle, stays

is a stable “this,” as Platocalls receptacle ensures some stability. It ndin which the it, by being that in which something movesa

€7~-50a2) and by ensuring elements change into each o ther (49 processes to take place. the continuous possibility for these

different functions is 4 What allows the recep tacle to fulfill these

: rather peculiar combination of features

se shapes that the things 1. The receptacleis free of all tho . This allows it to appearingin it possess; it is amorphous ving saw that to be all-recei receive all elements equ ally well; we 49 core functions. (pandeches, 5148) is one ofits edin (anoraton, s1a7) and inde 2 The receptacle is itself not visible

is important, because no way perceptible (s2bz). This


and hence possess a certain it would itself be a sensible thing, the basis

is meant to be is rather form, while what the receptacle being and which thus in which the sensible things come into

enables them to be perceptible.


as some seems to be trea red 3. However, the receptacle lf moved. We some degree, since i tis itse physical and sensible to ain the initial motion. saw that this is necessary in order to expl Form-coptes it can allow the Furthermore, this seems to be how


: Plato's “A Likely Account ofNecessity ophy 50 e points, see my paper ilos thre ofPh ory ese Hist ofth e on ofth al ussi urn disc r fulle “8 na a tacl ical Basis of Space,”Jo r ecep e as a Physical and Metaphys (2012): 159-95. ents and itsel fis not one ofthe four elem er that we need something thatsee wich Anaximandfour ms to be in the backgroundalso with 49 we saw This idea lly. equa ents elem all co rise give °us can th since we are told that in order for ¢ receptacle to allowfor all rie account of the receptacle, . ' ; have any of the features of not fcan c) be in it, the receptacle itsel the clement into being and to

i. cle is nothing but what appears in it; ontolog cally, howoventhehenomenologically the recepta ion for the existence of the sensible world.

$0 We can say th


receptacle is also a necessary condit


to becomesensible things: the receptacle guarantees their physical existence by beingitself physical to some degree. 4. Finally, while the receptacle is also treated as something physical, it is, first and foremost, a metaphysical genus and as sucheternal,

In order to understand the spatial characteristics that this entit and its content provide us with in the Timaeus, we should take into accountthat this part of the Timaeus is not concerned with any old sensible ching but actually with the elements of these sensible thin s For Plato the four elementsofthe philosophicaltradition, earth water air, andfire, are constituted by geometrical bodies; they are cetrahedra (fire), octahedra (air), icosahedra (water), and cubes (earth). We see

howthe physical and the geometrical world are closely linked Each of the geometrical bodies is made up of one of two basic kind of triangles. Thus for Plato three-dimensional bodies are composed ou of two-dimensional surfaces by moving the surfaces and chan in he angles between them. An imageoffire, for example, can a — in “he

receptacle because (1) there are basic triangles in the receptacle


(2) they can be arrangedso as to form an imageofthe Fotmof fre, If we now look at the spatial features wefind in the Timaeus we that in fact most of them do not comefromthereceptacle but rather from its content, the elements and their traces. The latter present us

with three spatial aspects: (1) regionalization, (2) distance and (3) ex-

tension, which implies a certain dimensionality. It is onk (4) cont nuity thatis actually provided by thereceptacle, Let us! , : h | these features in turn. nslookareach of 1, As the shaking motionof the receptacle leads to a basicbased ord r wealso finda regionalization of the traces like tolike, th ‘are here,regions Basic (53a). together elements gather of the of fire thedistribution on as the traces established earth there.


ere, those of



ga notion of >. The traces and elements also allow for introducin re traces gather distance. We maysay, for example, that thefi he traces together near the traces of air and further away fromt would be ofwater. A rough way to de termine such distances

traces: twotraces are close with the help of the amount of the

ween, andthey are far away to each otherif only a few are in bet

es. With the traces we can if they are separated bya lot of trac

, not the exact quantity of determine only degrees of nearness because they are ofirregular the distances between the traces,

, when the elements possess their shape. However, after creation

ld be used for measuring full geometrical structure, they cou e be

triangle couldin principl _ distances in the universe: one s to e out whatever distance need taken as a basic unit to measur

be determined.

siders the elements and 3. As for dimensionality, Plato con ee explicitly claims they possess thr

their traces as bodies and

of two-dimensional triangles. dimensions,*»! made up, however,

is amorphous; it does not By contrast, the receptacle on its own ic would keep permanently. possess any determined form that elements andtraces Butit receives either three-dimensional ordingly,it must have the or two-dimensional triangles. Acc needs to potentiality for dimensionality, but this potentiality be actualized byits content. for the 4. The spatial feature that we can c Jaim as specific n different ee receptacle is the continuity it guaran tees betw a body so there are no gap s between,say, surfaces and bodies;




basic features of any space, dimensionality, which it inherits from the things in it. And the idea of'a metric cannotbe based onthereceptacle considered on its own, as we saw. Anyfurther spatial structureis also derived from its content.

But why, then, does Plato call it space (chéra)? The reason for this seems to be that the receptacle is the continuous and constant “this”

in which everything in the sensible world comesto be, exists, and can moveand change.Accordingly, we can say that on its own thereceptacle provides the basisfor space by granting this continuity, while all further spatial aspects come in throughthetraces of the elements. However, wegeta fully fledged notion ofspace in the Timaeus ifwe considernot only the receptacle butalso the world-bodyin the form of a dodecahedronthat the demiurge forms, and the contentofthe receptacle together. The universe, as madeupofthese three ingredients introduces a full geometrical space: it has three dimensions, is fully compressed and connected—thereare nodisjoint patches—and sso closed and bounded.®The continuity between the elementsis secured

by the receptacle, and the basic triangles allow for the possibility ofdetermining the distances between elements, and thusfor a basic metric

Accordingly, we can say that we do find what we would call a geometrical space in Plato.™ It is a space that is not independent ofthe bodily elements but also not simply reducible to them and their relations, And this geometrical spaceis also open to a physical inter-

pretation: the geometrical structure of the elements explains how the elements can be transformed and where they moveto, and in this wa explains the changes and locomotions ofthe elements, ’

here and a bodythere. tacle as space, Given these features, should we understand the recep on either geometrical or physical space? Probably not. The receptacle its own cannot be a geometrical space, asit misses one of the most 51

1 3c, s c. Asa consequence, the receptacle has to allow forat least three dimensions. : :

52 In addition we might night countasspatial . features of the receptacle before creation ‘ the e factface th thar, ‘ as we haveseen,it whic eae is itself movable and can in turn move—features ofiinterest fefor a possible sible 53 We© could coul even claimthatit is orientable with respect to the basic elements as the can h ha

orientation (e.g., they can be right-handed) once the order of the universe is given, , _ 54 More specifically, it is what we wouldcall a Euclideanspace, since the difference between the si of the triangles matters, as does the difference between the twobasic 3 triangles angles. “meee




Plato's notion oftimg; How doesthis notion of space nowrelate to

basis for space, and time are com. In the Timaeus, the receptacle, as the

is uncreated and essentially pletely separate entities: the receptacle e creation, while time j, connected with the chaotic motions befor nal order onto the worly. created by the demiurge so as to bestowratio

completely different onto. Time and space are thus situated on two ofthe face

logical levels. And this gap cannot be

bridg ed with thehelp

timeis e sense with mat hematics. For that both are connected in som as we sa\y, portion theory, while, connected with arithmeti c and pro in the metry. It remains unclear the receptacle is connected with geo when

be connected,for Timaeus how the two could possibly we want to give an account of speed.


parts of the body that occupyit are continuous.Butin the Categories weget no indication which role zopos plays for an account of motion, since motion is obviously not a focus in this work. But motionis, as we will see, the crucial focus for the discussion ofspace in the Physics where Aristotle gives his fullest discussion of spatial notions. Aristotle usvall uses the word sopos in this discussion, which somepeople took as an indi cation thatheis nottalking about space but only about place. However, we saw earlier that the wordin itself does not necessarily indicate that Solet us look at the main tasks that ¢opos is meantto fulfill in the Physics: 1. Toposis a necessary condition for locomotionto take place and for our understanding of locomotion: there needsto be


a “where;’® or a “where from” and a “where to”in order for

a body to changeits position, for mutualreplacementto take


place, and for us to be able to accountfor such a locomotion.”


2. Furthermore, topos is connected to some formof power, as . each ofthe elementshasits specific place to which it natural moves, for example,fire in its natural motion always moves i‘ earth always down. ° Finally, the continuity of space is meant to guarantee the continuity of locomotion and time: at least in the order of understanding, the continuity of motion and timerests on the continuity of space.>?

4.1 Aristotle

d either as not clearly distinct So far we haveseen ti me and space treate irely different


ey possess an ent or as separate to such a degre e that th ted how they may be connec ontological status and thatit is u nelear these otle succeeds in bringing at all. Before we examine how far Arist of lp he © £ motion with the two magnitudes together for an account on of space we will first loo ke at his noti his notion of the continuum, in somederail.

POS 4.1.1 THE TASK OF ARISTOTLE’S NO TION OF TO stotle already That spatial notions play an impor tant role for Ari ” as one of the becomesclear from his Categories, which names “where ers, is ten categories. The category of “where,” however, like some oth amentioned only briefly and exemplified solely by re ference to the loc tion of something (“in the marketplace,’ “in the Lyceum”). Later on we

find topos amongthe continuous quantities; it is continuoussince the

55 Interestingly, this order is inverse to the ordet we




paopoion Topos: The DevelopmentofArisotlesConcepratBaepe ‘ab ee forte idea of a clear development in Aristotle’s spatial account from the Care ri ?: Piguln act that toposis discussed under the category of quantity rather th

fh ne . mete

question of how zopos is related to the idea of locationin the Categories ofalse AMS rises 56 Infact, we need a “where” also whenan objectis atrest, thatis, without eh . ee ae 57 Sce 208b1—8, Space thusserves as a metaphysical and epistemological bai roan


8 These




acc s, » topos ing to thsome account ntly inwhile the: traditio preted differe ; to others, beeninter itselfe.owers: have accordin:ngaccord o attract, powert i some possessing itself is seenas ts in the powerres the ers, g elements.

59 See Physics z19a10-19 and book6.



These tasks show . that the main focus of Aris: totl as igs-—as we should ex igat‘ion Is e’s investig pect from his ve ‘on. r Y project in the W

» While Aristotl e argues aSainst topos being thoug htof as a

ee tocomo e would not as k what toposi Sif there were to Aristotle, Th no locomotion, accor us, what Aris totle we is interested in is clearly would think of what a5 physical, no tas mathematical space. 41.2 DEVELOP MENT OF ARISTO TLp’s ACCOU According to Ar NT istotle, the th re e tasks space (1) notitself a bo fulfills dy, but that (2 ) Spaceis also not c mean that PN from bodies in ompletely een the waya void is, and th

body or a feature ofthe b ‘tgue thatit

of his theory o


in the sense of that . whic h has ap rine ok 2, : kind of limit; (2)either form nor matter: (1) itit isis that ie

can also be inde pendentofthat be independen ught of as “surrounding” or“ t of the bod: limiting * Matter, on the ot Y in the sense thar Aristotle un he it can ae d r ha it nd is , ca it se nn lf limited, In Salso arSu (2o9br1-45), guiing ap arguing agains ot be thoughta r toposbeing Atistotle’s topo “My article “A yj ely Ac ainse pi lato’s Conceptionof me coun the s an Mistake space fo d Pp lato’s Teceptacle sh t of' Scessity” for a discus fecepracle in the Tima are basic func : sion of the degr r Matter, tions and wh ee to whic y, neverthele ss, Plato do es n ot

nt of n° ‘t



(215211), wo

uld thre

heisc nceptof a co inuatuemn. tTh ntin cal and a dynamicalside: from a mere logiical Point of view, the assu Mptio. 0 V W t aps


erse Ww ould be

n of the dif



: ferent Part uld not “nsured any s of the un be iverse wo longer,

depends, Asfa dynami is concerned, the assu tion of avoid woulrd as mp make ir inics explicable how an impulse can be Passed on within the Aristotelian framework, if th “uttounding body ere is n° ad that takes over the im pu ls e in th e voi ba OW bodies can ke ep a certain dire ct io n nd given that the void ig completely in their motion. hat an e renciated so We do nop have any ° resources to distin gu there coydt is h di ff er en tp laces, hen be No na tural moti;o n. For natur Notions to al m otions are a Specifi ¢ plac as e, , for exampl the m ion of to

her one he Id » Or another one wil] hold in the Similar proble ms Would arise if we as an independen sumedtopos to be tly existing 1 Ne rval betw Cont

the Center



tbodt . Fo r

tands ‘ v to e i

threat has a logi

future, §

ae 60 Inthe Phys ics, elple of motig

at there is n

f kinésis, the co

ions as itis nota body : if topos were a body, thete would be two bodies in th es AME spot,topos and the bodythat topos, Furtherm is in this ore, ofa body, si

een the extrem ities ofa ll Sadiastém,

indep ende

ch Aristotle unders

undifferenti ated nonbei ng

s has the same dimens

Aristotle ca

be the case th

ns, he also s

o body assumption of aunvoavidoidable tothink of itas ain it, an :d. The kind , whi

hat while topo

ainer, what


Would makeit


lation. -

ody it contai

cannot be c

then it could

at (3) the relation ship be

fa part-whole re


Finally, Ar


of the worl d,


otion o earth

quite in detail marks off the relation bee en

eyand its Place fromthe relation ith both telations One relatum

ofa part to its who he

seems to be in the other, e

Part in the whol e, the body in its place. Howe ver, as ne

Points Out, a part always moves with Move a body, while a bo om s in a place (211a29~ bs), The place is independentfr



is not the body which is in it.5 The notion of“in”atplay here in which the one in which the partis contained in the whole or ing to the the whole consists in its parts. Rather, the “in” referr we when e relation of a bodytoits toposis the same “in” weus say “somethingis iv a vessel.”°



T TOD TepleyovTos Tépas dxlyntov npHTOoY, Todt’ zotw6 toT0¢ The first/immediate unmoved limit of that which surrounds—that

is topos, (212a20-21) Thethreepieces ofdefinition ofplace,(1) “limitofthat which surrounds,” (2) “first” or “immediate,” and (3) “unmoved,” Aristotle elucidates im-

on for the relationship This understanding of “in” is the right noti as Aristotle spells outin between a body andits ¢opas. It is, however, not work, detail, not the only one, and so Zeno’s paradox of topos does one of the other senses of since topos itself can be “in something” in Aristotle chinks that ve “in” without thus creating an infinite regress. in one, the wouldget into a paradox onlyif “in” could merely be used s a mark of the spatial sense, because only then wouldi tbe true that it’ ce. Thus Aristotle existence of everything, even ofplace, to be i na pla that “whateveris rejects Zeno’s second premisein his paradox of topos, in somethingis in a place.”

4.1.3 DEFINITION OF TOPOS AND ITS TRADITIONAL UNDERSTANDING as matter, After carefully ruling out that ‘opos could be un derstood ies of the thing form,or an extension (diastéma) between the boundar as which surrounds,® the following definition remains for Aristotle

the only possibility:

mediately before this definition. By claiming topos to be like “a vessel which cannot be moved around”in 212a15-16, Aristotle makes it obvious how to understand “the limit of that which surrounds”: in the

. same way as the (inner) edge of a vessel forms the boundary of what it contains, Zopos has to be thoughtof as a surrounding limit or frame.

Andasthevesselis able to take up anything (bodily) whatsoever, there can be anything whatsoeverin a place. Topos is the limit not of the body thatis surrounded, but of the body which surrounds. Topos being a limit has twoattributes: it is to be unmoved andit is proton, “first/immediate.” Thelast is usually taken to mean something like “being directly adjacentto.” This attribute seems to guarantee the continuity in the universe andto prevent any assumption ofa void between a body andits place. In addition,topos has to be unmoved.This requirement seemsto be necessary for tworeasons: first, ifa bodyis in place 4 whichis itself moving in place B, then wewill only be able to locate the bodyifwereferto B, but notifwerefer to.4. Second,iftopos itself was moved, we would have to state where this place moves and thus would getinto a Zeno-like paradox of topos,according to which

each place has to have a placeitself, which again will have a place, ad infinitum.

Traditionally the definition of topos has been interpreted in such part, while if it ts isa a part, it 1s wner, i container, sh irs : : j is immediately contained 1f ich iiss contained with continuous is 65 If that which

: contact with ‘ the container, vaca : a place; ef. Ross, ristorlesPiHoete ‘etotle! ics, :For a Ppart ofa onlyin it is a body in arts continuum to move én its whole, it would need to be actualized, and there are inhinitely many p that could be actualized.

66 But once an clementis in its natural place, it behaves like a pare thatis separable behavesto its

a way that fopos is understood as an unmoved vessel. Since Aristotle talks about a prézon limit, the focus has been on a vessel that contains nothing butthesingle body assumedto bein the ¢opos. Although many

whole. (On the notion of a naturalplace, see later discussion.)

67 See 211bs~212a2.For an explanation ofwhyit is exactly these three possibilities which arise as possible candidatesfor place and have to be ruled out, see Mendell, Zopo#, 219 fF, Aristotle's rejection of

understanding topos as diastéma can in part be seen as an attack on atomistic assumptionsofa void,

- 68 Aristotle refers to this paradox in 209223~25.




ofhis treatis scholars of Aristotle’s Physics take it that in the course

of on place there is a shift to a morestatic definition than the image ng thay a vessel allows for, this does not prevent them from assumi of , ner contai Aristotle’s notion of zopos should be understood as the body, and moreprecisely the container whichis of the very sameext


sion as the bodyin it.

consequences, which This understanding has two problematic with sufficiently: first, Aristotle, it is usually claimed, does not deal not with stich an accountprovides us 0 nly with a theory ofplace but obvious from one of space—a shortcoming that seem s to be already of chéra to refer Aristotle’s choice of words when heuses fopos instead investigation. Second, itis probl ematic how such q

to the object ofhis

h Aristotle's focus on kinésis, definition ofplace can be reconciled wit dwill nor because a vessel of exactly the same size as the thing containe initial idea that allow for any motion. So has Aristotle given up on the help to explain mot ion? I do not thinkso, .




his account oftopos should rotle andits probA In contrast to the traditional understanding of Aris; . following thathis defin lematic consequences, I want to show in the ceive tion oftopos fits his conception of kinésis, since he does not con ld call a as being an unmoved vessel, but rather as what we cou


his theory of frame ofreference. As a consequence, we wi Ll see that

topos is not only a theory ofplace butalso of space. t happensifthe The secondary literature has been worried about wha can do th atina surrounded body moves, since it is not clear howit vessel of exactly its own size. By contrast, what Aristotle worries about

in his investigation is what happens if the surrounding bodyis moving. Lookingatthis case in the following section will makeit clear what the core of Aristotle’s notion ofoposin factis.


4.1.4 REVISED UNDERSTANDING WITH THE HELP OF THE RIVER EXAMPLE Let us first have a closer look at Aristotle’s definition of topos.”° We have seen that topos as the limit of that which surrounds has two attributes: it is to be “unmoved”anditispréton,“first/immediate.” We sawthatthelast is usually taken to mean somethinglike being “directly adjacent to.” Taking it like that, “unmoved” and “first/immediate” exclude each other if the thing which is immediately surroundingis itselfmoved.”' For then the thing whichis directly adjoiningis not unmoved, and whatis unmovedis notdirectly adjoining. To understand these two pieces of the definition better, let us have a look at how Aristotle continues immediately after his reference to the vessel in 212a16-20: “So when something movesinside something which is moving and the thing inside moves about,as a boat ona river, the surroundingthing functionsfor it as a vessel rather than asa place; _ place is meant to be motionless so thatit is rather the whole river that is topos, because as a wholeit is motionless.” In this example of a boat

on a river the two features “unmoved” and “immediate” seem to exclude each other. So what does Aristotle do? Heuses this example to

pointoutthattheplace of a body need not be immediately adjacent in the way theside of a wall of a container directly touches the contained thing. Place ratheris the limit ofthefirst thing that is unmoved. (We can think ofit as an adjacentlimit insofaras it is not separated by any discontinuous gap from the thing moved.) Thus, place is not (necessarily) something which directly touches the thing moved, but rather the nearest thing resting, as the river is in its (moreorless) unchanging extension.Placeis theprdtonlimit ofwhat surroundsa body,notin the sense of immediate but in the sense ofthefrst thing that is unmoved. We are in fact not dealing with two different features, unmoved and

69 Cf,eg., Myles Burnyear,“The Sceptic in His Place and Time,” in Philosophy in History, ed. Richard Rorty, Jerome Schneewind, and Quentin Skinner (Cambridge, UK: Cambridge University Press, 1984), 232015: “The refinement [i.c., thar Aristotle is now taking the surrounding body to be static] does nor threaten the condition that the place of X is equal to X (211a28-9) and contains nothing bur X.”

70 This section is heavily influenced by Ulrich Bergmann, Unverdffentlichte Studie zum Aristotelischen Ort (Fassung Minden 2004).

71 Cf, eg. Ross, Aristotle's Physics, 5756.




first, but with the feature “the first unmoved”limit. Accordinglytap, should not be thought ofas an unmoved vessel; this is just afist pass of Aristotle when he develops his notion of topos in the Physics, and the

sopos. Rather, topos iyy Passage just quoted clearly contrasts vessel and testing limit—a frame ofreference—within which the bodyis situatyd


Asplace is taken to be the area iz which something moves and notasa point to which movementtakesplace,topos itselfcan in turn ensure the continuity of the movement. But why, then, does Aristotle choose the term ¢opos for such a frame of reference? We saw earlier that in contrast to chéra, topos can also

can be made v0 and which, under certain, but not all, circumstances,

be used to denote relative location or position in relation to a sur-

also explains why Aristotle at the oftopos as such a frame ofreference

this body in relation to another body, to the surrounding body, this term seemsto have been appropriately chosen.” The focus on the imageof a vessel seems to have led to the opinion

Precise as to contain only the one bodyin question. The interpretatin

end ofhis treatise on topos can suddenly say that sopes can bea king

ing or the mote of surface (21228). For in order to locate somethtances, ° er

of Something, it is enough, under normal circums two dimensions. (“Whereis he sailing today?” “In the Nile de tr ' is

answer need not refer to depth; it would also be understandable if we

lived in a two-dimensional world.)


itself a certain Place is this resting thing not insofar as it is

> body,

°r example, the river, but rather insofar a s it forms the boundary aryof a.

rounding, Given thatfor Aristotle the topos of a body always positions

shared by most Aristotle scholars that Aristotle’s Physics develops only a theory of place but not ofspace. Accordingly, it is claimed that Aristotle’s theory lacks the possibility of accountingfor directions,for the relations of positions, and so on.”> However, Aristotle’s example

of the ship shows thatall these determinations can be derived from his conception ofsopos as the nearest resting system ofreference: loca-

Thus the resting thing Certain atea in which the movement takes place. a certain area. In this way rms the limit in the sense of delimiting es Atistotle conceiv of topos as the nearest resting frame of reference , the Cartesian co. Ut in contrast ro modern frames, as, for example

tion is possible with the help of the nearest resting thing,and,if appro-

i.e. as given by the rth Atistotle is always thought to be physical,

Therelationofpositions can be established, for example,by therelation of two boats on theriver, Directions can be defined with thehelp ofthe relation ofthe place of a body to anotherplace, for example, the ship

place asa frame of reference Ordinate system, which are purely logical,

that for Aristotle ‘dies of this world. The reason forthis restriction is over and above bodies.” “Piss is, Ontologically speaking, nothing oses the physicaeen of hens definition of sopos presupp ; and he since without this continuity the ram € thing moved in this frame could not be directly connected.

As th

te ‘ . i ro be a, bo ne ate no further spatial entities with Aristotle over and above bodies, place has being cing’ounding g the moving g body.body. Nevertheless, its being ga a body body is noc whatis important for it

73 Borge ta


- the one'y to the objection that the passages in the Physics which take place to be the same size as Sible seaunded thing seem to make the interpretationofplace as such a frameofreference impos*tmy book manuscript “Ancient NotionsofSpace.” chapter5.

priate, ic allows for a further tightening of the frame. For example,a ship is on theriver, near Cairo, wherethere is the landing stage, and so

on. (Going further in that way wewill be able to distinguish the loca-

tion ofa boat on theriver from the location of any other boat nearby.)

74 WhenAristotle gives his ownview concerning place as the location of a bodyrelative to the imme-

diate surrounding, he uses the term topos; he uses chdra and topos interchangeably in places where he talks about opponents. 75 To take just one example, Wagnerin his commentary on the Physics, Aristoteles, Physikvorlesung (Akademie Verlag, 1995) thinks that with Aristotle the principles which are essential for a theory ofspace are missing: “Gre”(size), “Richtung” (dicection), “Lage” (position), “Lageverhialenisse” {relations of positions), and “Ortsbestimmung” (tocation) ofspatial things. Jammer, Concepts, 19 claimsthat Aristotle only develops a theory ofposition in space. Cf also Ross, Aristotle’s Physics, 54, and Hussey, Aristotle's Physics IH and IV, xxviii, for the claim thar Aristotle does not have a notion ofspace.




e moves toward the moyrh may be on theriver at Cairo and from ther

and Aristotle's thoughts of the river. Thus, the example of the ship fe *


fopos different things share—ma abouta topos koinos—the common. pace,’ also a theory ofspace. it obvious that his theory of toposis




Fsote a whole? Can it, according to But what about the world as ny a place or space? No. For OB accountof topos, itself be in 32 ee en

yis in a /opos (212031that which is surrounded by a bod e thee lf not be ina place or space sinc the universe as a whole can itse Ns a whole. Aristot surround the world . aw. could that body further no is e heavens as “But, as has beensaid, the is explicit about this: body , no e reisis j . . orin a Z0po5, since there [i.e., the universe] is not som where adition ).”” In the philosophica -10 2b8 (21 ” em th nds rou sur hat e's oe Aristtotl nt “Bi ins io at s aga

tae ack i ns and hezavy att thi cl iim has led to confusio thii ss cla by s most viviviidly expres sed of the universe? Either h out m y am at the edge stretc I nsif happe e s f si en there to happ e must be something I cannot do so, which means th at ther has to be in which case my arm prevent me from doing this, or I can, mixes up dif. er, this question somewhere and thus ina place.””® Howev e’s ;notion ofthe limit of ns of space: it confuses Aristot] ferent notio t ity thar ofspace as an 1n dependen ent surrounding body with a notion j nded“ or moving.It is as ythii ng bein ing g e exte is a necessary condit itiion for an

. i in order to ocate some. ifwe employed a Cartesian coordinate system is i you r coordinate system thing and then somebodyasked, “ Where in the coordinate system situated? e



So far we have dealt with topos as a freely eligible, although bodybound, frameofreference in which a body can movein any direction. By contrast, what has cometo becalled “natural places””? are fixedly given destinations of a movement. Theyare the places which the four elements will move to andrest if not hindered by someexternalforce, and they thus determine the direction the motion of the elementswill take, All elements,all simple bodies, have the potentiality to perform a natural motion,i.e., a motion thatis not caused by somethingexternal, and their natural motion will be to their natural place: earth will move to the center of the universe,fire will move up to the circumference, and so on, This idea of natural places goes hand in hand with the idea of a fixed matrix of the universe; there is an objective above and below. For us human beings whatis above, below,etc. depends on howweposition ourselves. But in the universe above is wherefire will move to naturally, below where earth movesto, and so on.8° Accordingly, the universe as

a whole can bedescribed in its setup as follows: In the middle there is a sphere of earth; this is surrounded by a water sphere, then a sphere ofair, and a sphereoffire. Beyondthe sphere offire, and that means

beyond the moon,thereis a sphere of aether, out ofwhich thestars are formed. Thisidea of fixed natural places notonly introduces what we maycall anisotropy into Aristotle’ notion of topos; it also gives an objective otientation in the universe and thus enhances the knowability of the spatial structure of the world.

24 that his account oftopos 76 This interpretation is also supported by Aristotle’s claim In 212 b23h would notbe possible on ensures that sopos does not have to grow if the thing in it grows, whic ; init, ere with the thing rywh eve contact in be to has topos where on, terpretati the traditionalin

body there can be neither sopos “97 Cf. also De Caelo 279a, where we are told that beyond the world nortime orvoid, since they all depend onbodies. framed as whether 78 CF£, for example, Lucretius, De rerum natura, 1 963~83, where the questionis we can throw a spearat the (alleged) edge of the universe.

79 Axistotle himself only talks aboutthe édias or oike‘os place—theplace belonging to some element, or that in which the elementsrest naturally (276224), or about“its place (autou topos) to whichit movesif nothing hinders it” (208b1t),

80 See Physics 208b14-22 and De CueloII, 2, These directionsare objectively givenfor Aristotle, By contrast, in Plato’s Timaeus above, below,right, left, back, and frontare alwaysrelative co a certain perspective—either our own or oneofthe elements, so thatfor, let us say, fire to move downwill always be to moveto wherever thereis lots offire, but the place where there is lots offire could vary. See Timacus 63e.




4.16 COMBINING SPATIAL AND TEMPORAL MAGNITUDES ForAristotle,time and spaceare clearly distinguished magnitudes; space is importantonly for locomotion, while time is connected with all kinds ofprocesses (also,for example, alteration and generation). However, ifwe wantto give an account of motion, wealso need to bring these twodifferent magnitudes, time and space, together; we wantto beable to claim that a certain body covers a certain amount ofspacein a certain time, of that another body wasfaster than this first body becauseit covered the same spacein less time orit covered more space in the sametime. We see thatin order to account for motion andits speed, we need to be able to

bring time and spaceina certain relation to each other andtodivide time

_and space with respect to each other. The way Aristotle can explain this connection of time and spaceis by showing that whateverelse we can say about time and space,their

basic structure as magnitudes is the same: both can be understood as continua; ie., both are magnitudes that are always further divisible. Aristotle spends most of book 6 of his Physics showing that time,

space, and motion are continua and how we can understand such

continua: they are wholes that are prior to their parts so that (potential) parts are acquired only by dividing the whole; they possess potentially infinitely manyparts, notall of which can be actualized at the

same time; and their limits are such that we have to distinguish between division points orinnerlimits, which mark off potentialparts, andouter limits, which mark offone continuous thing fromanother.®! This understanding of time and space guarantees that we can measure motion in terms of time and space, as we see it done in

Aristotle’s comparisons of two locomotionsofdifferent speed.It also allows Aristotle to respond to Zeno’s runner paradoxes: it explains how a finite whole can contain infinitely many parts and how an

81 Fora detailed discussion ofAristotle’s accountofcontinuity, cf. my book The Concept of Motion in Ancient Greek Thought: foundations in logic, method, and mathematics (Cambridge, UK:

Cambridge University Press, 2020)



infinite numberofspatial pieces can be covered in a finite sie. And while both time and space share the structure of continua, this does

not mean that space does notalso possess its own, specific features; while time and space can be combined,they are also magnitudes independent of each other.



as merely a stage or a container where things happen.Instead, geographers today commonly understand space as anactive agentin social, economic,cultural, and political phenomena. Atthe sametime,


they emphasize its constant production through practices and thereby its incompleteness and fluidity. Doreen Massey, for example, theorizes space as an ongoing process, always in the making,and inevirably


Veiling as an Embodied Spatial Practice

plural. She proposes that we imagine space “as a simultaneity of

Banu Gokartksel

stories-so-far.” Her approach emphasizes thevolatility ofspace rather thanits fixity; the boundaries ofspace are notrigid (despitepolitical actionsto imposerigidity by, for example, building up and militarizing


borders) but changing and porous,and therelations that produce or

are producedby space’stretch out geographically and temporally.!




reveal aboutthis p What would a spatial approach to veiling roduce anew . . a space and P Howdoes veiling create the body as

Feminist geographers have taken this view ofspace to destabilize the binaries that dominate geographical thinking, suchasplace versus space, private versus public, and localversusglobal. In such binary


constructions,place, private, and the local are rendered feminine and


particular and studied mainly by those interested in grounded and ~

e these ese q space that the veiled women traverse? L explor

situated research. In contrast, the approach to space, public, and the global continuethe tradition of disembodied, supposedly rational,

disciplines about spaceas a relation of im womens veiling as bodily experiential practices. Lapproachh Musl ates in the production of embodied spatial practice that particip

and inherently masculinist views ofthe world and their claimsto universality. Sallie Marston and others have added a problematization


ofthe conceptofscale to this body ofwork, calling into question the

© fwomen as they move the body space and shapes the experiences ul, Turkey. In Turkey, across the city. My empirical focus is on Istanb

treatmentofscale (frombodytoglobal) as given and not as a social construction.” Woodward, Jones, and Marston extendthis criticism ofscale to challenge the eagle’s point ofview ofthe world,instead

ve political issue, veiling has turned into quite a significant divisi

regulated by thestate.It especially in the 1990s, andit was actively arf has recently becomerelatively more accepted(e.g. the headsc ees), banis no longer enforced at universities and for state employ butveiling remains a contentious practice.

advocatingfor the fly’s,° Insteadofspace or place,they propose

Doreen Massey, For Space (London: Sage, 2005), 9; Doreen Massey, Space, Place, and Gender

Banu Gokariksel, Bocly Space/City Space In: Space. Edited by: Andrew Janiak, Oxford University Press

(2020), © Oxford University Press. DOI: 10.1093/050/9780199914104.003,0003


Recent theorizations haverejected the treatment ofspace as a fixed entity with clear and stable boundaries, as simply a metaphor, and


(London: John Wiley & Sons, 2013).


Sallie A. Marston, “The Soelal Construction of Scale” Progress in Human Geography 24, no. 2 (2000): 219-423 Sallie A, Marston, John PaulJones, and Keith Woodward, “Human Geography withoutScale,’ Transactions ofthe InstituteofBritish Geographers yo, no, 4 (2005): 416-32.

Keith Woodward, John PaulJones, and Sallie A. Marston, “Of Eagles and Flies: Orientations toward the Site,” Area 42, no. 3 (2010): 271-80,




better captures the actly using the conceptofsite, which they argue are often rendered invisiby, materialities and situated experience,s that res this emphasis on the by dominant ideologies. My approach sha


Some Muslims acceptveilingas a religious requirement. This practiceis often seen as an important componentoftheIslamic code of modesty for women. Writing about otherreligions, a volume that examines clothing,the body, and religion mostly frames the relations amongthese three areas in terms of social control: clothing becomes the meansofregulating and disciplining the body. In western Orientalist works on Islam and the Middle East and North Africa,this aspect ofveiling has dominated the way this practice has been simplistically (mis)understoodhistorically. In western and secular modernist eyes, the veil hides women’s bodies or makes theminvisible in public spaces. Thisinvisibility

ace. material and the experiential dimensions ofsp key concepts ofgeography As part of this critical rethinking of the

ing the body asspace. there is also a recent interest in study. mt closes, y, or the geography Described as the most intimate geograph political,

re social, cultural, in, the bodyis not only the site whe ced but also where they are. and economic processes are experien ersare often credited for . materialized.4 Humanistic geograph as they explore phenomenological putting the body into geography ;“Bod ites, example, ‘Tuan writes, understandings ofplaces. For


In with the sentient body.” d of the bodyis recognize geographies ofreligion,the sign ificance howembodied acts and emphasis hasshifted to understanding raphies.® From this and bodily practices produce religious geog surface’ receptacle or ‘inscriptive perspective, the body is not “a mere but “the eae ofreligion, rse iscou cum-d tionsenta repre of work the for central to the enactment o ae body and bodily practices are an no longer be taken asa given, space.”” Thus, the body can a space comes to the fore. ere question of howit is produced as to the practice ofveiling. I examine this question with regard implicates space; space co -exists

Gender, Cartographies 4 See Gillian Rose, “Geography and

-is seen as a sign of Muslim women’s repression within a modern

secular understanding of freedom thatvaluesvisibility.? In an attempt to counter such views, Lila Abu-Lughod proposes that we viewtheveil as enclosinga private space akin to the private space of the home for women." She takes Papanek’s studyoftheveil as “portable seclusion” to develop the view ofthe veil as a “mobile home.”Theveil as a mobile home aptly demonstrates howthe

practice ofveiling actively creates space. Fromthis perspective, the creation of a secluded private space inside the veil does not so muchrestrict women’s spatial mobility, and hence freedom (from a westernliberal point ofview). On the contrary, veiling enables women’s movementacross the city. Yet this reframing of the veil as mobile homedoes not challenge the

ities,” Progress in Hum an and Corporealities,

assumption abouttheveil as enclosing a space within(asif other

6 Julian Holloway and Oliver Valins, “Placing Religion and Spirieuality in Geography,” Social e Cultural Geography 3, no. 1 (2002): 5~9.

7 Julian Holloway, “Make-believes Spiritual Practice, Embodiment, and Sacred Space,’ Environment and Planning A 35, no. 11 (2003): 1961-74,


Bodies: Exploring Fluid Boundaries (London: Routledge, 2001)s Yi-Fu Tuan, “Humanistic Geography; Annals ofthe Association ofAmerican Geogrrapbers 66, NO.2 (1976): 266-276, cited in Longhurst, Bodies, 15.


$44-483 Felicity J. Callard, “The Body in Theo ade ma

Geography 19, no. 4 (1995): Robyn Long ust, Di) wa ‘ Planning D: Society and Space 16, no. 4 (1998): 387-4003 Geography?’ Gender, Place & Culture 2, no. 1 (1995): 97-106 Robyn Longhurst, i on , ‘c Geographies,” Progress in Human Geography 21, no. 4 (1997): 486-508 Robyn Longhurst,


Linda B. Arthur, Religion, Dress and the Body (Oxford: Berg, 1999). Malek Alloula, The ColonialHarem (Minneapolis: University ofMinnesota Press, 1986). Women Really Need Saving? Anthropological Reflections on d, “Do Muslim Abu-Lugho Lila Others,’ American Anthropologist 104, no. 3 (2002): 783-90. RelativismandIts Cultural Seclusion Abu-Lughod, “Do Muslim Women,’ 785. Also see HannaPapanek, “Purdahin Pakistan: and Modern Occupations for Women,”in Sepanite Worlds, ed. Hanna Papanek and Gail Minaulr

(Columbus, OH; South Asia Books, 1982), 190-216,





gan interig, kinds ofclothing do not do the same) andcreatin

house, Futthe, invisible to the onlo oker,just like the walls of a


ling creates the there is little room for thinking abouthowvei that veiling does create of the body from this perspective. I argue l and not discrete g the body as a space butone thatis rel ationa a porous b oundary . between stable. Veiling, instead, delineates and the public. The yeij the interior and the exterior, the private

in the visual economy ofpublic is also visible and participates

the veil does not complete spaces. Just like any other boundary, that is removed fromsocia] close off an interior or private space

the body, becomes and political processes. Instead it acts upon e endder i relatii on to gen the body in part ofa process that produces duct, and

moral con ideology, social norms, class expectations, rough which the body, religious practice. It be comes the means th

its physicality, as well as emo ti

gulated, ons are assembled and re

d , and certain others avoide certain desires and habits are cultivated ce bodied spatial practi or eliminated. Approaching veiling as an em oduction ofthe body, brings into focus howit participates i n the pr g as an embodied bodily behaviors, and emotions.!3 Viewing veilin e spatial experiences spatial practice allows the examination of th rk ofveiling includes of and through the veiled body. The bodywo

the cultivation ofpiety and modesty through

everyday practices,

Through veiling belief is formed, enacted, and embodied. in Istanbul, whom The case of one of myresearch participants

veiling in I will call Neriman, demonstrates the need for viewing



such terms that emphasize embodiment and spatiality.4 I met Neriman in 2004. She explained to me that she grew up asa nonpracticing, secular Muslim. When she decidedin college to begin wearing a headscarf and modestclothing as part of her newfound Islamic commitment, manythings changedin herlife.

She becamesubject to the headscarf ban that was enforcedat her university at the time. Her family disowned her. Andherrelation to her own body completely shifted. She understood her bodyas not accepting her new mode of being. The most poignantsign of this rejection was her sheddingoflots of hair. She re-created her body spaceby starting to shave her head (which she saw as

a disciplining tool), trying to learn how to walk andsit in skirts (she was used to pants), and watching her body constantlyfor any unwanted exposure, even of her wrists. Thus Neriman had to work on her body andproduceit anew. Her example showsthat theveil is much more than a mobile home orportable seclusion.It does not simply enclose a space within.It certainly is not merely a shell, not only becauseit affects the experience and presentation ofthe

self but also becauseit transformsthat embodiedselfphysically and emotionally.


Veiling emerges as a productive lens through which to understand

space notonly becauseofits significance for the creation of the body. Veiling deeply affects the way veiled women can and do

moveacross, experience, andparticipate in the (re)production ttre

nce and Ethical

The Mor al Ambivale 12 See Banu Gékariksel and AnnaSecor, “Even I Was Tempted’: ofAme:rican Geographers 102, no. Association ofthe Annals Practice of Veiling-Fashion in Turkey? the Gaze: Turning the

of different city spaces, Writing aboutthe experiences ofveiled women in Istanbul, AnnaSecor arguesthat, rather than a single and

Veil, Desire, an d 4 (2012); 847-62; Banu Gékartksel and Anna Secor, “The

40, 00.1 (2014): 177-200. Inside Out,” Signs:Journal ofWomen in Culture and Society

in the Sacred: Religion, Secularism and the Body 13 Banu Gékanksel, “Beyond the Officially Geography 10, no. 6 (2009): 657-743 Banu Production of Subjectivity” Social and Cultural

Mall, the Neighborhood, Gokariksel, “The Intimate Politics ofSecularism and the Headscarf: The 1 (2012): 1-20. and the Public Square in Istanbul!” Gender, Place, and Culture 19, no.

14. See also Gokariksel, “Beyond the Officially Sacred.”


malls have become more smoothly incorporated into daily urban

uniform city space, when we approach thecity ofIstanbul throug,

life and have become the popular destination of a newIslamic

g/unveiling» the lens ofveiling, we find multiple “regimes ofveilin

that govern this urban spa ce.'5 Secor’s analysis showsthat the urbay meanings and experience ofveil ing/nonveiling varies across the norm, while in another one district nonveiling may be

space; in use thre, veiling may be the common and expected practice. I will

city sites to examine the embodied experiences ofveiled women

mall, in contemporary Istanb ul: the upscale Akmerkez shopping

e complex of the religiously and socially the main street and mosqu where Fatih neighborhoo d, and the Beyazit Square,


take manyprotests over the headscarf ban andother political issues place.’ three upper Akmerkez opened in 1993 at the intersection of

middle-class neighborhoods of Istanbul. I conducted interviews

at this shopping mallin 1996 and 2000-2001. My research

r modern demonstrated the construction of Akmerkez asa secula ofmall patrons, space due to its management and the daily practices Akmerkez’s advertisements and management contributeto the k’s pictures construction of the mallas secular by displaying Atatur and speechesand,by doingso, visibly aligning icselfwith secular

modernity. One of the main components ofthe secularity ofmall




bourgeoisie and an aspiring Islamic middle class. Those whoare partof the Islamic bourgeoisie assert themselves as deserving and


legitimate members of the consumerist public constituted by the shopping mall. Today it is common tosee fashionably dressed

headscarf-wearing womenin almostall shopping malls (perhaps still with the exception of Akmerkez). The prime minister’s wife,

Emine Erdogan (whowears a headscarf and whosedress style is frequently discussed in the media), closed down the expensive

British-based Harvey Nichols store in the fancy Kanyon mall (opened in 2006)for a shopping spree one Saturday in 2007.

da . In contrast to the contested secularity of mall space, wefin

simultaneousplurality of styles, sartorial practices, and ideologies

on thestreets of the conservative Fatih district. Fatih, taking its

nameafter the Ottoman conquerorof thecity, is the paradigmatic and Muslim mahalle (neighborhood in its spatial organization historical demographics. At the centerofthis areais thelarge and beautiful mosque complex, completed in 1471 as the first monumentalproject of Ottoman imperial Istanbul. Today most

of these structures have disappeared, but the mosque remains

space was the rendering ofthe veil as backward, lower class, and not

very popular. The square surrounding the mosqueserves as a

often madeto feel that they do not belong in Akmerkez, and they expressed their discomfort in Akmerkez and similar spaces in the mid-1990s.

nature than that of the privately owned, closely monitored, and meticulously orchestrated spaces of shopping malls. On Fatih’s mainstreet, Fevzi Pasa Boulevard,is a concentration

modern, and therefore outofplace in the mall. Veiled women wete

Yet this was not a secularity that went uncontested. Thevisibility

ofreligion in shopping malls gained more prominence especially

in the 1990s with therise of a conservative Muslim elite. Shopping

playground,a picnic area, and a social space of a very different

of Islamic financial institutions interspersed with other banks,

foreign exchange bureaus, clothingstores, tourism and hajj (Islamic

pilgrimage) agencies, restaurants, andcafés, The main street of

this area is lined with stores that feature a mix of styles including ’

the latest fashions for veiled women. Women on thestreets are 15 Anna Secor, “The Veil and Urban Space in Istanbul: Women's Dress, Mobility and Islamic Knowledge,” Gender, Place and Culture: AJournalofFeminist Geography 9, no.1 (2002): 522. 16 Foramoredetailed discussion, see Gékartksel, “The Intimate Politics.”

dressed i ssed

i ; In an amazing range of sartorial styles, including a diversity

of veiling styles, from the completely enveloping black carsafto





t close-fitting shirts and jean, little colorful headscarves worn over friends at cafés, get money from They walk around, work, meet park, and go shopping. The veil i, the bank, take their kids to the ets of Fatih. Rather thanfeelingy very much“in” place in the stre discussions ofveiled women oy of discomfo rt that dominate the

feel in Fayh, about how comfortable they ak spe en wom ez, erk Akm ally Istanbul. Since veiling is gener

ir” They define this area as “the a store becatts, refused service ata café or accepted (and no one is various styles ofveiling

differences between of the headscarf ), the in veiled vidual style come to the fore

and the crafting of an indi

ih. women’s discussions about Fat

s publi,

Thi m Fatih is Beyaz t Square. Just a short walk away fro ne times but is co nstantly being city square dates back to B yzanti monumental main

anbul University’s remade.It is delineated by Ist y old books, street vendors that carr entrance, the Beyazit Mosque, uented a few coffeehouses freq nd 4 , um se mu hy rap lig cal an Islamic to political ces have been central pla se the of All s. vist acti by t ways and and se cularism in differen mobilization against the state ersity of y, the first modern univ sit ver Uni ul anb Ist s. t era eren in diff consistently n one of the stric test.and the Ottoman Empire, has bee g the cially followin the headscarf ban, espe forceful implementers of university became gate of the

hi storical 1997 coup. The imposing andclashes demonstration s one of the central sites of nu merous and 1990s. and police in the 1980s between students and the gu ards

with the entity called “the state” rs unte enco had e hav n wome Many them to take +


at this gate, embodied by security gu ards who asked off their headscarves.

waveof protests in the Middle East. These protests have made this square famousnot only as the hotbedofIslamist activism butalso for theescalated police presenceand violence with which the state hastried to suppress such activity. In the summerof 2004,as I was returning from a day of

research in Fatih, I had myfirst encounter with a Friday protest in Beyazit Square. The topic was not the headscarf ban; the demonstration was organized to oppose U.S. president George

W. Bush’s arrival in Istanbul for a NATO summit. First I noticed the police and panzerslined up on thefarside of the square. Then Theard Allahuekbers(lit. “God is Great”) coming from inside the mosque. Demonstrators started to pour out from the mosque Others joined this crowd, carrying posters that pictured Bush, British prime minister TonyBlair, andIsrael’s prime minister | Ariel Sharon side by side. Women,dressed in differentveilin styles, participated in this demonstration. They wore scarves their heads and draped kaffiyehs aroundtheir shoulders as


ofsolidarity with Palestinians. They were clustered on one dec f the crowd,following impromptu gendersegregation rules “That day I witnessed only one of manyinstances of an expandin feld ofwomen’s activism—in thiscase, grappling with re ional d globalpolitics. There are many moreprotests that scl 1 i d and formulate connections between women’s da “to-daylk ; d events in Egypt, theIsraeli occupation of Palestinian cetivories, US. Middle Eastpolicy. Beyazit Square becomesthe place wher

transnational imaginaries of being Muslimare formedthe h " collective action, and veiled women take cent er stage in Otiege i these se


Across from this gate stands the highly ornate Beyazit e is dating to the early sixteenth century.In front of the mosque ther

s. Crowds a protest almost every week, often following Friday prayer spill out of the mosque after the cua, joining others (mostly

women) who have been waiting for the demonstration. Theissues vary in scope, from the headscarf ban in Turkey to supporting the


Aspat spati approachto .


veiling shows howthis practice transforms even

| the most ost intimate inti and smallest ofspaces: the body. Veiling participates



e by organizing this space enoin the production ofthe body as a spac

s produced notin isolarion tionally as well as materially. Body spacei emotions,political motivayons butrelationally: other bodies, objects, thy,ust


the production of bodyspace and spaces all come into play in processes and

l, social, and cultural veiling practices. Complex politica theveiled boyy, fn ning of veiling and spatial relati ons shape the mea practices of secularism, Islamist po) Turkey, class relations, contested for lived Islan, gre


ety, and a concern itics, a growing consumer soci

Marife Mariiin

e veil and the veited

encounters ofth amongthefactors that impact th e quite diger ences different city spaces

body. A veiled woman experi to Beyayst rney from Akmerkez to Fatih ently, as this Reflection’s jou oun simply enc ter thest veiled womenrarely Square shows. However,

the remaking ofthese ce also participates in spaces; their very presen ce thus Opens g asanem bodied spatial practi

spaces. Approaching veilin tive of spaces and lore how veiling is produc exp to ons sti que w ne up ce-making. This discussion

participates in the continuous

process of spa

through daily tinuously made and remade showsthat spaces are con permanent, goes unchallenged nor remains

practices, and no boundary



Contrary to what one might expect, amongthe Platonists oflate anof conceptionsof space or tiquity wefind a rich and-varied collection ;


place,! due not just to their inherently rich ontology butalso because of their dialectic interaction with predecessors,? Questionsraised by or in responseto Plato and Aristotle, concerningthe relation between space and place, matter, bodies, motion, vacuum,divisibility,Ȣ@and the cate gory : gs to, to, led to pl ory p place belongs plethora of answers.3 The metaphysical 1 I discuss the terminologylater, 2 For surveys of ancient concepts of space see Kei



‘and Hellenistic Greek Philosophy,” PhD lisertaton, UserofUse “i988, forLa . Neoplaronisms see S, Sambursky, “Place and Space in Late Neoplatonism? Studies in Histor rnd pilosophy ofScience 8 (1977): 173-873 S. Sambursky, The ConceptofPlace in Late Neopl. om (Jerusalem: Israel Academy,1982); Richard Sorabji, The Philosophy ofthe C nbatore200.oo AD: A Sourcebook, Volume 2: Physics (London: Duckworth, noo),

onimentators 200-600

3 The most influentialpassages in this context are Plato, Timaeus, and Aristotle, Physics 1V 1-5 Dad UineriDeas Marije Martijn, bnagine a Place In: Space. Edited by: i i yp. y: Ai AndrewJaniak, Oxford University Press (2020). DOI: 10.1093/080/978019991410.4.003,.0004



e,led to an elak,aris presuppositions of Neoplatonism, furthermor evels: at the levyy oftt of then otions of place on three ontologicall thatofthe transcenyn¢in physical realm, at th at of the soul, and at

this ropic is Simplicius, y14 14 tellect. Our main ancient source on ce and pats

of Greek theories ofpla us a fascinating critical overview

wefind placeasa kind ofcontajne,,03 At the level ofphysical bodies, h

a mold or measure, a power whic containing cause (cf. Plato’s chéra), and Damasciuss views), oreven akind sorts the elements (Simplicius's e theor rding to Simplicius, “innovativ of body. This last and, acco Wewil} re

lus.> a kind oflight,is that ofProc which holds that place is the Neoplaronic n 3. If we go one level up turn to this n otion in sectio influentj,] n0es to souls, we find the ontological ladder, from bodi alexten

igible matter of incorpore tion of space as the so-calle d intell ects (phantasia, Syrianus, Peoclus sion orthe place ofmathematical obj at the Jevel discussed in section 2. Finally, cf, Plotinus). This notion is souret we find intelligible space, as the of the transcendent intellect, ace, analogoys to

ak, of lowerkinds ofsp: cause, 3 the Form, so to spe e and the locys, t0 intelligible space 1s also

s eternity as the cause of time. Thi ndin of this kind of space are fou es eti Vari t. llec Inte of ely, putit vagu ead but zor in Proclus, who inst s, Damascius, and Plotinus

lamblichu criticizes it.®

of s’s understanding of the “place This chapter will discuss Proclu cially on a problem

will focus espe ideas” (dA III.4): imagination. We object we imagine a three-dimensional ofits use in geometry: When j take place??



mental space, can be solved on the basis of the correspondence and relation between physical space and imagination, which is actually a kind of space.’ Phantasia, or imagination, is the capacity of the soul used in figurative thought such as in geometrical constructions. Discursive reasoning (dianoia) and imagination together reason about figures by

“projecting” or “unfolding” /ogoi and performing operations on them. It is not entirely clear, either in Proclus or in the scholarly literature, what kind of extension imagination uses. Most scholars assume that it uses a two-dimensionalscreen, but we will defend the thesis that it is instead three-dimensional space.8 The question concerning the nature of imagination is interesting for those who wish to understand Proclus, butit is also relevant in light of, on the one hand, the claim that Proclus'’s geometrical imagination was the precursor of the Kantian notion ofspace and, on the other hand, current theories of picture perception.

Help in understanding imagination may be foundin Proclus’s “innovative” notion ofphysical space, which has been considered a precursor

of otherlater views(e.g, the medieval theory of the empyreanas the place of the bodiesin the universe).? Proclus, as mentioned, argues that space has to be a kind oflight, corporeal but immaterial, consisting of very fine matter, and thatit is established by the World Soul as the place of the world.'° Close parallels between our soul and the World

Soul suggest that it too has recourse to somethinglike imagination.

we do? Where does this or an object in motion, what do of ons, which concern the possibility

Myproposal is that these questi

, See on Space, Jn Phys, Got-44, at GOL 4 Simplicius presents his overview in the Coro lary nt Commentators on Ancie on, Urms O. J. ns. tra Time, Simplicius, Corollaries on Place and uc: is to be found in Sorabji’s introd Aristotle (London: Duckworth, 1992), 3. A useful summary : tion to the volume. s, J Phys. 611.10-614.8, and $ Proclus’s view on space as a bodyis further elaborated at Simpliciu critically investigated at 618.7ff, See section 3. intelligible matter, see later 6 For Simplicius’s overview, and Proclus’s criticism of Plocinus’s discussion.

7 This is not a revolutionary proposal; see, eg., D, G. Maclsaac, “Phantasia between Soul and Body in Proclus Euclid Commentary, Dionysius 19 (2001): 125-36, who shows quite clearly why imagination has to be a kind of space butglosses over some differences we will address later:

8 s D. Rabouin, otGe Conception of Geometric Space andIts Actuality, in Afathemuatizing space: Bikhioeet Cpiscl The Objects ofcous)Geometry ropeae from Antiquity to the EarlyLy ModernModern Age, A, ed. Vincenzo i De Risi isi 9 P, Duhem, (Paras Hermesaps) Lesystéme du monde: Histoire des doctrines cosmelogiques agiques dede P,Platon & Copernic, i vol. 1 ( Ne , > 2012. 1o M, Griffin

Proclus on Ph a ace as the Luminous

i ehicl Vehicle of f the Soul}?

f i Dionysius 30


The evid ence £or

this Suggestion : : ine intel World Soul, wever, Divin is thin, + afits) and Nature ho in ; coHaborate in shaping i the c osmo $ by ional Creat iy : Proclus as charactetiZé 4 “Projected © principles (/ogot), ” w h It i is therefore temp ch sume of the Wor ting hema sothul2 ld Soul is at1tne i ! ¢ °mparable to th Some kind at o f the hu of imagin ; ati on is in play. In the fol lowing, L -ctnation al vlsh , are indeed spt o extent a analogo "S n The tert and in which se “OMParatio nse they “ i n;. wild e t h e so-c S

alledvehicle of th


e sow cis r © notion of ima gination ‘4 Soul anatio” €Xtension , § €ction 3 di d ml | teal space scusses the Wor ‘o and the P n, And sect aral] els bet ween it and imagi @ short c natio onclusion «ong : with some considerati ons for fwee

+4 Git “Y 4 rema s teferri rk on termiinology: the m ng tO sp a n o ! a o g c e y , place, a: di 4, “°SPect Sy and extension TL ively, but are ere and, these a Fe m° not the only *€ often terms used, nop u sed to, ‘vocally $ Se mean ‘ nd un fyoc systemiatically ing has t . a 0 o bed etived fromthe n G



endell, In Proctus's

place ay be bee of

rhe, case, tton, i (Leiden, Pos and Dyp ysical Necessity, See G. this ia “proclu os PLY ofNature Van Beil}, 2009), in Greek Neoptoni€ , Inge,



For Pro clus





fs the use of b oth diissccursivive e r realid’s plane geo mis com men tary on Eucli

, 80 metrical roo ion. p In i q h


soning a

nd Imag. inatio . : ination

etry, he des

. gest a io tsu two-di in terms mens nal tha ie t s wires t e d r eometry esides the obvious fact req ; th a f Eu . cl

cribes ‘hea

Screen. Ho wever, ; .

im of E id’s Elements three dime . Ss (an d that the ,0 nsion is u l t i m . a t e t o ne construct tonic so SO Proclu the Pla li ), eve. s states, ordinds n in g re is to p Proclus, la a ! geome try thrre ee dvn imeonl siions are needed * Cc ; a com try scientific p . roofs in ‘ ane 8 e use motion oflatnhee (e pl.g an.efoe h r e se Other cx so O li o f poin the Pp are

amples dtthe uw ts oursid e he use of in . fi niite te lilinnes. ioTnhese demands sugge st ars) an io al space. We wiil srar n itself is th : ensl ll t inatio r instead t i e e d i m ks ar . byclarif‘Cyheiantgmagi SS’s V.views inks e properly ce n ti . P o r f o c w l u h a t tr . he : w y: k inematic conth LIC tions, , proofs in str dso-cSubseq: ue ra plane eometry: alled intenlltig“4e d nature o , i i. f imagiinnaattion an an wil discu io perpendic' ul

ss 1the role ofgeometr : j bes i ical pro ols, han ntindi ionsi ides the requirementhsreethat geometr casatProclu1 s sees it indeed uses a thr dimensio: nal sp ace. y


ime OR a “Ording © Proclye™ Psychic Body; Appatitions eat soe in G, and Geor of the rn nD atonton World: Bosit and the Late Anci 607. ancien: ™inology L (Leiden: Brill, and Concepts 2013), 595~ and plac0, oP nae finig Bometr ‘lat in Classical related to once enry 0 with Greek Mendelj, “Whar’s athematics” ae aff Location Got in Mathematizing to ce: The "D Modern PrahiusehOY ofS? »€d. Vincenzg De Risi (Basel: vil alse P, Schrenk, Light?” fn ial NOC occuras “Proclus on Space telated to space as in geome ing to or vaste




2.2. Constr d ic Constru ctions i ucting ing a Helix: Kinematic ‘ What is a he ay to find o lix? And how ut is fe andoofwaetrconstruct ee and poin * uta squirr “t e e ‘ aahenu aytUm liex.wBu Pm at the back el at the ‘vill ae , The pat fo ll ow u h p «l th h e t r e ledge a perceptible, s T opPy initiono,obtain propereS as well as a univ ek vaoflidthme he sd we need a univ etho ersa a The Poetics 13 Forarecent overview is noti in ancie i ofthet history iloneof this notior bury AcaA. Sheppar d, «Blooms steathonnes > (Lot in don: ancient thoug he, see of Phantasia: She 2 014). demic, la: Imagiination in F

14 In Encl, 283,13—19,

. f mathemattics (An drew hilo sed in contem late ddPissue which is also ” Review of porary Plane sophy or me tr y, addres lati aolo Mancos an Review d Solid se onship betwee u, “Onthe e 5 boliLogi der n wh in ic to h we will not he re e » : (2 us 01 in 2) g : 2435 i pr evically or logi in plane geom cally)p rior fromthe p etr yin ens, for 3Pr)oclu s, oving the (ontologi terior, ‘i

wh i h di Oe Ss not render » WNic. a the pre oof f sciientific i 0 T exp. anatory.



of constructing it. Proclus’s first introduction of the construction of along a straightline that is moving aroundthe surface of a cylinder,

the simple causes which generated the complex world welive in.” Likewise,it is in the generation “before our eyes” of a mathematical object from simpler mathematical objects (a complex line from a simple

This moving point generates a helix any part of which coiracides

one, a triangle from lines) that we understand whatits true causes


a cylindrical helix is that it is “traced by a point moving uniformly


homoeomerously with any other.”!6 He then rephrases the method in

are.20 Harari’s example is the kinematic construction of an equilateral

slightly butsaliently different wording, namely that “the very mode of generating the cylindrical helix showsthat it is a mixture of simple lines, for it is produced by the movementof a straightline about the axis of a cylinder and by the movement ofa pointalong this line.”7 Besides the emphasis on the cylindrical helix being homoecomerous, which is not relevant to our purposes,'8 we see a shift here froma passive to an active formulation. Whereasin thefirst passage we read thar the helix is traced (grafomenén) and generated (ginerai), in the second passage instead we read about its generation (genesis) and begetting (gennatai). With this shift to slightly stronger terminology Proclus in face introduces an improvement on Euclidean proofs. A description or g diagramof the construction ofa helix would bedeficient. To constrayey a helix, we need a cylinder,a line on the cylinder, a point on the Line,

triangle.”! Proclus there identifies “the motion ofthestraightlines, the one moving towards the side where it makes the interior angle, the

and (uniform) zzotion. In other words, we need a kinematic consto-zgc_ tion. And forthis construction, we need three dimensions.

As Orna Harari has convincingly argued, Proclus sometimes presenrs

other moving away fromthe side where it makes the exterior angle? as the cause of the generation ofincrease in the exterior angle and of decreasein the interiorangle, respectively. And from this, he says, “you

can infer howthe generation of things brings before oureyes the true causes of whatis sought.’As Harari says, “Through the kinematic. construction, Proclus does not merely show thatcertain attributes

hold for a triangle, he also shows that these attributes are derived from the triangle’s being. ... The kinematic construction accords with this metaphysical conception; it grounds the triangle’s attributes in its

modeofgeneration.’” So the kinematic construction,accordingto Proclus, provides us with

perfect and scientific proof, as opposed to prooffrom nonnecessary attributes.”4 In the example, the auxiliary line used in the Euclidean proofof prop. 116 is notessentially related to the geometrical object in question. Therefore, although that proofgives us a true conclusion,

kinematic constructions as alternatives to Euclid’s constructions, be_

cause of Neoplatonic metaphysics and epistemology: we need to Ang 19 O.Harari, “Methexis and Geometrical Reasoning in Proclus’ Commentary on Euclid’s Elements”

16 In Eucl. 105.1-5, trans, Morrow: thy nepl toy KdhivSpov Ekta ypadopevyy, Stay eddelag KOUEy. nepl thy Emodverav Tot kvAlvdpou oypelou suotaxiis en’ adriig xivifrar. ylveran yap EME, Hg SpoonE‘ navro. te pépy maarthappaler.

17 In Eudl, 105,18-22: dydooy 38 rig xvkvdprxiig thks thy plthy ex tay dehy Kal adTIY yy, yeveory, yevvdirai yap THis uev edOelag Koxh ivovpdyys nepl tov dkova tod KuAlvopov, Tod SE ONES, depopdvou ent tis evdelas, 18 Proclus in the broader context defends the position that a geometrical object. may


homocomerous without being simple. The cylindrical helix is introduced as an example. Te 3g homocomerousbecause every part of it can be superimposed on every other part. But it is ney,

Oxford Studies in Ancient Philosophy 0 (2006): 361-89. 20 For physical generation,see section3.


21 Proclus provides this constructionas proofofprop, 16, that in anytriangle, ifone side is produced (using an auxiliary line), the resulting exterior angle is greater than either of the interior and oppo-

site angles. az In Encl, 310.3~8, modified by Morrow: atsla 8 tovtav 4 xlvnoig tay eGeudy, ths pb 84& novel Thy evtds ywviay xivoupévys el tadita,, tig 8b ed& noiel Thy keds yovlav and tovTwy deponevys. cal

Exeis bx TobTav ouMoyllecOat, nag al yevicels Tay mpayydtioy bn’ Sw jyiv ras &nOivds dyouc tiby tyroupévealtias.

‘simple, There are three types oflines: straight, curved, and mixed. Andthe helix is the prime ey

23 Harari, “Methexis,’ 385-86.

ample of a mixedline,

24 In Eucl. .06.1.-26.


i { }} i



inciples of geometscursive thought co or didisc Platonic form ntains te princi s, but cannot ples “wrapped rical entities, d exposes th up” in a kind as lower . ion. Instead, em it “unfolds an of unitary ‘ tibule,” It then agination sitt ing in the an ts them to t and can hence expld icPratesesenw “i n im ag in ines ation or


it does not Bive us any ‘ : co.nt information : sion or of the of chat regarding the obje ce. | L. An importan . “before 0 * question, then, is where, and w.hat, Is cyes”? And, | equalY imp t< |fi: doin

ortan f


t or our purpos who do, inndgi kinemati c es, or have in re c onstructi P COnstructig on? roclus does noh Nin s an the p‘at eofl d or in wax, line itself Andit is no that m t so m Oves;it is us, Fo cone js r example, the e nus truction ofi f: formulate s) of f das follows: « U fwe thi. nk (noe santes acitele re . ‘ n e project 5airg fL POint abovej t andf Moving in the



thou g


Constr U

rom the poi nt P e in revo ce of the ci lt 1 rcle and set t h e lin ixed”™ . : @ surf is so 5P mixed. ace, wh ich is n » in che sens¢0) ' js



We think” rformed “before our eyes overs ang*” of a circle. A in th nd we are t Ought h » Webroject a Produce a Co e m and thee os line and set it ne,26 in sotton ht”




from sensible with becase es ec

ion, and P, J


fe outs

: is not, o" Y discursive fanoia).27 We nteas of: thought (dianoia). ca Iversaq]

mendes apt for receiv

ing its fo

rms.” This description ofimagination, ibule”(ie, (i.e. jjuse inati hich “sits “s in the vestibule” which outside) of diiscursive re ib ut at are the attribute:es making scu d oning, sh : : evenasin ows wha : dispensa —and ble—assiis tant to th eve s e ge si om ble: it offerseter's i ee ercep a redisce ursive though it is separate 5 t: ic is sep ; d fromthe p 4 ) 31 a specia ial kind of ma he unfolded ge ch é tt er o r Ceptac. le Uspo ine nefoth itiesdo metrical theemt . A tooltool,, because entities,ct asa zas atoolfo r n wv ity”e 8andes ecial d “formati i hasa ve activi kind of matter it ty, ty produc tive , “in anand with P it “alw ays prod » th e body? and ence uces it exists “in < con and shape? be woca n,use I icture ” nsion k s that have di_. indi vidua pictu




a al

‘Oro 2014,

atus of

the Tesu

are ment al Which iso

lting pr duct Real??) e lacer, (Is it it imag imaginary? For thRe


of Perceptible qualities?”



recep Bacetaele,


the receptacle,

Aagomings mathem: and in Pr in Plato's un d Aristotle inboth ith Proclus, Commentary on im, equating receptacle (Jn rim 11 10.479), W! matter an the nd trans. D. Baltaly ke Part r: Proclus he World's Body, on vill not creat ed. a arallels separately. (Ct Times Conia abri (Cambridge, : these pa UK: Cambri ge Wersity Press, 2007) we v On the relevanc the iscussion. Ty e o t ic rec e¢ ept lat acl et er e dis fro cus mPlato's Tima bp daveoola gduaros Kell dy 3 eus, Sid te thy w . xlynay well 7 p dn Eu, re ahy kal yay tTitey ol eo com a TO xn pa vAT thy dndora ti atty, on owy tygey ie del kak Syjpyutvor



8 The



P, DMs a 64c, a nd d is use d is an allu sion to Plato, .

76 (1994): Lh Fhe Claessens argues Prods would na . f of all, the fact that inationis a kind matter, . in eiate Fst ecessity vie Praclus,mmnecting however,Being of nd, there is an an And in Plary followsto in Proclus’sinterne atics, As Proclis iverse

* Parallel with


aS “creative teasonprincip. ¢ cs because inciples” ne on Lowel orms, whichcreate entities



velo |

08 Ke ov real sue? Y Repidepely kal nepieMEavtes av rhy m

The veestibule”

as a metaphor


i a on


a Ee el, $4.27 27 55.6, . tran: s. Mor tow,

. C. en “Limagination che) as Plato's receptacle a Copadedl lus presented in the es ft ' Classique et “ttice: Kant” Llisératures 45 (2002 cient ” Descartes entre re Pro FOC nas a Symptom, mvt: ion Philosophy : “Proclus: on Fir Geschichte au G. Cleessens aha Imagiination a “eels Space,» Aehi on , Corp Dor oi ett) ore noe had two reasonsto draw Der Philosophie samt CE os Schrenk, che clus


Surfac ™PogexBanyYAP KUK)oy bY rg ey of th a . Ovtes O . } Uma‘ © cone is , Mixed, 5 cause it is both fb ave + ngle and 8elay , V O a ria ry ber e o e 8 r e a d dy oxy kiko kal onuefoy Kere , S7 ap gical St


it ¥was the p lace itional space between . ient Greece,it t. (In for a veallowed seat BE inside and ne ancien . where young , bev men w ere allowel to court maidens) brow dtdvois Tov hae ite thy ei imexriGerat wentvypéveng adtods xed bom Keen 1: by m0pby toy dd Sei, dyarmbone . Tpocyer kal ev ety dverhot ao beelng How adtiy, TOV dy dvehieres thy . exelyy H rank : wig elddv. indo cane alo Snriry Xapiaydy, ddyy ebtpent] ebpotion ty “ pds Ashas beenpointedthy 8 ae lars, there are femarkab’stx allels between Proc "imagination "imagination parals3 see out by sc and

necessarily on es; ho Maged” (to physic “imagine9 . avoid » “imagined” inciplesin 1 © Principles in imagination or ia)

¢ Ntolo

visible exte


essence . f geo. in er S and essential ; proper ties O * We can Not of Perform any op tong on them eratio ns a

Mages, Not




Involved in * S€ometrical





Kel eel mr ay 8 yr YvooKer ’ Tolautyy

6 TEA 5 PSI. fayev URAPS






: m odef The imagination used in in geometry is. not what ecame ofiti b 10 times: it does not Produce : its cont ents seives fiction, because it‘ rece its a higher source,34 percep ion Imagination, as intermediary i betw'een .

10th and intellect, js no t “moved”fro m outside the body, a i except? but receives know ledge from in tellect. Since the bod, i it is is b boun oat, aethe ow: it ho cs 5

wever (more on

divided center

whichlater) it “draws its knowledge d

and into the m edium ofdivision, extension,

For this reason everything

thought, It thinks th



tém “rézo¢is a reg a8 Wherea construction ton}is to take pla ce.

. . : As wewill ial side materia see, however, the standard in terpretati onofthe . : : jected: . rojected; hich ‘ oe in of imagination is chat figures are p it is ascreenon to which fig image ‘bed des : he relation betweenrational inciple and imagisis describe a ee the rela princip L a n d as that between a novel its film.3? We will first e Jaborate this view, an

the imagination, as th lf This is why thereis more ‘ i se nse worlds ere is more than on e circle in t for with extens ¢ io n there appear also differences . ‘ e and nuit1 in siz and triangles,35

and then show it to be

unnecessar. y and uuntenable.

ce ,

the circle is Not exte


te 1's me

nded, burin th e ins extens io e g te not ac ol n, M o r e o ver, although the ci Poreal entity, rcle ‘ stanteh the exte nsio n is accompa “intelligible n i bya e d sp Matter” Thati s Why geometr ca ecial x egn? , tality o inationit Ba

d shape;sit 1

imensionality 2.3. |Matter (Intelligi igi ble) and Dini

free of external ma tter, it poss r prov “ esses an intellig by the imaginat ible eecitei a ionitse

Theessence of

1 lurality,

his ne differen “ : discussion te. ion of the the € tentative a Proposal of Mendell,. in his. disc ion ’ (dare I say space) uses of f the th term érog in Proclus:

its sons thatit : icture ora shap eof thinks is a pictu

sh this- cicirtecle i e circle as extended ,and althoug



an «° intellect into extension, division P telligible ‘ matter.¢ nth b dy; and it makes vith in 4 isin pictures in and ve the body; ° Th€ prese nce of extension, intelligi‘ble matter has led shape, and ‘odlus rh 4® the forerunner schola hol ‘ of modern philosophy of vitisi to see Proc’ interesting to compare hotions of absolute spac 37 In this : contex i +



f Circles, and y n work wit . ne esse of different size h of circle, wit s, e v e n if there is a uni hout m agni q To sum 4 Iimmacine:t.ude, a 0" creator of cog +t ductive MOtio a ePe ies; ighass nitive entiit nof its Own, ti ided univel ts with which it * brings the un divi 34 Note, ho weve 4 Second typ r, that as Sheppard, Poet ics, e down?rathe of imagination May also has shown, this same im r than up, aginatio be used

) in Iamb!



for Imaginin n of 38 In Enel, 52 ie g nonexisten .2 t en baurie xn} ™p0—53.¢5 We od Savrwcte 23 eo pé oB co aN y e: kE vtpoy Karkyo 73 YWoordy, kell Sdoree Ipesat e usn ta bre ey ge y chdv e ey Cat s 06 HP almn Hop) VoH BA Kal ont Tpodyer ry 3p ox ef TAuores oben ke y yroaem Tod dep : TES, el TOY y TE Kbihoy Sa ork cutis, kell 8h tobco may epads THs ie a , Sep ay oe ont ea THs Kel By ‘tobto orariig veel v "ig e pbv burbs Sng rabapeiov t gnpr0te ME, kbihes, donep o¥B k dy Toig a ey pry alt tides rev re xbK hwy Ket .



losophy and Mashematis

36 D, Nikulin, Matter, ural Philosophy an \ thematics Imagina ination iYes and Geometry: (Ashgate, Ontology, 2002), d “quasi speaks 0steulisable extension eee longas w0 ainmain owen €xtension” andnate mero a as “appearing”to o geometric beets 41 og Spe! isualized, is nonet heless real. 4 Ba re Knowin athens 37 See, ease a eg,ee Proclus: Geometry, Imagination M, Domski, an

“Newton 1): 89-413, Southern Journal ofPhilosophy 50, no. 3 (2012): 4 38 Mendel, “What's Location?” 39 See Maclsaac, kworth, ji, Matter, ion (London: “Phantasia’; Richard SorabhMe of Space and Motion Quaestio Duc: ): §31988). Cf Christoph Inteligible Helmig, Arise valdbold but 7 (2007 into projectedi sensibles, as Ar 7H These Objects ve are as a projection nos Phantasia by discursive oa ia), Hence, iy 38 no.t phantasia / elas? nason( ros Ancient Philosop ” and Mathematics tianoia? CED, indivisin Proc’ fa non-extended Nikulin pagination an yTActoy ood): 1230721 “This icture-like ‘visualization act or Dvons a peomenrically 4 divisible figure ile mathematica object extended Proclus’ eanstory May be compared Account oforExp or héyos > 66) See : ho. 2 to a novel also ©, Harari, hichte on fe t Context?” der Philosophie Archivfiir Gese 9° 308).sonsin Non ive inte intel- . if ist he so-called ‘passive that onto whichit is 008): 197-64. CE. £ 7.4-6)"s Poie In Encl, Oe” Morrow: lece’ (x8 33 £7 6 8 mpoaNbuevoy Oytixds obtos Kahovpevos Classical * “Autner, “The nan Phantasia in Ja Timsateut Cambridge Distinction Peeween and Doxa in Proclus’ Proclus: dn Introd (Cambridge, Quarterly so (2002): UK: ttn and “on/onto.C Rabouin 257-69, R. t “Imagination? or seem nversity Press 2012); have both t eometric and nike ‘cle on space ae found Proclus’s conception * discussion.) eee ninety nsio So far J. (O'Meara, only one shee ane screen and space. tection See#ee imagination: consistently speaks D.J. phagonas Revived (Oxford: Clarendon of projection 89). Press, 1989)




Intelligible Matter seafyle 00 . is -s su suitabl! : is like i that it sat tt Sy the reception of universals perceptible matter von and their raulepliae into individ i ‘but unlike it in (imagined 0" in the sense that it receives image y inary) objects .






rather than perceptible : This is and is nF telligible Matter ones.340 This as w,

the distinction between inten sensibleant ina “One type of matter is intelligible, one sens! inition ; . the circle One part is always matter, the other nenualiy a plane figure” eb (Met. H 6, 1045233~35).42 “ ae Matter:

» where “round”is The pare 4 “bronze the form “lane” of t “ph

“Uggests that in the

definition ofthe circle, figure”is the intelligible ‘ scribes int Matter. Elsewhere, Aristotle Bible matter ag the de sible) ching | nonsensible substrate of

(otherwise + ak pete? As Helmig emphasizes, +. the igible matter 1 b ein the

s Mathematicals

background and imagination i i already j n se :ems: ° + por” Aristotle, who sible Witho king }is imp d ! states ut images, thar thin Which ination 4 ; Without implies ires that it requires ima a Conti 1mag Similar to intell nuum » Which in Turn implie. meth? : s thar it igible re qu ires SO Matter,44 Toclus, however, of intlig” Criticizes the Ar . Matter beca istotelian noti use he ta . on ke Sit to be Pass . ive,45 although onsiidd ering , ¢

40 In Ene) , SLO 20, 41 For At . istotle’>s No . tion

of intell igible

Matter, se g Helmig, “n h tei llltecGBt At angudesthefoLiramiUnitar “Aristotle fere 's Notiio . ts OfNeyttnourtialon oF inte onn,," wiwith c re & nce! vo : ll is ig ny h Johanse™ ib (O matter in Aris le xf ro or du d: : ce *z O s mn xford Univ totle, See also en eisda oe iow *O ersity Press, T. ‘€ Notion of jen 2012). ie compome maintain in igib th le Ma wlat P rt n . ae e Unity oftea ll tt g ice iner in this co: nt spite of its de exe tosolve a p ec ™ and (intel , ligibte) Matt a fi 88 Platon 43 th Thee oessence, ic Forms) er(rather th nition containi olrowoee 9? nsen €Aemicirele . this apor .. gical an, €-B- tw ia aijs solved exists, o On F , ; ” Ot beca. Parc of 193726, Se the us essence . Fa circ e itis use g also Mer, of the circle acbecal” 103008 | 44 Helm st, 1Osobile, whi a ie » “Ati s16 Sh utin half. Met. Z 10 ean be ° 1036a9-12 4 Ons> Wiwi th ref to AM 449 aah 6 vice Insp 5 1 4 5 9 and 450a7~ ir Ng the Cq 9 p18 be uation of in 8, cf . d A IL ri* telligible Ma as the 4] le isp’ tes tter with ASt echo” of 8° La Ti pass

i : ing hat discursive t shape aIso accord isc thought works ;withou ter: i matter should ni to Aristotle, Ose le, we need in forman of active, “knowledge? shaping ie, ae; capable of somehow even have a certain eter is



eetinus weine tethe vo j


© geom


also too has an orton _ intelli oF in alinbh matter, Jess, formwhich ence of Proc fa shapeless, 7 iticized, m. ng d, ma mainly because hesuggests the pres the he locus of jss, d inde the intellect. i determinate substrate of the Forms, ware e and cause of lower Thiis SSpace (capital capi S) would have - y beasthe to the caus so e of time. However, i kinds of "Pace, heanalogous ne Eternity a Intellect,tosays ; indeterminacy is presen Proclus, indet the “hcie substrate“s tas he ts er ofi the he determinate are (the \e Forms), onsnoas Plotinus, Plo nots e.g., TOTS me usd. es us e the same d2, 517bs) Platonic expression meoou and vont témog trepo dviog (Phdn 24703 i, $ “ esp. 509d, . ensively 5 but explains by Proclus at them iscussed ex TP IV 22 13), i

Aristotle in troduces

bronze” (


as metaphors, 48

But Proclus’s‘ own description de ic as well: is problematic .

. ma { Soul, exercitossing . ginat ion, as ‘e her capaacit y to kn ow, projeccts on thei .s . . d the imagin ©n a mirror, the ideas ina ation, re ceiving ideas of the figur es; an in is wiiththin,in, byby theiir r means in im 4 age form these reflections of what iniswavtd from the imag affords the soul i es and an oppo rrunit y to turn in \ {fin a mirr irror an d attend d to herself, It im se . is as ifa m

an looking at h




ive roc Intellect,n, 244.20—2 an ste 2 d IIT 158.51 0, wh €




ce. i ive intelle f i nteliig q Li ible matter wiith1 pass ive i oft 1on i due tot Ne equat itici s Mm 1S ssc injan scription : hylomo: : iption o f Being

would be p rphic desstence 0 f Be ¢ the Plotin 1540.8 we read correct in ne ity tha in Proclus’s ing and “maiftte and existe they P oclueeview r iew only unity d thoyov if “form” Plato ar i of) itsrefers power to(Sdvapis stati the ) “hePlotinians : a misunders being. Limic level of intelligi posi af is the dd and but as existence indindeterminate nature at the s matter and form nln ahs unlimited formes but relate to one another ate present not a there, and power, teuat” (Hs 48 The “xPtessiont “seat \ opos idedn (cf. Ari Id not be tal ken to re! fer to a 8pm, a) Parm, st. 930 10-20). Cf. Jn dd TIL .4) Ti, I 385.30-386.2, sh to a “gulf (uke ) ou the Orphic ic : where reByreferen 1xbmc) without plac h o a K tay a l ’ place or space ag YUPA TO) eloayd Kal yny bers a nTh to by Procclus as 6 ah pun). (2 aere e cpe e in is anoteehedr,e ners kindofabsolurespac eferred e we (tn Tim, 1 1615ff more . space of y : .), More on this later. For a neral re kon Greek terminolo 6 get d 1 cl See earlier discussio





Matveling at the

wish to look

Power of

nature and at his own upon himself ; directly, 49 .


€ arance sho

which reflects proj 1 intelligible M ected he reeS;fosu r gé kin ematt atter is flat —w hich would be ©oNnstructio a pro ns, Other p as Sages ts:

Our geomet


er [ ice Euclid] f . rent has chosen one ‘ of [the diffe he plane,... fo fer wil r his inquiry c an proceeedd malo re ca siely his With any oth : so hegives i, as er surface, Fo F


Surfaces] , t

work the su , ,, rthisre ko btitle “plan e h m i u n s t e t S c o Plane as Proj metry.” An gt d oe the ected an understandi Writing everything d Lying before our eyes B an t : Upon it, 1p ‘ . ; like a plane the ene, ion beco ming som et Mirror to understanding t a g e ef which the § down impressj ideas of Sof themse


This Passage and others Predecessor. like it are ‘OC:Jus’ . full of refe “Befo Te f ferences to PF Our eyes” e 22), and t ’s choes Aris he teflectj theSoul (42 io" t o t l e o ' s O 7 Nina x Not the Mirror is q well-known ic imagEs i treacheroy atonic S Pla Use of the Uta more ch . Mirror that ‘ POSitive ere, howe" here,” : ed Platonic is echo “ft example: Timaens (700° eo iver the liver in the in — 49 Proclu s, In E kath rg W ngh TM4hda1) tr o ans, M, dbyoug, A rry, YEPY “s modified by SE V el8cdX otiog TOR adoe, O'Meara, Pyth ea ras er e neépl Repbvere o, af $ obey E agoras Revi bg .TOov 7, Ay Pavtaa y, Thy el X i le K V IR! ao y E Ig ne V p é x&to POY by B Keun TPG Ke lo TTPOdAY ica} T Kol a Hodes Exovon tiy EvBov ntpov TO 5 US TH v afyt? yo: Kal apo el Bory eds & AUTHY TOU a , Sva Thy dnd ray elBebQeoy Bvtwy oavy» ofov n Tolade ete SO In En 6t U £06 OOventy ny ere c]. Ro kad Thy taurot ici . PeitEDa. 31219, trans, wopdiy V kal dpardy $ VSS ip etre P1 Morrow, by ev 8 YEwp droreec Brive, erlnedoy Et E TC LANoy n Pootlo pye bcheEd al 7 uevoc, why ncey Ke Kelzevoy Hen? ane ate nine Sov s+ , Tehvey, l Enidavelac wv Sa ™ obey bei Voet TPoseiK ee , r ++ re o laev bty Thy S Ka TS ll e, EVNS,t y b a ac y p e" avory Y thy voe éby bby 5' For th pd UtTralneSoy oloy al mpe diavolg Y, THe Be mpofeB is im AMyeoy T sen y davra Anuevo troip ep age, a s e c e a l f r vrey Eu as 2 rom Pub mbdy 62 Otr: ddoere tl Arab M lat?!+ TANSMis lic ang *phist usu ston D , see 1), D s exelvo Maras 18 Nove ly, 6 Aci i n Thé e d S m b, G e * . t m , D e Platonicien, M, Sebti, Collogue e C pit °S Vol. ios 48 (Leuven, Taternar; an Des Alexandrin Onal Tiny 4 Leuven Leuye ta-Lat




ns A La PO

et Louvain La-Newe

ir o JH in like a mirror that it receives nee fomthe rational sou 52 Such in ational and through mae soul can whichthe study se satisfactory Plato is r to the pethaps question not a ‘i a imagination lain as it . evo-dimensiona




seem to con firm


does not is P o ohosins exp her mirror. the planeness Two fa now one o ins uoted Proclus the second passage hheatane tl ‘hat ‘lane geometry, the subtitle ee lidsck and an € acits a ieae of Bucli of the images. rs to the general vert reliabi robably ah irtors of not very 8 and age ne Pro in in which would ones as Nii ions, but not by allow ae “Imagination . empnasin and wether in Proclus” kind of “smooth ee Proclus Pons nes which distorts Our whyProclus


Acsning imagination that it is unlike eo mirror of bodily


the Projections of discursive son.” urse, valogy The downside of this ‘ch doesasa aaa is that we ‘chee with very not combine with thet flat figures, whatwell matter, vena to som:e of dimensionality figures the kinematic i seome Proclus on i ne geometry.” constructio findsin e for Proclus’s p . tion of the duct The same rsboth “ ee dese «notion . vorines OFphantasia echoes of pictures. a Oper rid he the “708, mentioned i st now, and Tim. nati swan is compared to 500 its imprints ts he fa signet ring eeentively a Now although the Tntion in wax are not flat, the picteare is im enti


——— ‘sian ie Traditio _ . tt Forme ation in the Platonic and Concepts. Colnncep oS ne ‘on: The In fluence ©! f Mirror of Imag agination: a); Sharples an ‘a A, Shepp ard ten mA A. Berlin: De Plato's Tim ae Sheppard, Timaeus 70 gruyter, . a naens,; ed. R. R,Arnzen, fF" aches we in to don: Institute ae “Wie Gegenstand Mi rand inne of Classica deen, 125 Den stlchen Kreis? eantasevermbgen, 2001), Hung trischer ation Raumvorste und Geomet nA LHlaythams —EiKEL Raut und Ibn Mahan Kei in LN ‘Maier Seat Verlag, 2003), 115-40» Theorien ous Kreation: j Das Kultursehaffende igen Der Phantash Vern,

53. CE Claessens, “Proclus?

anto, ani + 54 “T Imagin ation, bo th by vir individua

in th the istence with andin . ivity and becaus and eve wisible e ith ictne ures (¢ypoithatha Ody, always produc Py virtueof l vie as asi ext on erex af +n, nd14 es individuaitli sha43pe, 9 to! dah s oe fordi maton 1. v §1.20-52.3 c ive i) a ° ea tythin 3)s t ne hav g en ib i thac iti know. s hasc e23 e) Cf inchaa npi sntasia bec 20-5 has. hi wont a. ,iby abe aus avtaclay, didn per k xb mov Kolloe the latter works wit h typos and morp!




still that of forms Prin

Mirror is q

mage,to the extent th

the three dimens

ions pr

at we tyne


ojected on the P sof Bc And Bain, Pr om With flat Platonicochys at times describes the eee ot a paintes " images as being analogous to thos . g drawin e figures (schém , ar + . + a) i n , maginatlvation sf bu thou . t ou gh in being a Painter, . t, th o r gh t, of im while com. mon ag . o bis ‘ clet ‘ pe

Finally, ther ‘ as e is the famili ar te


rception (hoinéA isthésts) ) 1s ae a eactliyi ology, used primaril;y for th ima with im in

also forits acti ng upon an


al -t acti » are also used in contexts wh ich seals ‘ rge 15 a Painter, dimensions; for example, : a nd in [. the d emiu. s h a p i n g 58 though a the universe he unfo An lds the rems. uf: 4Ypos

a lifelike

is the imprint of rei eli : 4 seal, it is also a mo ld, a fig (Children ar e Zypoiof th eir parents les used in .) describing rimply* at ima M

i Mage,



gination n

eed -

ludes the n involv? "0 need in this *espect umber ofdimension s to speak of ion or intell Projection fpet igible matt of © e r , 59 culty in un Suring in «on alitl! d e r s . tanding t Proclus’s nsion® he kind of theory is the f4 ct that it is unclear extehat v0 4



in, “Imaginati on” 165, 6] Remp, 1 233.8~16; oy _ ® Sov Ev huty 7 [p 39 6 LAV GIN 8 ve fs FTeP OV yey algetata ray mad MEATY,dy elvat tog YPduparéos rob diddy 2 oats it af aloOnceis ray alo! Kal? ate SVB VTE tobedraryeMouaw, ober tobTov yep ov by dnt Exelvenv romoug vay ane at WPatuariory Kard évBelevuta thy xowhy atonavy, cet TYi erepat : o elv Sata, Pyyp,, Rous Yoo, dvedlrre Oras Revived, + Cha [se Tobe dtavontixots 166), Déovs al Bs Le Tim, as Helig, Forms M2812, inspired is unfoldin by Tint, ssc; for his un lay Yoyz





. description ‘ Projection, really of an action roject: jecting, ( of The means. As a range ssion is quite clear. throwin hrowing, or thrusting forward) ¢ the expre f bj 8 that may

. nerspecti™ ted onto something. Fr om this .st hav More sy, Ccessf e uli

Perspective o f



be projected—e.g.,

eat, :a shield, a tongue o i aot very informative

a e

land land, aajjutting rock, and an elephant’s trunk’ —i ;ing g rock, the nature : of that in . hat at is i which ard ends up. to w isithprojected probolé as putting te The. it e combination onb, ion of the imageof theflatbrings to forward a weapon or : ‘ se mirror bring w m ind the image of ' sh shield in defen er, thea fat screens.s. However, A at screen, esnecialy to us who are so used

to haded cluded. | examples showthat other i options are notdeexby the word proboléA in the Helmig elmig argue arey s that the main point made ‘5 ‘hsthat o f the external Specific ift co : nex of Proclus’s presepistemology° is t t entation ‘ ion of f our innate hey are present in our /ogoi, i.e., ;the forms ‘.us co nsists in unfolding intellect, lect. Recollection (anamnésis) for Proc out vo oneself’! Helmig the logoi oi an and resenting them, holding them se *h s that, coout, llection, translating ha in in general, , in the context ofre sarreceptacle of some robolé ob i ae" as “projec tion” is misleading, as it ; ee wher ewe do find such a ‘sort. Th The exc ception ; he concludes; is a-eyphantasia. pasia *eceptacle, » namely tka the intelligible oe ie os

Instead, I would like ' to p ropose



We ‘could ld useuse th d ¢ sense of“projection”


Pa all instances of probolé, as . “holding ou vif the recep0 é.

orprimarily “hold ly to 4allow us to put tacle in question is os a type ofempty sp oh hi projected. The Some distance i one i betwee n us and tha t walc nivocal notion, benefit in general would and mo respe cifically, Ist beau sntellicible matter, ewentyOf at le would ould fit o our understanding a of intelligi five instances ofprobolé he Euclid Commentary /é an and in the us reg Only five have a preposi cognate;termsinforming egarding oe that he itional attribute Joaoi. i In those ®nto/into which the se five cases, AV it is nein soul projects the logoi. “in (en) passive intellect, Context which hich determines de the preposition: “in — 60 Liddel an dScott, sy, 61 Helmig, Fo rms and Concep ts,





') imagination” and even “ eq ei. s) nirror 162 Only i . “asif in (eis) a 0 the explanation ° of the “pla ne” of plane geometry, with sect Metaphors of 4 it h pr o) ec Mirror and drawin g, do we end he 2geome “onto” (epi): “We mu ‘ t ‘

rhe c

’ rf

st think of this pl ane or “sncursiv . emphasis] as Projec et in ted and lying before th e eyes, ng all things onto ape it, and ination imaginatio as likened © ft Mirror’ So i

: t is the Plat On ic images used th mine rhe ret at de te the appeal of the ive imaginra tion its ree images th tion as a ser fit cen, but all we need . at. give imag. is, strictly speaking; som e 0 p

nology and


Ween us and the

obje. ct of projje Cottoboration eccttiioo n. for the Suggestion . :

that in Pro clus’s’- g geoln etfjcali? oh;jects are furthermore . ensi 0. Space® can projecte be fo und i sor 50 n th

‘gination the


jected in three-dime™”

of inspiratio n, . Syrianus Offe rs

e work of hi

s teacher, w

an enlighte

and their Places, Hi

ning present

ho w

asam ayo


ical ob®

athem atic’


ation of m f i s focusis Proclus, 4 ¢ different f r om t n Aristotle, Sp ecifically on ie crihticaismt o O fa Plaronl¢ hi

“ommentin go

s crit

icism i Maticals as generated simu imultaneo usly wit’ ‘ os

, Syn

Particulars6 6

92a17~21), Interestingly, pt 9 tt Atistotle’s lead by defending a Platonic bu P, « Jocatio” * instead

8€Ometrica ] so


replies by pointing to

ds as ©PPosed to Physical

2 In Enel 56.16 ~18; 9 8b by bag ™POBADSuevoy

the speci ne hat

bodies on the 0


nadnrixds obras aor 8: aye KaOvMEVOS ee ep} Thy d Paipoiue Og; ve $77 Thole oh kalovvnBéveva . opi |: €P Ele kdtontpoy Ohware mel 3 In Buel, 12L2~6: chy Pavraclay mpofepanrats TOUS Tey oxNBdTwV Ber fv td dy Loyoug; cf. 1376-7. eon ent toby Thy éniredoy oloy dtdvorg Yeadouray, TPOP EPAnusvoy niae: kal npd dupdtoy 64 CE thee THis pay Favracies Phasis KE nt Prutone lamblichus,elmig, F Sand OnceptS, oloy bmnédq Kardntpy por opti i ing 0 i 299 inspired Interestiss :

ly, oWever, in lambliches by Mueller’s pe Mely o iy cate we find animage trae miss os : objects ag shadows falling ema Oe sare { oat ence of ; on the ground: » dC? cal enti “a 34.448, lam ®PPearan c ¢@r blichus’ pe ties (, ¢ fi , of 0} {ca p s i mary intere: rst a *ara i themati @ stisthe ontologi® cal Obje nce), but inet! cts in . f h © is certainly also discu iscussing thee soul (the, secon d appear ance),


is the plac her:67 he holds th e (topos) other: at our Te oco ies, just as matt f e r maa red forms. er is the place ofmathemat ical bodies, j n o re bueti s entirely mattered for Physical ms from t

forms on the

matter ‘ona lan

o ms, where as th

d cannot rea e lly po Passive and th imagination ecene i ¢,, is a receptac ssess me vvatical body fro Panther m le for] mat vese knowsit andi rve it to the exs able th

to p t hi

e high

er soul tne ” erarchy of four tent that i Syrian t is a yt ely us then presents a :i the Kinds of places e place o (topo), ve - place of physical ° dihesd,tth he place of mathematical bo enmattered fo f rm


s, the Place 68

reason prin

dies, a

ing in a numb ciples. ds o er of f place is in teresting in Sytianuss disb r e o m a i Ways, First is ght use das his broa ¢, o e0 saps rather in the sai ne might sa n semantic locus; this y,asit Sugges is hee t. That said n passagel , howess informat iv looseness, wh vaterial recept i acle c h ei, us ever, it look s

like Syrian is taking plac ral place, wi e here im th properti : propriat Of sorts, or p c es Svans’ e to the raps “cues distincOntologi

n. cal level in ue stion.NNootfe,m however, that dies a tion of imagin athematical b n physical ation as'the at o sit apd pa rently tered formsis Matter as th conte e Place of enm ith m bo dies with atter and Puts imagina ma

thema ite n on a paw the fact that forms; thelatteirodi le ve ls is clearer, despi stinction °F eu i s Syrianus is not w re hichfour places g to. very esplict a he is wo feSyrrriinan by appe us

aling tothe n The confusi ee the mes, on me Pe hat we besides (nat may call ma tter and e ure and the Source of th higher so e forms ul, resp ctve ). In that ee

67-Note thar Beometry use ‘ nof s superii mposititio figu res. 88 Sytianus in P , Met 18 ar caves ghoavtes ev mepl T0¥ Below Th 6.16~28: of : 088dy Hog eldect Thyohyys He i y ls da do vr ro ac xd la y y hoy Ttéw stbrots waGnyarn r ro v [Bo&a rnep ccig oouaor thy * g tm TOlous ev odex pyes ot aolddeevLa6uea does H thy 18 Ev vn xa wdov vt 7d pa Ur

e Bincardyey ai

Onuarricdy snk guorktys do t

dudarrey i abt y] Si va n baoyfe oupdtoy, v Adyeo. There 5 EvAty eBay adhog icon fo , og pad pa ride 4, wh riis mi is We follow Kro

Seopel abed rel Ba

rs Soverrau, ae

ss ere the subjec Pe t ° n, see also ing). on, Phantasia in (tel heleunrel ante d Mu ee vSrlann: no G. Watson, byauadsediif ng gv. ti on of On im ag yri ination, a Classical Though t (Galway: Galw iver Press, 1988). ay University sity








case, the four “places” § tianus ‘0 distinguishes are, I prop Matter as the f matte place for natural ; bodies (as the comp ound 0 form), nature re as the proper em 4 matter (i .e, imagination, place for enmattered fo rms, math the 4 ct ar , iving aspe“? Place for mathematical as a whole or in its rec etving ‘ he e “hig bodi es, and soul) as the place discursive sai of the immaterial reason principles renders imagination ie, ai P Parallel to the place of matter, burt mathematica] bo dies gather bodies parallel tonatural natural bo athedl tO enmattered forms—which makes more Objects are here sense, because iny,stigated as bodies. + anus Jase . We will First, however, returno Syt! antt | e Cus turn to ic and, a t first sigh! mystifying a rather poetic Variety

his thirteenth

ess ay



ss hier archy i

i In Pro lus’


Gs On Plato’s Republic, of the Ut / on the sp eo Proclus describes Nitive ACtivities a genealogy of \ of five “mowate oO the soul, Presenting . t tive subject rl in, thelir their shared o ve post : 18 Matt “ts Ot methods . of Operation, On the €pi : i ' a their stemo| Ogical vn the i" ladder, and the .

(Republic 8),

correlative leve 5 . lis ho" In three Phases «y o¢ dimensions.” ions.” First, . sou tO intellect. First, . o(v 7 This is ca a imen sio8 sd lled its " "+ Gastéma), . “first 9 line—the Stra . jon, n ight line of em datio ? ff lin eof Teve a1n rsion,69 S self” she d é “ oes in econd, wh evelops @ if en the so s e c o n d herself» by though, u 6 ” “sS| quat ‘ and “becomes plane; She « dimension Moves” » in q en $ pe® : rectangular 3 € wh { es g squar ) thoughe, bu t in an oblong ens he qt Opinion wh valteled Hf » Unequal Cogn PY itive capaciwt e ties being p Omes the thir e! d di

«oh is W mension, whic h is v

88 that come after


her, [soul] d ,; NE a cube u ettin8 t of the squa re life, and be g eepe®

69 orta tion is 6 the "sult : the Metaphysical ing Produ Pr ct Perfec oces. S of &neratio «the pot ting ir self by n fro ry imitating ma higher cause; reversijon 1S its Origin,


Im imag agin inat atiion



i f passive int e.HHect, (for be imag n,ward too, acin tiat is veioin a Ki nd ik th s, ro ugh th es but is wea ding «o th e fall Sand [making] e analothe oblong e into the soli ’ r [ie , with unequal ; similar solids . inking into diis [i.e., ss fo i c h is knowledge f m vide MM]edbeet percep si des— tion, wich is kno a


sal ys! 4 ‘ ose; P


dissimilars, the bodily and the in : co

rpor eal.

Thisi Paseage doe




cation direct descriptio shonrotbeofgifer aclear ininddi ns, butit .isicsa ‘yo: Proclus fe to say the rafofo llowing: ends an i etaP ll el to a hi. erarch y a hierarc hy oftypes of cognitattio th is pa n ati m Ta e te llectual acti‘vvii ty Of tys pes of"e mathh vata en ‘ em tities or Cien ions. ircula sion ur : 'S: compared t o a (s traight and circular) line, disc sive thought and ., aoination opinion to sq‘uare an d p perctangular p Ja2 nes, an d imagin at id aacu ndbere , and an oblon ively, Whatev:er that ception to so g, ee echytisi veca lyl, and m lids, 4 * athemat. m ea : nsns,, thneethe cclose re OF Pnysiea onabnedtween the pl ialanuti s, e between theace *dimensi e teal bodies in Sy thre e- he soonality of pe rian ss 5 “s lid” : :

. “eption and im ag

is suggestive,

in{ ati on a.

unk?or “falle

n into t

in Proclus

ite Sp 2.4. Impossible an d Infinite Spacace e

So far we have been tion.


i of im agina‘ ionality investigating the three -dimension Fintelligible matter a brief “

Before w Before we move on

to further seneyo ; Side on the difference between aside

‘ona Lextension and spa


—— A dpepeiay 7° Proclus Jy Remp. 1 52.4-11, wo of pby \ mpdg voby teTapeévy ped THY ‘roe 8 my trans, ocakAve) tows 38 Bub Kal iymeta ubv ebBeta obr0g yap avow THY voepay +8 Tpatdy tort nee a ep * in ampeodov, wires clovovom 82 ele taut tpey Biavolas

Yeauuh Kurktoutvy Sek thy émor ° lotanéyy Terpayovller 7p25 re Sidvout elva bee ‘ ee 3 tabtdv rung] evel cal Spoiov Bb per ldvoury kwountyy ert oatouoa,“yes Siavolagevovpylba tH ea ravetran Sov pews eK vy in acelveov wariy ad TepOuM ‘ yicw owy 58 obady UREVeOY, dol ment oe ‘ y xup t ceywviichyy Lar lfo vea Kall a " a wv, kabeveoy “avis ls 88 x her’ bive A éninede, THY nelly pnovse"pu ni e yey eve ta vob tls eotiv r 1gaclay yeron (ea eeyo mpoodopyev tv aoe dvopolous l yp ; a 38 mp ele yal dowpderon, ouny Kerde my e Udvlavovey OTEpEONS dvopoluy, THpATOS Ker kell bauriic elvan Ly Suvapeyny, 71 This hicrar chy maybeinspi rean tecraktys. red by the Py

‘thon eette thagorean



' '


For imagination

as well and

to be

a kind of ( imaged) space,e, wen eed oo the dimensions of , or the spatia. l re * ith, the fe . $ : oincidi ng W) h, ce pt 6. ac le c construed—wwe h, possibly boun dle need the dime nsions as su a Rabouin has co . el y w h a t nvincingly show prestl vi n, th at is preci In i some Seometrica ing l mo






constructio ns: constructions i requ iring lines, sucha req need 0al. s Prop. 12, F con : oss str ibl uct ion s of the imp ible ’ infinite 5Pa . jf waft

ce which al so 72 This . ibi allows for actual im poss litie:

Fe the object of study

becauise only che® ip

a ‘ . ited. infinite is thus inferior an to th e limi ‘ofl . omy of imagination, by its infer? . and th ereby discursive 0 ssi

thought, accounts for th ep of “onstruc tions ysi ng ¢ eft infinite lines, Using of t ‘ fecognizing analogy the a rsa icrute®feft darkness without

seeing it, Rabouin imaginatio n Picturin p en finite Jines 8 that which escapes thought, i

Other imposs


. in the “opacity ofspace. 74

dence Upppor ort t o of un p Nent of imagin in Sup derstandin i g the omat erial’ ¢ be: ation as a kind of s tween human che P Pace is to be found in im §ginatio thehe Pa” paru ag n as j activity of th orm d the © Operates e World in: Soul and between in geometry . andacet intelligible matte Matter, vont a : Wnic hich IS is in j fact fact space. L, we will brie pace.”> To best app fly turn cmation ap to the re lation “called Vehi between imagi c] € of d the soul; en We will Mo that which y,Co connects sou | wit ri \ : nt© the the Worlg Soul universal counterpart nt an so' ehu » tO find Out of imaginalof whether 1.

it has something .






72 Rabo uin, «

Proclag C onceptio n» cf be E “ Proclus C vel 284.21 867-25 onceptio , , , MW te da Evel CE bn Enep 286.9. p 285, 5-17,of Proclus Ju Enel, 2911, See also Mm

73 Rabou in, 74

Prehdpg pea Levy,

7s Sce Riel, P



8 4.



7 é TEP OV Oro Sern hey " Ths Gavtaclag 4 THs aoplotig TH tot & re

pe tis

to evaluate

ligible matter.

the candidates


. fora cosmic

derlying assu

terparto Oke . .ctiof intelns, that

mpti an level, we find on o is mutandis, at th Whatever we i , onan sics e of the soul cosmic level, nd * sda fr om Procks na is derive n) "e as

The basic un

set outin his


p. §§184~211) . as used in ee inte lligible mase For Proclus, a r ) the soul. ginst he so-c

Elements e

Brometry isa function 0 hcl(ocalhélemad )p,reoutmans sbele body,’ as th The notion he yscal body, is beat of the soul’ * oreal so st ul with a p which connects Papbors the va

e ves bat is to be known from found also hi lamblichus ts n Plato (Tim. 41 found inspir hicle, for whic ¢ ation nt) is taken either and Phdr. 247bh ama (G to en . 736 ) une ‘he de miurge (I,amAn bit by bit by be created at blichus)i.. lo once by (P wer and mortthale orphyry). It is considere real function eatse in is de s of the kind oflf f the lower, more soul, or thscent co rpo hich facilita e vt tes the after purifica tion, that “euon the one f o n e mortal, pneu ant discingu ishe two venie ss ascent” P s< The latter is throds amortal, e lu Th

minous ven fo senile an rmer consists in eter one ott ofthe ra nal means ofda onal soul; t ° a soul needsintior der to obtain 3 position in the layers of accret ions Sensible world, 78 ibed by Proclu hence pure Imagination is s as an in e“ the dese except pneuion, which tak and est es pLand co a ype im agination nt emplates Matic vehicle: remains in te }


” ‘ eption, osed to pere the shapes a nd the figures in the pheuma (as opp

in na 76 See also Hel ristotelianian Physicssi migig, , FoFe rms an ; J. Opsomer,, “The Integ dComecep h diey” in Phy ‘os. an . .we eetss ofN oveJrs and Indiis phy sic s « ond ; Phieso ivistbil n ioeity, ’ oi a ne Trabattoni in meGree , vl n: Brill, 20 ek nCop ide an (Le tatoni eu Chi sm,nodBl e ara donna an ed, 189~229,

—-Entwicklung 77 H. Dértie a:nd d M, Baltes, ihe: Grundlagen —Syste Baltes, Der Platonismus in Der Antike ” ls De Ursache Aer “Seele” A ismaus: Von Der Phi mten fe sche Lebre Des Platon ISMmUS: ie pwc inamore, Lamblichus an i zboog, 2002)\ hE Fins uife, vol, 6. Frommann Hol=be . ‘ the Toes ofthe Vebicle Sc ess, 1985). ofthe Soul (Chico, CA 78 In Tim, WH 236.31-36; In Remp,Il 107,14fF. Cf. Waiatson, Phantasia.


which contem plates the external ).”? This, vehicle » Consists of 'a kind of air and fire.

c pneumaa,, of the pact

: Imagin ination ion . iissa cognitive cap : soul, wh ich coop erates oning. In the acity of the on Fuc lid , the co mm i relevant en tary As iscursive reas . such, however, ; itie s. As suc h, vn is the all he huhu: man


3- THE Casp op NATURE AND THE Wo RLD S oul In the Previo us sections, Wwehave + peteen retion : istinctio the disti ignored “pace and matt gible ig er in ¢Xp . .

laining the theeejnteo Matter, The fa dimens‘ ionaitylY of ct ofthe Matt sp eaks ° ig er is, of course telligible Matt Pr oc . , th at I er rar}her than Space. ill comp are In the following, Activities of the i "World human soy! to Soul an those of nature argue that for Proclus, e and t ol . a kind Space is . of matter. S ubsequrenuy, argue that intelligible : to matter isoslikewise .

: a kind of sp ace. himself draws a pa rallel between

8S “moulded

by nature” hylomorphismn h 88 objects and matter— Consisting

“ the

t »


fato® (Jn Eucl. 7) , Neop”’ ; prindl™ ;

of °F perceptible a universal nehche app matter, each wit of the” fs s these Principles



are “ature,respectively,descendansF inv So the pat ed and geometrical between in into 4 tellig ible objects, m is a analyv? ter the logoi o and physical han wo r reas ©n principles deriving matter, on the on heat ture, on the : from discursive oth er! Tr i Worth our Oe roe Parallel, while to paycloser attent

davreaty, . TAS vo




avee FTES


}y ody F9

. Karl é K

t voElo Oe

es Hyatt Gewpotioa tig y TO


. ag Kepet ort


D 6

ig alo s 7B by t a. y ernzie fng, 6, 38 trl oyuKhindés V4 25 Bitt o el cag l 4 Tpryovoig moMois,, 2b ub aer etonay K parte O y $ e ix v r tl de ‘agl N o oxhuy bro oOnrie PS 2% r Kel wart t @ s s é voso ;of otdene kal obciy VS te n th g top ay adTo by u eco ie te Bo Suavole Abyos k iy, CE ue 14 j n b © At ws, 682 r B61 istote ‘AN co her: a nt y aor Natural and «exe Of the discussion c o Cat-irNe raryrari ne oncerni S¥chic” objece 1 ‘ d cay. fact ng geom s, Na i-hypostas ig below, he natural wo ture here may be uesterical he Oe ml rld” ” Ir maya d in Soul. Th ¢ a tr is is h ow Lernould,lso beeo” a ,w pends t it. “Imagination, o Ka



soul and its; capacitie tandis.; So these same sou sols ulshave the | sam e capacities, albeit mutatis“< s” muta Wo rl d Soul, Capacities also occur'at . 1 the level ofth. Id So ul —a s bei ng ert D! e ins des ‘ escribes Soul—that is, the Wo— r Il the way through into th the center of the an cosmos, permeating clus i ‘, in in his commentary ; coveriing n it on the to outside (Tim. 34b). Pro an entity intermedi he Ti ‘ ate diate beb explains this World Sou l as a it is both indivisib meen twe en e e the tra le den (in nscend t and en thesible immane i ic ic powers), Present (in nt; its it encosm ) ish ™ ic aspect) and divisible (in its ¢ logically sup eve erior. It has m1 ryw heher e in the re ° he bobx dy of the cosmosWo burltde Soul l asordering the been suggested that Pro cos“ma moss rocclu lus sees the "TOLht ion . is. based d p rima estion athematic:ally by her i maginatio ima : imagi i A.} '8 SUBol E d Soul as “un ily on the rol ra unr o lin } g , h face that Proclus descri rinciples.®3 bes of projection as such The the and Projecting reason menwi tion‘em erely asa synony p m sim es use s it me is not decisive however, ae as Proclus at different contexts. °F f processio ionn.or emaemnat a ion, and in many d i or re quire imaginnat Uso atiion, | ot been a rgued tha t the World Soul does n 1, whereas among othe h r. reaso ns becaus imagination a e it is notirrational, is a capacityof the irrationalsoul. 85 ee


Proclus , Nature, see also i ‘Nature an d Its »Metho ds in « Martijn, mn e: Philosophy TverProclus on Natur 4s (Leiden: of. Commentary on “Integration: : Brill, 2010); Op:Opsomer, Plato’s “Integ win Al h,*Proclus onthe Psyche: Id -idual 2 LE Finamore Wor Soul and the Individual Soul” in and E, Kurash, “Pro: iversity from One: A Guide to ? : Oxford Univ Proclus, tus, ed, ed, Pieter d’Hoine and Marije Martijnijn ( (Oxford: Press, 2016), 122-38, ; Tot6 évrog dua Thy Gayy dneiplayreal SS te Tin, t nT Besge Bary, otk I 79: roaien yap eat f Ge postibly ytwich iou xord peréBacw kal 7p voa av Roy For Plotinusso8t i vb thy kavtiig Lory. Exoveu Benda thy dneipoy toyPret duarho néou y mapotoay. del obv avehtrto View ofthis Process, ve aetvey 2 see Dillon 2013, Ghat, World Soul 4, Cf, Jn Tim. U iy 84 For example, so the : We Z e u s fhm applied *0 Zeus j bn pa m. . 104, Cf, te ct HI 30730-3, no Projecting divine activity hen Timl 79: ORM B96: ating is called ha ‘im 6CF oflogo by thesouls sou.Ob pnTin Hearne(ats eran misses ¢ € preye ie, delivery the sou

Y the projection o

the subtleties, but cf.

Gt Le,

trans, at 148).

85 See Maclsaac, “Phantasia’;

tion, which Proclus does

Helmnig, Forms and

not adopt.

of speeches


cepts, The irrational wo



ul is a gnostic no-





In the followin ton 0 be jprion 8) wewill investigate whether escrip nature of the Wo the d rinciples seh | rld Sou land its projection of reason Pp 4S any inform ation 4s to the e, chit receptacle, the mat ter of spac which is Projec ted, What, in turn, can we conclude on the basis 0 with regard to

the i Magination?

3%. The World Soul’; Pro jection

: . a Lik e the human soul, the World Soul pos n Pp finer($ sesses the “uaso logoi, Prior to their e obj ect

“un Possessingis Cognitive:rolling” into, in this case, sensi although

ivet the World Soul does not ercelV? sens*Sibtb],les as suc w P it+sie ), s) 2f h(itojs hotaf fected by on have knowle ic al ly lower en tolo dge of them “a fter the fact,” gica: lly it dodoes kno" Sensibles be so to speak, cause they are it “causa :

lly included”in+ itte 86 . sactt 4 The World Soul, we read, ’ ojec" ‘ “has the logoi of the sensibles, from itself» (mp0 P is Eauriic, Predicatj ePnuevove Exel tods Adyoug iy Participle).87

TAY aloe

What is this projecting ny gg its of

at this notion of projecting ‘al esis?

(and perhaps in theories of ° frond! of projectio 1 in

Proclus’s epistemology theory of pe ° e by ‘ception,

for example, as des¢

var. ton 85. unitear fits “A pane YThis lambhi { tht ; Lea notthe bef 8 sensibleChean that the gods know the sible © i¢h f reali cir but astheory ™ Ys un ates fitting sen their own essence a Sternally, ete, Onthesour’s i ca “sist! mode,” Now and Ey knowledge, 50 if those things whereof see also ELTh. $195* the cxg ip is the Pre-existent cause it pre-embraces Pre-embraces all Sensible "8 Of material causes poe vel things immaterially,things after the manner of a¢ te of bodily things incorpored) 2 YU) nde do] ly, 0 a tt sh : ay ltl PO 1 sa Th mpdyuare. ... WE Kal ene tT dy wav of cli Bey dpe tloSyra " clus t4rY on Plato} imac maven kor’ alclay wate Honda ane aye re p. el cay Besortasy : ye


ind South ce Part 2: Proclus on the Wo em” “sity Press, 2009), : refers in this con10 a coy individuals (ch de world 5 Pars 970.23~24). Ont Concepys, 226, 239, ar)1

lane IYBook 3,

al-forming Principles

ibles on is" afi 14

of sensib ¢ hones I prefer my translation—and, to

‘ , Simplicius, the soul projects Jogoon . or gan t o match th e to the renory e e.

Sense impression. A


iti oO fit results in recognit ion his context,. of course, as

This is not what Pr oclus is thinking of in t 'h s the the World Soul ha parallel is ges no sensory organs

. But per. a pro metry: the World jects rational So ul , lik e the human ph“e " Principles in a move'that is not pr tim an arilrt co itisve but cteative—and y gnit ; importantly, creati; ve of space. 88 sation (aphantastos)® vst and jus Nature is wi

is without reason or imaginati tos ng'es the reasonputs the /ogoi into act tcnha ang ion: “Tf, then,it is Natu . re ms {ly formed actuality Principles of the sper , m frompotentiality to the ft ey therefore it is Nature ’ , al* a





that has the reason -pr



inc ipl es in ac : theless a e though without reason the cause of or imagination, itis m e tur e does the *eason-princ Hcintipl ac esitn natura ings.”?° Befo l things. ThNae “daiogits work,» monic place ?


" howeves .the Wo. rld Soul aan mes ndofs.the Republic (ers) : in book 10 \epe iot) mentioned in lace in the genuine sense (AdAe aléaléthds Pro clusst“e elim lsonUs, s is extension,i.e., plac e topos). The World Soul, ; : Which has the form ivine and whic al principles ofall h ti ngs infec pa depends on what rts comesbefore itse lf

, impose

s and certain sy l affinity with di mbols fferent ee spaceis suspen orders among the ded

of space a specia

of the various immediately after her, an

gods. for ‘ nae instrument, So d functions as her conn at

ee 88 And like the h

89 In Pary 02: 902:

later. which it more oO n v hich| atics, 1 it h has a ser ‘on g connection with uman soul, s, abit i mathematic Li

ature {is not Nature just unintelligene

inative (0(ob tp BOY ov imaginative irrati nd unimag| but also irrational _ It seem: 5 ant - aand Dec Dub ae ; ere Syrianus

perteshas aa xa Hoyos Proclus copied ea poate). wae eke the term ais aphantastas from Syeanus Pclus devaluates its meaning. Feaing to what Uses it itiin 4 positive sense, it as refe for pure, unimage i orms, Comes below

Phantasia, . 9° La Parm, 792.20~25, trans, ‘ et i"a ig hey 4 Yuet wera! Ram 08 Hhoyes tot Suvapet l ous Morrow: El rolvev 4 bong ontp Heros ele th ee ? dytovera ye bg Adyous: y kar’ evepyeia Si y TOUS y dnhaow, abty av ota cat dddvracr bol wart EVEPY os bpws tor) Adyualtle duor kiiy. 9 See next note, We will ignore the contextof the afterlife ife.


she, being a ra tional and psyc. hical cosmos, brings this 0° "° perfe cl Cosmos br.‘yine 2 . of Space and life through the divi tokens. pat eit: As this exten, sion is immobile, . t f illuminated b “the unmoved rensio n yer9 V the 80ds,”» on ex e could call it ‘ a ace; ir does ot m F 1t ci spe in ples1 as 0OP)ip 8 change, bur it does always Pa rticipate in rational p tension o ve and chat f materia are at time l objects, which do s illumina oes not? * J ted,

© 8eneration of

i, “» pa

rticipating, a

this space is con th the different 'Y comitanntd r*eh adienggeofnetrecodt P€SOf souls, as w e Structi on find in Proclu of lower typ s’s no es of souls b the mixtu ; trOom the remr y f t h re of the i e d e m i ur nd ivisible The and the divisiblege ©m the indivisible (intellect) to (Tin the intermediate the divisible oD ae (p aistinctio” of soul, But € made, f within following

Unit, between

92 In Ti,

the developmentthe realm nn 0 br of mathematica * Aypercosmic "YPe of soul, ires only mh which requ involved in the ¢ ofp sph is hence not involve sons (P roe : requiring two im ft en sion € “Nstruction d of souls, in other words, is at Fhe

TOL sar,

(138, ange, 139 9), trans, Tarrane, slightly Modified: i bet oa dideiipay, Yael Kit8 dg Anise te 33 4 thts obr06, aang aus Oyyf Woadrine § THKoTES oy Tonog Too TevTde i Kerd ce yep ToOTOV buy) Y9US Po Tay Kad ypdyoy Exougy TOY Bdioraybver. ebertov ee Beleay drdvtay me [email protected] eye) ne K Kel eEnptypdvy Hoploy te a ad on Uvauere {718 one UY oboe tt Sade? os nony as "do 1p olkeidtyra ogang xa} kat deUXIKde Map Ea abuora totic drove) as toGroy Btdeornue kell ory drra airf oF 4 ja “ er kel formation aby kbopov Spyavov seth mo Uta also as The ade vine SiaoraTdy inciples? art i: ; tokens Yetsion, are the

as the oy logosatewhich alwaysfunction connected not only | as p provi ine” Binate j and hence DT seems y . t Sia is thy thar Maclsa ring Ost ac, “Phant atadigm 9 TO of fause t to the as Bide Ot ia ” is eke of physic orld Soy 4~10 Kt al space. nos and henc 3) ka or This m e e to phys MPnREvoy, aTeng Klvntoy ka d rey ToROV Ob th ic al Space, y Viiv 088 Veep ayes & T l del Seab aim ees tnd h et Tore pd y dé a md we pa tobtoy ravey Getiy y eer Se ON Tor He ficovari Tpd¢ orreooha rebyey Hyun 94 In 7 todu Elvan ry de rt ev Oedi oyy,weoalrt Ot on Alnas a } Bo so w lo 1 neab ntreae ebnp *5 i°Bton-I L29~a56 esp, rfa ah: r: ba t ngtan, 2 221 o Sedrv. C Ds tag9g drs E with RP Jn R Tira. 1138 , Winnington-I e 129 Ss” In Pro ngram, clys Co Not ¢ Du Preiye" m2 ™mentair e Str Le 1 Timde, vO Ls (Paris 2

* "ensigndivine 0

3 > 2




ith th of dimensionali . s ty, with e fWoorrmiId Soul, Ww)hich‘tha d a n ng the transitiio encosmic a hypercosm ic el on one, ic ement an time the generatiion on

: from one to . enilsions. three dimib it y, which w . : An interest“ her hehere re, e will not i: nvestigat e futurt ing p vk SIDLICY, r Ju s : ac “c tu on al na ly te b Spac m e a n in » s strument,’ Proc e as th is that v e P 1d Soul.9° y h Nature, whic is also the That woul he Wor d instrument b° , nt er o pa f rt ‘ orld ib of le in at te ur ma e ll igi "1 ttther, and the cou mak er, theythweouW be oe Soule thNeatcou inter art of dian ld oeti v »



thought willgetner; P part of briefly consid asio imaginsat the cosmi‘c co ider er a ann. Wee unte r i c l e , other option, the lumin ous vehicle, further on. ?


3.2, Space (diastema) whic h is the a instrument of th The snrension e and coul e World Soul is (dias d be le te ex te ns io n of the c the comp or ne, k bu t On Simplicius Space is not called space,% 8 Sy aces" Proclus's ” Proclusis ively in the Corollary on Beno his view ' the s Si the only phil osopher, . am licius tells us, to mana to innovative be a body, al=di mensional Position that plac extension) e (t hr = ee. as Siorvane “Corporealli beit an immate ght rial s


ph one, Or st ate, place.” rases it, ain argument begins with , Prociuss ms body in us f the distincP "he Aristote lian framew Which he e or. tremities, ne a is that tion between and extension s ape, matt between ex er, ee 95 On this see Mat tijn, Proclus.

96 CEL, Slorvanes, inburgh Un iversity Proches: Neo-Platonic Philosophy and Sciefence (Edin inburgh:: E Edinburg! Proclus: Press, 1996).


si lus’s‘ ownwotrks, se e later discussion. impl nts are Presente d by Simp icius ata fom = Simplicius, Proclu 14.8-6618 i Phys, S 18..7. For . en ius s’s position and thearg retary Nito critically disc ument gi ve n usses a f Good Summar ~ 614.8 a oc space, see C “ “Proclus Sorsbj y see Griffin, Proclus’s po eis on Proclak,s”*«On Proclu fate Space andM on s on Spacti Co © he ’; e as ad PrSc otion en nc k, thhiistJuopis, ochr luen Space”; Siorva s on es betwee s” insePheys C.tca nes, Proclus. On s ae ‘i ‘ACompari n Plotinus Plotinus and ee (Leiden:Brill, son benwe lo ‘lo e r sepnhye oon n an d F, Trabat Natua k ee reiinne e ed. R. Chiatadr Gree 2003),145 who, ¢ oppo however, overemph ‘ . 99 CE Siorvane as iz the op es s, Proclus, x42. en fi the pc e °®





of th © Nece otf. ssity of ¢ . lace. Af qu al it y be tw g een a body andi«.tiss P thin ‘AMantitatively e another so! sist qual


a solid, Proclus ee : h AvIng . to postulate Interpenetratio mari bo ion oO assumes an « j Mmateria] mo body, which isa ‘al kind light: acide special kind Oteover, To prevent

0 5 Scause it sygg ISIr . tuivi no bein has to be su and serve g pe ri or +. mosofu”t as i ntermed to theli |, ng w i h a r i y c h is a between the lessly alive, Wo

and the bodies r , whose life cons ld ormotion: sp , ii ists 0 wdias¥ petw yet immobil e i" : e!02_ 1 _ph

. perfect inte : rn uncerp at Pe These Proper ties are the co expenct™ Matter andits of. all el

Comm en ow lary on the the Nature n elabort Republic: , Pr anceimo of the li ta l C oclus h se. ght

in questi

ific ex

eger” on, albeit in a ention of . spec of Er, Repu g light see cea n by the so blic x 61 uls of the d se 6B),103 e is light, ‘ i! stretching farth, like f heave! of a Pillar; Ve from above throug ry similar h and more to the rai out all vote 0 if P u r e n , b ’ ow, but m Plato Porphyry clus ov) in identify calls “the bond of ; N itas heaven.”!P YI and the COsm . th ro wor ‘6 4 ic Ounterpart e first vehicle (ochém Is a) of the Text: Plato? ‘M

Of the luminous


vehicle o fhum an






; World Soul brings th with the forth itself, wee‘her 2 vehicle whic ,asit holds h the ”106 Tt is superior to he aven “cmiurge, ast 1s vehicle, not to hee isible, bu t on

ly to our mine me meeihen me Thi.c s World Soul,!08 the n osmic vehicle depend physica eye. is

s on t

immobij I € and in i di V. isible aand the seat (hed:7%a) or P I a ce (topos) QO f sta-

i onary ‘ngs, ngs, }09 and moving i thing . intermeThrue space, then, hen, is the vehicle i of the World So ul and aninter. — di ary between soul and oe oses the unity of the vehicle Proclus (apud Simplicius implicius) opp muleItiippliciticyity of of the bodies in it (Ty pe tiv 88 8K meol haiy v Po the Corollary 612.30-31). Inrte wvords the fame PORATED Simpli as the World So u nous vehicle cius, sive with the wh ole, wns nalo is coexten gi of es it be . tween On the basis Projects its logoi of t a e and into hk, man soul ma th ematthe World , between sensib Soul and th e ce ll ig hu ib le ted to conclu ical objects, we matter de that me may be “an in ect of geometry is the vethe spana vo ie by the hana vtant potential icle of the hu n ul restingly, man s so for thInistevi ew in Aristhtoertlee,is vo aces an analogy Source of insp of iration for who “There ive or potent is another light to yck ia l invellect: all ee MePas things: this it is by virtue (intellect) wh is of making < ic Is he 106







W i6aaq 7 ceta, ‘0 ving Se also J, immaterial bodies Passing through jal one cy fl inamore, “lamblichus ehoTeanspae on Lighe and the 5G. cl" "48 OfGods, ed, Hans Blumenthal an 5~64, . 1340, Onspiritual mo tion ton


c 8 Siory: nes, Proc bs Nay kus, Points is impor”f je tion fro even “ONC of t out, this he Main passage 1S} {us if! : etee, m s Ou eaan vesting aMViews, while i i atall rces of Proclus’s theory ‘or Pro as ! . For ica {niet ° Material lower levels1s of the metaphys! 104 Plato a? 6 ity chat “°tporeality, SD,e. 6 , biden. > ichus mo : “™onizin famblier ea and the possibility . obey By"stotle and Plato on tha Tavera ey “TA sh olpayes¢ xal is matter, see yg Terastvov Firnamo eur m8 T™Potepoy g2 105 In Re ‘i ae °ou"a : kal xaQapdrepoy

We wilt


++ €lyeu yeep


Ot go in to tl Ne puzzling : Reumatic suggestion fa ise v thicle as well.


2 Tin “L4t &Q ny ay TY) y TI poeNboto ouToLp4 m. I 2OLII143 8 Pavel 1Te TO7 pl Tov tautii olkov,

Cunpe. bovriig,

wdidrov 88 7

Jn Remp. 1951-1

1, 199.1821,

, no. t 1g 6.2 .9, 197 .14,See als ané-29, 198.2, 109 4 Kemp, o the summary 197.21-198.2, at Jn RempRemp. 11 198.2 No In Cut, ’ anes, Proclus, 3112— 5 ul Phys. 613.7-103 0132729 wich Siorvar ry Prbetween 25 so 56, See also Sio mplus, InP che ways is int i "3 erm in wh edi rvaan ic spa ary be nesd, Si h c Pro Pr clus, 256, and body, 108

Siorvanes, Proclu s, 253-54, is is ug) right to poiint nt to hiloponus’s

the many


ts of convergence i b erween

impoin and Proclus’s's hotions indicates ( 255-56),ichthere is an noti °Fithe space,an has only the heent but stv alsoity, as in , tein not “empty force” 4i voldvoid (whic ron nseane to Proclus, Pr “empty foe of pulling somet roel

shin ing in).New OnDefinition Philoponus’s ime Matter a ofA

eiden: Brill, 1997 69; P, Mueller2 nN Neoplatonism De rasan ae Auton de La ani the cincent | ieee Line XI Du ry Tradition Contra Prvm Jourdan, Gloses ot Commentaire de Jean Philop Dit Matidre Premiére

: . iden: n:Brill, 2011). (Leide ophia Antiqua 125

Du Monde, Philos


MARIJE MA RTIJN 4 Sort of positive . . statelike . light; ‘colours : into actual


: for in



a sense

ote ig makes PP" light

Le jan Syriat may be found SPOnse to Atistotle’s f . criticism of faron ist “! mathematic al , place 4” ar which mu st be be 1 hind Proc lus’s the risto o r } y . A et Occupythe s . ime is CO ne Sytianus b ame place at “est y referr the sam h

Some “Upport

for our j Nterpretation

Tightening g


eneral discussio n des cai ay .






mathematical consciousness, he aging faculty hel and * ° imna )i t e so this spherte rial bulk [o sphere, endo nkos]), j in wed with co-operation were ical extension in the ney f possessesits subeeance Intellect, whil with the will e Foteecits contempla-

and “en ma ig all bodies kes it capabl , e of ™ on tion of " ne wall es yand the totality of them both each indivi as a



ing to “cert o posit tha ain Philoso t” des the wh p h e r s . ” ole univer w. th is Vague?) te isa (PY se,"13 As; fetence to we rill see, the Platon Soul, l4 ist view o he vehicle of # € v Syrianus 8°€s ' gf : on to refer to . ” whi ch is? a Mathematical this “place and tensio Ny s . bo Y but is . ex f chrvedi angii) . . solid 2 if i (in the sense oO sion Perva


€ Mathematical

‘ble, immo

in being intangi 20 € from Fesi. st ance and of oLets hen f a as n d si any pa bi5 lity, 3 Ctiticism He chen P rel oahr of interp ‘ °B MathematYicbaetween the existence l body, w COMpared o f m e and no hich is bot to Pro h help i e Mathematica} body comes som lo to be, w.hen the 1 $5) isi pu fo rth (Probléthe tion (di 07 ve inte . : ntos), in a) in Virtue dis curs! ve we f oF Principle

the vital Spirit imagj (preuma) on d the €rent d in the spirit thes ss Tason-prin.: (for at on c “Principle of i a sphere, es jn . for instance, € com

TOleLy, che s Bi te, lov ie XPou' aTey,

Dillon and oF

wat Syrians scot ah he cosmic sphere with ed and 4 that body is constructed that therean4 moewe NO explicit mention,ve cosmic level, of proj ion, | vehicle, a imagination,17




ination as ‘Let us sum used in a uP Our ma in findings. Hu in lements n sahough : discmurasi ve tho eele projecting e end con igible mat e and incell sist maw& rvel ter igible m y r e c e i v i ng them; :“ ater , sional; inate wMaee fwO- but t tic vehicle ‘aecine “he W imagination Sche in the p ne ‘version oft

the velogo ge orld Soul, on the en r hand, e nerating three si ¢ -dim onalspaceine ,

he same

ee N6 In Mop 85.4~15,

gas el”

di Tponov hp tive Ket “ew SeMer, yap , and, ano at Si anes,A rela SN 1076058 967 Se alsg » "v s, 5 , ck us ro an a 84.277 ense ri rj Sy te a and Prochus’s ma p. ™M 84» ph che d Jz s 5 mort Meta Ing Pa ‘ *St bo argument e ont al Vehicl ssi dy”op Vehicle for the n B ; ob 196) gf gi! e, Wh Ote ° ic h is « Fefetences in SPecificalt for each u &,

the complete and rallel between the existence "he spheric extension ofthe smos and the genera of a mathemadica ° nes tot of choug hr (discursive “(i susie thoug thot : Pody: both have their rly) Two problematic oe vathematical bo teech Yi spect sila ses clea



patticipated Ore materia» so PorphYY (S$205~10), 196. ya et i Cara 1913;9,see Carlier discussi a on of in Remp . 5 On Syti anus, Ta P hys 618. 25~G19,2,


describe ' inous ve.hicle, w hich he ur lumi . ir moves, and ma th” aa eitete ions a ¢ difference t it Ing Syrianus three-dimensio i na. -ccotelilian between ematical vensional extension, ; ematica body, - from age isisa lees and less the aetion, relevant, a s . les . it 0 as i c h Hotion do (of ie of a geometrical kinematic (lo not, For . 2 devel lune, % body,us,ve ob Feats &g.): 18 ze adyoedks Baotarby : el 716 ee vl i fit Foire ky Hiv exne 1 toga tobto Buoyupltere Yeouetpotuevoy or elvat i rae KIVOEOS 7 dvtiKpug Che oe nas mies kal bm ore thvtiy Hpty wos twytotato’ ari Thy owveyvoo evi Tn Met. 86.27,

Tg Ev tote Yewuetpouutvars






Process; Na B ture then . ec activares seated e spac the logoi; and World Soul the sp is its lumi seo fi nous vehicle, €re are ma hing, 2 pro» wif ny re Maining ini questions. identifying ions. For one ¢ human ; Soul’ projject” Magination yf

and of Spa ce


with the World

: , of Course, js Soicle, le, 4as onf that the luminous ot so call, was the; gina higher, immortal a n ime vehicle. Human '} r Moreove’ it : lower vehicle, ic one. the pneumatic hereas °° mer of the World if Soul is immobile, w inteee Tequire motio . nt n (which woul 1ean that i d n fw eeded anothe . in . 18 . r, i ace

mmobile spac to be e May be tw o so. lutions: one, that here

in which; respects,

ye ‘ Fe { g

jmp We “o fi ;

py ec S inferior to the World Soul. Here we m SSar in other


yin Seometry but : hysict impossible in the+4 P ag u ést Would Carlier, to th . $ mations be to relegate imaginati of: € or: Sider calling leve] of Nature, wh g e ich one coul r the Phe d then ed "Matic ve intelligib e A hicl © second


e of the W le M o Will leave atter to the level o f the so-ca rld Soul, d fot * the l l *S e d enmattere undecided Finally, t . wo t he: thought s on the r Jus’s view e l e v a n | ce of e iter though n at! Q e, Tthas “eN s u SS8Ms to “An ggested thar P ticipar © roclus’s view © 0 a A a n tian doctrin "8 (V e


of the s »YWhich link s the p;ure co chemahe a | © Intuit Mages a ncepts©ingte” ion of s re Bener pace an w _, ordi ng ated 5, Ometr . d t i me and r € icg Nce t é a t o c r s c o o pes r Schem nd id Com Me d ata which cor ntary, “PVts, 19 resp Jus’ u en prese °rrow, iin n hihis j “s nts this as s introd oo evide uction phoo weve nt, Schm ? ngo Th i itz, n aPnokweone O l otF hthe ‘

e PaparUesci pants jy O slo for P Vnee N idm ointin t lea e i r nD i nigso . og g his our t n 7 ““ Us chi o me. P nigg, Euk l a t * o n rk ischey Phi lids Geom Alloso ma etrie Un phia m Ortow s o p t h $ i emat d thre ha ,P "roduc ica 8 o e Des Pyoktes tigy, an rockas, te on! " o o , W m, d Notes h ) i r . a burg: “+45 , ind ¢, P. in Boo of Euclid’s% Elements, Transl i ON¢ First Book : "Nceton, of Elemons . , Ti” Nj: Princetonk University Press, 199



we against this parallel, pointing hat Proclu s does no t have out t o thinkers have different ro fi pureingintuition andofthat the tw : the image ions of produci an ieee i he says, goes € P parallel, n important . : role in saat no des deeper tha h n thefact that the imagination ays a play nt Proclus’s Philosophy =o of geome try.beyond eye is chapter, Deciding on thisss e goes the scope of this issu chap course, . be made here is A second point to the jews roclus’s relevance may Vv i be m ; P mat ematics—-and, vice h nitive scientists or philosophers os " Versa, “thet Proclus, not by reading the 1 he 1 we ay find in understanding mi turning t tby kc, What is this nonto contemporary work. More Proclus bu etry? In philosophy ‘1 doing hysical sical extension. geometry? ion the soul uses in doing 8es remsain Of fmathematics, of to be answere ; hematics, as g mentioned, many puz? ‘stem tegardin ology, semantics, on ding the mathematical 2 . PACHICEs a c rlog methedolow, dagogy logic, eck of using thre psychology, an vty »: : 22 In cognitive : dimensions : " s ) (including in reometry, geometry. ster

the cone. the one


: W hat i Wwe may th 1 ne i C25 sanental ro tation: i { of que stions regarding,

b ilding?!3 And does an architect do imagin when she im agesines rotating . . image, ’ whic ich is' wha t an imag is Itit even Possible pos . to imagine ara Bometers do ae considering the manipulating tion eee Ba supp osedSP of picture oe i from the, perception tion otor interaction? Ba : ide ion ofvisually m uide g .

for now, these questions mere ly serve to op en imagination,





new viistas






21 Schmitz, . a ists . f 197215, vpleuedividad the idea fig ure exis ft Schmitz argues that in o: a geome trical ivisible matte! Proclustf onto extended ddivisi existent . which is project an ofa geometrical inSforehandas terms of images, “Simple undivi n of a construction Kant on the he other hand, Critique ‘ pt itself has in pure the of Pure the none spatial Rene intuition nen, “Wie (seeMibr? nis ate conce Kants . €ason A713/Bog 2 32. Onthe comparison, tnzen, . 22 Arana and Mancosu, oo» see also i “Onthe Dimensional 1a ° revit Leonens. itrovié. « : pon ees ista Alberti,i Me nta Rotation, and the igins © f ThreeOmputer ofthe Society ors rectural istorian. hy NO. 3 (2015):1312-22. Mode! . iel of Ferretti, “Pictures, Architecturi His on cts, 14 As Proposed erties and Effects, Motor Rclated Action Prop by ynthese 193, no. Gabriel . Ferre ~817, ’ 13 (2016): 3787 "25 T thank the cdam, Utrecht, in other contributors B17, 1c to the volume and audiences in L Lisbon, Arte would specifi Unich, and Oslo, wh i o n s oft ereI presente di hi s papet. d fferent parts and v ersions 0 cally like to th ank P eter Lautner for hisi suggestio ions.



; e to know d an the pat terns methe°r the spatia hat area l area otal re d lative to thescale ar it e hi n m e a ningfu within tha re


Reflection ANTS IN SPACE

Nicole E. Hell



oeqe The Possibstil ity


. of in ecol

ogicical al aa nfsd evacolenu ti: oniagryhptfoc0t: erns of Organi tral insigh an sms in space is t ecological Scienc a fF evolution st e, Charles Darw in’s in's theory 0 from observations f of finches from the Patt

finch sp cies

Coexisting on

beak size, Finch

Size. Beak



in the Galapagos lands. Isla * the same island ntiin wiere div


acted 9 n “0



ources a nd that this p 4 eo existing p P “weld cter trai' ts {V' o u l a t ions such tha © Pattern o © ” sf t char f birds On the pa landscape revealed a evolution, istory ° the his . Darwin’; Sq


work taught ind that ecology n ion play 9 ut} and evolution Na wwhere p H o l O te r g ra a nisms in ct. How orga ism themselve st?ji” nisms Ppshoel s in sPac e allows infe and their re ncees abo relationshi ut ho p S With ea WE must ch other, ‘Th m w theyvi afere view Pat te ak

e spatia pattern o

©X within

e goo . PPropriate scale, M ost ec ologica Messy or u ies, whe n

rn Sat the

one in ar

eas With

c re Ust make arbitrary d ecisio ns about where t0 dravv Which servations fe

Nicole E, Heller, Ant, © Oxford Sin Sy In : Space, Univers: p. Un: rsity We Press,

will be made, Researchers stru

Edited by, An

drew Janiak ,

viivversity Pr oi ersit 0

Oxford Uni

t’s behavi


or fro evolutionary processes ist seek7 to infer, ecologists

Spatial arena

Workis d


hich ccurri. wien p processes are occutr ng. : Too often ms using ou r we mea sure the lives of f o other organiis sm ns wro,tebyin 8 own human scale of experi.en As John Wiie 1989, e n de ce al . “ ists deal wiwith phenomenathat ¢ intuitively fa iliar, ar nd study sum . an nde weea re thheerefore ch rc ei ve and mopo thro rece st . lint keril: yc to wn scalpes e thhaatt ac Phenomena a le cossrdcowrr itec n h toudrep0en di dency may experi‘ ences;”!onThais ng be more or tenden on vnethe t we ar tudying a mouse or a moose. The scale problem hanges in our ‘Iw ly eacutse as our sense of : daily lilivves icularly a c sp ace hanove cu ar ou . toe bn ol nd larger . . we m through tec nh ogical « innovations. As in it fo h gr rmation more free Spatial wen if a With 8 eater ease and share 1 o examinethe li‘ ay have the at global . v e te e s ndencyto sca €s, w may ha e: Organi: sm Jl s at larg le er of hiss as well. . short Refl In the remai: ofthis ecti.on, I iscuss how ho this s n di d e r hr ; opocentric . as bs th Beneral bile ward stu dyi' ng phen om ena at e ant adtoto bizarre ideas Scale can ea ism, ms Speciificallyk , about etanis w se of the an th.e methods t colony I discuss th devela ope d and ia°lly e case oh applied at which were to reveal th the sp atia initially e ant co ony, 7 scale of g scale of an. t lobal elled to tne movement, Ww h: ave no SWI . humaN moLv ts a nd aermgeunetth ading to new, problematic c , lise en onccep ecte larged sens di theories, d ha dis onn e of ‘t acm ae e thse ec f the an ologic . al d th

ANT CoLonizs

The ant colony is a superorganism.


ivities O fa as single ant are The activit rather Meaningless unles su rvival i $ un derstood as a functii on of the a I John A Wiens “S ” Funct al patial Scal Ecology ing in Ecol Bye Bunn ction



no. 4 (19





ofthe colony. The colonyis the unit underselective pressure. We can think of the queen as the ovary and the workers as the soma. The colony occupies a physical space the same way a humanora plant body occupiesa physical space. But because the ant colony is also a collectivity—a groupofdistinct ant bodies spread out in space—the colonycan bedistributed in a variety ofways at any time. Ifwe were to map every antassociated with an individual colony, and then drawa line aroundthe area that contained that set of ants, this could be considered the colony space in time. Researchers therefore study ant populations by mapping colonies, notindividual ants. Because ant colonies are underground, we can observe only a fraction of colonyactivity fromabove. When there are multiple holes near each other on the ground surface, with ants

of the samespecies coming andgoing,a first step is to determine if chose holes are part of the same colony. A common method to

detect colony membershipis to introduce theants to each other and see how theyreact,called an aggression test. This can be done

easily by taking, on the tip ofa pencil, an ant fromonenest and

droppingit near the entrance of anothernest,or by bringing ants from separate nests togetherina vial. Ifworkers ignore each other, they are consideredpartof the same colony.If they react in alarm, for example quickly backing awayor lunging inattack,the ants are

treated as separate. Early work, conductedlargely in England and the United States in the nineteenth andearly twentieth centuries, described standard ant colonies as a queen and her closely related daughters living together, sharing work and reproduction,in one

nest. Colonyspaceas a singlesessile nest couldberelativelyeasily delimited in space and time. . Over time, as morespecies of ants have been studied in more varied locations, new colony formations have been discovered,

includinglarge, diffuse, sprawling colonies, in which ants are socially connected across manyspatially distinct nests, sometimes

over hundreds of meters. These colonies defy standard assumptions


about eusocialty—social organization in which individuals forgo their own reproduction andaid in the reproduction of others— because they are not ideal-typic family units: a queen and her

daughters, whogain inclusive fitness benefits due to shared genetic material. Rather, these sprawling nest aggregations are morelike communes. Nests contain many multiple mated queensall living together and being cared for by unrelated workers, When a colony appears to be highly diffuse and showsa lack of aggression between ants from separate nests, it is said to be unicolonial.

Theant I studied most extensively, the Argentine ant, is a model

unicolonial species.It is also considered a “trampspecies” because

* has been spread worldwide through human commerce. Thefirst unicolonial populations were documentedin boundedlocales, such

asa large football field or upland meadow. Recently, through the ‘pplication ofvialtests across oceans, it has been discovered that

Argentine ants are nonaggressive over more thanhalftheir global Ley This observation has led to the widely popularidea of a 8 a single mega colony that has taken over the world.”

a coinoe nesitclony is notsocially linked, a key attribute of larger spatial s ‘i “ at, as vial tests have been conducted at ever colony h al scales,t ne relationship between space and theindividual y has become disassociated, The ant colony has morphed from P y b a delimited spatial enti

in reproduction nda ounded group ofworkers cooperating unbounded ato Browth~to an abstract spatial entity: the global, Information a. of workers that do not fightif’introduced. among ants livin out ant colonies that is based on aggression tests

ant colony.I a ‘patil proximity helps us to understand the daily lives. th ater ecause those antsare in contactin their *

Fcy are boundedin a networkofinteractions.? Their


For insta‘ance, M

, « Sws, July 1, seanWales, Earth News: Ant Mega-Colony Takes Over World?” BBC Earth

3 See Deborah M G *P:// “Sordon, Prin, Ceton Univere Niversity shut Pres Encounters: Interaction Networks and Colony! Behavior (Princeton, : y Press, 2010), for an elegant description ofinteraction networksin ants.




interactions influence

son fficane wap : their collective ess ina signif fitn . provide foddeup? or Positively, and na : which Natural thus may «foc ‘ n Selection can thislo ny spac eint . act.

either Negatively

Scenario “Me The colony sp ft s rges from t . ws on of oo catle yen nest h e re lative positi pe c the scale o f ant Moy em ent. is same telation d the indivi sh iyidual €0° NOt emerge f " ip between space r o m an r e sting what ha d ppenss wwhen two ants from. avial. nyjg 18nt kilo

meters apart ar e put in sa that € relative Posi wi! f tion of cooper ative nest arre er directly or r to the indirectly, sh ouldn't .g ny. This vial relations i test then appe tween nests ars to af that is Virtual These a rather than ma linked by phe Possibil teria ’ ofa a pes it y ofmerging 4t no

t because they

or the possibility wrin


TH colony ‘Pace, then, defined erdo not fig" by th When “Ampled of acrogs the glob e set of nests that raphic sc a e, emerges at Uman “Xperi the geog. ence, are in fact me

rging or coop

NOt ant €x perience, € information inferred from ial tests at the the spatial ecology ofvi Blobal] s cale, rather, tells uman trade us about then etwo rk 0 ; ‘ . © twentieth century, as these nts ere cathj W' commerce, a . +563 is Andthe pa fo: onf e ttern of ag 0 gr “€COgnition es si ou t how . an ts ants recogniz 'Si related e othe rd a - | to 8eneti c and env . ironmen tal cues: ww © evoluti iality. But i on andst “ne


ability o © 8lobal-sc f eusoc 1alilys ale Vial te sts tell us ch abou! _ this globa . mu l’ ist

single megacolony ists VvirtUr™ ex oha

n Magina inati tion. Pe

itiis n ot sucpsisit rhapsit Should ©mer eas ge at the s a din “na ame time gr t i o n flows ha 08 Sc eat ve taken o 4M guessinope in t «tetwen n t suc Scalin i e r h centur gP p “Nomena. To Ventureto roblems maybey,rather al, gene rwe ow Somethi inh ng about an other

MS Capital,


is ec it ourselves to ial Las as re resesea archers that w the em bodied NS essentia e commit o atte f th at be rns at different ing. When exp er iaie l ncescalofes, w we measur e t e ceas hing The bu e to, descri, Spatial scal be rden the saame thin es, es ang.d relat . * ionshi‘ps . fa of the mile is ha t rmin process g ict hissctoalde e—te toidenetiW,.f emerge ial arena at y the exte nt of he spatia at whic a t which the dram ‘ . as of an organism life un fold. 's1



‘ the fourteenth century another definitio ofpl istotle ace, vn e n e had considered but rejected, cameto fin d some support. “between definition ofplace as the space (spatiumin Latin) intercept

e “hing thesides of the contai ner that would be void if there were ne located there. The Latin word spatiu m, however,already mn col the


Concepts of Space in the

loquially a place within whic

h it was easy to moves Then fourteenth century hot the in technical contexts Meaning ofNewtonian abs sp at iu m roo Tesrension, olute space or three-dimens iona) c ne on but the Newscientific meanin g of “distance” or “incerval in e e two dimensions—this as part of the development of mixed Mann e ical sciences of motion . This was the mosts ignificant sane was ‘ury achievementlinked to tin spatium. Consequently done in the area of a mixedthemaLa , . thematical science of motion ™ill . be

; Fo ur teenth Centut ry


"ARLIER WORK FOR COMPARISON 149 Edit h DudDud leyley §SylI,la

dis cussed inside hers;d e ase wel or out l as co con of the helepos n oft smsid os,era Thtio ty of e mul y four tip livsib es ili of spa ceeme in "s ading teenth centurywill be ex

i tts ptaerpppeees

ege cease


% INTRopuction

J: ncording : vs ‘Oa document i Philosophical . Concepts, the describinging the seties is edi pl:to “think : thek 4planned as describing of @%° iit sh the , i

meget of «fe life”


*Ominent events itors; were askedfth to helife , concept Us ethaps one sh SinId the

ould mn Atistorel: “oncepe of “space” had not Sentury, ’ ™ HOt Otelian

One a nd ‘le

Methods oy analy


Whete pl acewoe


ge5 it


born. In the fou" ph; vance df * . sar sciener philosoph and the 4two Aristorcelian werelink d io


the ‘ame, and

lifeh of rheche

say that in yet ,been born,

theology by use © dt

se ft

Ytical tools, In Aristotelian natural ilosop"Y philos sth

Ciosely related to “space” was “place” (locus in . dish Ody in pl aken to be the Closest unmoying a Place, as Atistotle define surface surrou” oving ufjf.

ditin book IV of his Physics

DOK tp 1

\ , “tsity Press (2026), 0 f an earteenth Ce 1 97Bor59., 0wo ntury In: Space, Edit rd Univer ed by: A °4:003,.009¢ siey Press, © "cepts of:


amined within the wo rks ofthe outst

Parisian author Ni cole

Oresme. f space in the “re are many ways in wh ich the topic “concepts ° . a interprelater Middle Ages” might have been unders tood. The ‘ation might be to suppos e that the word “space” reffee roughly 0 ers

ic of this * Concept close to Newton's> the topic ° absolute spaces ¥ is at Middle totpe Ages had i ater chapter Would be what inklings people in the ‘rom of what Newton later called the Closed World absolute space. In ve ncept back to ‘0 the Infinite .

Universe, Alexandre Koyré

traced suc acof the Scientific

the Cambridge . Platonists. In developing : the concep to

. ©volution ofthe Seventeenth Century, ae

s Koyré wrot e:


; This scientific and philosophical . revolution . . toughly as bringi forth the destruction


bringing forth t

can be deser ibed

of the

the disappearance, from philosophically an ,

Cosmos, thatis, Co

d scientifically valid

“oncepts, of the conception of the world as a

f ite,


closed, and ,






hierarchically ordered whole... and i

in d its replacement by an”j le’ nite and even infinite universe he is bound which is bo together tity of its fundamenta by . allt in wh ic ! components ts and laws components are pl ,eian a d Eb ng.! aced on the same le . vel oF being. i + Absolute Sp mp l ace » With its phys icatio™*: imp ico-metaphys ica"olft ing indeed the Ne he cessar Y and in‘ev : uenc it ab le conseq the sphere” th veerizati. o? 0 ts e “breaking of the cir cle ” th e er smertia 08! [and]of the discoy,‘ty or as sertion ofthe law o i and foremo §


iom of motia on) .

,J To consider me we i diev . ectl i al Concepts ofs i have tw,© se pace fromthis pers P rious drawba of# . o ck s s, First ofall, it would t later Medi leave ou Ce et it w eval aut ps i? h o r s i c h h a d the t t

o say about p he relevant y Og lace ae term, rathe r than “spa t of medieya h ew -s ce”), Sa l Views on b P l o a ce to medie r ro N absolute Sp val views ace, re A second drawback to for understanding SPace in te Ce edieval conc? late [ ation to m ivi Newton's absolute s vail absolute space . it wo p ace as if Ei is ie that nstein's spec . a re ial and ge al the Pplanted ner ori Newton's absolute space, no to ment es of such T t «an e. The eh ecent ide ie en other a S as wrinkles . “t im Scientifi e, in sp ac e . c Tevolutions Seventeen after the Scientifi ientific Revolu rion th century , © Primary goal ap § Of this Paper, then,is t that to he diseesi0 t





o consider d t er Place in the later Middle Ages atin wor related to he

to the




oF , SPati ¢ in the Se en um, not i venteen n r e lation to what th Centu of telev ry but in isa gett i ant evid off their ow ; e n n right. Vide et lespace fe. T a Koyrg h er 4 g artic” Collected infini .


ay XIVve some siecle? whichof it ina Ton ts 4 et contains °


tothe Infinire Universe




imore, MD:J (Baltimore,

2 Jo

pa #



the works of He translations nry of Gneichard of Middleton, from n in a 1982 b nas Bradward Walter Burl ook ine. Li e ey, and Thon nel Parodi simila entitled T rly medioevo, empo e . m Massimo h rima er translates into ni ng ry-source time and exis ‘le Or Italian hen esme,Nic , John Buri Space by Wi hola dan, Nicole lliam of Oc : nd others.? of Autrecourt William o f , John of Ripa, n schola Heytesbury, ‘0 ic of r writing o space in © most prol n th

e vcles and then v ane and Vacu s books Muicsh llebon No um Ado “he Sciething: Theories u Jrom the Mi nt if ic Revolution (1 1) and Planets, ddle Ages o 9 ), Grant tr gal Cosmos, Stars, and aced 1200-16087 U Orbs: The u 92 from e ace in the a huge num ideas of place, ber o ). T void, an iH o su mm Ne ar wt ize on's Principi High Middle a ree im ‘ges sare possible in e Grant discus ses wo ou enhigh gh to Newton, a a8 ape tes b carrying both his b rant inevit s mie a y s absolu ooks he end point of me te sp voncep . ts of space, Ts On CO discussed N ished paewton, e s on h i s Pers cast a PSc efforts to ov inaingoncepts of space left overcome earlier authors. ONcerning er from ” — poss But this isnot . the subject he . re. han e pat breadth, I de vo p te a ey °F my d a ere sf th discnusosioonsto e Caeeeeo t ane works ny of ian Ni rea cole Oresme, many wheotwenti‘eth Ni in th eth c century. inehiMiddle es in

st lawo T ax


ific anton

, Ina long series

rie, esen

pn Sei

e second halfof th



3 Massimo Parodi,


+ Za

Turin: n: Loes cher urine:

di della Se’ 1€: 24 ( Tem, Pi Oo e §Spazio ELHneddsLOCVDO, i Is Ss toria 1 J nel Editore, 1982 ), 4 CE Edith . "in n Raum uned Sylla, 1 sccellanea ! Me diaevalia bleton and Isaac Newton, alinvorstellun “f , Miscellan ese: ge n im Mittelalter, ed, Jon mun blication, 25 (Berlin; Jan Aeand Walter Specs was awaiting pu theAndreas present Pe Made use ofi de Gruyter, 1998), ES Os at de Peibours o(Suisse). World t¢ Me “Fromthe Clo oe Paper based © prepare a talk for a 2015 blished on as Edith "spate, mower to the Infinite TEMA, has Verse: The oadofNico Evidence oe ble©.Oresme,” ™ttaphysique UniSharfal in Léett, Ribordy, Petagine, Collec and A. £tcosmotogie, ed.'T. Suarez-Nami, O. NO. 86 (Rom e: F] DEM 20:6), 207~28,


‘1 08



These Writings

“Ncompass most


relate dto spi?

i not allo f the top if a8 Word or concept, discussed : { phipilosop™ by late medieval and mathemat n atura, relian ideas"¢face icians, Oresme both explicates Aristo © space or pl ce differe™ é. ace and Proposes understandings “ those of Aris *P essive natn totle, By,ery now and then, to avoiid exc without Clai vecuss e5p¢ ci ming to be comprehensive, i I wilill discus esting relate d work d one ew! tne by earlier thinkers, In section » oe hat Aristot , of wh at follows, after describing w. about place os what mt . in boo K IV ofhis Physics, T will Wrote about en Place j an hereas Aristo 1 hisi questions on that section defined o che tle argued 4 ioio ) that ics re.







Most unMo v

that it migh

ing sur f

t be

place was best de esme © surrou ft define it as t nding concaines Ov e! rhe! he spa

ace of the

best to Sides of th e Contain er that would be void ce intercepwere thet¢, taf introducin if no body g the word tio ion “space” int 35 I will d p l a c e . o iscuss Wha the defini t Oresme tion o ce 1? ° et ©OSMOs in Says abou outs! a his comme t imagina ntary in F ry he r e nch on eattui I will discuss what Oresme Aristotle's On nantill! f did id with Matizing the : with regard to 9 se ‘ence y pio : Meant .

: ists i of motion in ation at that ti ; me not to spat rel but distance 0 pre terval in on three-dimensional dii mensio hisese if i space i n °F Mo re rarely in . two di$ mens 1008S. ¥ Concludi ng temarks , ys Tistotle’s din ma Were Prim P h y s h i c s preser arily Con the ‘Niv ve n ersity f e c t r e e a d c h i aculties tg und of arts, isputation ergra 06 f includin s °n Ar d ° i" g Prepari istotle's W Masters ng s o

of ap

against a


nsequent l



y, thes e



works includ NsWers t ed Oodt o the ques € y | tions they ra offering si is

gni ed, a age Cause a Significan ficantly newer and better eo! : issues and th ei, Tesolution, t shift in the ac “ By the mid-fo ademic consen in POP urteenth c eneuy if im i alj aaa aofOnOc sidera ambl s eTein spfl onue senc s ebaon sedOron onrolo esme as g™ W'

other comme ntato tors son o Aristotle's Physics, WwW, h ether or not thes e later f ommentators agreed with it. O ckham.


I start with Atristotle’s ‘ctarle’

ve . . concep tion of plac € since this is bac keround for ics Aristotle . k rk discussed here. Phy Ar In bo OK 4. of the Physics ich of the ; r " h : —against the mists’ iverse f sists view CON: ato that the a 0 atom


‘les ty .space or void—that

s empty space


or there is .no such thing a : Void, 5 lik a He conclude in) . - ded that be identified nt s “place” a (locus in : the innermost tatin)® nding ‘ body.., The place unmoving and su rface of the surrou o fish Sh in ding in a | lak,ke is the it, an water sur rounding innermost surface the place of the terrestrial of mehe elements ; realm, the —

earth, water, composed o s, and air, and fire so forth, is, the in-; : minerals, , n. , , plants, animals, bun Nermost surface of the orbors hell of ether that rries the moon ‘oun ca arou . the earth

Here the water, and the ether sp here, in “ited the case of the terrestrial in the case of the fish, at realm, are real corpor { substances, 7



i at issue . According to




Ss Whi provide the place: S i ; if People , theyare overlook speak of the empt ing y space within ° otempty at all bu tis Fee With air Ip Su pposing that the worldis tomists had fille hiscommo based scientific n mispercep fve “sory upo Aristotle ha d chosen n me an wh in relation at “place” Pretheoretica| shot : (1) characteristics it is the fir “thing “pntaining that plac st nace i

le’s viev


e () a that of which thing’s prim Should Not it ary ace is th e be less than P s e s ob je or greater ct Should be left in pl th ac es an the ; (4 behind when ) place is (3) hing everyt ally an object which

we Say it is “above m nd(6 ea ” or “below”;aan ch body natu Moves to and remains in its r proper pl pl ace ha d i ace. 5 Thus for Aristotle j i n i n s Many functii ons beyo nd sii mple locat ion. 5 CE Editt Sylla, &s

and Spirit in the Transition from ce, in A ian Scien ee Sylla, Pace risto telian to Newtonia ¢ Dynamics ofAristotelian Natural Century, ; teenth i the Seven Philosophy . tijenhorse, frfrom Antiqu Cc.

iquity tot

: B si“ill, 2002), ), 253+ i vhs Autethy, eth and J J, M, M . H. Thijssen (Leiden







After Tejecting the

three - dimensional ,

Possibilities that

gnat tel

1 ight init place mi h be :* eso the con falwt;

€xtension between r ih, the *h body (since an fourth and i empty space would ie teristics, it is NOt a good candidate not have ningo f“place } Jatt for the mea Concluded th at ataa body’s plac une bl e bo e is “Peth bof e innermost o imh e, movathny” ontact s e contal”ecit wit. at whichit is that is Said inc t © bein plac Heargued that a¥ a e (Physics 212 soldit if it existe 457), d » Might be 0 defined as body that has h an empty space suitable for ct o bodyin i : tle, >

t, butinfact, lr NO vacuainsi imdinosgmo tos. Aris de °F outsid toll I! y Aris wo Fi na e th e fn te ite spherica defined ti me as “ the number aoe .



of motionin resp cto before (220a24-95), Th ec here cannore ig is de fi ni pe ti on of ti. me me unless th hat t.

ere are bodi ant t es Or substa inds t0 humbering nces in m tion J o of mo "Hon. and min : i a : Analogies . lead some emo d spac between time an '0 conclude ay depend up on asf that Space,like : : thoughe, SO m ent? time, thar j t does may not exist in human Conceptions indepen the external orldi

w: £N Aristotl . e' s lt" P hysics bec of medieya ame a standard art of ecuric l UNiversi th p eleai? ties > Most his defin co


ition 0

fplace, but they alt ” ac tive Possib also continued wiseus the Fr) ilitie S$ to that Aristotle fae) te: had considered, 5h dieval Ch In th e most tistian God cepte d an

Aristotelian : worldvi; ew,

ang What Go



II tha

a aS Created, th ; S finite e na tural world an n els. ins!dt ; d ansph and Contained ermost g ere ip of Which Within is ° 5 an out side of ich With no vacua, and out Pty

ural World


22 So

me Medieya

] Mo,

difications ofAristotle's,


Concep tOf

jas tistotelia n conc €pt f s elabor4 te y i of plpl to deal Wit a ace wa la . seser it e h Problems that wereintr jnsic tO ihe entire Osmos “Ncerned i the Place ¢ jg Nor ire cosmos: of the entire “Urrounded 5 atl inc

Vy anything,

: there is noiinner bow



of the surroundi. ng

“etnedthe pla ce dium. What,

dy nd problem conconstitute bo ¥to sup posed toits place. A beat rest vichin . mov of a bo ” of the cathing edralho the place oe

for instance,

bl ws

of the towers


by them, which that Notre Dame 'ychanging?the inner as boundary of the voting the air surro them consra blems suis Aside from ch as thentyOn were Particular se al Ari two e rn stotel *emic and intert ianism wined deve ments within an he acceptance definition of of Aristo

tat had an impact

upon ¢ isemologia

between mathPlace. First, the re a n “Matics and ‘L natural phi ophy as - autonome! Aye rroes, while ”° :on Aris “isciplines, In his mn“Arictode’s totle’s Physi"4 om the claim text, moved .ur always “ming of

ee demons

and mathematica) book 6 of to explain Ax Il magnitudes, the P.hysics that ° Must be contin physica ive e way, to uous claim that whi the alternativ: in the Scometrical = continuous nit udes may u magnitudes physical ad . Fa are

OF Batts that ost continuous. are at m contiguous and analogy betweenn mathematics 4 follows that hy may describe ‘. there isan the physical bod ies and rypercea natural philosophers philosop! , both in a hilosophical Duseenth-century i ofSentence WoroY ks and

in their rat commen es on Peter Lombar f mathematics (a theological ntion the rel in textbook), ations “Ockhornioes asics, The generally or mo vec agree true in mathemati der& not be tru With Avetroes e in cs that w n Physics an Vice hysi isciplines may be base don d vice versa:' different scientific ientific discip disciplin

different Principles,’

. Oxfor d pilosophy (Oxford: ing Point in Medieval ued to as Nedvaae Turning h century, phen ey an ‘la, “vinfnity the Roureenthcent Thoms See Edith Sy’ ford and Ontinuit University ignggehism and phyaration y: Bradwardine ofallanndHy for Contenponax an Ox is Tess volume in prepa edited by Stewart inuity. nina aural a Shapiro onthe historyof f ¢ concepts of continuity. 7 CE Edit h Sylla, : “Averroes Ya, ion: Iteratio “Averro: and the D and dlts . Theories of istinction between mareentheCeneu tics and uty Philosophy tee sics,” in Averroe. Physics; ception a Pel J. M. en: LewyNatal ity Press,oss) en in the Latin University i Bakker West, ed. Paul (Leuv J. j. s I41—g9_ ae StoleA




For many medieval Christian Aristotelians, then,there was nothog (no thing) within the cosmos that corresponded to the word “spy.¢’ (spatium). The things that exist are substances, both material substay4s such as bodies and immaterial substances such as angels or prime move, Substances haveessential and accidental forms, andthere is no subst,¢ or substantial or accidental form that correspondsto the definition of “space.” Whereas for Aristotle and for thirteenth-century thinkers li¢ Albertus Magnus and Thomas Aquinas(herecalled as a group antige,;}, accidental forms include quantitative as well as qualitative forms, for

William ofOckham and mostofthe moderni, termsfalling into the cage gory ofquantity do not correspond to formsin the external world. Nominalistic notions of place or space were affected by noming| istic conceptions ofthe distinction between mathematics and natyy philosophy, because, according to this view, geometric concepts, sch as those of point, line, and surface, do not correspond to really ey isting things within the natural universe. Rather, they exist within uhe minds or imaginations of mathematicians or otherscientists. Whete,s -we could suppose that a late medieval philosopher might posit tha outside the cosmosthereis space as described by Euclidean geometry, nominalist mathematicians would hold that geometric concepts exjst in the mindsorimaginations of mathematicians,not in reality. As far as Aristotle’s definition of “place” is concerned, then,this nominalistic distinction between mathematics and physics might be

taken to meanthatthe ideaofa surface did not belongaspart ofa def:

nition ofsomething physicallike “place.” In manydifferent passages of his works William ofOckham changed Aristotle’s definition of“place’ to refer to the body immediately surrounding a given body in place insofar as it touches the given body, doing away with the notion of 8 See Edith Sylla, “John Buridan and Critical Realism? Early Science andMedicine 14 ( 2009): 21I~4%

See also David Sepkoski, Nominalism and Constructivismin Seventeenth-Century Mathemuticd Philosophy (London: Routledge, 2013); Edward Grane, Much Ado about Nothing: Theories of Space and Vacuumfrom the Middle Ages to the Scientific Revolution (Cambridge, UK: Cambriclge University Press, 1981), 16, 232-34.




its innermostindivisible surface. He claimed, moreover, that Aristotle and Averroes, whatthey explicitly said notwithstanding, did notreally believe that mathematical indivisibles such as points, lines, and

surfaces exist in the natural world.? Intertwined with the ontological minimalism of the nominalists, including their denial of the real extramental existence of anything but substances and qualitative forms, was the logica moderna, used both by nominalists like William of Ockhamandbyrealists like Walter Burley. From the point of view of the logica moderna the truth orfalsity of propositionsis to be analyzed by examining the reference or “supposition” of the terms in a proposition, where there are categorematic terms with supposition for things in the external world, and syncategorematic termsthat affect the supposition of categorematic terms but do not have their own supposition. One mightask,in this

context, whether the term “space” occurring in a proposition is a categorematic term and,if so, what thingsorentities in the physical

world it has supposition for. One possibility was to say that space and timeare real, but they are not things, neither substancesnorqualitative forms inhering in substances. Sometimes the term “mode”was used to characterize such propertiesof things that are notfull-fledged qualitative forms. Anotheralternative wasto say that “place” is a connotative term, referring to relations of many things and notto an individual substance or quality. But the mostsignificant response to make was to say that propositions involving the term “space” should be expounded

9 William of Ockham, Expasitio in Libros Physicorum Aristotelis, in Opera Philosophica, vol ed. R. Wood etal. (Allegany, NY: St. Bonaventure University, 1985), book IV, t. 35, 68 See ase

Ruaestiones in Libros Physicorum Aristotelis, Opera Philosophica, vol. V1, Q. 72. Utrum locus sit aliqua res absoluta distincta a corpore locante; Q. 73, Utrum Philosophus posuerit locum distinctum a corpore locante et locato; Q. 74, Utrum locussit spatiuminter latera continentis; Q 75, Urrumlocussit corpus continens; Q.76, Utrumlocus sit in loco; Q. 77, Utrumsit idem locus numero corporis continue quiescentis quando corpus citcumstans continue movetur cira illud; Q. 78, Utrum secundumveritatem locussit immobilis; pp. 597-611. Also Quodlibeta Septem: Quodlibet I, Q. 4, Ureum angelus sit in loco per suamsubstantiam. William of Ockham B er :

theologica, vol. 9, ed. R. Woodet al, (Allegany, NY:Se. Bonaventure University, 1980), 24. pes



into other techni lly exact + ca propositions, ing itit ¢clea e thae spa making being used syncat e gorematically,10


Beyond the distincti on between the se rate disciplin f mat es ° ro pa matics and Physics and an analysis ined later) oft (to be explai ofent

"Of propositio

ns 5y teference i syne ateg+f qs0f'Hy to ca tegorematic g an dnatura terms an d ny Lp ilo if. their i supposition, ition, fourteenth-century wasaffec inge™ ted by regulations at the University ity o ofParis concern lations o fmas n ters of atts to mast ers of theology. In thecondem a Paris in 12 ined th a ;

77, the bishop of at masters if Paris determin e of arts were vel : not to deny that , lute p™ inf God could, by anything that i God’s s no tal oO gical contradiction. This i absoied 3ssu t ° i is im 4

that the

existence of 4 vacuum is not a logiical ca L

Master should



con contradictiorot ® € al 1st; a

that God could cause av:acuum toeX’” there are naturally deny . 0: or infact . no vacuumsin e cosmos. os. Even. be wh cond emnation : th ‘orate . of 1277,it was established by e Churchh jer God can Cause th rao mira cles to happen : the C rans we s tton of the and does so in , att de

bread an d wine rte enth cen! tu in the mass. In vats modern; Of the fou ten dealt w rose a that a c e of th © assu ith problemsofplace or space m

ption of God

’s abso lute power.


To sum

marize i : There were t least re ideas that migigh a three sets ofuty ide ht t hav h ide affected what Ores phi‘lo lossop op hers sa me and 0 ther fourtee nth-ce ntury about Aristotle's definition . of place. First idea that of all, there was the idea Mere are man . Y autonomous or semi-aut mous scientific discipline based on ex on

perience but orga


nized on theomodel of Euclidean geom onclusions on th e basis

“try, “"PPosin

g of those Princi principles and clemonstrati ng vcometers ha ples. On th ve their ef is Ometric Ptinci ples, which m Lnderstanding corr ay or may n es po nd o that exist to bodies “s Physical world, in cheir mindds and s


have vent ofche , the main requ irement 0 fomdiscipline of geome cing that its theo geometric princi try rems follow necess ples. ar Second, there w il y : an alysis of the trut as the logica mo h or derna and ‘ f falsity of Prop termsfor things ositions usin in the g the supposit World or in ion ominalis the mind. U ts li ke v i i nder this of Ockham ass umed that the on umbrella, n really exist as ly things that “tities in th d qualities. Ansep arab : e external d third, w o rld are there Ww. Or imaginations

accordi dition

subs nces as the impactofth e jostling beta os tween pshil d toopadhymiatndthsh 8 ' which ateoinleadgy-


to the actual wo ilosophers were sup rl r © Potentia Dei ordi d that Godhas createedbysidehirswor hada t in nata), one should Ve don also C

e of could do byhi

af ie mo doe i + tl!

w the word 9 aliquid? alde is used Syncategorematically ing large, quod non “place” : iting, “Sed alio >P on if est. Significabile wri per nomen nec emoweraile eef ego" it, c" vel lad ) d Nec complexe fn ys . nec incomplexe, sed solum vib ‘ He ie i tumest fingere demonstrands ‘hic’ quod veritat? as vel is by? Secundo iine : d non Sicuc patet quarto Seen est, ice, ubiee quo ye < himera mn OF ita est Sicut . metho i" Negativa ignifical , ut ista vera i signi iam mathematicus teideo ta lia pos Sus’ "vacuum! fingit lineam. icend? gn ete: et minus lt ialiter dicendo di d alld $ny proprie in obliquos ‘hic’ : ! Qod locys vel “hi? Ex uw Videtur Universalite d hoe sequ irur but Miraculo usly Possible, thar God could form ee Lah ig body, thatis bj 8ger than this world. Anditis true an at y Would not bein t

hat small bodyor place circ umscriptively or co This should be believed, beca use within the small ara st and in its small place there is

the body of Christ a 4s at the last Supper (cen a) and as large asit is in nnd confeured in paradise, the same way—inde ed in any quantitative part of . the Host, ho age as it Ww

wever small, the re is the who le large body of Christ, optimally Configur ed, But this Magnitude of the body of in the Host in Christ ,net aw

‘y Commensurable to the magnitude of Me “. "0 less could God make a bigger body in the place an Ne ° Bnitude of a millet seed. On account of this I also conc ‘ “a “Within the “Oncavity of the orb of the moon God could also it abo Ya hundred

times larger than the world,


sranging " MaBhitude and figure Wete a bo Yina citcumsc of the orb ofthe moon. riptive way Indeed if t “ measurable by He magninade It could not be larger than it now is, because it wou d pe Or its diameter to be the third part “proxima


Oncavity of the orb of the

tely 0


"Very hi,ms then,

Buridan first assumesthat 8 inside ¢ © sphe if God snnilaeed re or o rb of the moon, what was left int re mension and bodies of allsizes coe "thin


i, although ¢ hey would be there definitively ra

* Ibid, 1g n » 2015), 133 ,


it is difficul


th c ’ [una Uridan Writes: distance between the sides of theevac ated


t to satisfy the Ways appears im agination becauseit alto the imagination that there is space there, as “Pears to an sense th at the sun is not larger M







Gad than circumscriptively. Further on, however, Buridan writes that hering jp could also, if he choseto, put distances in the vacuumnotin any substance:


I thinkit is safe to say that when Oresme and Buridan were writing there were open questions about what would transpire if God annihilated everythinginside the sphere of the moon. Would there be intrinsic dimensionsor not? For characterizing late medieval concepts

umin two ways. I say therefore, since we could imagine a vacu in two ways. Thisis is possible by divine powe + for a vacuumto exist reason; therefore neither believed by me and notproved by nat ural n which this appears do I intend to prove it, but only to say the wayi

ining a vacuumto be possible to me... as far as the first way of imag without erned, I suppose tha t God can make an accident

is conc

eir subjects and conserve subject and can separate accidents fr omth

make an a ecident without any them separately. Therefore God can m it. Secondly, it seems that substance or any accident distinct fro not impossible. Indeed hy for God the penetration of dimensionsis er in the same place without can make several bodies to be togeth that is without one being their differing from each other in position,

(situm). Therefore God outside the other with regard to position sub-

can make simple dimensio n

or space separate from all natural

, natural bodies

outit receding stance, in which or with which, with

imagination of vacuum can b e received. This will be called the first nd way of imagining & previously des ctibed, Then about the seco because Godcould annihi~ vacuum, I believe, as previously argued, heaven and the magnitudes late this inferior world conserving the have. And then the con~ and figures just as and as great as they now cave Of

b of the moon would be evacuated


ofspacein relation to the history of concepts ofspace, one mightsa that in the middle of the fourteenth century a lot of attention was paid to place as a physical concept and to alternative ways of understanding place. Someof these alternative conceptualizations involved

consideration ofthe possibility that a good definition might be that place is the space intercepted between the sides of the container (of the

thing in place) that would be voidif there were no located bodythere This was problematic because the word “space” in this definition did not supposit for any thing (substance, quality, or quantity) but had a syncategorematic effect, referring to a numberof things and their relations. Although this was complicated: from a logical point of view,

ultimately “space” came to be part of mathematical expressions, as in saying, for instance, that in a uniform motion space traversed ‘ vuals velocity multiplied by time: s = vt. For this outcome, space was onder

stood to mean distance.This will be considered in section 4. To sum up: In his questions on Aristotle’s Physics, book 4, Oresme reviewed the arguments for and against the definitions of place as the innermost surface of the surrounding body or as the space between

the sides of the place that would be emptyif the body were not there Heconcluded that, all things considered, the latter definition would

probablybe best, but he explained howto defend the other proposed definitions, both the definition supported by Aristotle and Averroes

and the modification ofthis definition made to take account of the en

ru m, Libri IN-IV, 168-69. “Dico igitur Mod 46 Streijger and Bakker, Quaestiones super Physico quaestione, possibile

dictum est in alia cum duplici modo possemus imaginari vacuum, sicut est mihi creditumet non rattont est utroque modo vacuum esse per potentiam divinam. Et hoc dicere modum secundum quem hee naturali probacum; ideo necistud intendo probare, sed solum maginandi vacuum ess ego apparet mihi possibile, Primo igitur quantum ad primum modumi

potest accidentia separare a subieeds suppono quod Deus potest facere accidenssine subiecto et creare absque hoe quod cum a ionem em dimens simplic potest ideo are; im conserv suis et separat

quod non es sit aliqua substantia vel etiamaliquod accidensdistinctum ab ea. Secundo videtur impossibile apud Deum dimensionumpenetratio, immo ipse potest plura corpora facere simul

esse in eodem loco absque hoe quoddifferant ab invicem secundumsitum, scilict ab h quod unum sit extra alterumsecundumsitum,Igitur Deus potest facere sim licem din vwioners sive spatium ab omni substantia naturali separatum, in quo vel cum quo abe ue ho quodcedar recipi possunt corpora naturalia; et hoc vocabitur vacuum primam imaginationem vi gud


Deinde de secundo modoimaginandi vacuumcredo, sicut prius arguebatur, : 'D. tiebean annihilare istum mundum inferiorem conservando caelumet magnicudines wefet saurleset quantas nunc habet; et tunc concavumorbis lunae esset vacuum.” guras quales et




nonexistence ofindivisible surfaces. Along the way, Oresme describe the basic structure ofan Aristotelian scientific discipline,in which On, presuppo ses principles, some ofwhichare definitions while others ay based on experience, and th en demonstrates conclusions thatfollgy y from these principles. In the case of place, Aristotle had propty} heused x started with commonlyaccepted truths about place, which ionsof place criteria in choosing between alternative possible definit shapingthe While admitting the nominalistic ontological minimalism ts such as spice, discussion, Oresme allowed that constructed concep the external world, which do not correspond to things really existing in understanding thar can be helpful in streamlining scientific discourse, than denotative and such concepts or terms are c onnotative rather is nothing, but it syncategorematic rather than categorema tic. Space discourse, always assuming is a useful term to: make use ofin scientific as equivalent to itions including space can be expounded that P propos

complex syncategorematic propositions.

ions on the Physics So much for what Oresme said in his quest

am’s modification ofit to about Aristotle’s definition ofplace, Ockh the third definition avoid reference to indivisible surfaces, and about

enthe sides thar that defines place by referring to the extension betwe h he thought might would exist if there were no body present, whic that Oresme has be best, Lest one immediately jump to the inference absolute spice movedat least a tiny bit in the direction of conceiving

time of Oresme, the 4 la Newton,it should be kept in mindthatat the ce or more rat¢ly Latin word spatium meant in most cases linear distan have a conception two-dimensional area, so Oresme probabl y did not

of space as we thinkof it, thatis, as involving no body but havingan intrinsic metric in three dimensions. This will be investigated further in the nextpart, but before that, we might look at the evidence ofsome

others of Oresme’s works to confirm that he was notin the habit of thinking ofspace as three-dimensional. In his work On the Causes ofMiracles, Oresme denied chat there

could be action at a distance through a vacuum or emptyspace. Against







Avicenna’s suggestion thattheintellective soul or the imaginative fac-

ulty could move bodies at a distance, he wrote: I say that[ifAvicenna wereright] it could happen through a vacuum

since through a vacuumI would imagine and wish in such a way that a certain thing would fall or ete. ... I say that even through an infinite space [spatium etiam infinitum][it would happen if Avicenna wereright], since there could not be imagining and wishing through just so much and not through mote, etc. I say that he contradicts himself, [which is evident] if one looks carefully at what he says

earlier and at how the soul moves the body, for the mover and the movedare always together in some formof togetherness (see the Physics, Book 8).47 Andin book 1, Question 17 of his commentary on Aristotle’s On

Generation and Corruption, Oresmelikewise discusses whetherall action and passion occurs by contact such that the agent always couches whatsuffers the action.48 He concludes that theteis always contact and

that no medium can separate the mover and what is moved—neither a plenum nora vacuum. Somepeople say that spiritual agents can act with no more than metaphorical touching, such as when the prime

movers move the celestial spheres.“ But no natural agent can act on

a distant body except by multiplication ofits species, which cannot happen through a vacuum; if it could happen,thenits action could extendinfinitely since thereis no resistance in a vacuum.In this version

of the discussion Oresmeexplicitly uses the word distantiam where, in

47 Bert Hansen, Nicole Oresme and the Marvels of Nature: A Study of His De causis mirabili with Critical Edition, Translation, and Commentary (Toronto: Pontifical Institute of Medi eval Studies, 1985), 354-55. I have added some wordsto the phrasesin brackets. “ve 48 Nicole Oresme, . Quaestiones superDegeneratione et corruptione, j ‘ ed, Stefano Caroti (Munich; der Bayerischen Akademie der Wissenschaften, 1996), 142. (Munich: Verlag 49 Oresme, De generatione, 144. $0 Oresme, De generatione, 144-45.



On the Causes ofMiracles, he had used the word spatium.In section 4, I will return to the point that “space” as we understand the conceptis not a technical term of medieval mathematics or natural philosophy,

but before that, I wantto look at what Oresmesays aboutspace outside the cosmosin his commentary in French on Aristotle’s Ox the Heaven, 3. NICOLE ORESME, LE LIVRE DU CIEL ET DU MONDE

Oresme’s Livre du ciel et du monde was written in French for the king and his court rather than for a university audience. In book 1,

chapter 24 (by his count), Oresmediscusses extracosmic void space in relation to the possibility of multiple worlds.” The first Aristotelian text and Oresme’s glossare:



considerthe truth of this matter without considering the authority ofany humanbut only thatofpure reason.I say that,for the present, it seems to me that one can imagine the existence of several worlds

in three ways. Oneis that one world would follow anotherin succession oftime. ... Such a process will take place in the future an infinite numberof times, and it has been thusin the past. But this opinion is not touched upon here and wasreprovedby Aristotle in several places in his philosophical works. It cannot happenin this

way naturally, although God could do it and could have doneit in the past by His own omnipotence, or He could annihilate this world

and create anotherthereafter. ... Another speculation can be offered which I should like to toy

with as a mentalexercise. This is the assumption that at one and the same time one world is inside another so that inside and beneath |

Text: In addition,it is clear that beyond the heavens or beyondour

the circumference of this world there was another world similar but

world there exists neither place, nor void, nor time, for in every

smaller. Although this is not in fact the case, nor is it ar all likely,

place there can be a body. And those whosay there can be a void say that a void is where there is no body whatsoever andit is possible

nevertheless,it seems to me thatit would notbe possible to establish

the contrary by logical argument.®

that a body could be there. But it is the third way there could be multiple worlds that is of interest here:

Gloss: Outside the heavens there can be no body. ... Consequently, outside the heavensthere is no place, no plenum,and no void.” Soatfirst Oresmesimply explains Aristotle’s text. Note that Aristotle and Oresmearewriting aboutplace, not space. Further on Oresmesays: Now wehave finished the chapters in which Aristotle undertook to prove that a plurality of worlds is impossible, and it is good to 51 Oresmehasdivided Aristotle’s chapter 12 into many chapters. 52 Nicole Oresme,Le livre du cieletds monde, ed. A:D, Menut and A.J. Denomy (Madison: University of Wisconsin Press, 1968), 161; Aristotle, Physics, bk. 1, ch. 9, 279217-18 (only the first part of the text).

The third manner of speculating about the possibility of several

worldsis that one world should be [conceived] entirely outside the other in an imagined space (en une espasceymaginee), as Anaxagoras

held. This is the only manner [cest seulle maniere] of another world existing that Aristotle refutes here as impossible. But it seems to me that his argumentsare notclearly conclusive.*4

53 Oresme, Le livre du ciel, 167. 54 Oresme, Lelivre dit ciel, 171.





After giving several arguments against this third possibilicy, Otegne

not conclude thatit is impossible otherwise, as we can easily see by

comes to an argumentinvolving a vacuum or emptyspace:

whathas beensaid (desquelles ne concludentpas que ce soit impossible autrement, si comme ilpuet apparoir legierementper ce que dit est) 5%

to If two worlds existed, one outside the other, there would have

be a vacuum between themfor they would bespherical in shapg, andit is impossible that anythingbe void, as Aristotle proves in th, fourth book of the Physics. It seems to meandI reply that,in the first place, the human mind consents naturally, as it were,to the idg,

In this situation it follows that outside the world there is an empt py incorporeal space:

that beyond the heavens and outside the world, which is not ing,

quite different from any full and corporeal space (ume espasce

Then (donques), outside the heaven is an empty incorporeal space

nite, there exists some space whateverit may be (azcune espace quelj.

wide incorporelle dautre maniere que nest quelconque espace pleine

que elle soit), and we cannot easily conceive the contrary, It seen,

et corporelle), just as the extent of this time called eternity is of a

that this is a reasonable opinion,first ofall, becauseif the farthes,


different sort than temporal duration, even if the latter were per petual, as has beenstated earlier in this chapter. Nowthis space of

heaven on theouter limits of our world were other thansphericali,

shape andpossessed somehigh elevation onits outer surface simila, thi to an angle or a humpandif it were movedcircularly, as it is, hump would have to pass through space which would be empty~,

which weare talking is infinite and indivisible, and is the immen-

yoid—when the hump movedoutofit. Now, if we assumed thay

above. Also we have already declared in this chapter that, since ou

the outermost heaven was not thus shaped or that nature could nog

thinking cannot exist without transmutation, we cannot pro erly

make it thus, nevertheless, it is certainly possible to imagine this and certain that God could bringit about. From the assumption

comprehend whateternity implies; but, nevertheless, natural weston teachesus that it does exist. In this way the Scriptural passa e, Job

that the sphere of the elements or ofall bodies subject to change

26:[7], which speaks about Godcan be understood: Who wrenches

contained within the arch of the heavens or within the sphere of the moon were destroyed while the heavens remainedas theyare,ir would necessarily follow that in this concavity there would be a dis.

out the north over the empty place (vacuum). Likewise, since apperception of our understanding depends upon our senses , a are corporeal, we cannot comprehendnor conceive (comprendre ne

sity of God and God Himself, just as the duration of God called eternity is infinite, indivisible, and God Himself, as already stated

proprement entendre) this incorporeal space which exists beyond the heavens. Reason andtruth, however, inform us thatit exists (nous fait congnoistre que elle est). Therefore, I conclude that God can and could in His omnipotence make another world besides this one and several like or unlike it. Nor will Aristotle or anyoneelse be able

tance and empty space (une distance et une espasce wide). Such sit. uation can surely be imagined andis definitely possible although it could notarise frompurely naturalcauses, as Aristotle showsin his argumentsin the fourth book ofthe Physics, which [arguments] do

$5. This case of points sticking out ofa rorating heavenis in Aristotle, De Caelo, book 2, ch. 4, 2871223. See also John Buridan, Quaestiones super libris quattuor De Caelo et Mundo (Cambridge, valAcademy of America, , 19425; reprint, . New York:: Kraus Reprint, , 1970), , book 1 , MA:Mediaeval Q. 20, 92, 94-953 book 2, Q, 6.

: '

—_-_--7 du ciel,176, i els as elsewhere, Here, Here, as 17 . 176, Il. IL. 285~307;3177. Le divre 56 Oresme, I have modified ‘ in particular where I have inserted the French text,

ati @ modified the translation,




to prove completely the contrary. But, of course, there has neve rld, as was state been norwill there be more than onecorporeal wo



says what In this text there seems to be a non sequitur: Oresme first d, ust would be the case ifGod created more than one sp herical worl roa the word dongques to link his statement of what God cold do oreal spt" sult, but then he seemsto assert that the resulting inco*P outside the world és the case. i

In another passage ofLe livre du ciel et du mon de Oresme rep

this inference from whatis possible by God's omnip

otence 10" ‘

4 is fact in a way that seems intended. In this passag® Oresme is prom

compares the way Godis present in time to the way 6° in space: Text. The Ancients attributed to the

nehight gods the heavens and 4 ].

tegionsas the place which aloneis immortal [or imperisha™


Gloss. And Averroes says here that all laws and doctrines oertt one accord that Godis in the heavensand is eternal. Andthe? othe! the Eternal must be eternal, To aid in understanding chis if how

matters worthy of consideration, I wish,first of all, ¢ oP e pli Godexists in the heavensor elsewhere and thento discs , hell Occupied by other incorporeal things [choses incorporelles] a sit| mode of being. On the first point, the holy doctors have io wit

how Godexists in His creatures throughgrace oF glory 0 “ot fe and how they existin toes Him. Thisis not precisely the proP er ist'



ial ching®

J cop how God and immatetialt theirIr p| place but the question ‘ € Place must be considered; for as Averroes has stat°™ fol¢ say that G Od disis in j the heavens and in His propet pla ce.


57 Ore:sme, Le livre :

dy ciel, 176-79,

1.5 5O7—29, 2


I say that as God is necessarily present without succession inall past, present, and future time by reason of His indivisible eternity, which

containsand exceedsall other time andis its cause... likewise God in His infinite grandeur without any quantity and absolutely indivisible, which we call immensity,is necessarily all in every extension

or spaceorplace which exists or can be imagined. This explains why we say that Godis always and everywhere—semper et ubique—al-

ways through His eternity and everywhere through His immensity... . According to the doctors of the church, Heis in all places

in three ways: by His presence, His power, and His essence (presence, puissance, essence). And since God could not cause Himself to

cease to be for the space of a day or an hourorsince, in improper parlance, He could not not-exist for the space of any time (par une

espace de temps) whatsoever,just so is it impossible that He should not bein all places according to the three ways just mentioned above; nor would Hebe able to depart or absent Himself from any

place whatever, nor could any place be imagined which would be without the presence of His essence where one could say, “Godis

nothere.” This is true notonly ofthe places of this world, but also of imaginedinfinite and motionless space (en Lespace infinie ymaginée et immobile). For, if God made another world or several outside of

this world of ours, it would be impossible that He not bein these worlds, and without moving Himself, because God cannotpossibly be moved in any way whatsoever. Thus, from what has beensaidit appears that God is not more in heaven than elsewhere, in accord

with the three ways stated above, nor simply whenit is a question of Himself. Moreover,it is clear that He is not in the heavenslike someobject enclosed or comprised within the heavenly spheres, nor like something that does not exist beyond the heavensorearth. .., [Here Oresme quotes many sources on the glory of the heavens] God does not need the heavens nor any other place for Himself because He is everywhere—bothoutside andinside the heavens by reason ofhis infinite and indivisible immensity [par son immensité

14 8



3% The Place ofAngels


is bein shebs infinte et indivisible], as we have said; and,as “ai‘e e

no needofanything other than Himself. Tosay t. j so t in God who containsall things is more exact than that . aye heavens, and just as God’s eternityis not ‘ep nden ,dent cessive duration, so then neither is His immensitydepe ; sal ion, , didimension, local distance [ne depentde any extension or l dsuquelt pmlt it

extension, dimension ou distance local], butall depen

since He is the cause of everything by His abso lutelyfree



hath done whatsoever he wished .58

Here, among muchelse not guoted, Ore Low a smestates slsot

reasoning moves from the possibili cy that God could "y at orence create other wor lds outside the cosmos to the ee God is now and always

has beenin theinfinite imagin ary spy ast the cosmos. Here Oresme takes as given what theo logians ho eit about God.IfGod could create bod

ies and hence space anywher , side the cosmos, then Go d would alwaysbe in all such places a ; had He created something there, he wou ld be presenttheres " would not move ther e, because God does not move; therefor must already, eterna lly be there, 59

is Thus, in Oresme’s argument, infinite immobile imagined space ; outside the cosmos, the characteristics of which ment that starts from a Nat resu lt urally impossible Premise andfroman ie then reach?

a conclusion held to be true at the tim

about God, namely that God does not e ofspeaking because ofatt! move. Ag in the case ofthee nition ofplace in book 4 of the Physics, Oresme thus reaches a cond !” not unpreceden



After discussing the presence of God outside the cosmos, Oresme’s next topic is the place of angels,H e compares the position of ange ls in space to that of light rays coming in through the win dow. As immaterial substances, angels are not intrinsically of any corporeal magnitude, Angels can take on differen tsizes, but 2.9 all sizes; an angel cannot have a dimension larger than the diamete, of the world, no

A great many angels could bei n a small volume, as in theB ible legion of demons wascast out of a man (Mark $:1I-20), burt,f or Oresme and other fourteenth-century mod erni, an angel Would no t be ata point, because points are not real buc figments of nation. One might have tho ught that Ores humans with Wings. Even in Pi uring angels as like depicting the prime ctOv ers of the celestial spheres, artists typically sh owed anthropomor hi , angels located at the poles of the spheres tur ning cranks although ; © the text mentioned earlier from Oresme’s De Sen eratione, it was implieq th movers might not be in at celestial prime place at all.Oresme do es allowances for his lay reader Not, however, make s but assumes that q substances, and hence indivi sible or in place defin; Sels are immaterial With regard to the mot ion of angels, Or. Welly. possibilities, considering so [Sme lists different me as impossible. > fourth view states: tt,

60 For ideas of the Place and motion at Paris just before Oresme, Ste Reportatio, ITA (Quaesti:ones , ‘ in secundum libr um Seotenti: arum, Ro* Nani

set “monghis conremPOraties, alt hough it™

, William Duba, Emmanuel Babey,

and Girard J, Etzk

hi ©, Franciscimi de Marchia,

orn CLeuse 23~27, ed. 2010), 75-106, Q.16: Urr um angelus per se sit in loco vel tantuny, quomodo ange et Leuven

Tiziana Suarezlus sit in loco, Att. 2. Utrum angel University Press, ussit in loco divigy uccidens Utrumplures angeli Possint simu . Art, + Urrum et l esse in codemloc

o. Art. +Utrun, an, ili vel tinuo vel instantaneo.Ate, 5» Urru indivisibili; Are, 3, mangelus possit moveri in ‘Mstaney "&elus moveacur motu con-


61 Oresme, Le livre du ciel, 289-93,

58 Oresme, Le hiure di ctel, 27 7 85, IL. 41-6 59 Similar Feas 1; 71-87; 158oningis found i 65 (slight chan n other authors,

63 Oresme, Le livre du ciel, 293-95,

62 See note 39.

Bes to translation),







The fourth way would befor the angel to be moved suddenlyfron, one place to a distantplace, passing throughall the intermediates o, mean space between the two,in whichit would remainfora singh, indivisible momentorinstant. This Oresmerejects:

for such ay ‘This, I think, is simply impossible, for to be in a space

y instant or moment amounts t o not beinginthis spaceatall, becaus in reality, ng in any successive or permanent continuum, thereis, somy such indivisible measurecalled an instant orpoint, as I showed indivis. no time ago in my commentary on the Sentences. Thereis

way ible measure except the eternity and immensity of God, which

y-Four mentioned in the preceding chapter and in Chapter Twent of Book I. thatif angels In writing about the motion ofangels, Oresme is assuming the earth or ftom are moved it may be fromthe Empyrean heaven to If so, the exten. one place to another on or near humans on the earth. celestial orbs or by sion of the path of angelic motionis marked bythe the angels move, the terrestrial bodies (or media) on or through which

less sacred space than Hesays that wherever angels go, even if it is to a the heavens or the heavens, Godis present to them. God does not need

outside any otherplace for himself becauseheis everywhereinside and sity. Thus the heavens by reason ofhis infinity and indivisible immen

and space, angels provide anotherarena in which to reason aboutplace

In the case of Oresme’s definition of place in his questions on Aristotle’s Physics, the problems were, first, that Oresmepresupposesa condition (that no bodyis present between thesides of the space) thar


nothing. In the case of our understanding what Oresmesays about infinite space outside the cosmos,the problemis that the spaceis said to be imaginary andthatit is said to be infinite. How can we understand what is meantby “infinite imaginary space”? Wouldit be better to say “imagined space”? The meaning of “imaginary space”will be discussed furtherlater. In the same decades that Oresmediscussed infinite space outside the heavens, Thomas Bradwardinealso argued for imaginary space outside the existing cosmos. In his De Causa Dei, Bradwardine also emphasized God’s relation to imaginedspace outside the cosmos.® Like Oresme

Bradwardine wrote of God being present throughoutan infinite space in an unextended way: Therefore Godis necessarily, eternally, and infinitely everywhere in

an infinite imaginary position (sitv), whence he can besaid to be truly omnipresent as he is omnipotent. For a similar reason he can be said to be in some wayinfinite,infinitely large, or of an infinite magnitude, and even in someway, although metaphysically and im-

properly,extensively, For he is inextensibly and undimensionallyinfinitely extended. He coexists all together wholly with an infinite magnitude and imaginary extension and with any ofits parts. For

this reason he maysimilarly be said to be immense and not meas-

ured, nor measurable with any measure. Here attention should be paid to the words “evenin some way, although metaphysically and improperly, extensively”; “he is inextensibly and

undimensionally infinitely extended”; “Hecoexists all together wholly with an infinite magnitude and imaginary extension.” Is Bradwardine

is naturally impossible and, second, that an empty space seemsto be 65 Heis the main author whomKoyré commendsfor advancing the conceptofabsolute space. 66 Bradwardine, De C. 64 Oresme,Le livre du ciel, .95. No copy of Oresme’s commentary on the Sentences is known.

Spacey” 214.




Pelaci Dei Contra Pelagium, 178~79, as translated in Sylla, “Imaginary




saying that God and/orinfinite space hasintrinsic dimension? Mo,



Notall fourteenth-century philosophers concluded that there \, infinite (imaginary or not) space outside the cosmos. In contrast ty

If God annihilated everything inside the orb of the moon there would be nothing there, not even a vacuum or an empty space. It should be held on faith that God could miraculously create there in the magnitudeof a millet seed a huge body, greater than the existin

orary Johy Oresme and Bradwardine, Oresme’s older Parisian contemp

world. That body, however, would not be there circumscriptively, the

on thislater in the paper.

Buridan concluded that probably thereis no space outside the cosmoy,

waybodiesare in place, but definitively, in the same way that the lifesize body of Christis in the Eucharist, totally in every part. Meanwhile

If Gog bodies there, andit is unlikely that he would do such a thing. wanted to create more things, why would he not have made the ey. isting cosmos larger rather than creating another world? d, This is in the same question of Buridan's Physics already discusse firs; The where Buridan asks whether there is an infinite magnitude. Buriday principal argumentthat there is an infinite magnitude (which ang rejected) was that God could make a space outside the cosmos, make t canno if God makes a finite space, there is no reason why he tejust larger spaces ad infinitum. Buridan respondsthat Godcancrea within ble as large aspace as he chooses.°Just as a vacuumis not possi inside the the orb of the moon, even if God annihilated everything existing orb of the moon, so analogously it is not possible outside the

the orb of the moon wouldretain its spherical shape. Ifwhatever might

d to creat, because God would create such a space only if He intende

infinite power can cosmos. But what of the argument that Godby his

e to do these things? Buridan answers that althoughit is not possibl demonstrate that God could not have created an infinite space outside

of the cosmos, he (Buridan)is of the opinion thatthere is no space Or magnitude nor another world there, because God would have had no reason to create one ther€. 69

67 John Buridan,Quaestiones Physicorum, book 111, Qu, 15 (Leiden 2016), 1333 (Paris 1509) f£. s7vb.CE

Sylla, “‘Ideo.” 68 Buridan,Quaestiones Physicorum (Leiden 2016), 1403 (Paris 1508), f. 58rb; Thijssen,Joba Burdian’s Tractatus de Infinito, 21. . 69 Buridan, Quaestiones Physicorum(Brill, 2016), 141; (Paris 1508), f. 8b; Thijssen, John Burdiar's Tractatus de Infinito, 16-17.

subsequently be put within it werein place circumscriptively, its largest dimension could be no more than about one-third of the citrcumfer-

ence as is true for any sphere, butthis is not the case for things in place determinatively.”° Summing up the evidence that has been surveyed up to this point I conclude that Oresme and his contemporaries were testing and

modifying what Aristotle said aboutplace using the tools of the lo ica moderna and especially the method of counterfactual hypotheses According to Aristotle the cosmos is a finite spherical plenum out-

side of which there is nothing, not even empty space. If this viewis

confronted with various truths of Christian theology, such as that there are angels that move, there is transubstantiation of the Eucharist such that there is the appearance of bread inhering in no substance

but especially by the assertion that God byhis absolute power can ot could have done anything that is not known to include a logicalcontradiction, whatare the implications? Onelesson that emerges from

examining what Oresme and Buridan argued is that one cannot just change,add or subtract, one premiseorprinciple ofa discipline and expect to find the unquestionable implications, because in reasoning additional assumptionsare usually called for. Oresme and Buridan were affected notonly by the injunction not to deny to God any possible

70 Buridan, Quaestiones Physicorum, , book 3, Q. 15 (Leiden i ‘ ti 2016), 135-373 (Pari ; Thijssen.John Buridan’s Tractatus de Infinito, 18. o75 (Pars 1308), & XXX:




action that does not imply a logical contradiction,butalso by their acceptanceof thedistinction between mathematics and physics and},

ofplace shoud ontological minimalism. So, for instance, definitions as surfaces, such not involve nonexistent geometrical indivisibles

quantities fap Moreover mathematics is seen not only as abstracting o-physiyg} matic mathe natural things but also as creatively constructing as a mathemg,. subdisciplines. Space (spatium) could be understood


required for a journey from oneplace to another, as in Genesis 30:20 “Et posuit spatiumitineris inter se et generum triumdierum” (Then he

put a three-day journey between himself and Jacob). In fourteenth-century English, the word “space”also had connotations of the ability to move: ifyou have space, you have room to moveeasil In “The Parliament of Fowls” Chaucer wrote of thesituation when “i

the birds gathered on St. Valentine’s Day:


ical concept concei ved by mathematicians. This means that of as causal agents jp ical entities need not immedia tely be conceived

For this was on seynt Valentynys day. Whaneuery bryd comyththere to chese his make Ofevery knyde that men thynke may, Andthat so heuge a noyse gan they make, That erthe & eyr & tre & euery lake So ful was, that onethe wasthere space for me to stonde, so ful wasal the place,7!

expression “imaginary the outside world. Looking at the uses ofthe that imaginary space space” within this context leads to the conclusion

g conceptual within the minds of _ for Oresme andothersis somethin

stood to be somemathemat icians or other theorists. It is not under space neverthelags thing wholly in the real or outside world. Imaginary d mathematicalscienge, has an important function a sa part of a mixe cal entities as imaginary May However, this classification of math emati sional! Indivisibles with no not apply to things that are three-dimen on e or two dimensions dimensions (points) and indivisibles with

whathas (lines and planes) are imaginary (think mathematical), but

to be real andto exist three intrinsic dimensions is a body andis taken dimensions is also in the outside world. The ironyis that whathas three a body. This leads to thelast part of this paper. AND USE 4. THE EVOLVING MEDIEVAL MEANING OF THE LATIN WORD SPATIUM

I want to reemphasize that before thelate Middle Ages, the word

spatium was nota technical term of physics or natural philosophy. In nontechnical use it meant an uncrowdedplace or a place where motion is possible. In the Vulgate Latin translation ofthe Bible, onefinds,

Here in Chaucer's English, the words “space” and “place”are usedi close proximity, but “place” is a rather neutral word connected to loca tion, whereas “space” connotesthe possibility of motion as opposed to .













i Onealso finds in i fourteenth-cent : bbeing g hemmed din. ury English texts

space.of time.” Wha t about mathematics? ‘ ratics? In the Latin i Euclid j one does not find

what wecall “Euclidean space.” The word “space” (spatium) appears “


only rarely, and when it does appear, it most often meanslinear dis tance. Forinstance, in the Latin translation of the Elements ascribed

to Robert of Chester, some form ofspatium appears (in addition to appearances in proofs) in one postulate, one definition, and two enunciations of theorems. In Heath’s English translations the Greek wordstranslated into Latin as spatiuvmare not translated into English

for instance in the book of Isaiah, 49:20, “angustus est mihi locus,

fac spatium ut habitem” (Theplace is too small for me. Make space where I may dwell), Distance in space might be measured bythe time

71 Geoffrey Chaucer,, The ParlementofFoulys, ed. D. S. Brewer : , lys, ed. (NewYork: Ba p. 89,Il. 309-15. Onethe or ‘uneathe’ is translated “scarcely.”

tmnesand Noble, 1960),




or some other word.” Jp as “space” bur as “distance,” “complements,” -dimensionalextension, noneofthese cases does spatium mean a three passages in which the Worg Somethingsimilar can be said about the spatiumis used in the proofs of the theorems. an original fourteenth. In Bradwardine’s Geometria Speculativa, spatiurn like. century work modele d on Euclid’s Elements, the Latin

e-dimensional extension. Bradwardin wise means distance, nota thre


s meeting

Euclid’s.”8 Concerning line Postulate 2 ofbook 1 is the same as

le space (totum spatium) whicmh,h at angles, Bradwardine adds, “The who. . . pointis equalto four right angles." any : nds rou sur e fac sur ne pla any in n of book 1, Bradwardine writes. A Concerning,his seventh conclusio 7

angular and equilateral. To fi figureis said to be regular whichis equi space (spatiun)

to occupy the whole place (docum) is here said to be e.”” Sinceitis plane angles that are that surrounds some point ina plan three-dimensionalandin fact seems at issue here, spatiumis clearly not

ine first conclusion ofpart 2, Bradward to be angular. Concerning the

Chester’s Redaction of 3 [2]; Robert of Chester, Robert of 72 Four cases: (1). Euclid, bk. 1, pet. rt L. L. Busard and Menso Folkerys Hube ed. n, Versio I ard Addl lled Euelid’s Elements, the So-ca umiibet occupa “Item super centrum quodlibet quant (Basel: Birkhiiuser Verlag, 1992), #15: nts, ed.and tans, Eleme ofthe Books cen Thivt The d, re”s Eucli spacium [Siaerjert] cicculumdesigna center and dis. any with circle a ibe 199: “Ty descr Thomas Heath (New York: Dover, 1956)» 15-44 que ctea eorum spacii logrami paralle , 129, “Omnis tance” (2) Euclid bk 1, proposition 43: Busard necesse est” (This andthe supplementa equa sibi invicem esse diametrumsunt paralellogramorum Books, or perhaps mismatched); Euclid, The Thirteen following texts appear somewhat garbled [in equal are r diamete the about lograms ofthe paralle 340, “In any paralellogram the complements spacti logrami paralel “Omnis 131, , ii: Busard

def. area, spatium] to one another.” (3) Euclid, bk. 2, m diametrum mediumsecat paralellogramacirca eande [xwplov] ea quidem que diametros per trum consistuny diame m cande circa e umqu amor grs consistere dicuntur, Eorutim vero paralello Books, 379,

nominatur” ; Euclid, Tbe Thirteen quodlibet unumcumsupplementis duobus gnomo ofthe p arallelograms aboutits diameter “And in any parallelogramic area let any one whatever Euclid, VL.23 [24]: Busard, 180: “Si in 80 with the wo complements be called a gnomon.” (4)

mosimile spacio paralellogramum parciale distinctum roti para lellogra

atque secundum suum


writes, “Circles whose diameters are equal will themselves be equal Andbecausecentre is overcentre,so also will circumference be overcie cumference, andthe whole space (spatium) over whole space... wherefore they are mutually equal.’”° Here spatium means “area” In part 4, Bradwardine comesto thefilling of place (Jocé) by regul solids, so there, if anywhere, one might expect spatium to be the c. dimensional, but Bradwardine uses spatium analogously for the plane andsolid cases:

Nextit is necessary to see what is said aboutfilling place and which of the regular bodiesare fit for filling place. Both mathematicia and natural philosophers are concerned about this, as is evident from Aristotle in the third book ofDe coelo and fromhis commen. tator Averroes and through this a more useful skill in the matter 5 argued.It is necessary to takefilling place in solids analogously wi h

filling place in planes, which was spoken about in the cha ve 3 lines in thefirst part,for just as thereto fill place was to acct


wholespace that surroundsa certain pointin the plane, which ol nes aboutby four right angles in formorvalue, as was said there soalse hereto fill place is to take up one whole corporeal space ( at mn corporate) that surrounds a pointat which threelines mall rine tersect at right angles. And Averroessays that the paucity ofwath oe filling their places is the cause of the paucity of bodiesfillin chei places... on account ofwhich Averroes seemsto posit that onl the cube andthe pyramid amongsolids fill place, for the cube in cot : realfilling correspondsto the square in superficialfillin


therefore that according to truth the cubefills place, butaccording to Averroes’ opinion the cube and the pyramid.’” “

Books, 251: “In any paral illius esse fuerit, circa eitusdem diamecrum consistit”; Euclid, The Thirteen



both ro the whole and to one anothen™ lelogram theparallelograms about the diameter are similar George Molland (Stueegart: Franz trans. and ed. iva, ia Speculat Geometr dine, Bradwar Thomas

Steiner Verlag Wiesbaden, 1989), 22~23. 74 Bradwardine, Geomerria Speculativa, 26-27. 78 Bradwardine, Geometria Speculativa, 34-35.

76 Bradwardine, Geometria Speculativa, 61-63, 7 Bradwardi I space”ardine, j Speculativa, j 134-35. In the translation “place” al ala Geometria : place” always translates /o,

ys translates spatinum.






In the chapter on spheres, in part 4, the word spatium again appear,

Twelve equal circumposed spheres touch a single sphere... Becays, there is space in every direction (spatiumest utrobique)against the sides ofthose six spheres arranged in a circuit ofthe principal sphere. it is easily shown that just three spheres can be taken in one space and three in the other. Andsense indicates this, for, when we make ap. thirteen equallittle spheres from wax wesee that twelve can be

plied around the thirteenth in such a way that each touchesit below and together with this touches four of the lateral spheres, so that


then they will meet in someintrinsic instant of the hour andin any suchinstantonly a finite space (spatinmfinitum) has been traversed and the whole radiusis traversed by the two together, therefore that whole is composed of twofinite lines, and thereforeit is finite. On the samebasis it can be arguedthatit is impossible for an infinite space (spatiuminfinitum) to be contained between twolimirs.® In a similar context of his questions on Aristotle’s Physics, book 3, Question 17 (Whetherit implies a contradiction for an infinite to be between two limits), Oresme writes, “Either two lines starting from

the Latin

points Cand B [on theinfinite circle] are parallel, and then by definition they will never meet and so they will not meetbefore the center, or they are notparallel, therefore by definition they will meet ina finite space(ad spatiwmfinitum) ot before they are extendedin infinitum! In sum the word spatiumin the mathematics of Oresme’s time most

translation ofEuclid and in Bradwardine’s Geometria Speculativa,t


often meanslinear distance. Sometimesit may refer to an area or sut-

is no special linkage ofthe Latin word spatinm with three-dimensional

extension. What about Oresme’s use ofthe Latin spatinm? In his Questions on Euclid’s Geometry, Oresme rarely uses the word spatium, and when he does, it seems to mean distance. In reply to

face, but it rarely refers to something three-dimensional, and never to a three-dimensional “Euclidean space.” Interestingly, when, in fourteenth-century Latin mathematics or physics, the word Spatium

five points,78 the contact of any of the lateral spheres is accordingto

Here Bradwardineis clearly imagining a three-dimensional situation, andyet the word “space” does not have a clear technical sense differene

from space in a plane or angular space around a point. Thusin

jr Question 5, “Whether according to mathematical imagination

the sense should be conceded that there may bean infinite circle, in that from this a contradiction does not follow”7? Oresme writes:

Let Socrates be at the center [of the infinite circle] and Plato be on point B, and in thefirst proportionalpart ofthe hour let Socrates

or spacium occurs, modern translators into English tend to translate spatium as “interval? recognizing that the Latin does not connote three-dimensional extension,as does the modern English “space”82 In quoting English translations of fourteenth-century Latin texts, then, I have modified the published translations to use the English “space” wherethe Latin hasspatiumso thatthehistorical usageis clear.

traverse a foot of this radius, and in the secondjust as much,and so

forth, and Jet Plato come in the same way moving toward Socrates,

80 Oresme, Questiones super geometriam, 114, my translation.

81 Oresme, Questiones super Physicam, 419; Kirschner, Nicolaus Oresmes Kommentar zur Physik, 291 a ‘ mytranslation. 82 This is the case, for instance, in the English translation in Thomas Bradwardine, Tractatus de 78 Bradwardine, Geometria Speculativa, 142-43.

79 Nicole Oresme, Questiones super geometriam Enclidis, ed. Hubert L. L. Busard (Stuttgart: Franz Steiner Verlag, 2010), 113, “Utrum secundum ymaginationem mathematicamdebeat concedi, quodsit aliquis circulus infinitus ira quod ex hoc non sequitur contradictio.”

Proportionibus: Its Significance for the Development ofMathematical Physics, ed. H. L. Crosby Jt. (Madison: University of Wisconsin Press, 1966). Where Bradwardine writes “spatium superficiale,” Crosby translates “area” (128-29), and where Bradwardine writes “spatiumlineale” Crosbytranslates “linear interval” (130-31).



4.1 The Word Spatiumin Oresme’s Mixed Mathematical Works: Commentaries . But if one does not find in the fourteenth century spatium Meaning y, in mized three-dimensional space in natural philosophyor geometr lsense meanj,, g mathematical works one doesfind spatiumina technica on. one-dimensional (or rarely two-dimensional) extensi starts with Oresme’s commentary on the Sphere of Sacrobosco (especially geometry) two questions related to whether mathematics or things(entiti.3) concerns concepts in the minds of mathematicians the first question, in the real world (in rerum natura). Concerning onein which it is said “Whether the definition of ‘point’ is a good Oresme saysthat there that a point is that of which there is no part,’ that a point is notsomeare two defensible (probabiles) positions, first, tion thing in rerumnatura but only some thing feigned by the imagina ipan s pointi d, that a (solum fingitur per imaginationem), or secon divisible accident which is placed in the category of quantity tedueof the view that a tively, 8 Oresmefirst gives twelve arguments in favor those who want to point is not somethin g in rerumnatura. Then, for twelve arguments. hold the second position, he argues against those

d’s definition.®* Bur at The question ends by seeming to support Fucli definition ofline is a the start of the second question, “Whether the extremes goodone, namely that a line is length without width, at the ie, in above( seen of which are two points,” he writes that it has been surfaces are question 1) howit may be sustained that points, lines, and

certain accidentsdistinct from body in the genus of quantity, and that

can now(in the second question)it will be shown howthe other way be sustained thatposits that lines, points, etc. are nothing,butare only


imagined to be.® So that this position may be better understood, he argues againstit. To the argument that, if mathematicians imagine things, such as points, lines, and surfaces, that do not exist, then their imaginations

are false, Oresmereplies that to imagine somethingis different from believing or having the opinion thatit is so. Mathematics is a hypo-

thetical science (scientia hypothetica), which asserts, for instance, that if there were a line, it would be a length, etc. As Averroessaid in book 2 of the Physics, things need notbe as they are imagined by the math-

ematician. Mathematical propositionsare not to be takenliterally (de virtute sermonis). To imaginea line is different from imagininga chi-

mera, however, because by propositions aboutpoints andlines in geometry, we can know the measures or proportions (commensurationes) of things really existing.®6 From such hypothetical or conditional

propositions we can come to know categorical propositions,as is the

case in astronomy, andasis true in Aristotle’s De Caelo, such as that the heavenis notinfinite and thatit is spherical, and similar propositions. The terms of geometry can bein the category of quantity even though they signify no existing thing; for instance,if there were only a thou-

sand things in the world, nevertheless a number twiceas great could

be said to be a species of quantity.8” Things can be imagined which cannot possibly exist; even so, they need not include an internal contradiction because in showing the impossibility of their existence additional assumptions are made. In De Celo, book 1, the Commentator

calls such things possible false (possibilefalsum).88 Here Oresme, in pointing to Averroes’s commentary on the Physics, book 4, comment72, raises the issue that some things may be impossible even though they are not self-contradictory. This means

85 Droppers, “The Questiones,’ 44.

83 Garrett Droppers, “The Questiones de Spera of Nicole Oresme: Latin Text with English Translation, Commentary and Variants,” PhD dissertation, University of Wisconsin,1966,12, 14. 84 Droppers, “The Questiones,” 42.


86 Droppers, “The Questiones,’ 46, 48. 87 Droppers, “The Questiones,” 48. 88 Droppers, “The Questiones,’ 50.



1 that it Is


» e might have thoughtj * more difficult than on


4.2. Oresme’s Mj mes Mixed Mathematical Works: Treatises


Beyond ¢

or peat could not do based on whack bie What


in mind Aristotle’ : nes cannot bea Vactiumbe,has were one an an elemental bod ‘ ecause if chere

ody would move. might move in it, then #7 mua fast in a vacuum and

cannot be, given



©ase involves noti m es for the speeds of motion. Bit eos the¥ a taking a motion orate howfast the body moves in

al rofl that occurs 0 ap ° mobile velocit same the wi 0 propor yis that rule ating ise, using the bytes Ividedcalcul sect

medium won pt how much rarer the the to ond rtesP fheProposed case ; posed velocity in the vacuum: ft

rarer! ot the Mobile to move thatit wouldtake a medium

wt its supposed to move ne Averrens: the rarest at medium. This cas¢ thus} alls a false possible



the terms in § “

acrobosco’s Sphere that teferton 140


earch or

s, © saysetsS far as innermos,

as the point at the center Oo we a o am~ entities a image of w he vist

i" " ol °8 definition of placened, Of the surro ndin , 8C€ it ich Undernestood concer s bodyi g plac that ays, nn imagi thei Nermosr Surface (ah the surrounding body of "108 r? af oy) partemon corpus locansprop

2 in qua n0see eit . eeoic LCuur ess locus): ho ie ting ies Y indica ;ben tS can respond who takes 8° pile ? © terms ;nth to the arguments mae . € se ti way can easily resp: Prin the imag; Cond


8 9 Droppers, “Th

: 90 Droppers, “Th ic Questiones” $4 e Questiones” : 60

hematical works, Oresme writes treatises, shes on mixed mat dMotions, his Ox urations ofQualities an

the Proportion, On the Config

Commensurabi s, and his Treatise on the

°r Incomm a hd ortion


d to some sof the Heaven. Modele ion Mot the of '. dw: Bra ment on s in Motions the Proportions of Velocitie (this is especiall ardine’s On ns ofProportions),

sme’s On the Proportio the works set on true of Ore measuring motions(including alteration and sugme principles for well as local motion) and then demon In Ores

ntation-and diminution as

strate conclusions.

alities and Motions,

Configurations of.‘Qu ures or wane the


cerning local motion, represent theorems con s ion uat b tri dis tions. orm iff ies in space, and altera ofintensities or qualit

, In Oresme’s th tions eration, and illumination about local motion, alt ems cor 17 ce wo spa in to Lat the le uses mostly unrelated . td spatium has multip . quotations: se. This can be shown by a few

: €wton’s sen


and more of velo;city is gr eater cal motion. that degree antia] by meeans of which more space or; distance [de spatio vel de dist w.ould be traversed.!


In Io



Ine ar

by.-- the

motion is.- _measured . h would be . atium lineare } whic


e velocity of i w the intensity of th . s sp space [attenditur pene


ll be as ght of the moon sha li e th nd “A : ias as in Isa all be sevenfold Whence it is said light of the sun sh e th d d an se n, ea su cr e in y the light of th days,” for evidently the light of one da n ve se of t gh the li

s de sand Motions... Tractatu ry ofQyalitie ty sconsin alshGe alomlet Clagett (Madison: Universi of Wi drev e os e d M mé th an es , le Or co me Ni es le 1 Nico Or i Nic otitis 1 nibus qualitatum eF



Press, 1968), part 2, ch. 3» 277:

92 Oresme/Clagett, De configurasioniouss






intensively by sevenfold is as thelight which would be exte,ded through a space of seven days [/ucé que per septem dierum Parium extenderetur).”

If some mobile would be moved during one day with a certaiy 4. locity, and during the second day twice as slowly, and during thegp.-4 infinity | day twice as slowly as during the second day, and so on to

it would tray,ce never in eternity would it traverse twice that which

versed during thefirst day. But given any space less than twice thattra a space |seq in the first day, it would at some timetraverse as great transitumprin, quocunque spatio dato minori quam duplum adper O4 ) t e v ; iquando pertransiret]. die tantumdemspatiumal ce, and in one it In three of these texts, “spatinm’”refers to linear distan

refers to an intervalof time.

elikewise uses spapium In his Ox the Proportions ofProportions. Oresm

that fromg patio to mean distance. For instance, “It must be understood traversed or acquired, of times, and froma ratio ofdistances [spatiorum] ratio ofvelocities ”95 or any such [quantities], one can arrive at and knowa

or Incommensurability of In his Treatise on the Commensurability

spatium to mean disthe Motionsofthe Heaven, Oresme likewise uses Finally, he refers to an tance,” butalso the arc of a circle?” and area.”

ional spherimagined three-dimensional space (actually, a two-dimens traced): ical surface on which the path of the Sun’s motion is a space It follows from all this thar B describes daily a new spiral in ed; and it imagined as motionless, which it has never before describ



traverses a path whichit has neverbefore traversed. Andso,by its track, or imaginedflow, B seemsto extend spiral line that has already become infinite [in length] from theinfinite spirals that were described in the past. . .. In accordance with what has been imagined here, the whole celestial space between the twotropics is traced by B, leaving behind a web-ornet-like figure expanded through the wholeofthis space?

The “track, or imagined flow”is obviously not any kind ofsubstanceor entity existing in the external world. Thus most frequently for Oresme the word spatiumrefers to distances or to surfaces representing mathematical quantities in two dimensions, The efforts made by Oresme(like-those of the so-called Oxford

Calculatores, such as Thomas Bradwardine and Richard Swineshead) to develop a concrete mathematical discipline in which one may

measure motion with respect to cause (tanquam penes causam) and

with respect to effect (tanquam penes effectum) were among the most successful projects that were pursued by Oresmeand the other fourteenth-century oderni. That the word spatium was used in these efforts to meanlinear distance rather than a three-dimensional exten-

sion does not meanthat these efforts were irrelevant to the history of concepts of space. WhereasAristotle's definition ofplace as the innermostsurface of the surrounding body was nota useful basis for trying to develop a meaningful mathematical science of motion (scientia de motu), Oresme and the Oxford Calculatores constructed many useful

concepts that could be translated into mathematics, not the least of which were the conceptsof latitudes and degrees, wherelatitudes might be understoodby analogyto lines or, on the Continent,to triangles, !99

93 Oresme/Clagett, De configurationibus, part3, ch. 7, 409.

94 Oresme/Clagett, De configurationibus, part 3, ch, 12, 427.

Edward Grant 95 Nicole Oresme, De proportionibus proportionum & Ad pauca respictentes, ed, (Madison: University of Wisconsin Press, 1966), 287. 96 Oresme/Grant, De commensurabilitate, 197-99, 208, 210~11, 286 (quoting Pliny). 97 Oresme/Grant, De commensurabilitate, 260. 98 Oresme/Grant, De commensurabilitate, 241.

99 Oresme/Grant, De commensurabilitate, 176.

100 For somelogical works in which spativm appears, see Richard Kilvington, The Sophismata of Richard Kilvingron, Auctores Britannici Medii Aevi, XII, ed. Norman Kretzmann andBarbara Ensign Kretzmann (New York: Oxford University Press for the British Academy, 1990), which includes the sophismata: S12, Socrates pertransivit A spatium; S13 Socrates pertransibic

ae dt




Oresme’s work On the Configurations ofQualities andMotions was of the mostinfluential works in this genre. The role of the concept oflatitude in Oresme'sQuaestionessuper y, ted Generationeet corruptione exemplifies the usefulness ofthis construc beg, A concept for talking aboutissues that may be mathematized. and ANSWer tiful example is the way Oresme uses Latitudo in his analysis

has a deter); to his last question, IL15, “Whether any corruptible a,, nate period ofits duration,” where by “period” is meant a determin yp, by d measure as lasts time during which something corruptible in which heis mog revolutions of the heaven, and the corruptibles the longer or shore, interested are living things.!°' Thefifth cause of magnitude ofthe Jay. period of duration of a corruptible, he says, is the which the form of such itude of proportion of primary qualities with much variation in the a corruptible can remain, or in other words how humans the allowed proportions can occur without causing death. For for crows the variation is very narrow,so thelifespan is short, whereas This use of the word latitude is broad, so they might live 360 years.'° Galen, considering the “latitude”is close to the useoriginally found in

tude of health is a sort of variation in degrees of health, where thelati

corruption may not scale. Even a kind of animal that stronglyresists under which it can last longif there is a small latitude of complexion concludes that exist.Setting aside violent causes of death, Oresme spatium, et Socrates incipiet pertransivisse A spatium; S14, Socrates incipiet pertransire A

spatiumquamincipiet pertransivisse A spatium; Aspatium, et nonprius incipiet pertransivis se A later $27. Socrates incipiet posse pertransire Then sirum. Sis, A spatium incipit esse pertran a Socrate; $29. Socrates movebitur A spatium; $28. A spatium incipiet esse pertransicum

movendumsuper illud spatium. super aliquod spatium quando non habebie potentiam ad the Kretazmanns translate ir appears, spatium word the where ata sophism these Everywhere in f.37v. “Motuum

atum(Venice, 1494), “distance.” William Heytesbury, Regile Solvendi Sophism parte temporis spacium ergo localium uniformis est quo equalivelocitate continue in equali pertransiretur equale.” f. s21, commentary, “latitudo motus localis habet attendi penes spacium.”



there is an approximate maximumdurationfora given species and that for humansis perhaps more than seventy years.!°4 He will define the

period for a species and the mean ofthelatitude between the max_ imum and minimumlifespanas the period ofthat species andsays that it is a pretty speculation whether the mean should betaken arithme-

tically or geometrically.! Of course a personcanlive longer than the

period for the human species (we would say thelife expectancy), as a humancanlive to fifty years old. Somewill live longer or shorter periods dependingon their location,diet, and so forth, even if they do

not die a violent death. This is just one example ofOresme’s use ofan extended or metaphor- | ical sense ofspace,i.e., a concept of dimension, extension, orinterval

(the latitude ofvariation in human complexion) to apply mathematics to natural science. There are many othercases that might be mentioned

in which Oresme and the other fourteenth-century moderni apply extensions or dimensions in a wider sense to a mathematical science of nature. As historian of science, I have been interested in the efforts of

Oresmeandother fourteenth-century moderni, especially the Oxford Calculatores, to construct mixed mathematical scientific disciplines. In this, many of them wereinfluenced by Ockhamistic ontological min-

imalism in combination with the logica moderna that analyzed the truth ofpropositions makinguse ofthe reference (the so-called supposition) of termsfor thingsin the real world. While a very simple case of this mightbe,for instance, to formulate the proposition that the earth is a sphere, where sphericity is a form inhering in the material earth, the most important achievements of the moderni involved propositions containing so-called syncategorematic terms. When a proposition containing syncategorematic termsis to be explicated using the idea

s f. s2v, Commentary of Messinus,“In uniformi itaque motu habervelocitas totius magnitudini motus.” cissime punctusvelo beret quamdescri lineam penes attendi localiter sic mote

ror Oresme/Caroti, Questiones de generatione, 293. 102, Oresme/Caroti, Quiestiones de generatione, 298. 103 Oresme/Carotl, Quaestiones de generatione, 300.


104 Oresme/Caroti, Quaestiones de generatione, 301. Then he adds that the time through which

heaven cannotendureis perhaps one thousand years. 105 Oresme/Caroti, Quaestiones de gencratione, 302.




t always exphy;neg of reference or supposition (their term),it is almos ns in which the yome as being equivalent to two or more propositio n. Thus one says that the propoyi, have simpler types of suppositio « ee ’ nye . “ propositions SOuates Socrates begins to run’ is equivalent to the two this Socratey wil} is not now running” and “Before any instant after have been running”’ .


se these two propositions are appropriate beygu ele


there is nofirst instant of any motion. moderni did not angy yre ‘To mathematize motion, then, the but instead nyuch simple propositions such as “All men are mortal,” irin g exposition into jul. more complicated propositions requ of their terms. When tiple propo sitions with varying suppositions ual, perhaps the best def. Oresme concluded tha t were it not so unus between thesides of a hody inition of place would be the ext ension w ere not there, he was testa in place that would remain if the body expositi on ofthe pronostively end orsing an opinion that required tion into a technicall y

correct form. Thus one should notsay thar


the moon were annihilated, there everything inside of the sphere of is

were, because there would be a vacuum where the bodies previous beinside of the evacuated no such thing as a vacuum. Nothing would sphere would be evacuated, sphere. What one should say is that the has supposition and che Then it is the sphere for which “eyacuated” sphereis in fact there.'°° Aristotle’s Physics, his _ From the examples of Oresme’s questions on and his mixed mathecommentary in French on Aristotle's De Caelo, matical works, [have

ding described how what Aristotle had to say, inclu

a.M. 1964), book 106 John Buridan, Metaphysicales Quaestiones breves (Paris 1518; reprine Frankfurt ‘vacuum’


huius termini “4, Q r4ff. 23vb, 24rb, “Tune ad propositum videamus descriptiones significat omnialocade oratio hec corpore: repletus non locus licer indicantemquid nominis,sci sed


its weak points, motivated Oresme to come to the conclusions he

came to. What Oresmesays wasalso motivated by the nominalisti or Ockhamistic or Averroistic separation of mathematics and ph vies

whichcalled in question any positive extension inside the ovacuat d otb of the moonor outside the cosmos where there was no bod Th

influence of logic on what Oresme had to say aboutthe possibile f three-dimensional extension without a body was interrwined with the effect of theological doctrines or condemnations particularl ‘i

cause ofthe effect of the Condemnation of1277 to require su spor

of the proposition that God can do anything that is contradiction.

logical nor soa

Medieval students, as part of their undergraduate education peatedly practiced exercises called “obligations” (de obli ation 2 involving competition between twospeakers. First a proposition “

posed (the positio or positum), which the respondent wasobli dio accept as given for the exercise to begin. Then other pro ost ns were posed, which the respondent had to accept orreject in li htof

the positum, always trying to avoid self-contradiction, The its , itself could be false in fact or even naturally impossible, Whe mattered was that correct or valid inferences be drawn from he positum together with anylater propositions that had been acc i

This led to careful consideration not only of what was ossibleb of what might be compossible or simultaneously possible 107 Gi mn consideration of God's absolute power, as well as of God'sordained

power, careful distinctions had to be made between what was han

rally possible versus what mightbelogically supernaturally vossible In the passages from Oresmediscussed earlier, his reasoning abo te what mightfollow if God created multiple worlds draws upon such

omne repletumcorpore; mundoindifferenter per illum terminum‘locus’: et significar etiam

mododivisivo ... propter quod ista oratio sive mentalis sive vocalis ‘locus non

repletus corpor’

illa oratio non pro nullo supponit nec per consequens illa dictlo ‘vacuum’: quiailla dictio et

differunt nisi secundum vocem. ... Sed tamen ego nego quod hoc nomen ‘chimera’ significat

chimeram. Immo significat omnia composita et omnia impossibilia componi: et etiam hoc nomen vacuumnon significat vacuum si impossibile sit vacuumesse. Immo significat omnia loca

et omnia repleta corpore, modo tamendiverso qui nonhabet correspondentiam inre.”

107 herieeepers ofrules for disputations Deobligationibus. In one set ofrules, each new propositionpur forth had to € comparec only to the originalpasitum, and not alsoto the replies thathadbeer Biv

o previous propositions. See Paul Vincent Spade and Mikko Yrjénsuuri

wu cories 0 igationes,” Stanford Encyclopedia ofPhilosophy, ed. Edward N. Zal inter 2017, htcp:// =obligationes,



modal logic. Many centuries earlier, as mentioned, John Phitopons had also proposed drawinginferences from false or even impossible premises. Such reasoning can be found in many places in Oresme’s Work. For example, with regard to the possible rotation of the earth,in his com-

mentary on book2 of Lelivre du ciel et du monde, Oresme describes the motionofthe heavensand says: So,it does not follow that because the heavens move in a Citcle the

earth or some other body must remain motionless at their center, for, supposing that this is so and that the consequentis ttug, still the consequence is not valid because circular motion as such does

not require that any bodyrest motionlessin the middle of ah,dyso moved. It is not absolutely impossible nor doesit imply a contradic. tion,ratherit is possible, to imagine that the earth moves with the heavens in their daily motion, just as fire in its sphere and a grea part of the air participates in this daily motion, accordingto Aristoele in the first book ofMeteors. Although nature could not move the earth bje and thus, it is however possible in the second meaning ofpossi

impossible in Chapter Thirty of Book 1.18

5. CONCLUDING REMARKS the later Middle Ages might This paper on the concept of space in

cr, given have been thought to be addressed to an unpromising subje

al philosthat “space” was not a theoretical term in Aristotelian natur

the works ophy. I have shown that when the term “space” appears in

g of Oresme(or other late medieval authors) it rarely means somethin s one similar to Newton’s absolute three-dimensional space, and, unles



an extent oftime. It is not that medieval Aristotelians could not conceive of three-dimensional space; they knew thar the ancient Greek

atomists had thought the universe was composed of atoms and empty space. But they tended to hold thar space,as usually defined,is a self-

contradictory concept. Once it became customary to consider what God might doin his absolute power, most commentators accepted that, even if there were

no naturally existing empty spaces,it is possible that God could cause a vacuum or empty space to exist, for instance, by annihilating eve-

rything inside of the orb of the moon. But what then? Philoponus had explained in the sixth century that one could posit that some-

thing impossible is true as a way to clarify the understanding of reality.°? Philoponus himself had followed this program to revise or

overthrowsignificant parts of Aristotle’s science.""° For fourteenthcentury Aristotelians, physics and cosmology (the doctrines contained in Aristotle’s Physics and On the Heavens) were potentially demonstrative sciences based on principles and demonstrating conclusions. One cannot expect simply to insert an exception into the demonstrative science of physics and then proceed. By the fourteenth century, Aristotelianism was supposed to hold that the world contains substances qualified by accidents depending for their existence on the substances. Ifa vacuum or empty space was supposed to have a definite

measurable extension, this would meanthatpresentin the vacuumwas an accident not inhering in any substance. Already, theologians had concludedthatin the transubstantiated Eucharist there were accidents not inhering in a substance. Thomas Aquinas and those following him hadsaid that in the transubstantiated Eucharist the quantitative form (i.e., the extension) had been given by Godtheability toserve in place of the substance of bread as the basis for inherence of the other

considers imaginary space, most frequently it means distance or even

109 Philoponus, Corallartes on Place, 36-37. See quotation above at note 33.

uo Richard Sorabji, ed., Philoponus and the Rejection ofAristotelian Science (Ithaca, NY: Cornell 108 Oresme, Lelivre duciel, 367. He refers to 211-213.

Universiry Press, 1987).









[low!ing qualities. On the other ose fo 5, 59 th hand, Ockham and i ive fort” 5 quantitative of View concluded .ering™ : that thereactually are hs re without ? the qualities of the Eucharist | are simply the substance, lll do $ gece?"

ce Thus, for the fourteenth-century ; on one ha 00 erni, O moder that Go d could 0 ing insi the sphere 5 e108" annihilate everything before drawing ins Id be the inferences c48®


jn about what wou+ inles perhaps? 1wold suppositions or principhe said that” fic! Scientific disciplines hati as Buridan indicated when (Ideo gua" Necessary,

have to admit additional :

SO to speak, for for the the



intellect to borro in

*portet intellectyr, f humanum), elt Place and Space abot rs were not indepen dent awit? concep that cO Or subtrac te cienee» d from the otherprin . ns li a as ciples of aeheedas germs‘k jbo! & many other concepts, whet her de one mi (quid nominig ght ¢ svio® | or quid rei), W the Concep hat yhat was t o fempty Space was de gc gard to prim pendent on e matter, ¥ ion, GO% " ma th ematics, quanti 80 forth an ty, mor stand th ’ pte d s © on. This e ™ jee™ is why one cannot under eval scholar on j e h out the concept of space W asonin whole context g' wt within which or imaginin g

he was re rote If, for inst . ts ance, it is nt Science that of Ar is a principle have alLand only mn corporeal substances a three @ post”¢ Where does th at leave an Aristotelian dto reason? Place of an . yao who is aske MCOrporeal substance iin? € an a such ng °F empty Space? as God or pe ather than Ying Lo ple J to solve such ical pro” eu Seenth-century metaphys Stlence of hematic© py Modern; chose cat the “asurement at to develop a nh 0 of motion re spect both wit



jnas a cn Emery Mura,the py sous and Handmaiden uinas WYSics of th Science: St. Thomas i Aq (Dordsechn Rech and E dieval ed rncic?cer ef j and E, gi eucharise” in The Crls ur al Co ley nt ext ofMjlosophyY ud e Y#8, Boston Studies 75) 349~96 fs ‘, in the Philo

( Fanquam Penes causam)

. and with


effe (tangu respect ectum) ion and covering al an d diminution teration and r 8S Well as enee the ect to

AW peneS

.developed ne w of quality, ve-

local motion. As part of this scienc Wantitativ e, t e conce

pts such Ocity, acc eleration, Propor as latitudes and tion, and so Subdise forth plines

“grec demonstrative

h they imaginativel on the basis of thes y e concepts, W Mi teatively develope sets of conclusions, d. Then they demonstrate Such as the conclu d motion will sion that a cover uniformly seec a muc *Pace in a le ra te d its middle given time as a un iform motion degree, Which

could then be compar

ed to observations

if theyfit.

« chematies, such clieving that math as em at ic s, ev en concrete 5 rt kine Mathematical astr matics, exists onomy, mechanics,

dynamicsoF as quantitative fol in the minds of ma themat ians ‘h aticians co OrMs in €xternal real uld conity, the nominaliic stic mat em dcould “truce Mathematical th en demdefinitions and suppositions ssit nstrate “Onsequences on ions I think that the basis ofthe the NOminalistic

se oppo thematizing na

ture c approach to ma sociated by Koyré Platonic

and voluntatisti

"aS a valid alternativ e to the

approach, eh, on e sets out repeat: In the nomi nalistic oevation Principles, Often aris , that are at ing from re

With Galileo 2 To


ection on onse “ast Physically Po n mathematissible and then derives conseat cally, Later °, One ma resulting structur y consider whethe e Matches physi r the e but motion sth at ar ca e l re al it y, Typically, it is not 9 ON ening the structur ncerp, A e of i nd atte

ntion is paid

"Mm iftheplaces or forms ga to problemsof ties are suppose to be inedorlost in “Ontinuoys (or no | t),



; £ scientific discipli though We might th nes ink that once pieces o ared to observation feVeloped, they shou ld have been comp :

inale et “ wine istotre doctrinale "era ine d, Koyrd, “Le Vide ives d'histoire © et Yespace infini au XIVE idcle? Archives also Sepkoski, Sep site money Age 17 1949): “Mina . nall. icm vor ; Clo cheat Koy sed ¥ ré, Fr N 45om w 91; the s 9 Koy e Ma ré, an Constructivism, . ( ature: Barro wv on t Antoni Malet, “Tsaac IdeasCas 59vg eolo Bical Voluntarism and “Isa ical Optics, urnal ofthe History of the Rise of Geometrical (1997), ,” Jo





the texts, Whig to see if they fit or not, this is not what wefind in

ing ,¢ are most often connected to university teaching. Then thetest many genrey ¥¢ the doctrines or analytical techniques is donein the one might suppg,. disputations engagedin at universities. Abstractly each other, that when the analytical techniques were tested against drop the infer, might discover which ones were superior and might

with differg,,

ones, but since the disputations tookplace year after year in which competitip, classes of students, there was not re ally a venue Asseen in Edward Grany, could lead to theoretical winners andlosers. still going on through yp, Planets, Stars, and Orbs, similar activity was . seventeenth century. by Oresme, the Oxford In the kinematics and dynamics developed structures of theorems Were Calculatores, and others following them, might sometimes become developed into small subdi sciplines which university students whostudieg parts oflarger works. Meanwhile, the

sophisms using mathematica] works on proportions and disputed using measures

such as those techniques (or analytical languages), developed habits ofcritical and of motions or first and last instants, continue to be of use in relation to mathematical thinking that might manyand varied subject matters! mathematico-physicalscience, Onthis route to the development of fromits position within a space is a concept that derives its meaning attempting to show empitilarger disciplinary structure. Rather than one develops a mathematical cally or by experiment that space exists, ether one-, two-, or threedescription of motion in which space—wh n a whole system is dedimensional—has a mathematical role. Whe d in that comprehensive veloped, then the conceptofspace to be foun

17 5

system mightbe accepted at the sametime to refer to something real

Rather than thinking that mathematicsis absolutely certain and ne | essary and rather than trying to discover physical sciences that wre equally certain and necessary, the fourteenth-century moderni tr


mathematics and physical science as equally hypothetical, often wal confirmed and in accordance with experience, but utely knownto betrue.!4 P ue nos absolucely Where fourteenth-century Aristotelians were most original withi the general range of concepts of space in the sense of dimensi was perhaps in conceiving of dimensions other than length breadh, ind width. Then intensity or latitude of heat could be a dim vine could the intensity of motion orthe laticude of velocity b “adim . sion. Time could be a dimension. There could even be a lt de“ol proportion within which proportions increased and decreased (1 hi ;

would,in a sense, be a logarithmic scale in our terms). And for ihne dimensions one could inquire whether they were continuous : 4 h. mogeneousor contained discontinuousparts. “ve

reted so that the dynamics de-

reinterp 113 Inthe Renaissance, however, the mathematics ofratios was

foundation, See Edith Sylla, rived from “Bradwardine’s rule” lost their mathematical

“The Origin

in Relation to the and Fate of Thomas Bradwardine’s De proportionibusvelocitatumin motibus c Revolution History of Mathematics; in Mechanics and Natural Philosophy before the Scientifi

he: Springer ed. W. R. Laird and S. Roux, Boston Studies in the Philosophy of Science (Dordrec Verlag, 2008), 67-119.

One: eh say that they took geometry to be empirical, 114 4 Onemi iri anattitude that was taken up again in th eteenth century as people began to develop non-Euclidean geometries. sens


perspective, painterslike Piero della Francesca(c. 1415-1492) and Leonardoda Vinci (1452~1519) surely pursued pictorial


Art and Architecture Linear Perspective in Renaissance

Mari Yoko Hara e

y, through which the soul contemplates and ‘The eye is the windowof the human bod ang the soul is contene in its human prison enjoys the beauty in the world. Forthis, nt. torme its is prison n huma this withoutsight Painting LEONARDO DA VINCI, Treatise on

technique in applied

e?! Or was it an idea that geometry, of practical artistic valu

vity—a “revolution conveyed a worldview and a modern subjecti in Panofsky Erw n oria in the history of seeing?”? The art hist

influential 1924 publication, advanced both interpretations in his


truction in as Symbolic Form. 'To Panofsky, spatial cons

much a consolidation and the pictorial field was ambiguously “as

n ofthe domain ofthe

systematization of the world as an extensio

1 The epigcaphis from Leonardo da Vinci, Trattato della Piteura, capitolo 28, Biblioteca Apostolica / Vaticana, MS.Urb.Lat.1270,htep://‘cocoon/leonardo/chap_one/vu specula

per la quale la suavia CID28/o1#chap_top. “Questo [Iocchio] é finestra de humancorpo carcere et sanza questo ¢ feuiscela bellezza del mondoper questo l'anima si contenta della humana essa humana carcere é suo torniento.”

2 Hans Belting, Florence andBaghdad: Renaissance Art andArab Science (Cambridge, MA: Belknap Press of Harvard University Press, 2011), 13. Mari Yoko Hara, Spice, Vision, and Fuith In: Space. Edited by: AndrewJaniak, Oxford University Press (2020). © Oxford University Press. DOI: 10.1093/080/9780199914104.003.0007


self.” Consideringtheartists’ motives in employing perspective providesa different avenueof inquiry into this question. In developing techniques suchaslinear, color, andaerial


Was pictorial perspective, a conventional

verisimilitude pragmatically. But their rules, which were elaborated upon,codified, and widely disseminated by others in numerous treatises throughoutthe fifteenth andsixteenth centuries as practical instruments ofthe trade,also allowed the artists much more than the meretranscription of the world’s appearance. In the wordsofthe celebrated humanist Cristoforo Landino (14241498), “perspective bestowsreasonto art”becauseit is “part philosophy, part geometry.”4 In the representational tradition, space andvision (and therefore sensory perception) were bound together intricately, in part becausea virtual space in a perspective image was logically constructed as a single viewer's field ofvision, and every object within it was portrayed in geometrical relation to an imaginary perceiving eye.’ Antonio Averlino,called Filarete (c, 1400-1469) explained the system succinctly in his Treatise on Architecture: “The

centric point(in a perspective painting) is your eye, on which

everything should rest just as the crossbowman always takes his aim on fixed andgiven point.”¢ (See Figure R3.1.) A perspectival representation embeds the beholder’s vision within its own structural constitution and acknowledges the physical presence of 3 Erwin Panofsky and Christopher Wood, Perspective as Symbolic Form (New York: Zone books 1991), 67~68. 4 “Arte idest la prospectiva: che di questo assegna la ragione. La prospectiva & parte di philosophia 1 Cat : ct parte di. geometria.” Cristoforo Landino, Comento sopra la Comedia di Dante Alighieri poet fiorentino (Commentary to Purgatorio canto XV) (Florence, 1481)


s David Summers, Real Spaces: World Art History and the Rise of Western Modernism (London: Phaidon, 2003), 431-89, on virtuality. Theissue ofthe ideal viewpoint is much discussed

in theliterature on perspective. 6 Antoni : . 3. ntonio Averlino [Filarete],




/iarete’s Treatise on Architecture, trans, John Spencer (New Haven , CT:Yale University Press, 1965), 1:304-5.


Played with th

is ¢

The © FI Morentine

Ne Ne of Of the the

t i

archi hitect








i i Oo nan d inatl

-1446) was

(1377 Filippo Brunelles f tst toexperiment ; tion ics atial representa ased on Euclidean with esinole i is imology, geome try, oph thalmo : wa, Stedited and optics. . with : ive . ‘nventing : a erspectiVel in two famo us pan linear p ine baptistery Paintings int} that ° the showed alazz d t the Florentine and Ppof fiona. “8 Signoria | building)” these ba While ‘ (the city ile neither counc il bui Wo Paintings ee tive is i's expertise rspe Survives, Brunelleschi’s in persp still much evident has the in hishi in | projects, >

itectura extant architec suc hi ; BruneLleschi R3.2). re IN}. in Florence Used Projectiye . Scometry (Figu . ‘tticulate mo and a ane dular the interior system > the diameter of of this akin, the Column 0 al church, ta om ag its basic ion The PP ‘lationship, unit of measurem “tweeneach the een each part and Peeveen Ole are Marked part an directly onto c—by ine s the lin °N the floor, . the Ceiling coffers, the built bels along the or Meade,’ foy vt and the c es instance . oordinated : With these visua cu atchitecy carefully . *eveals his . c principles. . Visual ¢ cts mathematical esign of this Space d en PetSDective to those Theynear as a whole

silica of

San Lorenzo



“Opper plat

Jan (Hans) Vredeman

e “ngravin

g, plate 30,

de Vties (1526-~1575) , Perspeetiv,


the Observe , bef re it. This One-to-one Ce e es. Was argua rapp ort with th bly wha i ‘19 ile discove ™madethe technique mo stappealing tO ri (15711650) e$ of binocular and conic vi : by Johanfn‘soxe ne! ‘ si a on nd oth €ts ti allowed seventeenth-c ate Sig -centu. ryp rac h tin their id argh

to sj ul


S More convinai ly, I wot appeal of cing atlia Perspe Ctive remained inin itte its pote e Visual d nce a rtist® ialogue with iss a viewer,

© ma "2

ofa rly and

are i. to where space recede is show ont (here regula othe set abov' Converge at a joh} ; vanishing oO

Mmed} at


al Oblique lines i altar), tunellesch?’s p temarticulates W Proportional titical N otio Ns: sys ble: second, ¢ 7 fist, that space that was mea surable;


tune leschis “xperiments with inear in Kemp, lence, shi p interpretation : on. Slenes, i and e, see Martin ” Art literacur i © Hh TinearTheperspective, Nonsense: at nee sspect ive? 7 Osa, (1978), In on, from i's Pe ter Pr Reflection, Dainting f Brunelleschi’ an De ch ‘34~61; David Summers, apa Unellesch HiMi Wersity Western Pain nag 61-67 of North Carolin (c ivers fe of lina Press, 2 Ant ista Alberti, WEtih sta : Leon tser er i a is e ta St P Bat ate Univ t: aOrnk Pai UK, Mbt 1970)3 mbriiddgge, ersity P e tn S ‘ainnti ition res Crti doe "Wersi ng 4 t Tran a Rudo (Cat nal Edit slatio ty Press, n an io n 201) Brunelleschi and ‘P rg an d in roportioion nin Perspective?” Journ: nal of the Warburg Pe r Pp & 16, nos, 3~4 (1953): 275-91,




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fully in the service of devotional practice (Figure R3.3). Heavenly figures—the Madonnaandchild,saints, and angels—occupy a fictive throne room,depicted to scale as an extension of the architectural chapel space where the beholder would haveactually encountered the work. The perspective in the altarpiece helps the painter create a visionary experiencefor his audience, directing the beholder’s devotional gaze towarda celestial apparition, and to a realm that lies beyond whatcan be seen corporeally in nature, By suggesting directaccessto the divine realm,the pictorial representationalters the nature ofthereal space the viewer occupies as well, transporting himor her to a supernaturalplace. In LeonardoParasole’s (1570~1630) woodblockprint after Antonio Tempesta (1555~1630) of Christ Restoring Blind Bartimaeusto Sight from 1590 (Figure R3.4), two narrative

), Basilica FIGURE R3.2. Filippo Brunelleschi (1377- 14.46

of San Lorenzo, Florence


ing subject’ eye.” With his measurability was relational to the view

blurs the boundaries linear perspective construction, Brunelleschi experience, and the between mentalspace, the space of everyday well as spiritual realm, representing the harmony of the cosmosas

our place within it. rendered The space that Renaissance visual artists meticulously sion.hit in perspective often pert ained to the spiritual dimen served, i Astonishingly, the mathematical rendering of space y.In his anything, devotionalart richly rather than impair its efficac r graceful San Giobbealtarpiece, for example, the Venetian painte ctive perspe GiovanniBellini (c. 1430—1516) enlisted pictorial

momentsare present withina single representation, breaking the normative relationship betweenspatiality and temporality. Linear perspective is employed to symbolically represent the sense ofsight that Bartimaeus miraculously regained." The sceneestablishes a literal analogy: the spatial rendering techniquesignifies effective vision. Butif vision is synonymouswith an orderly perspectival vista here, it is hardly a neutral instrumentfor thepassive processing of sensory data. Rather, it is an exalted form ofspiritual seeing, a divine gift that leads mankindto true knowledge and wisdom. It reveals not only the appearance ofthe universe but

also its ordering principles, and,as the open-armedgesture ofthe awe-struck protagonist suggests, the results of this revelation are

10 Paul Barolsky, “Naturalism and the Visionary Art of the Early Renaissance” Gazette des Beaux

we i. that sets down asa science ion * isiinventio Li ofBrunelleschi, 42. Manettii saw Brunelleschi’ 9 Manetti,i, Life perceiee. as objects distant and near of enlargements and reductions the rationally and “properly by the eye” (emphasis mine).

Arts 129 (1997): 57-64. a1 The blind man’s nameis given only by Mark (10:.46~52) as Bartimaeus,which literally means “son

of Timaeus.” This could be a referenceto Plato’s famous dialogue by the sametitle, where visionis

discussed as a meansto acquire true knowledge.



aN ‘

FIGURER3.4. Leonardo Parasole (c. 1570-1611) after Antonio Tempesta (1555~16

) Christ Restoring Blind Bartimaeusto Sight, 1590-91. Woodblock print, ioomm 30), 123mm, Copyright, The British Museum, 1868.0612.592, *

arresting. Perspectiveis figured here as an important characterinit s

ownright within the narrative, one thatarticul ates our relationship to nature as well as to the divine.

Galleria FIGURE R3.3. Giovanni Bellini (c. 1430-1516), San Giobbealtarpiece, 1487. dell’Accademia, Venice.



Geometry and Visual Space from Antiquity to the Early Moderns Gary Hatfield


portions of the carpet that can be seen lie on imaginary straight (or rectilinear) lines running from faceto carpet.! From antiquity, the rectilinearity ofvision has been a foundation ofoptics (consideredas the science ofvision). Thefact of rectilinearity admits of various theoretical explications. Rectilinearity does not ofitself reveal

its causal basis or the character of any causal interactions between world . and eye. Does the eye send outa visual powerthat touches the objects, as ancient extramission theorists held, or do the objects in the world impress themselves somehowupon theeye,as intromission theorists hold, or do both types of causal process occur? A significant characteristic of medi-

eval theories ofvision wasthetransition, within geometrical optics, from mainly extramission theories to newly dominantintromission theories We will follow this transition, althoughit is not our primary focus

subject-to geometrical analysis The phenomena of sight have been n this study has been that wesee in from antiquity. A core principlei ough a n open window, wesee those straight lines. In looking out thr drawn from eye to object. These objects for which a straight line can be n us and every visible point, that straightlines may be imagined betwee If we turn to face the is, all the points (or s mall areas) that we cansce. can notice

on again applies. We objects in the room, strai ght-line visi visible that are not occluded that those parts of the carpete d floorare

t. An object can block a by the table or by booksresting on the carpe w.hat is behind it. There are line of sight that would otherwise reach r instance,if further regularities. If the eyes change their position,fo

into view and other we stand up, new portions of the carpet come

e portions that werevisible before are now occluded by the table. Thos

Gary Hatfield, Geometry and Visual Space from Antiquity to the Early Moderns In: Space, Edited by: Andrew Janiak, Oxford Universicy Press (2020), © Oxford University Press. DOK: 10.1093/080/9780199914104,003,0008

The fundamental insight that vision occurs along straight lines permits the use ofgeometrical constructionsin describing how wesee We can easily understand the occlusion of portions of the carpet by tracking a line from our viewpointpast the edge of the table vo che carpetitself, Other aspects of ourvisual experience can also be mod: eled by considering the relations between lines of sight. As a friend walks away from us, the angle formed by lines from our viewpoint to her feet and the top ofher head gets progressively smaller. Orif our friend is standing next to a young child, so that both are at the s distance, the child takes up a smaller angle.


These simple facts may seemto explain a pervasive aspect ofour visual experience: that we perceive things as having different sizes. However.

the simple fact that the person walking away takes up a smaller visual angle, or that an adult and a smaller child whostandat the samedistance 1 Complications arising from that fact chat we have two eyes were knownfrom antiqui noted as needed; here we might note that, for many purposes, the linesofsight fromthe wes can be treated phenomenologically as having a unified otigin at the brid . of he howe,Other subtleties, such as those arising fromthe diffractionof light at an edge,are not furtherwoved,met





take up respectively larger and smaller visual angles, does not explain 41

aspectsofourvisual experienceof size. Considerthis fact. Supposeyq,, ofy,, are standing on the sidewalk speaking to two neighbors whoare peng, same height. Oneis five feet away, the other ten feet. The nearer r. yy, subtendsa visual angle that is twice that ofthe more distant neighbo the farth., the nearer person is not experienced as being twice astall as direcyyy not person. Accordingly, phenomenally experienced sizes do on. Astronome,, follow visual angles. Or considerthe classic moonillusi

the moon ove, have long determined that the moon on the horizon and been obseryeg head subtendthe samevisual angle. Yetit has frequently moon overhead, that the moon on the horizon looks larger than the angle. Again,size perception does not simply follow visual used in describing be Dothese examplesentail that geometry cannot hat the geornetry the perception ofsize? Indeed not. Butthey showt a direct covariance wigh of size perception is more complicated than ption does ng visual angle. Moreover, they suggest that size perce not onlyis the sun always track the objective sizes of things. Thus, timeslarger than perceived as a small disk when,in reality, it is many moon) may appear adif. the earth, but the same heavenly body(sun or . ferent size on the horizon than when overhead


geometrical framewor This chapter examines the development ofa

ts ofvisual perception, for understanding and explaining spatial aspec positionsof things which includes the perception ofsizes, shapes, and


We have noted that geometrical constructions were applied not only to instances ofveridical perception, in which visual experience matchesor corresponds with physicalsizes, shapes, and positions, but

also to cases ofillusion or misperception. In thelatter cases, the experienced spatial properties do not match the physical objects. These illusory cases might then promote a notion ofa spatial experience that

does not simply copythespatial characteristics of objects but includes aspects arising from the subject’s manner of perceiving. This in turn mightyield the notion that visual experience creates its own visual space, whose characteristics can be described and explainedusing


metrical constructions. This chapter asks whetherthe notion ofa, ecifically visual space also cameto be implicit (or indeed ex licit) f

veridical perception.


The first section examines the ancient backero i Euclid’s and Ptolemy’s accounts of a visual ventea formed by visual rays emitted from the eye and on Prolemy’ intro.

duction of additional psychological factors into his explanations The second section examines the sophisticated intromission theor of the

medieval Islamic natural philosopher Ibn al-Haytham. He scticulat ' a coherent bodyof psychological explanations for visual experiences of spatial properties, which was taken notice of by authors in the Lati West. The subsequent three sections consider the accounts ofs ati 1

perception in Kepler, Descartes, and Berkeley.


r narrative thread will be

in the field ofview. Within this framework,ou

actors only selective, focusing onsize perception and bringing in otherf

ive examination of howa as needed, This focus allows for a comparat


single problem wastreated geometrically by various theorists, ancie medieval, and modern.”

problem ofsize 2 Historically minded readers may question thelegitimacy ofseeking to track “the have been there Can millennia, two by separated are whom of some authors perception” across common problems and comparable approachesacross such a long swath of time? I find no priori reason why there should not have been. Optics as a science had a continuous textual tradition

(later texts referring to, or clearly drawing on, earlier texts—this continuous tradition constiwting

only a subset of the texts that onceexisted), Even though optics was placed in differing social and

intellectual contexts across this time, theoretical constructions remain comparable. Probl evolve, but similarities maystill obtain between newandold, Further, there's no at fori reasonto suggest thar, at a gross level of description, phenomenological observations cannotbec smpared, This is not to assume that phenomenal experienceis “the same”in every respect in is soooea intellectual contexts. The claim for comparability must be sustained by the pla bil rofthe a Plaunblny ofthe spe cific comparative analyses offered.





the geometrical part of their subject matter. As he famously explained in the Physics, “While geometry investigates natural lines but not gua

Within ancient science, astronomyand optics pushed the furthestin using geometry to describe natural phenomena. Ancient astronomy recorded the positions of stars accordingto their place in a presumeg celestial sphere, as observed along a line of sight from observerto star,

Ancient Greek astronomy adopted the explanatory principle thay

only uniform circular motions should be used in forming geometrica]

structures that could explain the observed motionsof the sun, moon,

stars, and planets. The celestial sphere of stars was thoughtto rotate once a day aboutthe earth atits center, also yielding a daily rotation of the moon, sun, and planets, with these bodies further changingtheir positions in relation to the fixed stars in monthly, yearly, or longer

periods. The ancient Greek cosmologist Eratosthenes (c. 235 BCE) sought to

determine the size of the Earth using measurementsof the lengths of shadows at different latitudes: These efforts took advantageoftherecti~ linear propagation oflight from the sun to the Earth, on the assumption

that the sun is at such a greatdistancethat the rays oflight reaching the

Earth are effectively parallel. Somewhat earlier, Aristarchus had used

the same assumption to estimate the ratio between the distance from the Earth to the moon andto the sun.Evenearlier, in explaining the location and shapeofthe rainbow,Aristotle had appealed to therectilinearity of the lines ofsight from observer to water droplets to the sun

natural, optics investigates mathematicallines, but qua natural, not

qua mathematical.”4 From antiquity, the science ofoptics was unequivocally a science of the principles ofvision. As a geometrical scienceofvision, the ancient Greeks dividedit into three compartments: optics, or the theory ofdirectvision (vision in unbrokenstraight lines relating eye and object); dioptrics, or the theory ofvision in refracted lines; and catoprrics, or the theory ofvision

in reflected lines.* Ancient geometrical optics was extramissionist: it posited visual rays emitted from the eye andintercepted by objects in the field ofview as the meansby whichthesense ofvision gains contact with andsois able to see those objects. These rays proceed rectilinearly, unless

refracted by a change in medium orreflected by a mirror. Nonetheless, ancientopticsalso examinedtherectilinear propagation oflight fromluminousbodies, such as the sun, the moon,ora candle, notonlyin thecase ofcast shadowsbutalso in the concentration ofthe sun’s rays by burning mirrors. In optics proper, the discussion ofextramitted visual rays was related to the phenomenology’ of visual experience. This can be seen in the

4 Aristotle, Physics, trans, R. P. Hardie and R. K. Gaye, in Complete Works ofAristotle, 1:315~4.46,

at 194b10—11, This passage does notexplicitly say that optics is “under” geometry in its treatment of mathematical lines qua natural. Still, Aristotle’s statementin the Posterior Analytics that optics

(or the moon, in rare cases). Althoughthis discussion occurred in his

looks to geometry for the principles governing its geometrical structures is consistent with this quotation from the Physics. For a recent discussion of the mathematical sciences in Aristotle, see James Lennox, “‘As If We Were Investigating Snubness’: Aristotle on the Prospects for a Single wn

Meteorology, in the Posterior Analytics he observed thar the study ofthe rainbow is related to opticsas thelatter is related to geometry—that is, each is “under” the other.3 By this he meant that somedisciplines, including optics, must look to geometryforthe principles that govern


Science ofNature,” Oxford Studies in Ancient Philosophy 35 (2008): 149~86. Hero of Alexander, as quoted in David C. Lindberg, Theories of Vision fromal-Rindi to Kepler (Chicago: University of Chicago Press, 1976), 14. Another ancient writer, Geminus, omits di-

optrics andlists scenography as the third branch ofoptics, as noted by Abdethamid I. Sabra, ed. and trans., Zhe Optics ofIbn al-Haytham: Books I-III, On Direct Vision, Vol. 2: Introduction,

Commentary, Glossaries, Concordance, Indices (London: Warburg Institute, 1989), lvi. 6 Wilbur Knorr, “The Geometry of Burning-Mirrors in Antiquity,’ Isis 74. (1983): 53~73.

3 Aristotle, Meteorology, trans, E. W. Webster, in Complete Works ofAristotle, ed. Jonathan Barnes, 2 vols. (Princeton, NJ: Princeton University Press, 1984), 1:55§-625, at 373422-3774273 Aristotle,

Posterior Analytics, wans. A. J. Jenkinson, in Complete Worksof.‘Aristotle, 1:39-166, at 79a10~13,

Aristotle’s works are cited,as usual, using Bekker numbers.

7 Here and throughoutI used the term “phenomenology” in a broad sense, to mean the descrip-

tion of visual experience. I thus do not intend a specific connection with the phenomenological tradition in philosophy associated with Edmund Husserl and Martin Heidegger. My usage has kinship with the phenomenologicaltradition of the Gestalt psychologists and their notion of a




ts seven first systematic treatise on optics extant, that ofEuclid, wholis “definitions” or postulates at the start of the work:

heeye pass 1. Let it be assumedthatlines drawndirectly fromt through a space ofgreat extent; in our vision is a 2. and that the formofthe space included with t thelimits ofour cone, withits apex in the eye andits basea vision;

ch the vision falls are seen, and on does notfall are not seen; that those things upon which the visi withi na larger angle appearlarger, 4. and that those things seen le appear smaller, and those and those seen within a smaller ang ear to be of the samesize; seen withi n equal angles app

whi 3. and that those things upon

hervisual range appear

hig 5. and that things see n within the higher, while those

er; within the lower range appear low

on the seen within the visual range 6. and,similarly, that those within chat on theleft appear onthe rig ht, while those


appear on the left; several angles appear to be more 7. but that things seen within clear.2

that th e relation between the eye and things These postulates suppose ys tely described throughlines orra in the field ofvision can be adequa eye (or from a point within the eye) that come from the surface ofthe The visual field has a circular character, and go forth to touch objects. ator in the eye andits base on the objects creatingacone th at has its apex

———_—— ience as we haveit, kept as muchas possible apart from description of“direct experience” or exper oughtto belike, on wh hich see Kurt Koffka, Principles ence experi what of s ption theoretical conce , 19 35), 73ofGestalt Psychology (NewYork: Harcourt, Brace Burton, Journal ofthe OpticalSociety ofAmerica Edwin Harty ?’ trans, Euclid of Optics , 8 Euclid “The c. BCE. fourth n the in writte ally origin 35 (1945): 357. Work

19 I

seen! Only those things upon whichtheraysfall are seen. Perceived size correlates with visual angle. Visual up or down andright

left correlate with rays upward or downward from, and to the in h o- a

of, the center ofthe visual cone. Things upon which more i “fl “f

seen more clearly. The many propositionsthat follow these def ; ‘on make clear that Euclid considered the rays to be geometricalli vesand to be finite in number,so that as the distance increases, the Vneohne gaps between therays increase (hence the “angles” ofpostulat os For our purposes we may focus on postulate 4 andits use in a

propositions, Euclid held that perceived size directly varies

with veal

angle. This has several phenomenally plausible consequences, Yisnal proposition 36, having shownthatforacircle perpendicular tc | ot .

of sightall diameters subtend the samevisualangle, but tha fa . vee obliqueto theline ofsight some diametersfall under a lar a vl ‘cic

a smaller one, he concludes,“The wheels ofthe chariots 4 me se . .circular, sometimes distorted,” by which he means “hatcher. onan

appear ovoid, He further considers parallel lines going into th lene such as the parallel ruts made by wagon wheels on a straight ‘i aa

As seen from E (Figure 4.1), the lines alongthe ruts can bein

in tee

transversals (such as KT, LZ, and BG). Visual rays (such as eee with

tersect the parallel lines at each transversal, formingvisual an | «oe KETand LEZ) that diminish as the transversals become in . a . Accordingly, the transversals are seen as smaller and the w won rien

pearto convergein the distance.

“Somes SP

Euclid’s results are phenomenally plausible, because things d small in the distance, and parallel lines, such as railwa tek ° look

away from the perceiver, do phenomenally converge . Bat But, veouenext 2 x

Tem 9 The terms“fiel¢otvision’ ision”and “vi visual field” are here used in a phenomenological sense, t

. . : i ees onsin ront of theperceiver within which things are seen, and outside ofply ‘i

thingsarenors n, assuming a stable gaze. . Later authors, , including i t Prol m1 noted that se thing ng Prolemy, thi edgeofthe visualcone are seenless clearly (see Lindberg > Theories ofVision on,une 17). 10 My example, whichis historically plausible.









rays but is continuous,he treated rays as elements within this continuous cone. Heheld that “luminous compactness”is intrinsically visible (IL.4), a term thatrefers to a bodyilluminatedby a sourceoflight and having sufficient density to impede the extramitted visual rays rather than allowing them to pass through unaffected. The rays are intrinsically sensitive to color, whichis a primarily visible attribute; except luminosityitself, nothing is seen without color (IL.s—6), The percep-



tive faculty uses its awareness ofcolorto perceive otherattributes, includingthe length of the ray: “Longitudinal distanceis determined b how far the rays extend outward from the vertex of the cone” (II 16).

Directions (up and down,right andleft) are felt in relation to the cD A


J. Transversals KT LZ, and BG are seen ricure 4.1. Lines 4B and DG are paralle ’s er visual angles, and so, in Euclid account, are seen

from E under progressively small d, Optica & Catoptica, trans. John Pena with diminishingsizes. Adapted from Eucli (Paris: Wechel,1557), 8.

ded, matters aare not that ma : +e It Prolemy (second century), 12 contended, theorist," simple.

he differed from Ptolemy was also an extramissionist. Although does not have gaps between Euclid in holding that the visual pyramid

ometry of the cone.Finally, the perception ofsize, shape, place, vetivin

andrest arises from perceiving illuminated, colored bodies in a direw tion at a distance and as changing over time. Thevisual faculty judges

such properties and thereby produces appearancesofsize, shape, pl . and motion(II.7—8).¥ ree Biss Prolemy’s discussions ofsize and shapeperception are difficultto in-

terpret, but several things are sufficiently clear. First, although heheld visual angle to be an importantfactor in size perception, he did not equate size perception with visual angle. In particular, he wrote that, in acase such as Figure 4.2: If two magnitudes, 4B and GD, have the same orientation and

subtend the sameangle at £, then, since 4B does notlie the same

on, severe ly reduced from whatonce existed; for discussi 1 The source material for ancient opticsis t Mathematical Optics: A Source Ancien of tions Founda the and my “Prole see A, Mark Smith, opcan Philosophical Soctety 89 (1999): 11-19. The Based Guided Study,” Transactions ofthe Ameri

distance a s GD [from point i E] butisi closer to it, i 4B will never appear larger than GD, as seems appropriate fromits proximity [to

t attention from recent interpreters. tical treatises of Euclid and Ptolemy receive the greates ion of an unknown Arabic translation translat Latin a is work ’s in. The extant version of Prolemy references are inserted into

de lémir Eugene de Claude Ptolémée: Dans la version latine dapres Lurabe French translation, and Prolemy’s with text Latin 1989), Brill, : (Leiden trans, Albert Lejeune ed. and trans. A. Mark Smith Optics, ofthe tion Transla English An Theory of Visual Perception: the first book, (Philadelphia: American Philosophical Society, 1996). The extant workis missing ion. Subsequent covering the properties of fight, the extramiteed visual flux, and their interact vision , books cover the basics of visionitself, vision in plane, convex, and concave mirrors and involvingrefraction. The third-handstatus ofthe text and its corruptedstate make close readings perilous.

13 Sometimesit seemsas if Ptolemy invokes judgmentin a way that respondsto giv


paragraphs, textual of the unknown Greek original. In the ensuing ue relevant editions are Claudius Ptolemy, L’Optiq the bedyof thetext, by book and section. The de Sicile, ed. and

and does not alter the appearanceto create a new appearance: of two things that > seartohavethe

same size, one may bejudgedto be farther away on other grounds(11.58). In other cas sitse me as ifjudgmentitselfyields an appearance(II.s9). For discussionofthe relation betweenj 4 mene and phenomenalicy more broadly in the history ofvisual theory, see Gary Hatfield, “Pe cotion as UnconsciousInference, in Perception and the Physical World: Psychological md Philsophicdl Issues in Perception, ed. Dieter Heyer and Rainer Mausfeld (New York: Wiley ‘and Sons,2903)


my? 194.




However, although Ptolemyclearly holds that perceived size increases with perceived distance, he does notdirectly state that it does so with geometrical precision. There are two questionshere.First, does he hold

perceivedsize to vary in direct proportion to perceived distance? The diagram suggests that he does, but he is not explicit.!6 Second, does



FIGURE 4.2. Perceptionofsize at a distance as analyzed by Prolemy, c. 160, Adapted

he hold that the appearancesthatresult from combiningdistance and angleare veridical? Thedifference in distance between the twolines in Figure 4.2 is said to be “perceptible, which might meaneitherthat the

dupres from Claudius Ptolemy, LOptique de Claude Ptolémée: Dansla version latine

40. Luvabe de lémir Eugene de Sicile, ed. and trans, Albert Lejeune (Leiden: Brill, 1989),

fact that GDis farther away than ABis perceived qualitatively, or that the exact distance between them is perceived. In fact, for there to be a difference in perceived size, full accuracy is not required. Moreover,

E]. Instead,it will either appear smaller (which happens when the

in noting theeffect of obliquity on perceived size, Ptolemy treats perceptible differences in visual angle, distance, and obliquity as qualita-

will appear equal distance ofone from the otheris perceptible), or it

(which happens when the difference in distance is imperceptible), (II.56)

butlying at differen, When objects subtending the same visual angle

distances are correctly perceived as beingat differentdistances, the one farther away appears larger.

With Figure 4.2 in mind,it is tempting to suppose that Prolemy

has here formulated an early version of the size—distance invariance

. hypothesis (SDIH), according to whichperceived size is a direct func tion of perceived visual angle and perceived distance." Thelatter two

are combinedtoyield perceivedsize according to a precise function,so that perceivedsize is exact at each perceived distance. When angle and

distance are perceived accurately, so is size. In this case, the diagram describes both the objective relations amongangle, distance, andsize and theperceivedrelations, whichachievefull veridicality.

tive factors that balance one anotherin an either/or manner. Thus,in

Figure 4,3, a difference in angle (line 4B falls under a larger angle than LM) can be outweighedbythe fact that LMis farther away than AB andalsois at a slant, in which case LM can beperceived as larger than | AB,If the difference in distance andobliquityis “outweighed” by the

difference in angles, then LM will appear smaller than 4B, andif the differences balance out, the twolines will appear equal. There is no clear statement that the amountofdistance and the amount of oblig-

uity enter into a calculation. The factors would appear to “balance” one anotherso that either (1) the pansof the balance are equal (factors cancel out) or (2) one set of factors makes the pan go down and so wins out.!”

16 Another ancient author, Cleomedes, was explicit that the relations between real and apparent distance and real and apparentsize are the same (Ross and Plug, The Mystery, 29). 17 There are casesin which Ptolemy's geometrical explanations do seemto indicate a precise perceptual outcome. In examining cases of reflection in the mirror, Prolemy’s diagrams and discussions

14 Location £ is the center ofrotationofthe sphere ofthe eye, on which,see Albert Lejeune, Excdide et Ptolémée: Deux stades de loptique géométrique grecque (Louvain: Bibliothéque de l'Université,

1948), 54-55. 15 Helen E. Ross and Cornelis Plug, The Mystery of the MoonIlusion: Exploring Size Perception (Oxford: Oxford University Press, 2002), 28.

at least suggest that objects seen in the mirror are seen at a distance thar equals the length ofthe ray. In this discussion, he says that objects in the mirror are seen at the samedistance as they would be seenin direct vision (which does notexplicitly affirm thar the distance is perceived veridically in direct or mirror vision, because direct vision cannot be assumed always to be accurate). Bur he also says that in both direct and mirror vision, “the farther removed they [the objects] are, the

farther away they appearto be from the eye according to the amount by whichthe visual rays are

en aaa





horizon: they appear brighter when overhead but dimmer on the horizon (I1.120, III.s9).8 Ptolemy noted other tendencies toward inaccuracy in perception, which hedid not evaluate with metric precision. Thushis generalstatement aboutshapeperception is that “the visual faculty perceives shapes by meansof the shapes of the bases upon which the visual rays fall” (11.64). As it turns out, this holds only for the shapes of surfaces that D





riGuRE 4.3. Theeffect ofvisual angle, orientation, anddistance onsize perception. If the orientation and distance ofBG are accurately perceived,it appearslarger than AB, By contrast,ifLMlies relatively near to AB,as in the diagram,it will appear smaller than AB even thoughit is perceivedas slanted and as farther away. But ifLAC

lies sufficiently beyond 4B andits distance andorientation are accurately perceived, it appearslarger than AB, Adapted from Claudius Prolemy, L Optique de Claude

Ptolémée: Dans la version latine duprés larabe de lémir Eugenede Sicile, ed. and trans, Albert Lejeune (Leiden: Brill, 1989), 44-

This qualitative treatment ofdifferences in distanceis also found jp Ptolemy’s treatmentofthe use ofcolor in painting, which provides the basis for his accountof the moonillusion. Prolemynotes that “mura. painters use weak and tenuous colors to renderthings that they want to representas distant” (11.124). As he explains(11.126),for objects thar are perceived as having the samevisual angle and for whichnodiffer. ence in distanceis perceptible by the length ofvisualray, bright objects appear closer and hence smaller (as 4B in Figure 4.2), and darker objects appearfarther way andlarger (as GD in Figure 4.2)—on theas-

are perpendicular to the axis ofvision. Objects that are oblique to the line of sight appear with a distortion: “Squares andcircleswill appear oblong,because, amongequalsides and diameters, those at right angles to the axial ray of the eye subtend a greater angle than those inclined [to that ray]” (II.72). However,if the length of the visual rays were accurately perceived in such cases, one could expect shape to be accurately perceived (thecircle perceived asa circle at a slant and theslanted length perceived accurately, as might happen ifLMin Figure 4.3 were perceived with metric accuracy). Accordingly, the perception ofthe

circle as oblong mustarise from a misperception ofthe length ofits diameters (a misperception ofsize due to obliquity). Prolemy doesn’t take up the othercase from Euclid,ofparallels receding into the distance. His remarks on circles and squares, while having some phenomenological plausibility, do not fully articulate and resolve questions aboutaccuracy ofperception of the length of the rays or specify geometrical structures that would fully accountfor the phenomena.

sumption that as an object becomes moredistant,it affects sight more weakly and hence appears dimmer. These considerations can explain why the moon(or sun) appears smaller when above andlarger on the 18 He maybe relying on experiencein asserting that che sun and moon appear dimmer onthe horizon

than overhead.Interpretation of Ptolemy onthe moonillusionis vexed; see Abdelhamid L Sabra,

lengthened” (111.76), If he means absolutely by the amount that the rays are lengthened, thendis-

tance is perceived veridically, We know, however, from his other discussions that he doesn’t hold that distanceis always perceived veridically, so that differences in distance might be perceived to be correctly ordered without being metrically accurate.

“Psychology versus Mathematics: Ptolemy and Alhazen on the Moon Illusion,” in Afathematics

andits Application to Science and Natural Philosophy in the Middle Ages, ed. Edward Grant and

John Murdoch (Cambridge, UK: Cambridge University Press, 1987), 217-47; Ross and Plug, The Mystery, 153-54.




be. lemy.” The reasons for this should we have seen in Euclid and Pro m met.

Ibn al-Haycha as we examinethe challenges that to a new intromission theory are Ibn al-Haytham’s contributions especially impor.

come cle ar

are he accepted about light many. Two points that

agent ofvision (as opposed to sometant: (1) that lightis the direct iumtransparent); and(2) that, when thing that merely renders the med opaque the sun or a candle) falls on an light from a source (such as in al] not a mirror), light is scattered body with a matte surface (so, 9-98).2

s(I.3.2 nt (small area) on whichit land directions from each poi is scattered from obj ect in the field ofvision,light Accordingly, for an ace ofthe eye. The var-

t o all points on the surf ive light rays from face of the eye therefore rece sur the on nts poi s iou sion theorist needsto find

eachpoint ofthe object

Unsensedrays Optic

Crystalline humor NN


f _\v--==

Vitreous humor

Radiallines (light rays


are sensed)


Cross-section of visual pyramid

FIGURE 4.4. The geometryofsight according to Ibn al-Haytham. Only y light li rays right angles are sens 1 that meet the cornea and crystalline humor at rays cnofacne ofthe lines or lines, folew the geometrically constructed radial i 2 « , :

crystallineSrathecenterof the eye. The light rays are thenrefracted at the reneofthe of the visual pyramid into the opti y a cross-section ip c nerve, “The cave chat touch cael forma visual angle which ‘ouch each end ofthe object ay Source: author’s dieu with perceived distance, enters into size perception.

ili follow th lightn that thatthe eye receives only thoserays ofdraw from thece e rectilinear paths of imaginary mathematical rays

to each point on the surface ofthe object. This imaginar co ofthe ve

as the visual uy construction exhibits the same geometrical structure sed (object to pyramid—albeit with the direction of causation rever ofthe eye (the but that story is not our focus retinal image to the ested in how Kepler related his new theory ofthe

t, as applied

extant theories of the geometryof the perceptions ofsigh

Hence, we mayattend to whathe says in his discussion of mintor co find his general accountofvision. Weare particularly interested in di. rection, distance, andsize (quantity). °

to visual angle, distance, andsize. ee




26 Kepler isn’t always consistentin1 hishis usage usageof “image” and “picture” but we can ’accept his disti istinc-

tion orour own purposes, as it distinguishes two fundamentally different concepts within his

25 For example, Smith, From Sight to Light.

optics, the purely mental imageas experienced and the purely physical picture ontheretina





In his account of direction, he aligns himself with the notions 4f

the pyt. a visual pyramid (or visual cone) and visual angles within thevisual fieh{ amid. Having noted that the eye is a globe andthat correspondsto a hemisphere, he continues:

objects to the whole It is therefore fitting thatcheratio ofindividual in the ratio ofthe enhemisphere be estimated by the sense ofvision, this is what is commonly teringformto the hemisphere ofthe eye. And the visual pyramid within called the visual angle, or the vertex of For in any single gaze, the the eye, whose baseis the objectitself. hemisphere. .. . Althoughall eye becomes the center of the visual is the centereitherof the eye the solid angles are in a point, which s, they nonetheless cannot be or of a particular one ofits covering surface is required, by which the distinguished in a point: thus, a angle,as is evident from the geometrical may measure the s olid


eyeis therefore itself sufficient to writers. This round sh ape ofthe

knowbyplain commonsens ¢ place among the principles we might set up aboutit. (ch.3, prop. 8, 79) that the eye has a sense ofthe angles

mid. It is for him, as for IbnalKepler in effect adopts the visual pyra that cannotbedirectly equated Haytham,a geometrical construction er’s theory form pencils oflight with physical light rays, which in Kepl lens so as to producea collecthat are refracted by the cornea and al pyramid hasits tion of points on the retina (a “picture”). The visu Kepler later conjectures,is the vertex in the center of the eye, which,

sec. 2, 184). It consists of lines center of curvature ofthe retina (ch.5, world, through the center of constructed between a point in the visual

notfollowthe paths of the eye, to a point onthe retina,lines which do spread out and the majority oflight rays in a given pencil, since these then are refracted and refocused. As regards distance perception, Kepler offered a new means for its accomplishment—a means that was available to pre-Keplerian theorists but (apparently) not developed by them. He explains that

FIGURE 4.6. The triangle of convergence. The converging eyesfixate



Interocularline segment LM, together with the angles of rotation forthove j

given fixation, forma specific triangle. Because segments LN and MN : te nde converge on the focal pointofthe eyes, the lines shouldbestraight, n ob ont ° the diagram, and eye M should berotatedslightly left. Diagramb ‘an ne he en, Source: René Descartes, L’Homme de René Descartes, ed, Claude Cle li na ted (Paris: Theodore Girard, 1677), author’s collection. Sellen andedition

the “commonsense” becomes acquainted with the distance betwe

the two eyes and thenlearns to associate distances to the comn on

focal point of the two eyes with particular angles of convergenceof the eyes (see Figure 4.6). The same method can be used with one ‘ by moving the head from onelocation to another, mimicking the oat tion of the twoeyes.”” He invoked the geometry of the ciangle which specifies that angles ofconvergence covary with distance for mgiven set

rethe angle berwnnn ae 27 Kepler also developed a geometrical a istance with wi one eye that depends onsensitivity " ; gecount of distance hen Forenlns ¢ ers ° " ¢ pupil and the projected image on the retina for a given Phtesphieent Gen) . ion,nssee Gary y Hatfield, “N atural Geometry in i Descartes and.K Kepler?» Res



, of of eyes, and then conjectured that perceivers learn to take account



this covariation (ch.3, prop. 8; ch. 9, sec. 2).78

Finally, in his brief remarks on size perception, Kepler continies the earlier theories in which visual angle and perceived distance coy, together in the perception of size. He accounts for the “legitima,. upon y,. comprehension of quantity” in the usual way: “Ic follows

the sense of comprehensionof the angle and of the distance, where

angle, from the sides that come togetherat the eye and the base of the pyran; d, up between them, makes a judgment about the

The “sides whichis the quantity of the object seen” (ch. 3, prop. 15, 83). 1¢ are the lines to the edges ofthe object, which form the visual angle, have Size the judgment of the base of the pyramid is accurate, then we

perception, nor constancy. Kepler doesn’t give us much more onsize squares or di._ does he examine the perception ofoblique circles and cuss the moon illusion.”? image reWe maynote that although Kepler’s theory ofthe retinal not require any quired a significant changein visual physiology,it did ion, asis inradical change in the geometrical account of size percept d, In dicated by Kepler’s own retention of the notion ofa visual pyrami eye the anatomyof the seventeenth century, Kepler’s new functional further discussions was swiftly adopted into optical writings andled to

of neural transmission. These also largely continued the geometrica] analysis frompre-Keplerian theory.

ac Mathematicis Usgiles 28 Frangois de Aguilén, Opticorum Libri Sex: Philosophis tuxta the pre-Keplerian tradig continuin although 156-57, 4-5, prop. 3, bk. (Anewerp: Plantin, 1613),


Descartes provided a comprehensive natural philosophical interpretation of the new physiology ofvision, including the physical character of light and the neural transmission ofoptical stimulation fromthe retina into the brain, topics that he covered in great detail in the Treatise on Man and summarized in the Dioptrics.°° Thelatter work arose from a

project to theorize lenses for telescopes and to make them, and so i emphasized refracted light in its title. But it also covered traditional topics in the theory ofvisual perception,including accounts of the er.

ception of“light, color, location, distance, size, and shape” (AT 6it30 O 101) and a brief discussion of mirror vision. Descartes’s accountof size and shapeperception again continues the themes andprinciples

have seen. oepiewe Descartes developed a corpuscular theory oflight, according to whichlight is pressure in an ethereal medium. This pressure aacts m echanically and follows ray geometry in being scattered fromeacl point on the surface of an object. Colorin light amountsto spit ‘on

the particles of the medium, and surface colors impart one on in other spin. The pressure and spin follow ray geome try in tlNat ami bientlight is focused by the cornea and lens onto the retina,aly whict cn . consists offine nerve fibers. Thesefibers transmit motion fromeach eye along g the optic nerves into the brain brain in i the point-for-point i patp tern we have seen before (as a two-dimensional order). The nervy . . . . . . c fibrils influence openings in the brain, also in a point-for-point pat

tern, causing subtle nervous fluid to flow forth fromaa gland gle in th e

tion ofoptics, described triangles of converging optic axes as a meansfor perceiving distance, If,m ect (Rome: Instituru as August Ziggelaar, Frangois de AguilénS.J, 1567-1617: Scientist andArchit theneither


with the Paralipomena, Historicum, 1983), 6o—61, contends, Aguilénwas notfamiliar unknown) pre-Keplerian source yet tas acommon(bu is there ous discoveryor we have simultane perception. for the theory that convergence of the optic axes can yield distance the moonunder various Keplerdiscusses the problem of measuring the objective visual angle of found himdiscussing haven’t I but 5), prob. 11, ch. 2333 5, sec. 5, ch. mena, conditions (Paralipo the moonillusion itself, The problem ofdetermining the objective angle was motivationalin his work on pinhole images, leading to his theory ofoptical “pictures” (Lindberg, Theories of Vision,


30 References to the Treatise on Man are to the Adamand Tannery edition of Descartes’s “ vol. 11 (with page number), followed by page reference to Treatise on Man change (Ar), abbreviseed ae ek Thomas Steele Hall (Cambridge, MA: Harvard University Press, 1972),

erences to to R the Dioptrics are to AT,vol. 6 (with page number), followed by page reference (N ere Descartes, Discourse on Method, Optics, Geometry, andMeteorology, trans, Paul J. Olscamp Merrill, 1965), abbreviated “O.” Emended translations are marked with an asterisk(* omeetobbs


dhguityerw iene oe





Pler did not. relate the neural processes that

$6), Ke ler

change to





that since Descartes realized t Bu . on ti ep tc atc with the ‘inMN processes th ion would covary bras

FIGURE 4-7, The visual system

showing in I as portrayed ved in Descartes'’s Treatise, ysiolo By and the transmission j of light si

pineal physi into the b

of the two-di -dimensional pattern


rain th tough a patte €

tn ofneural

motions. Diagram by van Gutschove® dirio?

iL Ho)m Mi eC de é René Desc,artes, ed. ’ Cla y 1677 * author’s collection

enter of the braj ain .

ins i This flowing again reca pat (the Pinealg. land). (5 tern gure 4.7),3! rtes’s accounts . n Fi both : rion sho¥ ist: ance, , size , and shape perce traditj Onal e h not so em el . mid dichoug. name pyra al su vi e th of . .. l] m) as we i on atti s He ut forward new! deas pa Innovyva ons. reulelyPyabhi o ut dii stan S epti The K . et on : m ar tc Pe hi c f leri an anatomy; 2S Kepl . ed,call . The in Kep fos self realiz ri E. ‘ focus ora mechanism h c

Point- for-point


ear and {4

r Vision,

‘ e to retain g0° é dation kno wh’ as «“accommo atl thee


s now e ¢ at perh Proces a é of th € whole eyeth(and thaps th l s act to change the shap sccle mu ye qn } {> . and h ence Nee its es cal len th) so as to achieve acco gth) . Cartes m s thi



h ystem In whic




s favor ls possibility, y, b but appears- to



odat . y e in‘ id ol ace,cosumm ontr ch proc esses cou Id prov the ey e of from lo or matitaonncab Pt o objJects mi e th ng wi 105-6) 4 ut distance, sl ance (AT 63137 ar che b nin to see dist th es ev li be so al ¢ ge conver ntrolling the with dsin processes co ry va co es ey e th Of he a ty e regulari c ogical fect posit s a ps hysioll ychop0 tWeen th case, heocinesef s se © brain pr s that produce acOmmod controlling the muscle . ation and conver of the

al experience and the perceptu , es fi Wt s ct je ob e innovative ideas . fo. the on. In addition to thes ti xa i t i d a r t s ke ception, sect Vo countfor distance pet ac to s an l me na io it ad Uch as Weak o tr reduce d visual as indicating distance of rs lo co ed ll du t je ob , ANgle for Ist:

nsize. n the town ofknow

ed. ements are plac el ve ti va no in e es th s, ic ramid nto a frames the Dioptr aped by the visual py sh ly ab iz gn co re is tion of spaand the SDine that account of the percep s hi ns gi be s te ar sc s “the di‘al properties , De , which he describesa on ti si po or on ti ca on to OUF body nen in hee lo e object lies in relati th of rt pa a 6 e with a i 6:134, O1 g of direction by the ey o sticks in ns se es ar mp co Treatise, the tw ind man holdin . He that cross. (In the ks ic st two ng analogy with ‘ouch a sing 11:160 H 62].) The T [A ct je ob an on on tne t our focus here's Abling noten, point in its own right. Bu interest scussing distance©s Way it establi is of framework for di e th as serv’ ea funcn io ct re di Size irecti i . and | ishes as Descartes proceeds, the directions pyrami For, ape. oF the visual which Yon . lines in Ibn al- Haytham _ Simi " £° in| Kep| lar to the radial


my a nd physiol em ina way ig the stiimulation f; ofth the nervous syst! val al optics j logy romth, the two An example a mechanical idiom. th ¢ pine? ew sit C0 of FS it cyes Tbr Leatternon the surface th brain to expe osited a whe.

common { ee speculates be y thar 5 tha(Ome— 10) mse P -.2) y cy qi tone aeHa n ain single Keplen Conjoinji NE ofth, ( Dut requires a dif se Vision, 80677 ch. ferent physiolo 180, ics, ° n yisio® is follower Witelo) BY Ofney ral ¢ € serves the funceion of “ingle ._Uerves Tansmission th . an that found in Ibn al-Haytham a 32 Johannes Kepler , Dioptrice ( Augsbur,gS David 1962), prop. 64, F,Tanke, 161), repri


Feprine, Cambridge, ge, UK:US Heffen

the skeleton is in effect of directions* of size and shape: system perception The the cr, istance is attached to yield

e center of th b not tO the be offering the ody, re a he a) escartes may s in ( Natural with direction directions 34 In th atfield, specifies the by ewo eye seen as Descartes vraaal world cye, Soto me nae hinted ar ofa Icon cess} ake phenomenology the to ee €ssion en 33 Se



inated with the center ofthe



ore detail Geometry,’ form







Descartes’s account ofsize perception advances a clear stateme,, Speaking of of the SDIH and the tendency toward shape constancy. objects, Descartes says:

we cannot phenomenally present themaslarger, because we do experience objects larger than one or twofeet in diameter. Rather, we

or the opinion, Their size is estimated according to the awareness, of theimage, that we haveof their distance, compared with thesize

This distance combines withvisual angle to yield the perception of a small disk. Theillusion of a larger horizon moonor sun arises because we experience the distance to the horizon as being longer, owing to

tely by that they imprint on the back of the eye; and not absolu while the the size of these images, a s is obvious enough fromthis: when objects images maybe,for example, one hundred timeslarger

are limited in the distance at which we experience them: we experi-

ence them as being at most one hundred or two hundred feet away.

the many intervening objects, whereas we experience the distanceas shorter overhead, where there are no intervening objects (AT 6:1 44-

they are quite close to us than whentheyare ten times farther away,

45, O 111).

their distance does no¢ of this, but as almost equal in size, at least if deceive us. (AT 6:140, O 107")

al-Haytham. We perceive shape by perceiving the positions (direction

meslarger because do not makeus see the objects as one hundredti

extent thar The size of the image “on the back ofthe eye” is the retinal treated as Kepler has already treated as equivalent to visual angle (here visual angle an area). Descartes denies that perceived size varies with

a function (see also AT 6:145, O 111). Accordingly, perceived size is

the sizes of objects as of visual angle and perceived distance. We “see” indicates that the “almost” the sam e whentheyare near andfar, which

ence ofsize and combination of angle and distance creates the experi utsize. This amounts does not merely support a detached judgmentabo moderate distances.35 to a claim ofnearly perfect size co nstancyfor ance, whether Descartes relativizes perceived size to perceived dist the latter is accurate or not, when he remarks that we perceivesize accurately if we are not perceptually deceived about distance. In n), he in effect analyzing the moon illusion (or moon and sun illusio sun as beingat appeals to the SDIH. We experience the moon and most one or two feet in diameter(he claims). The problem is not that


33. Descartes’s desctiption of nearly perfect size constancy suggests that he could accommodare

convergingparallels (as with wagon ruts) by suggesting thar, as things become moredistant,distanceis slightly underperceived, but he doesn’t make the point explicitly.

Descartes gives an accountof shape perception similar to that of Ibn

and distance) ofthe variousparts of an object: “Shapeis judged by the awareness, or the opinion, that we haveofthe position ofthe various parts of the objects, and not by the resemblance ofthe pictures in the eye; for these pictures usually contain only ovals and diamond shapes

yet they cause us to see circles and squares” (AT 6:140~41, O 107"), Theretinal imagesof circles and squares that are obliqueto the line of sight have the shapeof ovals and diamonds, in which the diameter of a projected circle is longer on one axis than another. In terms of visual

angles, the visual angle fromedgeto edgeis wider along the axis of £0:

tation of the circle, narrower along the diameter perpendicular to that But we “see” circles and squares. Whichis to say that, for moderate distances, we accurately perceive the diameters ofthe circle as equal andat various distances from our vantage point, as Ibn al-Haytham taught. Descartes thus continues the geometrical analysis of size and shape perception from the opticaltradition, as purveyed by Ibn al-Haytham

andhis Latin interpreters. He uses the sameprinciplesfor veridical size

perception asforillusory perception. In both cases, visual angle (visual

directions) and perceived distance combineto generate an experience ofa spatial configurationin thefield ofvision. Heeffectively describes a perceptual space ofperceived directions, sizes, and shapes, analyzed




ang sn accordance with a visual pyramid of phenomenal directions distances.*¢ OF SIGHT BERKELEY’S REJECTION OF A GEOMETRY

sought to minimizy Berkeley framed a theory of vision in which he in the visual percep. the appeal to the geometry ofsight and to expla gh a proces, tion ofdist ance, objective size, and objective shape throu In this discussion he makey of learning, with t ouch educating vision. gs from geometrical optics whilg explicit or implicit use ofs omefindin gards the perception of distance by rejecting others, especially as re, “lines and angles” (§12).”” published a year before hig Berkeley’s New Theory ofVision (NTV),

Principles ofHuman Knowledge,>8 was intended

secretly to prepare the

metrically described spatial properties found in

the immediate objectg

work. This preparation way way for the immaterialism in the second that there are no common, geo. to be done in large part b y showing ly quite radical,as it rejected of touch and vision. This move was real epted by both Aristotle and the doctrine of “common sensibles,” acc ial

size, shape, motion, and other spat Descartes, according to which

touch andvision. properties are sen. sed by both and tangible size are of different Berkeley’s claim that visible size , is counterintuitive. We'll come kinds, sharing no common structure ts that orig.

on, Berkeley posi back to it. More specifically as regards visi

shapeaccord with a correlate inally our visual experiences ofsize and System of Natural Philosophy, tans. Samuel 36 This conception is affirmed by Jacques Rohaule, describes vision as consisting

, 1735), whenhe Clarke, 3rd edition, 2 vols. (16715 London: Knapton the mind, wh ich accounts bothfor our seeing in the formation of an “immaterial image” (132.1) in ‘ ~23). (1.32.22 s llusion andfori ) (1.32.13 are they as objects a New Theory ofVision, towards Essay An , Berkeley 37 References are to sec tion numbers of George ), and (below) Zhe Theory ofVision Vindicated grd edition (1709; Dublin: Rhames and Papyat, 1732

and Explained (London: J.'Tonson, 1733). and 38 George Berkeley, 4 Treatise concerning the Principles ofHuman Knowledge (Dublin: Rhames Papyat, 1710).

2 17

of the retinal image and not with the objective sizes and issh apes of Visual magnitude originally

things, as determined by touch. with a counterpart ofwet ° cauated with visual angle, and visual figure | cnenin gure. perceived by sight. Through Distance is not originally we sizes, and shapes with ou ba come to associate tangible distances,

within the original or immediate visual experiences. Th with ' oun the the convergence of hee, muscular feelings associated with weaseyes, or by arm extension sociate tangible distances (measured associated tangible distances become the mediate obje paces). These

re nes The te. diistance. experience ofvisible objects visibl phenomenal yielding adistance, however, is not combined with value for

renade theorists from our previousexperience on). to yield objective size (as inas of dis> tolemy yield the

Rather, the same visual cues are di. size. A small visual ma Neate rectly associated with tangible associc convergence yields the gnitu ated with muscle strain for near object veeecePtion ofa small tangible (objective) size, as the mediate Berkeley wants to deny that the human body, its senae described by von organs, and physicallight inhabit a common space, (SSt45necessary connectionbecwene 51). Heis eager to reject any

the projectic tangible distance and visual angle, explicable through ; Le I do not question thi ical light onto the retina. While

Berkeley's motivation, I do find him

relying on geom ‘ escription of

angles in two . a ee arguments. I consider his use of lines and

for a given wbjeesconaria Berkeley writes as if the visible magnitude with tanknowthis? Second, he ieee gible distance. How does he gument from the geometryof sight to the conclusion the “die, ar oan. ofitselfand immediately, cannot be seen” (§2) that I wish

In the NTYV, Berkeley remarks that computations with lin tnalyze

ofrh gles can be usefulfor determining “the apparent rmagnivade (§78). Apparent magnitude varies with tangible distance - ‘ hee

of constant tangible size. Here Berkeley uses the lines and an , light (“ h ica “s of physical of path tangible the optics to describe much as Berkeley treated the tangible world as the world of ohysics),




Byparallel reasoning, the visual figure ofa tangible circle oblique to of Vision the tangible line of sight would be an oval. In the Theory

this conception Vindicated and Explained (TVV), Berkeley affirms visible sizes and of howtangible sizes and shapes are correlated with tive grid in shapes. He imagines the construction of a tangible perspec e eye to various points front of an observer, with lines from the tangibl

in the tangible scene (§$55—-57 ).

Theplaces wherethe lines intersect the

h track visual angles) and plane grid then predict the visible sizes (whic visible shapes. of the visual pyrIn this way, Berkeley in effect adopts the geometry

to scene. Hetreats a perspecamid as applie d tolines ofsight fromeye of the proper objects of sight tive cut on the pyra mid as a predictor ). Note that he denies any causal (original or immediate visu al ideas t the tangible magnitudes on theretina relation; he does not hold tha ousvisible properties. Rather, cause visual sensations exhibiting the vari various sensory ideasas he posits a regular correspondence among the Authorof Nature” (NTYV, 3rd. ed., established an d maintained by “the een tangible andvisible

tion betw §147). But how does he knowthe rela


line is here equated with distance? Surely not the line taken by light

rays as they enter the eye, since those form a pencil that expands ae

contracts, meaning that someofthe rays are bent(refracted), So re construction in sumably we havehere line of sight as a geometrical

wn information aboutits fixed own the visual pyramid. That line does notoffer fixed location of the eye and on depends it rather length; in the scene for its construction.


©c points

However, there is information aboutdistance contained evenin viey t th‘

(small area) in the field of light coming from a single point -. an single point on the retina as a projected through refraction

. to be itself in order AL, feet) fo of a few by Descartes, the eye must adjust distances noted for points closer or farther away (within the convergence of th so, as noted by both Kepler and Descartes, e two

with distance. Berkele eyes on a single fixation point varies

them. His argument t was aware of such explanations and rejected

seofcon of conthe c holding, byintakin authors such as Kepler and Descartes the asdistance vergence, that the mind calculates between therm on of the rotations ofthe eyeballs and the distance

he should claim an imideas? According to his own principles (§r0),

Berkeley obiewsch then applying an angle-side-angle construction, to

end noticeably smallervi he does note that things farther away subt

he claims, ma oe fixation lines, and interocular distance. But, common people) mn estly not all perceivers (including children and

And fvisible magnitude and figure. mediate phenomenal awarenesso sual

fication ofthis sort. angles ($44). But he doesn’t offer a general justi r ofthe visual cone. Instead, he appeals to the geometry ofvisual angleo NTV is §2: “It is, I think, The most celebrated passage from the diately, cannotbe seen. agreed by all that distance, ofitself and imme to theeye, it projects only For distance being a line directed end-wise remainsinvariably the one point in the fund of the eye, which point er.” Here Berkeley also same, whetherthe distance be longer or short appeals toa geometrical result, to establish

the (alleged) commonplace

ot be seen that distance, being a single line from eye to object, cann eye).2? What because it projects into a point on the retina (or at the Nova, and edition 39 The mostlikely source for Berkeley's claim is William Molyneux, Dioptrica

aware of therotation ofch s that calculate, one would need to be

these geometrical constructions or are able to




perform the needed

Berkeley's objections are guilty ofignoratio elenchi, thats, of refuti an argument not made by his opponents. In the case of secom ‘ode tion, and plausibly also for convergence, Descartes did not holdthe: 1709). ButBu in the ver y passage in (London: Tooke, » 1709). i which i Molyneuxsays “Dist:



to be perceived: for ‘tis a Line (or a Length) presented to our Eye with its Endton ods nae ‘ch

must erefore be only a Point, and thatis Invisible; he allowsthar, through converge , Ms,which cant icperceived for objects near at hand: “But as to nigh Objects, to whose Distancethe Inecrval ‘angle vss ears a sensible eyted their Distance is perceived by the turn ofthe Byesorbit




Axes” ( (113). He does not specify a psychology

of th

i iy

ndetiotne i

convergence,but he had available to him both learning and nativesacount underlying




the mindcalculates. Rather, he posited an innate psychophysiolggal


independent accountsofdistance andsize perception andso, as a psy-

mechanismby which the brain states that control accommodatio,, and

chological process, does notacceptthe direct combination of distance

convergence throughthe ocular musculature—brainstates that there-

with visual angle according

fore vary with distanceas they adjust the muscles—directly cause jy she mindan experience of the appropriate distance by an institution ofa.

ture (an innate law ofmind-bodyinteraction).It also seems that Kepler did not explain the perception of distance by convergence a5 4 marrer of calculation but of learning to associate a tangible distance wip, a degree of convergence (Optics, ch. 9, sec. 2). Even Ibn al-Haytham gid not hold that the mind uses geometrical calculation to becomeaware ofsize, but rather thatit learns piecewise the relations betweenyjcual directions and tactually learned distances. The theories of Ibn al-Haytham, Kepler, and Descartes differ fom

that of Berkeley in that the first three conceived of visual and tacpyal

space as of the same kind and so as comparable to one another (eyen in casesofillusion). They therefore could allow tactual learning to directly inform, withouttranslation, the experience ofvisual distances. In contrast, Berkeley contended that visual and tactual space are of completely different kinds, His arguments for the latter point would

need to be convincing if he wanted to dismiss the previous theories

without begging the question.

Be that as it may, it is clear that Berkeley, too, regularly availed himself of the geometry of the visual pyramid in his theory of vision, Phenomenologically, he would agree withIbn al-Haytham and Descartes that oblique squares andcircles, at least at near distances, are experienced as squares and circles (NTV §§139-445 TVV $59). For Berkeley, this would happen only in mature perceivers who haye

learnedtoassociate tangible sizes and shapes with the—for them now unnoticed—visible sizes and shapes. He thus describes a phenom-

enal visual world in which size and shape constancy obtain for near distances, in which case the SDIH holds as a phenomenological description. He doesso in conjunction with applying the visual pyramid, nowallied with a psychology ofassociation. His new psychologygives


to the SDIH.


There is : long history of the application of geometrical constructions to

spatialvision, including the perception ofdistance, size, and shape. As

a geometrical description of the phenomenology ofsight, this applica-

tion showed development, Thus, whereas Euclid equatedvisual size with

visual a fun

angle, Ptolemy and manylater authors described perceivedsize as

cton of perceived angle and perceived distance. Berkeley agreed

that siz “s constancy obtains, and so he accepted this; phenomenological desctiption, riuctions in describing visual expe wen Using geometrical constr subscribed to m, Kepler, and Descartes all tha Hay al{bn t e e som eis combined account in which visual angl vi rm of Psychological ey rejected this ox yield perceived size. Berkel bln ae distance to cues are used in dis-

thar, although the same naneeaneaneed arguing in the two cases. they operate independently n, tio cep Pet n le el “B ely accepted ch educates vision was wid tou t tha nt ume arg also i heat uries. However, there nineteenth cent now Sighteenth and

which eley, according to in opposition to Berk ory the undergo enmh innate mechanisms that to due is ion vis Spatial vironme : . contine isof Womental tuning.4° Beyond theissue of n ativism, ther logy visual ing 8 discussion dj 3 . ofhow best to describe henomenolo the p eee

40 This nativism Tourtual, Die §

century, in the works of Caspar Theobald Jseitigen Beziehungen ibres psychischen umd

Omas K. Abbot, and Touch: An Attempt to Aesthetic Disprove the cco! z aay Tronchen Lebens: Sight Ein Beitrag zur physiologischen idSOO 64). Onthis controversy 11 nineteenth-century German of Vision (London: Ls empiricism i as opposed to the empiticis nativism of Ewald Hering including 164" Green, the and psychology, * Longman, physiology i of Hermann vonFlelmetes sceGar Hatfield, The Naturaland the Normative: Theories of.Spatial

Perception from Kantto Helmboltz (Cambridge, MA: MITPress, 1990) and R, Steven Turner,



. . : remain open concer the wens space.4!


ption of ng is © ngoi PP es

theight. There nineg ofs deslecriobliique to the lin of a circ receding int

e w to siion of ho uss discus ‘.

parallels cribe the perception of ct ‘n distance The spplic structions © aspety geometricalAndcon of continui n ains ao sust it . phenomenalvisofual de experience continues

with a long past

relying of the pre ibing visual experience by scr obs l ica log eno As we 5 nom see in straight lines we t tha e w h with Descartes, wac lied roa vis s are app ion uct str con a pre metricor may not accurately sent che phy® n, eeo tiotie cepper may ofperpro ct iali.spat s ial the eyes ° objects that reflectlight to arc ha Properties and status en ese ofr topics is visual space remain

n Reflectio CER


Jermifey Groh


ain ons in the br

ty of neur ed bythe activi at di me past sixty is t ee of 'S one me. Over the

How thou

ti ists mysteries of our neu roscient Yeats, a suce breat has allowed s od il. th me ta bee of new level s of de g in as re nc to investigate -i er ev arn signals with much ¢ o le ut despite «brain of data, we have

ew bi e resulting sl higher cognitive ms create is an ch me al ic iyles w biolog abilIt anguage: hought and l ests that t as h c u s e i n nction sugg u O f n i a r b ui triv eory of echani sms h m t g n n i a i r u b g i e r : t tion. there m a n between th connectio le for cogn his b i s n o . y be a p s e r e t Und and those gnition,’ odied co erception p b m g subset n e i f y : e o l m t Now ounndkeidng ng so r ti g va as ti ac y sl riou tes that thi involves theoorr,ynpva ostula n© the brai hways of of s t a p and motor rela she S attributes ory or motor

about how



In the Eye’ yes Mind: Vision and th : University pnd: € Helmbolpz.Hering z¢ 1994). j ry 41 For ent y int o¢ . i

to™ CoControversy (Princeton, NI]: prince

he ongo going di b Scussionon, see M; , Geometri NJ:. Erlb atm, ace (‘Mahwa e }; s Gary He ield Philoso nek Wagner os l 26a @ i ries of Visual Sp h P Phi e? ac ” » epti ee af ology of Visu Sp rc haw en Pe . xc om of icE y en ph Ph eo so the in Spatial|e 2 (2011): 31~6 ield and itive Factors ception,’ in l Experience oa Hatf menal and CognB' ord: O* ord no he Universi ta el Sar: pe tf d xf Ha ) (O an y 12 id ed i . Gar 20 § > ed. vo Phenom y Press, nd Sarah Allr ath Eric chwitz enologi Reseearch he72 Do ThingsSu1pok ae ? Philosophy eh t al Contentsteal 389-99; eel “No Dual Hopp, ch Look: wt Theory Phooneoe and Cognit Ser Sci ve B53 3 Problems 42 The member: ofthe works hop ology Uae 12 (2013); i . Duke ° ideas at the back on j where for feed niversity level Versi herein, 3 gave initial fatle ch Telhank cane for helpful Thanks alo telaouise Daoust an oo mm ents o 2 the penulvimace ion, This re. search was supported by cFA- $7881-


6171617 8 59 (2008): Review ofPayChOl2GY Questions, nnetl mae” Research jon? 2 20108! nded CognitionCognition in the Sciences 21, 0. “ mabodied the Learning of rr ence Journal ' Forexamph Bars . 453 Rafaa oe, i nm a Testable Explanations Red ? ctio Appropriate



(2012): 324-36. ited by? ace: Ed

ought 1m Sp , Spacefor th 9009 Jennifer Groh sity Pres Oxford Univer 3 ? 3/080/978019 DOL: 10,109

Press (2020), al, Oxford University AndrewJani






auditory, smell, taste, and motor responses that would occur if you were actually doing so. Put another way, thought might


_ in yourliving roomdrinking a cup ofcoffee, that thought might be implemented bypartially activating the visual, tactile,


For example, when you mentally picture sitting on the couch

involve simulations run via the brain’s extensive sensory and motorinfrastructure.

Oneimportant implication ofthis idea is that the way sensory Less

howour thoughtprocesses unfold. That is, what these brain areas do in responseto sensoryevents, or how they guide motor actions,

neural activity

and motorareas function in their so-called day jobs may *mipact

mayspill over into how they contribute to thinking, shaping the

nature of these cognitive simulations.

That's where space comesin. The brain deals with space in nearly every aspect ofwhatit does. When you watch your child playing

in the yard, yourvisual system monitors the location ofyour child relative to thestreet, particularly as your auditory system warns you

that a caris approaching too fast from around the corner. When youcall out to yourchild, your brain issues motor commandsto move your tongue andlipsto the series ofpositions needed to

form each phonemein the speech sounds “Get out of the road!

Location, whetherof the external events orof the relevant parts of





activity FIGURE . ; ateactiverey pains have two mainformats for encoding information: Taieros Activity

ate Sensitive to he neve they are, In the visual and tactile systems, int he Ieation for that neuron € cations of stimuli and respond only to stimuli in Whee » ceimulus is located can bei, * Teeeptive field (indicated by the dashed circles).

tus This type of codeis tefe . qed from which neurons are activated by that weneurons A3, Bs, locations activateneore and C3, Ip othe, *O as a map. The image ofthe guitar would

areas of the brain must monitor and control in such everyday

of stimuly . systems, stich as the auditory system in the primate Df £ neuronsis. by knowing how active a common. population 0 illexhi . ample, “an j be inferred il exhibit Here,a fercing neuronsfrom


level of activity

the body,is the overarching theme ofwhat the sensory and motor

Studies of the neuralbasis of sensory and motor processing have

exof g2lei 4 Buitar is located to the right, “right-pre Fot level

“CUVity indicat



Atigy § how


ind is com


i farto the right the ¢ g guitar sovweype of code Known a

: " » “tves as the needle ona dial of “rightness,

established two important ways that neurons encodeor represent the spatial aspects ofthe sensory events or planned movements.”


a kind nces, information about space +1s s ate t in Oosted mecon s reflec I N some iInsta of neural “map” oflocation. In such a map, individual n ¢ Jennifer M. Groh, Making Space: How the Brain Knows Where Things Are (Cambridge, MA: Harvard University Press, 2014),

*€sPOnsive to specific, ‘discrete, and often small regio spa j ns of space. imuli stin visual to i respondonly might example, i neuron,for A visual i located within a narrowly circumscribed region,say, an inch or tw :







in diameter.? Different neurons respond to differentlocations, and

Vis wal map :

across an entire population, visual stimuli at all possible locationsin space influence theactivity ofsomespecific subsetof the neurons in the population. Often, such maps are literally map-like: they have a topographical organization in which neighboring neurons






i y meter Auditor


aS AS Fé




are sensitive to neighboringareas ofspace. Yoursenses ofvision and touch both representspacein this way. The pattern of activity, ie, which neuronsare active, informsother areas ofthe brain where the visual stimulus (such as the rambunctious child playing in the yard)orthe tactile stimulus (such as the mosquitobiting your hand)is to be found. . of asa “meter.” Instead thought be Another type of code can of involving a point-to-point correspondence between locations and neurons, such meter codes involve a correspondence betweenlocations and neuralactivity levels. Thatis, a large swath of neurons might respondfor any location, but how vigorously they respond as a group might convey information about wherethatlocation is. Movementsare the prime example. When you move your arm up, howfarit goes depends on how much activity your muscles generate. The same muscles contract

for both small and large movements, but they will contract longer or harder to movethe arm farther. Sound is similar: how vigorously different left-preferring versus right-preferring groups ofneuronsrespond helps you detect where a soundis to yourrightorleft.4 If sensory and motor processing have anythingat all to do withthought,thenit seemslikely that spacial maps and meters play some role. Recruiting these pathwaysfor thought-as-simulation

3 Or muchless, , or much more—it —i depends onthe brain : area and how far awaythe stimuli mull are, . . Kristi ‘Other’ Transformation Required for Visual-

Jennifer M. Groh, “The Porter a Kel * Audit tion:and itory Integra Representational Format,’ Progress in Brain Research 155 (2006): 313-23.

enee ecgg FIGURE R42,



type of stimulus andan the same population of neuronscan serve asa map

for one

observed in the Primates for another. ‘This schematic depicts the ac tivity patterns uli. Visual stimuli at diffe ~Petiorcolliculus in response to visual and au ditory stim (illustrated

as different colored tthe locationsactivate different neural populations Soundsat those same Positions activate hee Ms’ ofactivity inpanels A-C,left side). level ofactivity that witha but “ons throughout thestructure, depends on the sourDin location (A-~C, rightside). Source: J. Lee and J. M. Groh, “Different Stimuli for Oculomotor § ifferent Spatial Codes: A Visual Map and an Auditory Rate

Pace in the Primate Superior Colliculus? PLoS One 9 (2014) 85017:

mi hti


tivating he maps; ac

t “ngspatial patterns in ed, evidence graded lerdeons , or Perh. Inde

nvity in the meters coding suggests that th, ole both in en r l dua 2 ys pla rietal cortex the locations f. Pa din allowing auditory targets 49 d an es al su Vi OF d he that involv humans an t—a kind of thoug un co to ly s al ey nk nt mo s. Me s between quantit ie hip ons ati rel d re de wranin atically or to bein g carried us seems well suited th s er mb nu ng ti ationships Manipula at handle quantitative rel es th outby brain structur

tly a matter of of other types. Ie is presen

debate whether these





mathematical functionsof the parietal cortex are handled as a map or as a meter? Another exampleinvolves the brain’s map of the body and body. related language. Wordsrelated to differentpartsofthe body, such as “kick” and“lick” have been shownto evoke activity in different

sensation and thought and that the relationship between the two

locations in body-related cortical areas in a fashion roughly resembling the brain’s map of the body for touch or movement.$ In

used for thoughtbut that wouldstill show the fingerprints of

fact, even metaphorical usage of touch-related words, such as “I had

a rough day,’ can evokeactivity in the body-related cortex.’ Such observations supporttheidea that when understandinglanguage, your brain may simulate sensory and motoractivity related to that meaning,

The cognitive linguist George Lakoff and the philosopher Mark Johnson have noticed that spatial terms may be used to denote even nonspatial relationships.® For example, we maysay weare

close to someoneorthat someoneis a distantrelative; that we have

high regard for someoneor that someonehas been /oweredin our estimation. This important insight suggests a way for sensory and motor brain mechanisms to support not only concrete thought, for

which the sensory and motor connection maybe obvious,butalso moreabstract thinking,

A critical question that has yet to be answeredis how the brain simultaneously processes sensorimotor signals and thought. - Onepossibility is that there are in fact distinct brain circuits for

5 J.D. Roitman, E. M, Brannon, and M.L, Plate, “Monotonic Coding of Numerosity in Macaque Lateral Intraparietal Area” PLoS Biology 5 (2007): e208; J. D. Roitman, E. M. Brannon, and M. L,Plate, “Representation of Numerosity in Posterior Parietal Cortex? Frontiers in Integrated Neuroscience 6 (2012): 25; A. Nieder and S, Dehaene, “Representation of Number in the Brain, Annual Review ofNeuroscience 32 ( 2009): 185-208,

6 O. Hauk, 1. Johnsrude, and F, Pulvermuller, “Somatotopic Representation of Action Wordsin Human Motorand Premotor Cortex; Neuron 41 (2004): 301-7, 7 S, Lacey, R, Stilla, and K, Sathian, “Metaphotically Feeling: Comprehending ‘Textural Metaphors Activates Somatosensory Cortex) Brin Language 10 (2012): 416+21.

8 George Lakoff and Mark Johnson, Menephors We Live By. Chicago: University of Chicago Press, 1980.

is therefore one of similarity rather than identity. Thought-related circuitry might be similar to sensorimotor circuitry because

it may have evolved through a process of dupli cation of brain tesources, creating extra capacity in brain

circuits that could be

its evolutionary history, Whether the thought-circuits and perceptioncitcui tS are coextensive : . inct but similar, the or dist paramount npor Ceivin importance of per.

distinctive marl.

g and moving in spaceis likely to

leave its



Space in the Seventeenth Century


this shift from the “closed wor ld”t o the “infinite universe? in Koyre’s famousphrase, rpresents a profou nd development of the seventeenth century. Itis pe r aps as momentous asth . .ion from e transit a Ptolemaic ‘ toaaCopernica n worldvie . . . w, wi

th which it was .intertwi4 ned. 2 tis tempting to descri that mathematicians a she shift captured n Koyrés title by noting

thinking of space (o. . tonomers, and philosophers shifted from


as infinite. This descri ometimes the world) as finite to thinking ofit idea that the conce coe of the century’s shift is predicated on the change was in breakin, Pola itself underwent little alteration—the

Binning to think that . “se of old prejudices or assumptions and bethe Concept of space a.ace mightactually be infinite, Thatis, we hold But as it turns out hee» and then alter our notionofits fearures.

the developments swith 1s anachronistic: it involves a description view of the century’ in the seventeenth century from the point ° interpreters in ein end. Briefly put: it specifically involves today's that is Widespread ane an attitude toward thinking about space ™ Century's end and then imposing that perspective When the seventeenth century began, the natural world was finite


and the humanplace within it secure; by century’ end, nature had been embedded in aninfinite spacecalling humanity's special Status


€ ofPhilSophy j d the irinfini : in pottsno Copernican : . nfinite Universe, ap least of any :prominence appears to have esp ouse So” i i ; to Dey the onception” Public. AfeerThe Descartes, however, no OTT nan stronorty in the peepmens of Western Th h Omas Kuhn, Copernican Revolution: Planetary Bp, Thomas

into question. In the old picture, one inherited from Aristotelianism,

188s 'n thelate sixte

nature’s familiarity was well established and nature was amenable to

Galileo and Descartes had been absorbed and extended by Leibniz and Newton,natural philosophy underwenta seismic shift: mathematics took center stage in the study ofnature, and space’s infinity becamethe

“fundamentalprinciple ofthe new ontology.”! Thereis little doubtthat

1 See Alexandre Koyré, From the Closed World to the Infinite Universe (Baltimore, MD: Johns

Hopkins University Press, 1957), 126. As Kuhn writes in his Copernican Revolution, published in the same year as Koyr¢’s classic: “From Bruno’s death in 1600 to the publication of Descartes’

AndrewJaniak, Space in he Seventeenth Century, In: Space. Edited by: AndrewJaniik, Oxford University Press (2020), © Oxford University Press. DOT: 10.1093/080/9780199914104.003.0010



study through familiar philosophical techniques involving little mathematics. After the revolutionary challenges to Aristotelianism by

she (Cambridge, MA: Harvard University Press, 1957 ‘aye Steven

SPIN, The Scientific Revo century might be the exception that proves Kuhn's ritei

The two devel

Cvolution (Chicago: University of Chicago Press, 1996): 22 ty

article on

t Opty : : : 4 Yeronounces a Cartesianism, piments Were intertwined, but distinct. In an influentiet “tT the shift in thinking Nicholson distinguishes the developments and then De 9have noticed tseventeciathece he effect of ¢ © Cartesian MB about idea spaceofindefinite to be more extension significant:upon “Few,one however seem © nificant ofall of the moet«e fan infinite

umberof wy, twy Conceptions: the idea of infinity, the problem of the poss p to the characteristic forme ‘ OF of a universe infinitely extended.In this conception fethe ost profound ta in E one : changes which nin England, at least, by the idea of progress, and co in men’s conceptions Ot themse} oe ocurredin seventeenth-century thought; for the real rane ugh the expansion Ves and of ¢ i icanism than tar ages ofthe Oundaries of ¢ uni verse cameless through Copernican oP Marjorie Nicholson, “The Early Stag: of Cartesianig > thoughr through the ideaof infinity. at} Nicholson, then, it was not merely anism in England? Srudies in Philology 26 (1929): 370 rotation to its infinity—to a Univers, “ *Pecial attention given to the concept ofspace pv ee seventeenth-century thought.

© “infinitely extended”—that represented a profoundshi tionon spaceitself, rather than on +


to focus their atten!


Sofie they were willing to

began whereby Nature or theess world, was philosophers a complex one, andit took considerable effort befo

accept theidea that space wasinfinite, in possibility ifnot in fact.




on mathematicians, astronomers, and philosophers working at the

century's beginning. Ironically, this interpretive approach, tempting as it is, and despiteits attempt to describe a profound shift in human thinking, dramatically downplays how radical the transitions within these fields in this century really were. The transition was not merely from a finite to an infinite world, from the Scholastic to the modern, but more profoundly, it was a shift in which the concept ofspace, along focus for with its structure, became a singularly important analytic

mathematicians, natural philosophers, and metaphysicians alike,

Even the most revolutionary thinkers working between, say, 1600 and 1650 did not take the concept of space to be of anyspecial] phil-

osophical significance, if they contemplated such a concept arall.

of Concomitantly, they did not analyze the structure or the Nature space itself. Motion, objects, God, persons, even nature itself—these

were the objects oftheir study; not space. For figures working within

mathematics, metaphysics, and natural philosophy between 1659 and

con1700, however, it had become commonplace to contemplae the cept of space, to consider space’s structure, and to ponder i nature famous of space. Whereas an earlier thinker would analyze Euclid’s

of figures parallel postulate by considering the operations and features

might el and lines, a later thinker would consider what the postulate us about space’s structure and about the structure of a *pace in which

wou d analyze the postulate failed to hold.3 Whereas anearlier thinker rect motion by considering what would happen to a body moving consider vous linearly under no external constraints, a later thinker what the principle ofinertia mighttell us about space itsel , Whereas an earlier thinker would analyze extension as a crucial attribute of

andthefewedation: 3 See the fascinating work of Vincenzo DeRisi, Leibniz on the Parallel Pagilate be ofGeometry (Cham, Switzerland: Springer International, 2016), which ocuses on Lei nizs un standing of the foundations of geometry in general, and of the parallel Poste ate in Particular, i‘

which also has importantimplicationsfor our understandingofthe focus of geometriciansin the

century as a whole.



physical substances, a later thinker would ask whether the inherited substance-accident metaphysics could accommod ate Euclidean space.

In considering the trends that led to a new philosophical emphasis on space, along with a newfocus on the questiono fits basic structure, the danger of anachronism looms large. It is Precisely because figures

like Leibniz and Newton are so close to us compar ed with their me-

dieval predecessors that we are tempted to use our b asic categories to understand zhem. We are temptedto think about m etaphysics or theology, physics or mathematics, in our terms when analyzing Newton; wefind no such temp tation when

Leibniz or weread ancient or medieval

sources, for their worldis too foreign for such an approach. It seems obvious to us, for instance,

that there is an aspect of physics call ed

the theory ofmotion,which involves such thin 8S as the principle ofinertia and which has obvious consequences for our understandingofthe na-

ture ofspace. Equally obviousis the idea t hat

geometryis the science of space, which involves the notion that it Provides us with information about space’s structure. And so on. We t fashion—lookto the great philosophers discover what they thought about the th to space or about geometry andits relati

on. This is tempting, but interpretively dangerous: a great evolution ¢ ookplace during the course of the century,

presented theories of motion, al On g with ideas about wh at we would call inertia, without belieyi ng tha t such issues had any specia l consequences for thinking about Spa ce, indeed, without necessari ly having anyparticular view about sPace itsel£ The conceptof spac e, to the extent thatit occurs atall in s uch thi nkers, is decidedly unimportant. Similarly, we now understand thar Spa ce

can be conceived from Euclidean ones, but numerous

a variety ofperspectives, including non-

mathematicians overthe centuries analyzed the famous parallel postulate in Euclidean geometry by thinking about what it mean s for two





¢ struct stroure lines to be parallel without ever asking questions spouse ng

ofspace itself. This seems odd from our

perspect- the first mathe-

evidence that near century's end, Leibniz wasi n:a

about space’

matician to consider the parallel postulate by thin i ures. Space structure rather than thinking merely about lines ice and natural became an object of study in mathematics, metaphy . haps some. ry. Fernap: philosophyover the course of the seven teenth centu thing similar happened to time with the advent 0 fthe special theoryof relativity in the early rwentieth century. ch century began, The seventeenth century ended, and the eighteen Newton and his

with a profound andlasting debate between a his followers.In

followers andhis great challenger G. W. Leibniz on cher fields, the mathematics, natural philosophy, metaphysics, an d differential calco-discoverers of what we now call the integral an Leibniz’s famou s culus represente

d two dramatically opposed camp * d and colle

ague and extensive correspondence with Newtons reeny indicated

Samuel Clarke at the beginning of the eighteentn ¢ + understanding that the two camps differed profoundly on the pOe weonian “absoof space and time. Indeed, the debate between a onalist” conceplutist” conception of space and

the Leibnizian ‘relat!

hapter’s focus tion continued well into the twentieth century? iN Oe newperenables us to view this familiar philosophical topic fro Taibaia and

spective: despite their numerous and deep Bere he concept of Newton fundamentally agreed on the importance oft t only that Space for mathematics and philosophy. ‘They agre ed _ mathematicians and philosophers shou 7° ld ponderthe conceptt ofkein spaceg and the basic structure ofspaceitself, but also thatit was wor ed

these topics at the center of a major internatio nal debate



4 Lowe this point toa fascinatin g talk with Vincenzo De Risi. 5 See, eg, Larry Sklar, Space, ters! California Press, Time and Spacetime 1974); John Earman, World Enough and Spacetime (Berkeley: University ot 1989); and (Cambridge, MA: : more recently, Rob DiSalle, Understanding Spacetime (Cambridge, UK:. Cambridge idge University Press, 2006), .


ensnare numerous thinkers in England and on the Conti nent. To say

the least, it is difficult to imagine a figure such as Bacon, Galileo, or

Descartes engaging ina protracted dispute concerningspace. Thus, the Leibniz-Clarke corr ondence is a telling emblem ofth e century’ esp shift. Precisely becaus that correspond e ence wasso influential, Leibniz, Newton, and their followers managed to convince many of their readers th at space itself, andits representation, is worthy of special attention in numerous fields, The dramatic tise in th. e impo rtance of thinking about space in seventeenth-

Century mathematics and philosophy had sng continued to serveasa profoundly important obje ct study throughout the next two centuries, and its statu secure today, Many developm as such ents in the relevant fieldss—ine ach lndl ng t e i

Consequences, Tr

ominance of Newtonian mechanics in the nineteenth Century, the emergence ofthelate eighteenth an inth

non-Euclidean geomerty | the nineteenth century, and the developmentofthe theory of relacvity in the early twentieth century—ensured the cont

inuing importanc®

Space. This fact tenders our k difficult. Wesimply today that, “8+ Beometry tas take it ‘er fs ace} was always cons idered the science ‘ “hings

We presume that phys ics has always been concerned with su hysic aS Space and time, matter and moti ians

on; we assume that metap have always had to ponder whether ‘st a or whether it is space itself exists difficult for . merely a kind of abstrac tion or idealentity. It .is there oO US to ima .

made, a

jons had been Sine a time before all of these : assump "tmetaphysicians paid “2 geometers natural phi an losop hers,little space—and jt heed. Tha ’s the task

S Conceptual representation—

of history, and

ofthis chapter.

ur contemporary categories can confus us e

when we consider early ;

t might seem ob-

Nenteenth-century knowledge. Consider eof body involvesits eee contemporary reader that the hache conceptitself is there-

changingspatial position over time, and oe ace, just as it must be we inherently connected the ree . But as wewill see, many inherently connected to thetoconcept oftime.



seventeenth-century figures thought about motion by thinking abou; paths, trajectories, object relations, and a host of othertopics, without

ponderingspaceitself. Similarly, what we nowcall the principle of in. ertia is obviously at the center of seventeenth-century thinking about motion, and Newton provided what became the canonical formula.

tion ofthe principle with his first law of motion in 1687. For Newton, the laws of motion have important consequencesfor what space must belike, a point he emphasized in his unpublished anti-Cartesian manuscript, De Gravitatione.® Butthis late seventeenth-century fact can provide a misleading lens when we wish to understand the century as a whole, for Newton’s predecessors did notshare this notion.Itis true that some previous thinkers—such as Galileo, Beeckman, and Descartes—held various ideas that are relevant or similar to Newton's

law.” But for Newton's predecessors, the question is this: To what ex-

tent did any philosopher pondering what we would call inertia also consider the potential consequences for the character ofspace orfor our conceptofit? By century's end, this question was commonplace, but it was rare before 1650.

Descartes is an important representative of thinking about motion before 1650, and his work was central to developments in both England and on the Continent throughouttherest ofthe century. His canonical work in natural philosophy, Principia philosophiae (1644),



Jaws in sections 37 through 42. Hence on thesurface, it appears that Descartes is thinking aboutspace, motion, and thelaws in tandem, much as a contemporary reader might expect. But when we lookat

the details of Descartes’s analysis of space—and theallied notions of internalplace, external place, etc.—wefind that our expectation is hampered, for Descartes argues that there is no “real distinction” between space and corporeal substance (section 10), His argumentis as follows:

Thereis no real distinction between space, or internalplace, and the corporeal substance containedin it; the only difference lies in the way in which we are accustomed to conceive of them. For in reality the extensionin length, breadth and depth which constitutes a space is exactly the same as thatwhich constitutes a body. The difference arises as follows: in the case of a body, we regard the extension as

somethingparticular, and thus think ofit as changing whenever there is a new body; butin the case of space, weattribute to the

extension only a generic unity, so that when a new body comes to occupy the space, the extension of the space is reckoned not to change but to remain one and the same,so longasit retains the same size and shape and keeps the sameposition relative to certain external bodies which we use to determine thespacein question. (AT VIIIA:45)

presents three laws of motion,the first two of which are linked to the development oftheprinciple ofinertia. In part 2 of this work, on the principles of material things, Descartes discusses space in sections

We may conceiveof(e.g.) a stone’s extension as a feature ofit, and of

16 through 18, motion in sections 23 through 33, and then the three

the place it used to occupy neara tree as distinct fromit, but this distinction arises in our conception of these things; it is not a real dis-

tinction.In fact, spaceitself is nothing but body. In Descartes’s natural 6 See Isaac Newton, Philosophical Writings, ed. Andrew Janiak, and edition (Cambridge, UK: Cambridge University Press, 2014). 7 There isa lively debate about which philosopher—Galileo, Beeckman, Gassendi, Descartes, ete.— first formulated whar we would now callthe principle ofinertia, and also about the related ques-

tion of whetherthere is such a thing as that principle before 1687, which saw the publication of Newton's Principia mathematica, For an innovative and nuanced accountofsuch issues, see David M.Miller, Representing Space in the Scientific Revolution (Cambridge, UK: Cambridge University Press, 2014).

philosophy, then, the concept of space, and spaceitself, drops out; we can analyze the natural world using only the concepts of motion

and body, along with the laws that govern them,for spaceis identical to body. Presumably, that is why the concept of space—orofgeneric extension—plays no role in what Descartes regards as the proper






of m concept of motion, OF the idea


. otion in the “strict sense. Motion .




fined as follows: in that sense is de

ld be | understoog by nd, we consider what shou If, on the other ha ce with the truth of ;cordani

For Descartes, then,the idea of a space just is the idea of SO.


: ‘ : three dimensions. It is identicalical in extended in is j cont entmething that :

of a stone when werepresent the stone merel y . notas bearing any otherfeatures (such as wei

as bein

fo the idea g extended and oe

! ernninate ni atUre COit. , we ign a det our aim is to assi y the matter, and if ne piece ofmatter, or one bod

ght ore ‘ the content of our idea. Moreover, Descartes anal rs etc.). This is YzZes the reference of “space” in the nextsection, contendingthat the v to thesize, shape, and position of some body relative to oth wbd

-54) orum). (AT VIII-1:53 spectantur, in viciniam ali

are identical. It might be thoughtprejudicial to put things j : from Descartes’s pointof view, space does not4 in this way. After all, philosophy asirrelevant any more than bod step out of his natural

mmon motion, not in co

usa’ ge but in ac

ansfer ofo may say tha t motionis the tr ely touch it, and wh sp others which immediat e th of ty ni ci vi e th om fr cinia Corum the vicinity of others lex vi we consider to be at rest, to m nqua quiescensia mediate contingent & ta corporum, quae illu d im 4

f space 6; of is not understoodin termso The moving body’s vicinity h in ¢erms of the body’s relationship wit positions within space, but nge , properly speaking, to moveis to cha other bodies. For Descartes r one ies. Spac e itself, to the extent tha one’s rela tionship with other bod »

h a thing, is irrelevant.

can formulate aclear concept of suc

relevant for Descartes’s ynEven if the concept ofspace itself is not : What does

causes, one mightstill ask derstanding of motion and its

One might ex-

point of view? such a concept consist in, from his ) would reply that space pect that the author of the Géometrie (1637 magnitude. Or perhaps it is that is a three-dimensional Euclidean ous

which containslines and planes and

figu res of varioussorts in varj

relationships with one another. Regardless ofwhat Descartes indicates

refers consider meaningor reference in Descartes’s te or nodes. Whether we rms,1S) space s and body

one andthe same! If you like: if the current. mops Out, for they are States is crucial to my analysis of contempome, en of the United

Donald Trumpis crucial too, and he doesn’t fail global Politics, then the president. Indeed, heis rel “OP out as irrelevant just heis ly, becau ident.seSimilat space is crucial in Des “vant because heis the S's 5 yste m because bodhe n crucial, and they are one and the same, = yis But this objection misses . a deeper Point: in contemporary terms, we do not think of extended body; wethi : : y nk of a three-dimension I

tain features. Thus, we cannot use our te tm ‘ =" Magnitude with cer.. Or our contemporary ry concept, pt, to much effect when studying Des we mustuse the cartes; thar concept of extended bodyinstead, sinee

in his mathematical work, however,he is quite clear in his natural phi-

structure of Descartes’s system, Thereis n

11 ofpart2 ofPrincipiaphilosophiae that one can conceive ofan empty

Newtoncalls space in that system ’ either

losophy that“the idea ofa space”is rather different. He hints in section space, notin the sense that one can have a representation ofa space that

he reflect s the conceptua

l |; : . that system. More important, ae like there ig nothing what we call space in : 9


a as


ing lik.© what Leibniz or .

. What this means,finally, is that when » We will s ee. Desc

artes analyzed motion—

is actually empty of body perse, but rather in the sense that one can

conceive of “something”thatis extendedin three dimensions without

ipso facto conceiving of that thing as bearing any other features (such as being red or smellinglike a rose). And thar, in turn,is “just whatis comprised in the idea of a space,’ even a space thatis called “empty.”

tion. What is motion? It is a changein rel .





body andits vicinity, which is constitute

n Motion i ; ue in mocontin

re between the mov ing

y other bodies. Space,


spaces, positions, places, etc. are irrelevant. If Descartes hag begun part 2 ofPrincipia philosophiae by presentinghis three laws of motion, then this might not have been clear. But he wisely began by explaining that space is identical to body, and then proceeded to defn. mo. tion as a changein relations among bodies, and only then presented his three laws. When read in context, the lawstell us nothing about

what wecall space. To considerthatissue is to ponder a non-Carrestan philosophicaltopic. Descartes was not alonein this conception.In thefirst book of The Elements of Philosophy (1655), entitled “Concerning Body.” Hobbes presents a strikingly similar view. He distinguishes between two conceptions ofspace: first, imaginary space, the “phantasm” or imaginative representation of some bodyexisting without the mind, one in which werepresent only the extension of that body; and secondreal space, the extension ofa body, one thatis independentofour thought (unlike our representations) and coincident or coextended with some

part ofspace.® Toillustrate: An apple on my desk hasa certain extension (it occupies three spatial dimensions), which Hobbescalls a real space. But when I look at my apple and then close my eyes and imagine it, 1 amdealing with an imaginary space. Here is the kicker: since real spaceis identical to the extensionof bodies, there can be no such thing

as emptyspace.? Ofcourse, we can émagine empty space—thatis where we obtain imaginary space—because we can imaginethe extension of

a body considered independently of that body andits other features (“accidents”), but such a represented empty space is merely a representation, For instance, I can see my apple, imagineit, and then imagine thacall its features—its redness,its smell, etc.—other thanits extension disappear, and then I am left with the place of the apple, which Hobbesidentifies with imaginary space.!° Unlike the apple itself. the 8 Thomas Hobbes, Elements ofPhilosophy (London,1839), I,8, $4. 9 See Hobbes, Elements ofPhilosophy, 1, 7, $2.

10 Hobbes, Elements ofPhilosophy, Ul,8, $5.



place of the apple is not part of the world around me; it’s just a phantasm or representation. So empty space is nota contr

adictory notion—

it is not logically impossible—but since it is merely somet hing that we can imagine, rather than something that can exist, it is not a feature of nature. Philosophical treatments of space are an important exceptionto the general rule that the great “moderns” ofth e seventeenth century— from Galileo to Hobbes, ftom Descartes to Cavendish, from Leibn iz to Newton—made their names by breaking from Aristotelian and

Scholastic approaches. When the topic is natural change, causation more generally, the proper analysis of sensory perce ption, the structure of human knowledge, etc., the moderns broke with what they took to be the dominant approaches of the medieval period and the Renaissance. Notso with space. Forat least the first half of the centur modern thinkers were not more inter ested than their predecessors . thinking

ofspace andits structure as fundamenta l philosophical topics of inquiry. Like Descartes, Hobbes is an excel lent example of this fact, He was of course one of the most important modernsofhis time, a great critic ofAristotelian ideas, and

a Proponent ofthe latest

thinking about the mechanical philosop wos hy; but he did not take the analysis of space to be of any moresignificance than Descartes did, 22 11, Nowadays, we would probably say that for Hobbes , ° This S is 4atlONs—e g things, * B of as well. ut the laws -§. Ones cally na They indicate that

distance—to othe



thing impedes notent

of common sense—and Na e

that bodies moving in ne jectory. Yet if motion isnen : primafacie, that we can Ite vicinity, for that will cha i


could alter a moving b dy

re precisely, we withits vicinity, which ’ evar 8¢ its relations push its newly acquired| ;nies “s being rese un at » 6 with ly we body without impeding ie He ity alon £ it, therebe sudden the ng pi op te st y the Nee id in nsion If Newton is correct in ident ntifyi ng thiIs tensj avoid it in his own system? co ig cle .Pa wh at can he do makes He ch ¢ outset, t, in ch che i n ar from ften ignored “definitions” that beg; . begin D e at 0

24. Newton, Philosophical Writings, 35.

and other bod; es.




Gravitatione, that he will



think of motion as “change of place” rather than as a change in object relations. He addsthat placeis a part of space that a bodyfills.?5 Newtonis perfectly well aware that this idea involves a rejection not merely ofthe Cartesian theory ofmotion—indeed, Descartes takes the ordinary person to regard motionas involving a changeofplace, which he jettisons on various philosophical grounds—butalso ofthe identification of space and body. Forifplaces are parts ofspace, and ifspaceis identical to body, then the idea that motion involves a changeofplace just is the idea that motion involves a change in relations with other bodies, Andthatis the view that Newtonrejects in this essay. Thatis why he endsup explicitly arguing that space and bodyaredistinct from one another, so that he can preserve the idea that motionis a change of place and not a change in object relations. Indeed, he begins the discussion ofwhat hetakes to be the errors of the Cartesians with this admission, which follows his first four definitions: “When I suppose in these definitions that space is distinct from body, and when I determine that motion is with respect to the parts of that space, and not with respect to the position of neighboring bodies, lest this should be taken as being gratuitously contrary to the Cartesians, I shall venture to dispose ofhis fictions.”*6 Perhapsit will not be gratuitous ifNewton presents arguments to convince his readers of Descartes’s problems. Newton spends the next seven pagescriticizing the Cartesian theory

of motion.


There are two questions remaining, First, if Newton wishes to adopt

a theory ofmotion that lacks the tension within the Cartesian system, ° what notion should he adopt? And second, assuming that he can avoid

thattension by adopting a theory that coheres with the laws ofmotion, whatview ofspace is consistent with that adoption? These are two of the questions that drive Newton's discussion ofspace in the Scholium. Asforthefirst question, one mightask as a corollary: IfNewtonrejects 2§ Newton, Philosophical Writings, 13. 26 Newton, Philosophical Writings, 18.




the Cartesian view that the proper idea of motion—that motionin the

_ true sense—involves a changein objectrelations, then howelse should he think about whathe will call true motion? Hewill contend that true motionis absolute motion; thatis, true motion consists in a change of

place, where thelatter is understood as a partof space indepe ndent of objects and their relations. He thinks that the idea that true motion

is absolute motion should cohere with the laws of motion, since they seem to indicate that motion does not involve a change in objectrelations. But the laws also indicate that there is somethings peci al about constantrectilinear motion—the germof this idea,

at least,is already in Gassendi and in Descartes—because it require s no forces. One might say thatat least primafacie, this ideasits better within Newton's framework than within Descartes’s. The laws do not tell us that we can discoverrectilinear motions, nor that there are any in nature, But they doraise the question of whatit would meanfor there to be such motion. Newton asks this question of Descartes, and in the Cartesian system this is a seemingly difficultissue, because a straight-line motion would have‘ to be understoodin terms o fthe moving body ’s relations to

other bodies, such that those other bodies would have to define some motion as being straight. It mightbe poss ible to work out this idea But on the surface, the Newtonian System seems better adapted to handlethis idea: it tells us instead that a motionis straightin virtue of

whetherit traces a straightline in space itself, indep endentofobjects and their places and relations, The point here is metaphysic al rather than epistemic: we may notbe able to discover any such motion .

there mayin fact be no such motion inall ofnature +



but at least we lack

anydifficulty in understanding what such a motion would consist in When discussing questions about Space, time, matter, forces, and motion in Leibniz and Newton, not to mention their followers,it is of course tempting solely to emphasize their numerous, influen tial disputes aboutall of these topics, and muchelse besides. The famous Leibniz-Clarke debate obviously helped to set much of the agenda of eighteenth-century philosophy. Andyet, as with the discovery ofwhat





coexisting objects, just as time is the order of the succession of events, Whatexists in Leibniz’s nature are the physical objects (the substances,

we now call the integral and differential calculus, at a deeperlevel of

analysis, the projects in physics in which Leibniz and Newton were

at the phenomenallevel) themselves, and in virtue of these physica l

engaged had a crucial commonality. Naturally, each thinker grew up

objects existing, they bear various relations with one another (e.g,, they

in a post-Cartesian environment, and each decided to try to make his name by balancing a healthy respect for Descartes’s philosophical achievements with a critical attitude toward some ofhis views. From Newton's famous anti-Cartesian tract, De Gravitatione, and the details of his Scholium following the Definitions at the openingof Principia

are a certain distance apart). So the objects and their relations

are what

constitutespace.Indeed, spaceis nothing over and above these objects

and their relations; it is merely an abstract way of charact erizing the network ofall such objects and relations. An analog y: In the old days, the telephone network consisted of rotary telephones in homes and offices, a bunch of wires connecting them, and some switches in var-

mathematica, to Leibniz’s famous “An'Essay on Some Notable Errors

of the Cartesians” to his mature articulation of the nature of space and time, wefind a plethora ofcriticisms of Cartesian ideas. Two cen-

ious centralized locations. The telephone network in those days just was the phonesand the wires and the switches; there was nothing called “the telephone network” above and beyondthe se items. (One

tral criticismslie in common between the two thinkers: both Leibniz and Newton determinethat the Cartesian identification of body and space mustbe rejected, and both believe in tandem that Descartes does not recognize the significance ofinertia (or of Kepler’s laws, for that

can throw in theirrelations for good measure.) In that sense, Leibniz

rejects the Cartesian view becauseit confuses something that is a substance, namely physical body, with something thatis nothinglike a substance, namely the network ofrelations amongphysical bodie s,2” Questions about the identity of space and body intersect with questions about the application of mathemat ics to nature (all the

matter) for understanding motion and, in tandem, space. From Leibniz’s mature point of view, Descartes’s contention that

space and bodyare identical represents a mistake at the deepest metaphysicallevel. Bodies in Cartesian physics are stebstances in the specific sense that even if they cannot exist independently of God, they can exist independently of one another, and just as significantly, they are bearers of properties. (The same is true of their various parts.) They

rage in the seventeenth century, of cour se). From Leibniz’s perspec-

tive, Descartes’s physics is not sufficiently math ematical; even a cur-

sory glance at the differences in the vortex theo ries presented by Descartes in his Principia in 1644 and Leibniz in his Tentamen in

are also the focus of Cartesian physics, for they are the itemsthat are

1689 will confirm this point. It is not merely the case that a plenist view of nature might bedifficult to square with the application (e.g.) ofgeometry to questions about the motions of bodies, on the grounds that bodies and their parts are exceedingly complex and prevent any straightforward application of ideas about lines, planes, and thelike.It

subject to the laws of motion. For Leibniz, however,it is a serious mis-

take to think that spaceitself is a substance; indeed,thatis one of the primarycriticisms that he presents to Clarke, Hence he would object to Descartes's view in his Principia that the idea of space just is the idea ofa something withcertain features, thatis, a bearer ofproperties.

is also the case, at a deeperlevel, that Leibniz enables the application

For to conceive of space in thar way, of course, just is to conceive of

it as a substance (on a commonconstrual of the latter notion). For Leibniz,in contrast, spaceis nora real thing, a being, or a substance at

27 Of course, these points do not ipso facto constitute an argument against the Cartesian identification of space and body, Rather, they indicate how Leibniz conceive s ofspacein a radically antiCartesian way and,indeed,partly for the same reason that he rejects the Newtoni anview.


all; it is merely an ideal thing, an abstraction. For Leibniz,briefly put, space is the network—heoften calls it the “order”—ofrelations among



Leibniz of course is a special case. He not only accepted the substance/property framework; he madea career outofclaiming that the Aristotelian tradition from which that framework ultimately arose had been mistreated by the “moderns”of his era. From his pointofview, any being that exists mustfit into the substance/property framework in some way, both at the phenomenallevel, where we find material

objects and their features, and at the more fundamentallevel of metaphysics, where wefind,in his later thinking at least, monads andtheir intrinsic features. But as shouldbeclear, Leibniz brilliantly evades any

problemsin his thinking about space and time by denying that either is a being ofany kind. Neither space nortime, fromhis pointofview,is an entity, which meansthat neither mustfit within the substance/accidentframework. To use one of Leibniz’s favorite examples: Ifwe want to provide a metaphysical analysis ofsomethinglike the Prussian Army, with the goal of cataloguing everything that constitutes it, then we would write down a longlist of soldiers and tents, horses and swords, and so on. Once wehavelisted a// of the things in the Prussian Army, we would befinished; there would be nothingleft over, no remainder, called “the army” that would have to be accounted for in some fur-

ther way. To accountforall the items in the armyjust is to accountfor the armyitself. The sameis true ofspace (and oftime): to accountfor all of the things in nature—all the rocks and trees and planets—is to accountforspaceitself, since it is nothing more than theorderor network ofthe relations among these objects. The analogyis more precise than one might think because a proper analysis of the Prussian Army would likely have to includethe relations amongthe things that constitute it as well. We must write down that so-and-sois a lieutenant, and so-and-so a private, and also that a certain horse belongs in the

Fifth Company,etc., if we are to capture properly the things within the army. If we do so, we are taking accountoftherelations among which Newtonread in his youth, See also Antonia LoLordo, Pierre Gassendi andthe Birth ofEarly Modern Philosophy (Cambridge, UK: Cambridge University Press, 2007), 106-24.



the items, and not merely the items themselves. So the analogy with

Leibniz’s view ofspaceis pretty close.

We might take this line of reasoning onestep further. Ifone accepts

the basic substance/property framework, then one has a choice: (1) onecan follow Descartes and simply identify space with body, thereb evading metaphysical trouble; or (2) one can reject the Cartesi an iden. tification, but then one must deny that space is a “something” arall Similarly, ifone wishes to think of spaceas distinct from body, but one does not wish to follow Leibniz’s route for thinking about space then of course onecan reject the assumption that the substance/ propert

frameworkis correct. In that case, one can attempt to evade metaph : ical trouble, or rather one can deny that the analysis of what counts1s such trouble should be understood in terms of that framework Because of these very considerations, Leibniz’s theory of s ace and

time camein forfar less critical assessment than Newton’s For unlik Leibniz’s relational theory, Newton insisted thatspace is indeed some. thing above and beyond all of the natural objects and their relations with one another; it is not merely a set ofrelations, In that sense it is unlike the Prussian Army. But then, what is it? This is precisel the kind of question that Leibniz posed to Newto n and his followers, From early on in his career, Newton understood the fact chat if he wereto reject the Cartesian view, he would haveto articulate the ontological status of space in some way. Instead of concedin “chat space is some kind of substanceor feature of some substan ce, hedecided in.

stead to proclaim—as we havealready seen—that space has “% manner ofexisting.” What reason did Newton give for

vesting such

a radical view (evenif it was to be found in Barrow and in Gassendi/ > ‘ Charleton)? In an espec :

ially rich passage from De Ghravitatione,

Newtonexplainsthat space is not a substance

because it is not amongthe properaffections that denote substance, namely actions, such as thoughts in the mind and motions in body. For although philosophers do not define substance .as an entity




that can act upon things, yet everyonetacitly understandsthis of substances, as follows from the fact that they would readily allow extension to be substance in the manner of body if only it were capable of motion and of sharing in the actions of body. And on the contrary, they would hardly allow that body is substanceifit could not move, norexcite any sensation or perception in any mind

categorization, according to Newton).*# It also lacks the dependence


status ofspace, struggling mightily to explain how so methingthatis


on another item characteristic of accidents, Yet it is not nothing: we

have a clear representation of its features, which include uniformity

and infinity. So space is something, but not a thing thatfits into the standard ontology of Newton's day. In De Gravitatione, Newton evidences an awareness of the peculiar

neither nothing nor a substance nor an accident can be properly un-

But Newtonhastensto add that spaceis a/so not an accident inhering in a substance, for

we can clearly conceive extension existing without any subject, as when we may imagine spaces outside the world or place empty of any body whatsoever, and webelieve [extension] to exist wherever

we imaginethere are no bodies, and we cannotbelieve that it would perish with the bodyifGod should annihilate a body,it follows that [extension] does not exist as an accidentinhering in some subject. Andhenceit is not an accident. And muchless mayit be said to be nothing,since it is something more than accident, and approaches more nearly to the nature of substance. There is no idea of nothing,

nor has nothing any properties, but we have an exceptionally clear idea of extension by abstracting the dispositions and properties of a body so that there remains only the uniform and unlimited stretching outof space in length, breadth and depth.


of space, which involves the confus ing notions of “emanation” and “emanative effects”—that seem only to muddy the waters, It is prom-

ising to see Newton struggling to transcend what w

€ might nowregard

as the limiting constraints of ontological theorizin g in his day, butit is perhaps disappointing to see him conclude that sp ace “is an emanative

effect of the first existing being.”35 For the later ideais simply unclear to manyof

his readers. Fortunately, by the time of the first tition of

Principia mathematica in 1687, Newton decided to presenthis conception of space and time without recourse to these notions,

In the Scholium following the Definition s at the beginning of Principia mathematica—in the prefatory m aterial, before book 1 begins—Newton does nottackle what we mi ghtcall the ontological

status of space as directly as he had done in De Gravitatione, just as he does notdirectly criticize the Cartesians by name, as he had done

34 This viewrepresents another remarkable area ofagreement between Leibniz and Newton: see the

first paragraphof Leibniz’s “Specimen Dynamicum?”in G. W. Leibniz : Philosophical Essays, Roger Ariew and Daniel Garber(Indianapolis, IN: Hacker, ed POEEAE ASSAYS, CC 1989), 118, , 3 Much ink has beenspilled in trying; to grasp this fundamen: tally complex notion. . See See the th classic i debate between McGuire and Carriero: J. E. McGuire, “Predicates OF Pure Existence: New on God's Space and Time,” and John Carr ton iero, “Newton on Space and Time: Com ments onJ.E McGuire,’ both in Philosophical Perspectives on Newtonian Science, Philli Bricker and R. I. G. Hughes (New Haven, CT: Yale University Press, 1990). See alsoed,HowardSte in “Newton's


The negative view here is this: Space does notfit into either the substance or the accident category, for it lacks the actions definitive of substance (as tacitly assumed by philosophers who employ this

derstood by philosophers. Herelies on various Previous philosophical positions—including Henry More’s Canabridge Platonist





32 Newton, Philosophical Writings, 36.

33, Newton, Philosophical Writings, 36.

Metaphysics,’ in The Cambridge Companion to Newton, ed. 1. B. and George Smith (Cambridge, UK: Cambridge University Press, 2002) for a nuanced Cohen interpretation of the claim

about emanation.



earlier. Thereis little doubt, however, that he presents his view as an alternative to Descartes’s conception, and that he presents certain

aspects of space that have important consequences for howhe thinks it should be regarded in ontological terms. Newton begins with an

important proviso: the Scholium is preceded bya series of definitions that introduce “unfamiliar” terms, or concepts, to the reader, including

‘what would then have been the novel notion ofmass(importantly dis-

tinct from Descartes’s own quantitas materiae, which is equal to the volumeof a body) andthat of centripetal force. But space, time, and motionareperfectly familiar terms and ideas, so Newton neednotde-

fine them. On the other hand,he doesindicate that it is common to think about these ideas solely with reference to “sense perception,’ a tendency he himself wishes to avoid. Hence the introduction ofhis famoustrifold distinction: we mustdistinguish absolute andrelative, true and apparent, mathematical and common,notionsofspace, time,

and motion. As with many famousphilosophical conceptions, Newton's viewis shrouded in misconceptions. Foremost among themis the idea that in the Scholium heis trying to “prove” that spaceis absolute; similarly,it is often said that Newtonis arguing that spaceis absolute rather

thanrelative. These ideas are misleading. In such cases, it behooves us to bracket the historically important interpretations of a text and return to its original wording: “Absolute space, of its own nature and withoutreference to anything external, always remains homogeneous and immovable. Relative space is any movable measure or dimension of this absolute space.” This indicates right at the outset thatrelative space depends on absolute space, since the former is a “dimension”

of the latter. Indeed, he adds, “Absolute and relative space are the same in species and in magnitude, but they do not always remain the same numerically.’36 Hence absolute and relative space are the same



qualitatively—e.g., each is a three-dimensional magnitude, unlike time—butthey may bedistinct numerically, For insta nce, the relative space determined by the walls of my office is the same in species as that

portion ofabsolute space with which it coincides, and at an instantit

is numerically the same as that portion

of absolute space with which i

coincides. But as the earth rotates, the Space of my office comes t co.

incide with another portion of absolute space, which m are distinct numerically. Newton develops a conception

h ‘they mans thar they

of absolute space to assi i derstanding motion. If we have‘ reaso n to believe—as as NNewto ton nthink hi wks s we do, and as he makes clear in his arguments in De Grevitrri that the true motion of body does n Ot consist in a ch e ‘ e o change a betw een the body erwes 8¢ . of relations (v : and other bodie s (whether they constitute its “vicinity” or noc), then our question becomes: : Whar at, then, does it con sist in? Newton suggests that true motion is not ¢ e lative motion but rather absolute motion: a body thatis truly moving may o ay or may not alter its relations to

any other body, so its motion should b e understood instead as involving a change in its place _ . Butsinc i ‘: space that a body occupies mayitself move, we should not e btherelative Ot be tempte to conceive of the body’s true motion10n in ; terms of its j relat

ive : space; 3i 1nstead, we should conceiveofit as a chan gein its bsolute place . So true motion is absolute motion, And abso ; lute motiio Nn, In tur n, obviously pace e. Newton js Partly b rea ssuring us here: we need to think of space a Solute to .




in the right way, but we needn’t w


conceive of true mo tion


orry about thac idea because abso. : species th same in is the | ute space is and in magnitude

is not a radically different sort of thing re 1 rs.

as relative space. It

despite i iliari its unf;amili arity to

In regarding space as distinct

from body, Pace DescartesS an : : : and Hobbes, and in consider space to have certain f ing im bility—ind d ; eatures—such as movability—independent of anything external to it, Newton was * . . ’

primed also to focushis attention on the structure ofspace itself, much 36 Newton,Philosophical Writings, 84, for both quotations.

as he attended to thestructure oftime. He writes:





Just as the order of the parts of timeis unchangeable,so, too,is the orderofthe parts of space. Let the parts of space move from their

Although Newton certainly helped to ensure that Space andits structure would remain important topics of philosophicalanalysis for

places, and they will move(so to speak) from themselves.For times and spaces are, as it were,the places of themselves andof all things. All things are placed in time with reference to order of succession andin space with reference to orderofposition.It is of the essence ofspaces to be places, and for primaryplaces to moveis absurd. They are therefore absolute places, and it is only changesofposition from these places that are absolute motions.°”

For readers of Newton's text who are sympathetic to Cartesianism, or perhaps to Hobbes’s views, places within space are defined by the objects that occupy them, or perhaps by relations among them (eg, a place might be defined as lying a certain distance from the surface of some body). Newton's absolute space, in contrast, has a certain inherentstructure independentofany objects that occupyit. That structure consists in an infiniteseries ofplaces that bear essentiallocation:to bea particularplace just is to inhabit a certain location within absolute space, and sucha place is immovable in the sense that it would not be itself if it were to be located somewhere else. According to this conception,objects are not placed within space in virtue of bearing some relation to other objects—say, being a certain distance from some centrally located object—butratherin virtue of inhabitinga particularlocale, which latter is part of the inherent structure of space. We know that space’s structureis inherenttoit,finally, because space remains ho-

mogeneous and immovable independently of everythingelse, so there is no event involving objects and their relations with one another— causal relations or spatial—that can alter spaceitself. By 1687, then, space andits structure had becomea significanttopic of philosophical investigation.


the foreseeable future, that fact does not entail that his readers and

interlocutors were prepared to endorse his distinction between absolute andrelative space. Moreprecisely, although some were ha

concedethe usefulness of that distinction, many insisted that thevery

idea of absolute space is problematic. With such a view comes


of questions, even if we evade common misconceptio ns, Consider ony office again: We do not encounter much trouble in thinking about the

things in my office—the books, the desk, the chair, the pens, and so on—noreven when considering the air that fills my it office, buttherel ative space in whichall of these things coexist is a bit more difficul cult to grasp. Whatsort of thing is it? Newton does not directly tell .

unlike the extensive discussion in De Gravitatione , affirms . he neither nor denies that the relative space of my officeis a substa nc €, an accii dent, .





an emanative effect, or anything else, We

merely know that


everit is, it is the samein species and in mag nitude as ab ol ve space But that merely kicks the can down the toad, for wh sott 6 “ching is absolute space? Newton seems to thi nk: My veader vill he “much trouble in understandin g whatrelative space is —he me ions ti ne but does notclarify, such items as “th © space of our air”—a nd so if I indicate that absolute space is the 5 ame in specie : s and m agnitude, j

then they will understand whatit is, too. Unlike relativ. esp aces, however, absolute space is inmovable—since the things i

which hon can mone.° fice are definitive of the relative space of my office it remains L, including its walls, so too can the space of my office—and that “heheerste In this sense, we know two of the properties

pace has, but Newton decides against any explanation of its ont ological status, so we are left wondering whether it is a kind of substance, a propert poy , or something else. conj The conjunction o f Newton's’ bold proclamation i s about absolute space, coupled withhis refusal to articulate the proper ontological

37 Newton, Philosophical Writings, 86.



conception of such space,led to serious objections. Leibniz led the charge.



As we have seen, Leibniz certainly concurred with Newton that space and its structure require substantial analysis—and perhaps that they

had been neglected byearlier figures such as Descartes and Hobbes— but he insisted throughouthis long career that the Newtonian con-

ceptionofspace led to a series ofserious philosophical errors. Leibniz’s most extensive debate with the Newtonians concerning space and time

would not occur until the very end ofhislife: his celebrated correspondence with Samuel Clarke, Newton's parishpriest, friend, and supporter in London,is his most famous interaction with the Newtonians,

occurringright before his death in 1716.78 Leibniz fomented the corre-

spondence in November1715 by sendinga pithy, provocativeletter to Princess Caroline of Wales, one designed to provoke a response from Newton’s circle in London.?? Leibniz’s letters to Clarke are methodologically characteristic: he leaves much of his own systematic and complex metaphysical theorizing—including the monadology—in the background, bringingto the fore only those elements that are both necessary forhis criticisms of the Newtoniansandalso likely to garner 38 The exchange first appeared as G. W. Leibniz and Samuel Clarke, f Collection of Papers

which passed between the late Learned Mr, Leibnitz, and Dr. Clarke, in the Years 1715 and 1716 (London: Knapton, 1717, andit was reprinted many times in various editions.

39 Why did Clarke respond on Newton's behalf, and what was Newton's actual role in the correspondence? These questions continue to puzzle scholars; see, eg., I. B, Cohen and Alexandre Koyré, “Newton and the Leibniz-Clarke Correspondence,’ Archives Internationales d'Histoire Des Sciences 15 (1962): 63-126; Domenico Bertoloni Meli, “Caroline, Leibniz and Clarke, Jostrnal of

the History ofIdeas 60 (1999): 469-86. There is no documentary evidence, such as letters, between Clarke and Newtonindicating the contours ofNewton's role; thenagain, at this time,since both men lived in London and Clarke was Newton’s parish priest, the lack ofletters or other papersis unsurprising. That fact aloneis intriguing, for the theological differences between the two are salient: since Newron was a committed anti-Trinitarian—a fact known to Locke and others, such as William Whiston, Newton's successor in the Lucasian Professorship at Cambridge

(see James Force, William Whiston: Honest Newtonian [Cambridge, UK: Cambridge University Press, 1985]) he may have decided that Leibniz’s contentions about“naturalreligion” in England

would best be answered by Clarke, a rising star in the Church at that time and clearly a formidable theological thinker. On the other hand, Clarke himself was certainly not an orthodox Anglican

thinker—his Scripture-Doctrine ofthe Trinity of 1712 was read by some as showingat least some sympathyfor Unitarian ideas—so he was not an unproblematic figure in this regard. Perhapsjust as important, Clarke was a serious metaphysician, a more systematic philosopher than Newton,as was evident from his Demonstration ofthe Being andAttributes ofGod of 1704. He was therefore in a position to engage Leibniz on metaphysicalterritory, writing in depth aboutsuchissues as the principle ofsufficient reason, which Newton apparently did not take seriously.



support from Clarke. Thus the key to many of Leibni z’s criticisms is the principle of sufficient reason, which he know s Clarke will en-

dorse, although with a distinct conception ofits scope . Leibniz asserts while Clarke denies, that the principle demands that each act of divine willing requires a reason; for Clarke, divine willingitself is reason enoughfor somephysical state of affairs to obtain.

Leibniz argues in particular that absolute Space is incompatible with the principle ofsufficient reason,if the latter is properly understood .. I have many demonstrations to confute the fancy of those who take

space to be a substance orat least an absolute being. ButI shall only

use, at present, msone demonstration, which

the author here gives


. ion to insist : upon. I say, then, thar if occas space were an absolute

being, something would happen for whi chit would be impossible that there should be a sufficient reason

—which is against my axiom.

And I prove it thus: space is somethin 8 absolutely uniform, and withoutthe things placed in it, one point of space absolutely does not differ in any respect whats oever from anot her point of space. Now fromthis it follows (supp osing spac e to be somethinginitself, besides the order of bodies am ong themselves) that it is impossible there should be a reason why God, Preser ving the same situations of bodies amongthemselves, should have placed them in spaceafter: ” one certain particular manner and not otherwise—why everything wasnotplaced the quite contrary way,for instance, by changing east into west. But if space is nothingelse but this orderor relation, and is nothingat all without bodies but the possibility ofplacing them, .




then those twostates, ) the one such as it is now, the other supposed

to be the quite contrary way, would not at all differ from one another. Their difference therefore is only to be found in our chimer-

ical supposition of thereality of space.4°

40 See Leibniz and Clarke, 4 Collection ofPapers, L 335.





Leibniz is clever: he eschews the thorny problem of determining whether Newton's idea of absolute space commits him to thinking of spaceas a substance by presupposing only that Newtonthinks ofspace as existing independently of objects and theirrelations.If spaceis indeed independentin this way, then it would seem that Godfaces a choice: whencreating the world, why place the earth in one particular part ofspace rather than any other? Theparts of space, independently

differentiating features. On the other hand, as Leibniz says, since the places within space lack other features, space itself is uniform: there is no substantive difference between one place (over here) and another

of objects andrelations, do not differ from one anotherin anysalient

harmless. Similarly, there is presumably no reason for Newton to deny

respect, so it would seem that one could not even theoretically devise a reason for placing the earth anywherein particular, as opposed to anywhereelse in particular.#! But since space with all its places exists independently of everythingelse, then God must have some reason to place the earth in one place rather than another.” More important, just as the structure ofspace—forinstance, the fact thatit is constituted by places with inherent locations—is significant for Newton’s discussion in the Scholium,it is crucial in Leibniz’s ar-

gument against the conception of space in that very text. Forit is key to Leibniz’s case in his third letter to Clarke that independently of the objects that exist, space is uniform throughout. The conjunction of Newton’s idea aboutspace’s structure with Leibniz’s idea can be difficult to grasp. On the one hand, as Newtonsays, spaceitself consists of a series ofplaces that differ from one anothersolely by their locations, which latter must be essential to those places, since they lack other

(over there). But do the sole features that places seem to have—viz.

their inherent locations—count as a substantiv e difference? Leibniz is counting on the fact that they do not. Andif he is right, then he

can accept Newton's construal of space’s struc ture as philosophically Leibniz’s view that space itself is uniform, It therefore turns out that Leibniz and Newton actually concur

on thestructure of space. Where they diff er is on the philosophical

implications of space’s structure, Leibniz spe aks of the Newtonians as endorsing “thereality of space” which he regards as “chimerical” Accord to Leibniz, ingt ci mniz, whereas ordiina i na ry physical i i cts are disc te, i rete obje constituted by their parts with internal distinc tons amongthoseparts, space and time are continuous and homogeneo us, Leaving aside the deeper metaphysical level of monads, as Leibniz does when corresponding with Clarke, he means that it is physical objects that are the real things; space and time are merely “ideal,” abstract entities whose continuity and homo geneity signal thi s special status, For or hihis part, Newton accepts the idea that Space is continuous and homogeneous but denies that these facts signal space’s special n physical status. Space .







is neither a substance nor an accident, neeit supervene on nor does

material objects and their relations; it has 41 This argument does not depend onour having the capacityto refer to places within emptyspace.

42 Leibniz avoids this problembyasserting that space is nothing above the objects in the world andall possible relations amongthem(hence he holds a kind ofmodal relationalist view, in more modern terminology), For him, Godfaces no problematic choice, since space does notexist prior to the creation ofmaterial objects: to create objectswith spatial relationsis ipso facto to create space.(‘To create the Prussian Army, one merely needsto createall its horses andsoldiers and weapons,etc.) Clarke's reply to this argumentis disappointing: he blocks Leibniz’s inference by denying that the divine will must have a reason to place the earth in one place rather than another (Leibniz and Clarke, 4 Collection ofPapers, C 3:). The principleofsufficient reasonis not violatedin this case, according to Clarke's interpretation ofit, because it requires only this: if the earth appearsin one place rather than another, there must be a reasonthatit appears there, and the reason in this case is simply the divine will; there is no further question about whyche divine being madea particular

choice rather than another.

“its own mannerofexisting.” endorse this idea. The dispute between Leibniz and Ne Wwton regarding space helped to determine the fortunes of space in the philosophical eighteenth century. The criticisms of Newton w . hi ere widespread and nonpar-

Leibniz and his followers were never prep ared to

tisan: philosophers from Berkeley and Du Chatelet

to Kant joined the

chorusofcritics, The key to understandingthesecriticismsis to grasp

the precise nature of their connection with the substance/property ontology that many philosophers still assumed in this time period.

Berkeley, Du Chatelet, Leibniz, and Kant did not reject Newton's view




of absolute space because he fails to articulate a conception of how spacefits into a substance/property framework;rather, they contended that ifNewton were toinsist that spaceis absolute in the sense ofbeing distinct from body,andalso distinct from objects andtheir relations— not to mention distinct from any set of relative spaces—then he wouldface a special problem, one that his Cartesian predecessors and Leibnizian interlocutors did notface. If space is nota relation, a network ofrelations, a body, or even somethingthat supervenes on bodies



this volume by Michael Friedman.) This combination is moresi gnif-

icant than it might seem. Had Newton's conception been useless for further developments in physics, it might have died on the vine. Had it been useful for physics but metaphysically unobjectionable, it might have : merited no further debate. As it turned out, however, this concep tion helpedto ensure that the representation ofspace its structure, and its status, remainedsignificant topics of philosophical dispute for the rest of the modern period.*3 «


orrelations, then it certainly seemsas if it must be some kind of being

in its own right. But what kind of being is it? Few philosophers were satisfied with the possible answers to that question. However,it is unfair to Leibniz to portray him as historically and

philosophically importantsolely for his role as Newton’s (and especially Clarke's) critic. It is tempting to argue that Newton placed spaceitself on the philosophical agenda in 1687 with his remarks in the Scholium following the Definitions before book 1 of Principia

mathematica, and that Leibniz kept space on the agenda through his criticisms of that view in numerous venues, especially his correspondence with Clarke in 1715-16. Tempting, but unfair. It is more accurate

to say that Leibniz and Newton each found Cartesian physics to be hampered byits identification of space and body and byits failure to recognize the importanceofinertia for understanding motion; to

understand motion through thelens ofinertia, in turn, requires one to think aboutspaceitself in new ways. It is Leibniz’s recognition of the shortcomings of the Cartesian approach,as muchashis dislike of

Newton's specific method for overcoming those shortcomings, that helps to show the importanceofphilosophical analyses ofspace tohis eighteenth-century readers and followers. Nonetheless,it is obviously difficult to overstate Newton’s importance for these developments. Hearticulated a conception of space,

time, and motion that was extremely fertile for physics, but considered highly problematic within metaphysics. (The story of what happened to that conception in the eighteenth centuryis told in the chapter in

43 For ; very helpful conversati; ons that substantiall y altered aly al my argument in in thischi paper, I woul

like to thank Mary Domski, David Marshall Miller, Vincenzo tee Risi, and David SanfordAlt

translations are my own unless otherwise noted.



systematic interrogation of the material world undermined,


however, chemistry’s symbolic connection with the cosmic space. While theelite astronomer Tycho Brahe (1546~1601) placed a chemical laboratory in the basement of his observatory (see Figure Rs.1) to study the correspondence betweencelestial

andterrestrial signs, the humanistic scholar Andr eas Libavius (c. 1955-1616) designed his imagined chemical townhouse as a civic space that would socialize chemistry as public knowledge. Insteadof toiling amid furnaces in a private hideaway to discern

Mi Gyung Kim

cosmic signs, the civic chemist had to main tain an upright

household,strive for civic virtue, and particip ate in society as a free manliving in “a body politic of strictest piety.” He

Thelaboratory has become an indispen sable site for the production andlegitimation of scientific knowledge. It

symbolizes modern science and our capacity for an artful manipulation ofnature that prod uces an enyirorument built with hybrid things—artificial things that are ac cepted as "natural. The laboratory work sas a contact zon e between the natural and the artifici al to forge intelligible sciences. Its

historical developmenti n step with the material practice of early modern chemistry t hus

merits a serious consideration in

understanding representations

of modern science

and their claimsto the truth of nature, Th e architecture of knowledgeproducing spacealso reflects the changingidentities of scientifi c fields and their practitioners,!

The chemical laboratory was stabilize d asa well-demarcated space with specialized instruments and techniquesin thelate six teenth century,Its emergenceas the spa ce designed for a


1 Peter Galison and Emily ‘Thompson, eds., The Architecture of Science (Cambridge, MA: ‘MIT

Press, 1999),

Mi Gyung Kim, Chemical Laboratory and the Cosmic Space, In: Space, Edited by: AndrewJaniak, Oxford University Press (2020), © Oxford University Press, DOI: 19.1093/080/9780199914104.003.0011

would arrange furnaces and vessels to display the virt ues of their products in full light. A “truly libe ral art” had to reveal

“not their cosmicsignificance, but their bene

fits to mankind.”?

Where the laboratory was located and wha t it produced would shape chemists’ social identity and chemistr y’s cognitive status in ascertaining the truth of nature, evenifits 5 ymbolic design and internaldivision privileged alchemical tho ught and practice.3 Libavius’s Alchemia (1597), which soug htto reha


Paracelsian chemical medicine as publ ic knowledge, appeared just a yearafter Johannes Kepler’s Cosmog raphic Mystery (1596), whichsoughtto legitimize the Copern ican universe , albeit in mystical terms. In other words, chemistry as useful material knowledge had acquired a distinct soc ial status different from

that of astronomy by the turn of the seve nteenth century.

2 Owen Hannaway, “Laboratory Design and the Aimof Science: Andreas Libavius versus Tycho Brahe,” Isis 77, no. 4 (1986): 585-610, Also see Bruce T. Moran, Andreas Libavins and the Transformation ofAlche

my: Separating Chemical Cultures with Polemical Fire (Sagamo re Beach, MA:Science History Publications, 2007), 3 William R. Newman, “Alchemical Symbolism and Concealment: The Chemical House of Libavius,” in The Architecture of Science, ed, Peter Galison and Emily Thompson (Cambridge,

MA:MITPress, 1999), 59-77.






The semipublic laboratory developed in tandem with the purpose-built theater in early modern Europe to constitute a new“space-time knot”that reflected the changing socio political environment. Asa space of performance, the labor atory conceptualized nature just as the theater theor ized society.




Their intense locality made far-flung places and esoteric objects

interact in space “shaped by human relationships with thebu ilt

environment.”* The laboratory not only helped articulate the ideal of science as useful knowledge but

also configured an ideal polity

for Francis Bacon (1561— 1626), a statesman wh o wished to be “invested of that ¢riplicity ... a scribed to the ancient Hermes; the powerandfortune of a King, the know ledge and illumination of a Pri

est, and the learning and universality of a


”> His effort to ground matters of p olicy on natura l laws went beyond the observation and contempla

tion of nature to advoca

te am aterial intervention resembling the alchemist’s. Anideal union of England and Scotland to forma peaceful Britannia of sedition had to belike a chemical Milt


quality, rather than a physical Compos itio, 0 ra mere juxtapositio

in place.® In the context of intensifying

Bri tish imperial dreams,


Bacon envisaged a utopian common wealt h organizedas a res earch laboratory. In New Atlantis (1626), all inhabitants of Bensalem

devote themselves to improv ing material knowledge for the benefit oftheir fell

Sate eneNOE EP

raving from i FIGURE R51 Eng (Wandsbeck, 1598).

owcitizens, The Salomon’s Houses ets up experimental regions on the ground, underneat h the surface, and in theair to enlarge “the bounds of Human Empire” An organized yet

jae instaur t atae mechanica Tycho Brahe, Astronomiae 4 John Shanahan, Ben Jonson's Alchemist and EZatly Modern Laboratory Space , Journal for Early Modern Cultural Studies 8, v0.1 (2008): 35-66. ¥ Spaces J f $ This is how Bacon wished to characteri ze James Lin his Advancement of Learn ing (1605); reprinted in Francis Bacon, The Major Works, ed. Brian Vickers (Oxford: Oxford University Press,

: 1996), 122, 6 Francis Bacon, A briefe discourse, touchir1g the happie union of the kingdome s ofEngland, and Scotland Dedicated in private to bis Matesti (London, 1603).






helps manipulate nature secret commerceoflight (knowledge) , arts, manufactures, and for humanitybycollecting the sciences as an efficient

special instrument—the air-pump—lent credence to a cosmic space

al center to organize the entire laboratory lay at the hidden imperi ld serve Bacon slifelong goal of globe asa knowledge space that wou lding an ideal

Cartesian niatural philosophy. The Royal Society of London

inventions ofall the world,”

The ideal commonwealth

ity but a project of bui philanthropia—not mere char of the ough a fundamental reform commonwealth or empire thr ora mical lab tory as

bilization of the che knowledge system. The sta ate connection kened chemistry’ immedi a civic or moral space wea its social status in the

le rehabilitating

that contained nothing,or the concept ofspace as a conta iner , which would have resonated with the atomistic rather than the ran an active campaignto forge a vision of new or experimental

philosophy that would “increase the powers of all mankind” and free themfrom “the bondage oferrors.” The “absol ute perf ‘

ofthe true philosophy” would require a path of“slo w, and sure

experimenting” 4 \a Lord Bacon, for which chemistslab or (except n


to the cosmic space, then, whi

f or the alchemist’s Pp ULSuit i oft he Philosopher i 5 s Ss Stone) 1i

(avant context, leading scientists the English Restoration integrated : such as Robert Boyle and Isaac Newtonree athematical visions of la lettre) perimental or m ex ir the o nt chemistry i undation of Anglican phy to establish a universal fo

chemistry and alchemy, which muted the perceived political

en me the noblest improvements.’

ivic order.

natural philoso

theology.’ Boyle's Pp

s meantto forge hilosophical chemistry wa

emists and corpuscularians

a “confederacy” between ch

in order

le language pr actice in the respectab dress up sooty chemical speculative and therebyto transform of mechanical philosophy the e experimental philosophy. Th u philosophy into a true of at natural the create new phenomena challenged laboratory in its capacity to of natural philosophy.In order traditional space-time configuration .



“vacuum”— cussion on the existence of to curb the metaphysical dis redefined e material substances—Boyl the space entirely void of all

) a spacein which noair it operationally (or physiologically as laboratory with a existed. Theartificial space producedin the

ed. Brian Vickers (Oxford: Oxford 7 Francis Bacon, New Atlantis (1626) in The Major Works, ty Universi Press, 1996), 457-89.

and Science in Early 8 Onthe notion of knowledge space, see David Turnbull, “Cartography Mundi 48, no. 1 Imago Spaces,’ ge Knowled of tion Construc the Mapping Europe: Modern (1996): 5-2-4. 9 The term“scientist” was coined onlyin the 1830s. Thomas Holden, “Robert Boyle on Things above Reason.” BritishJournalfor the History ofPhilosophy 15, no. 2 (2007): 283-312.


is rhetorical differentiation between

subversion ofthe latter, allowedfor the fellows of the R. oyal Society

to appropriate the laboratory for a “true” philosophical project.! An assembly of gentlemen could direct judge, conjecture, improve, and discourse upon the experiments to “matters offaacct” t” 2 potentia and ntially l on a par liaame oit me ntar neanse Truthe nan ity.son e y poli philosophy depended less on the speculat ive ontology ofnatu j


than on the epistemic andsocial practices .



in the laboratory.


In Isaac Newton’s mathematical natural philosophy,? the

interatomic chemical space became hom ologous to the cosmi space. While the Newtonian Synthesis in the conventional historiographyrefers to his mathematical homogenization of .



the Copernican universe with the noti onof gravity, 12 Newton’s

ownvision included experimental sciences to forge an integrated domain of Nature—theabsolute space that was Go d’s dominion «


10 J.otr Andrew Mendelsohn, “Alchemy y and Politics as in Englan E olitics in d 16491665" Past and Present 135

Boyle, “Somespecimens ofanat tempt to make chymical experiments 11, Robert Boyle, useful the notionsof


the corpuscular philosophy?’ in The Works ofthe Honour able Robert Boyle newoda, 6 vols. (London, ; 1772), tt1:35.4~$93 9; Tl Thomas Sprat, , The Historry ofthe Royal Society of London, I

the improving ofNatural Knowledge, znd ed, (London, 1702); Steven Shapin reelotnon setter

eviathan and the Air-Pump (Princeton, NJ: Princeton University Press, 1985) 12 Alexandre Koyré, From the Closed World to the Infinite Universe (New York: Harper, 1958)




ing gravity and governed by uniform mathematicallaws. In offer

ipia (1687), Newton as the universal cause ofall motions in Princ

aof Nature harbored a wish to “derive the rest of the phenomen cal] by the same kind of reasoning from mechanical [mathemati

principles.” He hopedthatall actions in nature were governed by the “attractions”gravitational, magnetic,electrical, and

chemical—that would follow the same mathematical law.Just as the long-distance interplanetary actions were governed bygravity,

the midrange interactions of magnetic andelectric bodies might be caused by the attractions that follow similar mathematical laws. Chemical attractions at the atomic level would complete this “analogy of nature”—that nature as God’s dominion mustbe simple and consonant toitself.!? Newton's infinite universe was organized bya set of forces that determined the distances between the bodies, large and small. Their singular mathematical form—an inverse square law—evinced the unity of God’s design, while their material manifestations regulated complex interactions between bodies. The uniform law ofattractions would guarantee the homology between theinfinite cosmic space and theinfinitesimal chemicalspace. Establishing the truth of natural religion—one that would bolster the moral foundation ofthe British Empire— required rehabilitating space and time as God's creation. After a series of Boyle lectures that sought to reconcile Newtonian philosophy and Anglican theology, the debate between Gottfried Wilhelm Leibniz and Samuel Clarke took place in the context of the Hanoverian succession, whichraised the specter of Leibnizian theodicy in the English political scene. 13 Newton, Opticks (London, 1718), Query 31.

G. W. Leibniz and Samuel Clarke, A Collection ofPapers, whichpassed between the late learnedMr.

Leibnitz, and Dr. Clarke, in theyears 171s and 176 (London: Knapton,1717); Steven Shapin, “Of Gods and Kings: Natural Philosophy and Politics in the Leibniz-Clarke Dispute,’ Jsts 72, no. 2 (1981): 187-215.



Modern chemistry cultivated as useful knowledge(espec ial

for medicine) did not realize Newton’s or Leibniz’s theolo ‘cal dreams. Instead, it became a science of making new material

worlds. How chemists crafted symbolic spaces to repre sent the productsof their labor is a complicated story, one that should alert us to the historicity of the chemical (and perhapsall scientific) bodies that make and remake our built environme

and the ever-expanding chemical empire.” In Diderot and ° d’Alembert’s Encyclopédie, the chemical labor atory was represented by an affinity table that organized the o serati of“salts” —acids, alkalis, and their combinatio ns (ce Fi



Rs.2). The project of enlightened civic chemistry de ended on chemists’ capacity to organize and repre sent their soot practice in a rational manner, even if the relationships (nap ts)

between chemical bodies were visualized through whe I hen “I symbols that offered a shorthand. The downward nf — the well-lit laboratory that exposed neatly arran edch nical instruments to thetable of affinities traced an itineraty of Nature’s metamorphosis that produced useful knowled

Such anorderly representation of chemical labor cultivared a new generation of chemists, Antoi ne-Laurent Lavoisier’s

vision of “a revolution in chemistry and physic s,” now kno as the Chemical Revolution, did not simply consolidat ne

theoretical structure for chemistry,It also projected nex ‘de I of the chemical laboratory. A modern scien ce of chemistry ~ would depend on the metric (barometric, chermometric,

ww we. yptcs . 15 MiG Misme wSuena ee and Mirage: Writing the History of Chemistry” Studies in History and ceeel Seer 6626N99) 3565 Mi CrungK im, “Constructing Symbolic Spaces:

Chemical olecu ademie des Sciences,” drmbix 43, no. 1 (1996): 1-313 Mi G



divine omnipresence, whichis indeed very hardto avoidif one. assumes that space and time are two ete rnal and infinite non-thing s subsisting in themselves -+. Which are there (without there being anything actual), only in or der to containall actuality within themselves” (A39/Bs6)—ice., if one attrib utes to Space and timethe attributes of God's imm

ensity and eternity, !! Kant's description of the Leibni zian y iew, however, is consid erably mote puzzling. Are not the fu ndament al entities, on this view, nonspatiotemporal simple sub stances or mo nads, suc h tha t spa ce and time (as ideal “well-founded phenom ena”) ar e then possible as arising from the purely intellectual relati ons of coexi Stence and (in ter nal) causation holding among these substances and between their own (internal)

oflimitations), But with what right can onedothis, if one has previously made both into forms of things in themselves—and, indeed,

into forms which,as a priori conditions of the existence ofthings, even remain when onehas annihilated the things themselves? (For,

as conditionsofall existence in general, chey mustalso be conditions for the existence of God.) There is therefore no alternative, if one

does not pretend to make them into objective forms ofall things,

except to make them into subjective forms of our outer and inner modeofintuition.[This kind ofintuition]is called sensible, because it is not original—~i.e., it is not such that the existence of objects of

to God's knowledge must be intuitive,


an »

t because only this kin diate, Discursive or conceptual thoug " d of knowledge is éme‘ ot we matter given from without (ie. » In intu dy contrast,is always mediate, insofar as it: depends on ition) forits content, Ifsp ace had absolute reality in Newtoniansense, therefore, the it would be a (transcendental




ly) objective formof the existence all things. Bur since this leads of to absurd co nsequenc : es(the existence of God in space Kantian alternative remains, Note ), only the thar here—andin t he relat od'in ed Passag space e earlier ), in the Aesthe only tic {A39-41/B56~58)—Kant does not object to the Newtonian conception on space were (transcendentally) object the grounds thatif ive and independer it would be problematic; he saves this kind of objectio

i See the General Scholium added to the Principia: Mathematical Principles ofNat second edition of the Principia, 1. Newton, The ural Philosophy, ed. and trans. I. Berna rd Cohen and Anne Whitman, assisted byJulia Budenz (Berke ley: University of California Press, 1999): “[The true God] is not eternity and infinity, but eterna l and infinite; he is nor duration and space, but he endure s andis present, He endu res always and is Present everywhere, and by exist and everywhere he constitutes durat ing always ion and space.” The Newtonian doctrine of divine omnipresence is a centralpoint of

contentionin the Leibniz-Clarke

. : to theform rather thanthe matter ofthe tedies; In other words, Newtonian absolute space pertains . ed withinit; ba tetits structur therefore is in (Kantian outer appearances) contain princi e capa eo e, ,is in principl a priori cognition in Kant’s sense (see note3).


correspondence. For an extended discussion of therelationship between Kant’s trans cendentalidealism and this Newtonian doctri ne, see Michael Friedman, “Newton and Kant on Absolute Space: From Theology to Transcendental Philosophyy in Constitueing Objectivity: Transcendental Perspectives on Modern Physics, ed. M. Bitbol, P. Kerszberg, andJ. Petitot (Berlin: Springer, 2009), 35~50.



concerned states? And,if so, why should this view be understood as nce, alwith “relations between appearances ... abstracted fromexperie B56-57; emthough in the abstraction represented confusedly” (A40/

phasis added)?” Why, accordingly, must the Leibnizians “contest the a priori validity of mathematical docttines in relation to actual things (e.g., in space), or at least contest their apodictic certainty, in so far as... the a priori concepts ofspace and time, according to their view, are only creatures of the imagination, whose source must actually be

soughtin experience” (A40/Bs7)?”

Thespecific target of Kant’s criticism emerges more clearly in his re-

mark to the antithesis of the Second Antinomy, which concerns the

infinite divisibility of matter in space: Against this proposition of the infinite division of matter, the groundofproofofwhichis purely mathematical,the monadists have brought forward objections—which, however, already make them objects of suspicion, in that they are not willing to grant that the

clearest mathematical proofs are insights into the constitution of space,in sofar as it is in fact the formal condition of the possibility of all matter, and they rather view [these proofs] as only inferences fromabstractyet arbitrary concepts, which cannotbeapplied to real things... If one listens to [these monadists], then one would have

12, One of the best-knownpassages in the Leibniz-Clarke correspondence, $47 of Leibniz’s Fifth

Letter, describes how somethinglike the Newtonian conception of absolute space arises by abstraction fromour observationof the relative positions and motionsof bodies. Whatis explained

there, however, is nor geometrical space bur rather space as a dynamical framework for describing the motions andinteractions(forces) of bodies; it therefore concerns the relationship between space and time. This is important,as I shall explain later, because it turns out that the most fundamentaldifference between Leibniz’s idealism and Kant’s concerns the characterof the dynamical a

framework in question. On the view that space andtime arise fromthe purely intellectual relations of coexistence and

causation, holding amongultimate simple substances conceived purelyintellectually, the system of relations in question wouldconstitute a formalrather chan material element ofcognition in Kant’s sense. It would therefore be capable, in principle, of a priori cognition. Kant could (and would)

object that such a system cannotexplain the (assumed) synshetic a priori status ofour geometrical cognition; it is important to appreciate, however, that he does not makethis objection here,



to think,aside from the mathematical point, whic his not a part but

merely the limit of a space, also physical points, which are indeed also simple, but have the advantage, as part s of the space, offilling i through their mere aggregation. (A439/B46 7) “

These physical points—or physA ical monads are thus simple and elementary material substances, s material substances, they are what

Kant himselfcalls bodies and th us what Kant himi self calls appearances. According to precisely the arg ument of the Sec ond

An nomy, how. ever, all such bodies in the Kant ian an sense must be j i“ y didivisi infinitely isible/y since

they are possible, for Kant, only within our eee space—which, in turn, is necessarily infini . divisi too (Ately vg e bleinsecon accorddance vice

with the “clearest mathematical For the monadists in question, b substa nces or physical points sre wring enna #67). the ultimate simple

en “apposed to constitute this space

“through their hee ney

mere aggregatioare n” ve 440/B4 l “8 68). yeHe re,however the monadists that Kantis targeting run y Pro em of the composition of the i morespecifically, into Zeno’s metrical Paradox

of extension, Acco

thee .woeaes one never attain an extended region ofeee te ine aenum veo pwtended simple elements (points), not fore wuld beeen

such elements, The only way

out, therebody toca be cowie ©rs elements out of which the space filled by a than unextended points) ney very small extended regions (rather

and, as a consequence, to deny the infinit e divisibility (phys ical) secon pace, d hovnofofZen Forone would otherwise o's metrical pret i run into i the according to which aninfinite

numberofextended(finite) " cements could never compose a finite extende regio (a body) n d not only take the points ms“imronadises thac Kant is targeting here ich (physical) space is composed to be what hecalls appearances—thereby making our knowledge ofspace a posteriori. They even go so faras to deny the evident mathematical proposition of infinite divisibility—thereby “contest[ing] the a priori




«J: alld l dl


I 2 ematica


. . ‘ on to actual octrines in relati

: g


_ in ques argued in detail that the monadists Bs Vincenzo De Risi has of the Second Antinonyinclude tion in the r emark to the antithesis ian philosophy, tatives of the so-called Leibnizian-Wolff quite explicit, in the remark bee st eibr i himself.!5 Indeed, Kant is 14

of mat

hat the monadists in question imthe“hes of the Second Antinomy,t do not include Leibniz:

asit is necessarily piven I speak here only of the simple, in so far

ved into t me ts in the composite, in that the latter can be resol s races i g constituents. The proper meaningof the word Mona e which is inne Leibnizean usage) should only extendto that simpl

onsciousness)-an®net diately given as simple substance (e.g.,in selc

. etter “ ‘ as element of the composite—which one could ov seances And, since I want only to prove [the existence of] simp

| could call t : thesis in relation to the composite, as its elements, Fowere’ ecause of the second antinomytranscendental atomism.

this word

ic hasalready long been usedfor the “esignation© a part

ecu sum) ane ular modeofexplaining corporeal appearances (mol s] may e ve e therefore presupposes empirical concepts, [the thesi

68-70 the dialecticalprinciple of monadology. (A4 40-42/B4

concepti in cighteenth-centu ci ryry conceptions of Jayed an important role in d i discussions of including 8 eessions a rearmen Hy3 ; fefor an illuminating treatment, areene metric more generally; t4 Zeno matter ‘ Architecture ofMatter. aot xzn Dee nna his contemporaries, see Thomas Holden, The i chat these saneprro bemplay emphasized has Sylla Edith . 2004). Press, (Oxf di Clarendon ea i ; the me within loped develope were that ts : i i argumen vest onding role in holastic wradition ro the effece that the natures ofreal (physical) ‘ chola:

things cannot b

sbuoply of Space _ mathematically. »of » i andDhiaseP AnabyisSitus Leibmia’ Vincenzo De Risi, Geometry and Monadology: of Christian ‘ol f , Get s ‘iBinger ° a 1Birkhduser 2007), 301-14, which discusses the work t to #focontrast ( isi i De Risi i iis par ticularly concer! i connection, iin ¢ his ' 4Alex xander Baumgarten anWolf, no expounded those with continuum the of composition mee views on the .

1731)s vlog generalis methodo scientifica pertractata (Frankfurt: Renger, Casosmologi. me


sages from the i the passag along with i j cited, ion just considers this passage in the discussion De Risi1737 6 een i , i the following ider ji that I considerin : , Foundations ofNatural Science Me Metaphysical


Kantis clear, therefore, that properly Leibnizian monadsare not to be conceived as physical points out of which bodi es (together with the space they fill) are supposed to be composed . They are rather mindlike—and therefore entirely nonspatial—sim ple beings, which are given (at least to themselves) in immediate selfconsciousness. To be sure, both space and physical bodies in sp ace are in some sense derivative from these beingsas ideal well-found ed phen omena. In nosense, however, are they composed out of such beings. Itis even more striking, however, that when Kant treats the infinite divisibilicry of matter and space in the Metaphysical Foundations

ofNatural Science (1786), he not only distinguishes Leibniz fro m his Leibnizian-Wolffian follower s bu t also explicitly appropriates him on behalf of Kantian ide alism. In Particular, in the second remark to the fourth Propositi on (demonstrating the infinite di-

visibility of material substance) of the Dynamics chapter, Kantap peals to the

argument of the Second Antinonyy to resolve a conflict

concerning infinite divisibility between the “geometer” and the


One would therefore hav e to concludeeither, in spite of the geom-

eter, that spaceis not divisible to

infinity, or, to the annoyance of the ofa thing initself, and thus that matter is not a thing initself, but merely an appearance ofour Outer senses in general, just as space is the essential form thereof. But here the philosopher is caught between the horns of a dangerous dilemma. To deny the first Proposition, that space is divisible to infinity, is an empty undertaking; for nothing can be argued away from mathematics by sophistical h air-splitting. But viewing matter ° as a thingin itsel f, and thus space as a property ofthe thingin itself, amounts to the denial ofthis proposition, The philosopher therefore finds himself forced to deviate fromthis last proposition, however common and congenial to the common understanding it may

metaphysician, thatspace is nota Property

be. (4, 506)

Thus the errors of the “metaphysician” need here to be corrected by che (transcen dental) “philosopher”—ice., by transcendentalidealism.!” Moreinteresting, however, Kant proceeds to contrast the “metaphysical” view heis targeting with the views of a (not yet named) “




great man :

A great man, who has contributed perhaps more than anyoneelse

to preserving the reputation of mathematics in Germany, hasfrequently rejected the presumptuous metaphysical claims to overturn the theorems of geometry concerning theinfinite divisibility of space by the well-grounded reminder that space belongs only to the appearance of outer things; but he has not been understood, This proposition was taken to beasserting that space appearsto us, thoughit is otherwise a thing, or relation ofthings,in itself, but that the mathematician considersit only as it appears. Instead,it should have been understoodas saying that space is in no way a property that attachesin itself to any thing whatsoever outside oursenses, It is, rather, only the subjective form ofoursensibility, under which

objects of the outer senses—with whose constitution in itself we are not acquainted—appear to us, and we then call this appearance matter. Through this misunderstanding one wenton thinking of space as a property also attaching to things outside our faculty of representation, but such that the mathematician thinks it only in accordance with common concepts, that is, confusedly (for it is thus that one commonly explicates appearance). And one thus attributed the mathematical theoremofthe infinite divisibility of matter, a proposition presupposing the highest [degree of] clarity in the conceptof space, to a confused representation of space taken as basis by the geometer—whereby the metaphysician was then free

Pose se s space to compo

out of points i , and matter out of simple parts,

and thus (in his opinion) to bring clarity into this concept. (4 507) Thus it is clear, in particular, that the “metaphvsician” j sicia ian in question Physic i essenti:ally the s ame as the is representati 2 ve ofthe e LeiLeib bnin;zian-Wolffi phi

losophy targeted in the Second Antinomy

(A43 5/B467)—and also, appeaheti rs, ,cin(A3 the9~4 pass a ge assa tn theitAest cont i g Newton rast 1/B 56~ i. ing ¢8) onia } ns and Leibibniz nizii ans In the immedi diately following i i cussion Kant dis

goes ke it clear that the “great man in questi i woe ” himself: question is none other than Leibniz ‘ han Let b The

: aberration . eground foer this lies in a poorly understood

monadology, which has . a : as nothing at all ‘ to do with the th explanation lanati . of natural appearance ara es, but 1s rath . . €r an intrinsicall Latoni ally correct platonic } .

conceptof the world

not atall as object ccvised by Leibniz, in so farasit is considered, an object of “in c ° the senses, but as thingin itself, andis merely ‘ . dere the appenun erstanding—which,

however, does indeed un-

of the senses, . . , Therefore, ’ Leibniz’s ideaa [Meinung], soPearances far as | comprehend it Was not to explicate space through the order of simp le beings €xt to one another, it was

rather to set this order alongside space as correspondingtoit, but as

belonging to a merely int elligibleso tld (unknownto us). Thus heasserts nothing but what has been sh ownelsewhere: namely, that space, together with the matter of wh ich it is che form, does not

divisibility ¥ of space; i g to the second passagejust quoted spaces 1 moreover, accordin (4, 507), the “metaphysician” goes on “to ¢ The connecee between thedien Space out of points, and matter out of simple parts” (4s 507)17 Nore that Kancis perfectly clear that his transcendentalidealism runs counter to commonsense, but he takes himselfto be “forced”into this position by (amongother things) the argumentofthe Second Antinomy.


and the corresponding discussionin the Aesthetic i . heh in the Metaphysical Foundations

icis that, in both,the space ofthe “metaphysician” is taken ro attachto of be inherentin thingsiin themselves, with the result thar the representation the “mathematician ician’” becomesnecessarily confused,

Presentation ofthe





contain the world ofthings in themselves, but only their appearance, andis itself only the form ofour outersensible intuition. (4, 507-8)

betweenthetwothinkers Leibniz’s “oh and argue,in pes particular, for deep continuities berween Leibnizs "phenomenalism” and Kant’s.2!

Thus Kant here depicts Leibniz—against the Leibnizian-Wolffian

“metaphysician”—as a defender of the Kantian doctrine of transcendentalidealism. There are indeed strongsimilarities between Leibniz’s idealism and Kant’. Leibniz, like Kant, considers two essentially distinct classes ofentities: purely intellectual beings or noumenaandpurely sensible beings or phenomena. Space, for Leibniz, together with the matter

or physical bodies that appear withinir, is an ideal well-founded phe‘nomenonrather than an ultimate metaphysical reality; moreover,its phenomenalstatus dependsessentially on the perceptualrelationships among the mind-like simple substances that constitute ultimatereality, as each such monad mirrors all the others fromits own (perceptual) pointof view. There is also no doubt that Kant’s formulation of transcendentalidealism is deeply indebted to his assimilation of Leibniz.?°

Wevertheless, Kant is very clear in other texts——espec ially in the Amphiboly of the Concepts of Reflection in the first Critique—that

the p roperly Leibnizian conception of space differs quite fundamentally fromhis own. The most important difference comes underthe heading of matter andform:

In theconceptof the pure unders tandin

g matter precedes form and

Leieniz consequently first assumed things (monads), together with an. iraner power ofrepresentation ,

afterwardsto groundtheir in order exte rnalrelatio ns and dth ty of their states (namely, their the communi , epresentations) on this. Ther rep efore, space and tim i e were [thereby] ossiible— the former only through the relation of the substances the Jatter through the connection of thei


one another as ground and conseauence wattnFrethot woula

It is no wonder, then, that some of the mostinteresting and sophis-

hav€ to beif the pure understanding could be related ow vcd

ticated recent interpretations of Leibniz emphasize the continuities

to objects, andif space and time were determinations ofch sein

19 Robert DiSalle has suggested that Kant may here be describing his ownearlier position in the InauguralDissertation (“On the FormandPrinciples of the Sensible and Intelligible [nzeligibilis]

we determineall objects simply as appearances, th then th form of the“forin , gocuition (as a subjective icuti ats sonsttution ofsensibility) precedesall meacter (the appear Ppearances), and therefore Space and timeprecede all app oear anc e esfee the andapla llcedat(Aa a of 6y/ exp Bee eris)ence, and rat : her make

chesselves. If, however, they are only sensible intuitio ' i whi h

them pos-

Worlds’) of 1770. This makes sense, because Kant both characterizes the realm of simple beings

as a “merely intelligible world [bloff intelligibeln Hele)” (4, 508) and describes “Leibniz's idea” in termsthat he himself can “comprehend”(4, 508). Note, however, that the view of the Inangnral Dissertation is quite distinct from Kane’s matureorcritical transcendental idealism, insofar as he had notyet (in 1770) found a bridge between pureintellectual concepts and our(spatiotemporal) sensibility; the so-called schematism of the pure concepts of the understanding,I shall return to this last pointlater. 20 Kant'’s assimilation of Leibniz’s idealism underwent a long evolution, fromtheearliest works of

his precritical period,such as the New Exposition ofthe First Principles ofMetaphysical Knowledge (1755) and Physical Monadology (1756), through the Inangural Dissertation (1770), to its culmination in the critical period in the Critique ofPure Reason (1781/1787). Fully understanding Kane's

relationship to Leibnizian doctrines therefore involves not only distinguishing Kant’s attitudes toward Leibniz and his Leibnizian-Wolffianfollowers, but also distinguishing the stages of Kane's ownassimilation of Leibniz at various points in his intellectual development.

is describi . Ka ntee mn the properly Leibnizian view that space and .

rs +0 rie ti +of he purely relations between and ind-lilee intellectual ef dete* mina lons of the mind-like simple substances that constitute ulcma reality: space in terms of relations of coexistence amon g such substances, time firom causal connections ‘ between

their (individual)


set especially R. M. Adams, Leibniz: Determini q Idealist ; (Oxford:: Oxford University iversi erminist, Theist, pres, 1994), and De Risi, Geometry.




In thecase ofthe concept ofsubstance, in fact, Kant eventually arrives at an even stronger result. Not only must (phenomenal) substance be temp

asting this conception determinations or (inner) states.22 Kant is contr

founded phenomena of the sense in which space and time are well-

intuition with his own conception of them as pure forms of sensible

orally extended (as permanent); it must be spatially exte

nded well—and so, by the argument of the Sec ond Antinomy, i can neither be simple nor consist of ultimately simple (substantial) parts. Kant suggests this

zed in the within which alone any substantial real object can be cogni

non) first place—as what Kant himself calls an appearance (phenome or object ofexperience.

conclusion a few pages earlier in the Amphibol

y, where he cont contrasts hi * own conception i of the constitution of matter with Leibniz’s monadic conception:

) cogniThecrucial point, for Kant,is that we can have (theoretical

tion only of such appearances or objects of experience; (theoretical)

imposcognition of noumena or things in themselvesis completely sible, For, in the absenceof an already given spatiotemporalintuition

Onlythatis internal in an object of pure understanding which has

within which to order and thereby determine such objects, the pure

norelation at all (with res Pect

to its existence) to anything

different fromitself. By contrast, the internal determinationsof a substantia phaenomenon in space are not hing but relations, andit itself is nothing but a totality of mere relations. We are only acquainted with substance in space thro ugh forces that are active in space, either driving others into [ this space] (attraction) or stopping their penetration into it (repulsion an d i mpenetrability), We are acquainted with no other properties co ns tituting the concept of a substance which appearsin space and which we call matter. As object of the

understanding on its ownis capable of no (theoretical) cognition at all. Whereas I can certainly think objects of the pure understanding—

such as God andthe soul, for example—independently of spatiotem- ’

poral intuition, no such (noumenal) object can be (theoretically)

cognized,”? This is why, for the critical Kant, pure concepts of the understanding such as substance, causality, and community can play their properrole in (theoretical) cognition only if they are associated with what Kantcalls spatiotemporalschemata: substance with the temporal relation of permanence, causality with the temporal relation of succession, community with the spatiotemporalrelation of simultaneous

pure understanding, on the other hand, every substance must have

internal determinati reality. However, whetoe ”hich pertainaccidents ‘hose which to [its] internal um Se entertain as internal except



my inner sense presents to me—namely, that which

Hiteriself a thought oris analogous to it? Therefore, Leibniz,

after he had taken awayeverything that may signify an external rela-

tion, and therefore also compositi on madeofall substances, because he represented themas noumen a even the constituents of matter, simple substances with powers of representation—in a word, monads, (A265~66/B321~22) 2


we ~

22 Again,this view shouldbe sharplydistinguished from the (Leibnizian-Wolffian) view that spaceis composed outof“physical points” (or physical monads) as its parts. De Risi, Geometry, is especially concernedto argue that Leibniz’s more sophisticated approachto the problem ofthe composition of the continuuminvolves moving away fromthe traditional model of part-whole composition and toward the modern conception of abstract relations between elements in what we nowtake to be set-theoretic structures. See the important footnote to the (secondedition) Preface (Bxxvi): “In order to cognize an object it is required that I can proveits possibility (whetherin accordance with the testimonyofexperience fromits actuality or a priori through reason). But I can ehink whateverI wish, as long as I do not contradict myself—ie., if my conceptis only a possible thought, even if] cannot guarantee whether or not an object correspondstoit in the sumtotal ofall possibilities.” Kant indicates in the remainder of the note that one may be able to cognize such (supersensible) objects through reason from a practical as opposed to purely theoretical point of view—which is why I have inserted the qualifier “theoretically” in parenthesis. [ shall return to Kant’s conceptionofpractical cognition at the end ofthis essay,



sen is here alluding to his critical conception of matter (and ma-

terial substance) asfilling the space thatit occupies by the interp lay

of attractive and repulsive forces exerted at every point of the space





which Kant’s theory of experi rience . is ‘di . voaisN . : P ce Is providing a metaphysical foundation s Newtonian physics—and, indeed, f, dati i physics inwhich gravitational » a foundationforthis

in question. And it is precisely this conception that underlies the demonstration ofinfinite divisibilicy in the Dynamics chapter of the Metaphysical Foundations—whichis followed, as explainedearlier, by





enthusjast; is Senthusiastically ata distance) ;

ical monads.”4 This demonstration of infinite divisibility, moreover, plays an essential role in Kant’s demonstration of the permanence of mate-

to find a middle vo aert ofthe matter, In his lifelong attempt

chapter—the proposition thatthe total quantity of matter in the universe is necessarily conserved in all interactions of matter. And the latter proposition, in turn, plays an essential role in Kant’s conception of how momentum or whathe calls “mechanical moving force” is conserved in all interactions as well. The schematized category of

substanceis thereby connectedto the schematized category ofcommu-

nity or interaction, with the result thatit is a sufficient condition for the existence of a causal interaction between two material substances

that momentum(the product of quantity of matter andvelocity) be conserved in the process of action and reaction.” Finally, since momentum between twogravitationally interacting bodies is necessarily so conserved,it becomescrystalclear, at this point, that the physics for

We have now reach e

trans, (fromthe first Venetian edition of 1763) J. M. Child (Cambridge MA; MITPress, 1966).

Kant’s critical conceptionof matter, by contrast, is incompatible with the physical monadologies developed by both the Leibnizian-Wolffians andhis ownearlier self—and, most importantin the present connection, with the properly Leibnizian conception ofsubstance as well. For detailed discussion of Kant’s earlier physical monadology, in relation to both his critical conception and his evolving divergence from Leibniz,see Friedman, Kant’ Construction. For further discussion of the interconnections among quantity of matter, mechanical moving force, and momentum in the Metaphysical Foundations see Friedman, Kants Construction, chapter 3, Kant reformulates the principle ofthe permanence ofsubstancein the second edition of the Critique as a quantitative conservationlaw (B22.4): “In all change ofthe appearancessubstance is permanent, and its quantum in nature is neither increased nor diminished.” This echoes the corresponding propositionin the Metaphysical Foundations (4, 541): “Inall changes of corporeal nature the total quantity ofmatter remains the same,neither increased nor diminished.”


the Leibnizian and Newtonian “« between

— 2. ground’ positions—betwe

“mathematical and metaphysical” ; me© always to nat ure—Kantaly approaches tons en . than Newtonianrather embraced ys leibnizian physics. And he ¢ iod, ls thatassbren only ray scendental gradually to see,“the in thecritical pe. idealism,came in which form of a






JECtive consticuti10n of


(A267/B323), can possibly do




sensibility) precedes all matter”

yatural world that Newes bMstice to the scientific cognition of the .



: the Py;neipia y Hisi argument in

quetryis true of real (physic al) 1)

gytem. Onthe basis ofthis

in fa

: act a chieved.


' For Newton begins

Presupposing that (Euclidean) gea

Pace—ar least throughoutthe Solar

. resu et . Notion, which govern the (ph : Pposition andhis Axiomsor Laws of , . ysica [email protected]) motion, Newtonis th ) concepts of mass, force, and (true ous then ¢ : : givitation from the initial “Dh n ableto derive the lawofuniversal 1





na described by Kepler’ laws ofplanetary motion and, at the ¢ to establish the center of time, ame miss ofthe Solar System as th :

24 Thefirst remark rejects Kant’s own earlier conceptionin the Physical Monadology, according to which matter consists ofultimately simple monadsthatfill the space they occupybyattractive and repulsive forces exerted at only the central point of the space in question. Kant therebyarrived at a point-center atomismsimilar to that developed around the sametime in Roger Boscovich, Philosophiae Naturalis Teoria (Prostat Viennae Austriae: Officina Libreria Kaliwodiana, 1758);



ata distance (and thus causalaction embraced .26

a (second) remark rejecting the Leibnizian-Wolffian doctrine ofphys—

rial substance in the secondproposition of the following Mechanics





aljtrue motions therein are ¢ t ivilegedstate ofrest relative to which

‘ © oO be Kant,is paradigmatic of scientif defined. Newton's achievementfor tific cornir


: : 1tl ims: in the Metaphysical Foundations to 5 en of nature, and Rant £ eee ita metaphysical foundation— thitis, to explain, on the basi soft .

or pure concepts ofche understanding, how this kind oO ofk € Categories . . nowledgeis possible.?”

26 [sharp contrast to Newton's much n . physics involved an enthusiastic embr: tee nous attitude, Kant’s underst wtonian anding of Ne the inroduction to Friedman, Kant’s Consty,“e of Action at a distance throughout his career! see and Kant _ 27 I discuss Kane's “metaphysical” rej “ction, and compare Friedman, “Newton

pn in great detail in Eriedma

reretation of Newton's argument for universal [email protected]

Metaphysical Foundations of N. » Kanes Construction; see the introduction to Immanuel Kant ‘LK: Cambri dge Universis piural Science, ed. and trans, Michael Friedman (Cambe a cinterprets what he calls “ y Fess, 2004), for a shorter and more accessible account: ™

F be



$ “absolute space,” in particular, as a limiting idea ofreason—as th never

tobe arrived at ideal endpoint ofa Procedure for moving (in accordance with Newton’ argument,






Kant’s explanation substitutes his own three Laws of Mechanics—

the conservation of the total quantity of matter, inertia, and the equality of action and reaction—for Newton’s Laws of Motion, and Kant takes these laws to realize or instantiate the three Analogies of

also emphasized, the category of substa nce which is the most fundamental pure >

ly intellectual conre cept for Leib niz, can find nothin gat all ultimately simple andself- suosistent in its application to natural phe nomena.” As Kantputsi tin his Solu tion of of th

Experience established in thefirst Critique: the principles governing

phenomenal substance—th

the categories of substance, causality, and interaction or commu-

ject of (theoretical) cog sensibility, andit is nothing

nity. Whatis most important, as I have emphasized,is that the pure

conceptsin question are spatiotemporally schematized here: the quantity of substanceis given by the aggregate of movable matter continuously filling a given space; causality pertains to changes in the quantity of motion (momentum)in a bodyeffected by a second bodyspatially external to the first; interaction or community pertains to the rela-

tions of coexistence or simultaneity between spatially distant bodies throughout the whole of (physical) space. So universal gravitation,


e §

i Y> ccond Antinom

mentfrom geometry, depe nds on Kant’s synthetic rather than

as a genuine action at a distance throughoutthis space, isa paradig-

matic realization of the category of community.”* Similarly, as I have

fromour parochial perspective here on the surface of the earth to the center of mass of the Solar System, and fromthere (in accordance with Kant’s ownspeculative extrapolation) to the center of

mass of the Milky Way galaxy, the center of mass of a rotating systemof suchgalaxies, and so on ad infinitum.This reinterpretation of absolute spaceis reflected in an importantfootnote (A429/ 457) to the antithesis of the First Antinomy in the Critique, which concerns the extent of the

phenomenal world in space and time. I discuss this footnote,in relationto absolute space in the Metaphysical Foundations, in Friedman, Kant’s Construction, 156-59. oo


‘This is important, not only because Leibnizian physics explicitly rejects such action at a distance and restricts all physical interaction to impact, bur also because of the way in which Leibniz’s ownmetaphysical foundationfor this physics builds an analogousrestriction into the more fundamental monadic level. For example, in Gottfried Wilhelm Leibniz, “Specimen Dynamicum, pro admirandis Naturae legibus circa Corporumvires et mutuas actiones detegendis, et ad causas revocandis,” Acta Eruditorum, publicata Lipsiae, Calendis Aprilis (1695): 145-57, translated as

Gottfried Wilhelm Leibniz, “A Specimen of Dynamics,’ ed. and trans. L, E. Loemker,in Gozepied Wilhelm Leibniz: Philosophical Papers and Letters, and ed. (Dordrecht: Reidel, 1969), 435-52, Leibniz grounds the phenomenalforces of inertia, vis motrix (momentum), and vis vive (kinetic energy)—which together explain interaction by (perfectly elastic) impact—-in more fundamental (passive and active) powers exercised in the realm of non-spatiotemporal noumenal substances of which the spatiotemporal realm of matter and motionis a well-founded phenomenon, And at the noumenal or monadic level, more generally, there are no causal interactions among substances at all, but only a preestablished harmony that coordinates their causally independent unfolding. Kantdiscusses the doctrine ofpreestablished harmonybriefly in the Amphiboly, sandwiched between discussions ofsubstance and space andtime, under the rubric ofthe community ofsubstances


dition of the Critiq ue, that our representation of space is an ( @ priori) intuition rathe Moreover, given the extan : t Newtonian an * than a concept. d Leibn izian conceptions and time themselves, it is similarly reasonable at they just are the corresponding intuitive

29 The concept of substance for “ , the critical Kang, corre: sense either, thatis, in our self-cons sponds to nothing truly substantial in inner cio fundamentally from Leibniz (and Descausrtex bys of our owninn er states. . Here Kant diverges . Artes i apperc

eptionreveals, by the Y arguing that our pure self-c -cons onsci ciousn could be self'subsistent, subseateh ne ofthe Paralogisms of PureReason, noactual object that of all substance whatso see itimately simple. And by the argument of the Refuration to Ideali both sm, the .

Metaphysical Toundanonass be realized in space. I discus s these points, in relation an if f the Critique, in the conclusion of Friedman, Kane’

Construction. For my most 3 o Reconsidered”




inDienst aos issue See Michael Friedman, “Synthetic History “ vu ethod: Reinvicorats, ' : . Philosophy ofScience, ed. M. Domski and M. Diekson(Chicages the Marrtage of History and .


specifically 585~99,



Court, 2010), §71~813,




representations—so thatintellectually conceived things in themselves (if there are such) are not spatiotemporalatall. Yet this conclusion by no means amountsto thecritical doctrine of transcendental idealism. It was already fully present in the precritical Inaugural Dissertation (1770), and it is very close, in any case, to

Leibniz’s own doctrine of phenomena and noumena,ofthe sensible andintelligible worlds. In the Ixaugural Dissertation, however, there was as yet no clear connection between oursensible and intellectual knowledge. There was no basis for concluding that knowledge of purely intellectual noumenais impossible, and, more generally, there was no conception ofthe necessary schematism ofthe pure concepts of the understandingin terms ofour spatiotemporal intuition—the only kind ofintuition ofwhich we humanbeings are capable. It was only in the Critique ofPure Reason (1781/1787) that Kant was able to demonstrate (at least to his own satisfaction) that and howthepurely intellec-

Nevertheless, it is equally important to Kant that pure intellectual concepts can have objective meanin 8 and

reality from a purely practical point of view. According to the Critique of Practical

Reason (1788), in particular, the three principal i Principal ideas of pure practical reason—God, Freedon

oF 1, and Immortali alitty—all acquir i e such meaning andreality in relatio n to our own immediate :experienc e of the moral law as authoritativel y bindingon our will. The idea of Freedom acquires it directly fr omthe immediate experience in question, The other two ideas, God and Immortality, then acquire it inquire it in i ur ach necessary asiev directly, ofo ing presupp Ositions or postulates for the possibility 8 \or atleast continuously app roximating) the highest end of moraliey the realization of an ide

al moral community ofal l hehe ae (a Kingdom of Ends) here on the surface of the earth n the case of theoretic


these ideas, therefore, althou gh they forever elude our iti

tual concepts ofsubstance, causality, community, and so on function as

a priori conditions underlying the possibility of all human experience ofthe sensible world.*! Andan equally important negative conclusion then followed. Such purely intellectual concepts or categories have objective (theoretical) meaning andsignificance only when applied to objects of our experience in the sensible world; they do not have such meaning and significance when one attempts also to apply them to (putative) supersensible objects such as God andthesoul.”



famouslyasserts that he ha d to “de make room forfaith [Glaub en}” (Bxxx),33 So it is especially striking, finally, that Kane also finds a pure ly practical reinterpretation of the New tonian doctrine of divine omnipresence throughoutall of infinite space. I explained at the begi nning ofthis essay how Kant, in the Aesthetic, rejects the alternative

31 Thus the Transcendental Deduction of the Categories—which was completely rewritten in the second edition—is an essential part of the critical doctrine of transcendental idealism, 1 dis-

cuss aspects of the second edition version,in relation to Kant’s conception of space and geometry, in Michael Friedman, “Space and Geometry in the B Deduction,” in Kane's Philosophy of

therefore essentially ir ‘ast . by contrast, Kant, onalatet relation of a thing to us. On the present interpretation, is space and timeas pure forms ofse ib\ ™Is not only based onthe radically new conception of important)

Mathematics, Vol. 1: The Critical Philosophy and Its Background, ed, C. Posey and O. Rechter

the critical period, For Langton, then, the ignorancein question does not depend on any special

features of space and time, but only on the fact that sensibility, for Kant,is receptive—so thatit

on thedistineerels nsible intuition tharfirst emerges in 1770, bur also (and most

*nctively critical doctrine of the schematism of the pure concepts

~ ~

(Cambridge, UK: Cambridge University Press, forthcoming). ” 32 Rae Langton, Kantian Humility: Our Ignorance ofThings in Themselves (Oxford: ClarendonPress, 1998) interprets Kant’s doctrine of “our ignoranceofthings in themselves” as deriving froma fundamentally Leibnizian conception ofsubstance involving onlythe intrinsic properties ofa thing— properties holding independently of a thing's relations to others. She bases this interpretation on aclear and illuminating discussion of Kant’s precritical writings, which she then extrapolates into

"Thus us Kant’s Kant’s

ofthe un-

deni of the possibi atelity of theore denial tical cognition of the supersensibleis precisely what opensup pens up the p possibilit y for a distinct istinctiveive kind ki of practica i l cognition (Buxix-xxx): “I can thus not even assume anneh

men ] God, LB } reedom and Immortalit on behalf of the necess ary : + practic my reason, if tIdo not, at the Same time, me] specula


deprive [beneb

tii i i specularve reason ofis pretensionfo




Newtonian conceptionofspace largely because of its commitmentto this doctrine. I also suggested (in note 26) that Kant’s conception of the extent of the material universe in space in the First Antinomyis closely connectedto his reinterpretation of the Newtonian conception of absolute space as a framework for determining true motions. In an important footnote appendedto the General Remark to the Third Part of Religion within the Limits ofReason Alone (1793) Kant then describesa “sublime analogy” betweenthe (theoretical) community of all matter in space due to universal gravitation and the (purely practical) ideal moral community ofall rational (human) beings on the surface ofthe earth: When Newtonrepresents [the universal gravitation of all matter in the world] as, so to speak, divine universal presence in the appearance (ommnipaesentia phenomenon),this is not an attempt to explain it (for the existence of God in space contains a contradiction), but rather a sublime analogy, in whichit is viewed merely as the unification of corporeal beings into a world-whole,in so far as

we base this uponan incorporealcause, The same would happen in the attempt to comprehend theself-sufficient principle of the unification of the rational beings in the world into an ethical state and to explain the latter from the former. We know only the duty that

draws us towards this; the possibility of the intended effect, even when we obeythis [duty], lies entirely beyond thelimits ofall our

insight. (6, 138-39) Thus here, once again, Kant’s complex and multilayered conception of transcendental idealism has both a positive and a negative aspect

within the realm of theoretical cognition. His “metaphysical” explanation ofthe possibility ofNewton’s paradigmatic theoretical achievement amounts, at the same time, to a rejection of the doctrine of y

d vine omnipresence —which Kant te ke




mpt roa chive to achie ve“extravaga ; nt insi ‘ ght” into the supersensible. Yet Kant als o , a purely practical point of view, a legitimate (analogical) W

“ for this same Newtonian use i doctrine ine in i effectively directing us tow ard

every highest ends of morality,





example, if the angles two lines mak e with a third sum to two right angles, then the twolines never meet—they are said then to be

parallel. In particular, the three ang les in any triangle sum to two right angles, a nd the Pythagorean the orem can be proved. Nowif this assumpti on aboutlines is dropped,


can be proved, and §cometers are left with little to say. On the other han d there is no good reason to believe the

assumption, whi

ch makes statements aboutlines meeting eting ind ; efinitelyar awe what can be checked, " “y well beyond




The assumption, generally called the p arallel postulate , was madethe subject of many investigations thattypicall y hoped to deriveit as a theorem fr om the other ass umptio ns made by Euclid.

Non-Euclidean geometry grew out of a centuries’ long investigation in many different mathematical cultures of the groundsfor accepting the geometry described in Euclid’s Elements. The Elements is not necessarily to be read asa mathematical description of physical space.It is incompatible with the Prolemaic universe, and we have no evidencethatit

was intended as anything other than whatit is: a largely formal accountof elementary geometry in two or three dimensions, together with three books on properties of the integers. Some of it is very sophisticated, and otherparts are a confusing mixture of precise definitions andtacitly assumedbeliefs. The problematic feature that concernsus hereis one of Euclid’s assumptionsin book 1, thatif two lines cross a third and make angles on the samesideofthat third line that add up to less than tworight angles, then thoselines will meet (on the sameside of the line as the angles have been taken).In the presence of the other assumptionsin the Elements many usefulresults follow; for

Jeremy Gray, Non-Luclidean Geometry, In: Space, Edited by: Andrew Janiak, Oxford University Press (2020). © Oxford Universiry Press, DOL: 10.1093/080/9780199914104.003.0013

self. For example, one mi gh

t ywhere equi distan t toa straightline was

itself'a straight line, but this

too was n

ot self-evidently correct. Or one could assume that the angle sums of trian gles were two right angles, or that it was pos si ble to make exac t scale copiesof figures ofdifferentsizes, These ass

umptions, taken toge ther with the other assumptions of Euclid’s Elements, do imply thep ara llel postulate, but they remain assumptio ns, Nonetheless, at motionsare



for example? Islamic scholars (Ibn al-Haytham wedUne _ Kh ayyam) debated whether one may move a figure along a straight “dine and haveit trace out anothers trai ght line,

Western scholars (Wallis and Lagrange) wond ered j f the existenceofsimilar figures of different sizes was not a fu nda mentalfact about the universe worthy of being assumed,



The breakthrough came atthestart of the nineteenth century, when,for reasonsthatarestill only imperfectly understood, people beganto believe thata different geometry mightbe possible, one in which the angle sumsof triangles were alwaysless than tworight angles, and in which twolines that nevermeet cross a third and make angles chat sum to less than two right angles. ‘The two men who published accountsof the new geometry were Nicolai Ivanovich Lobachevskii in Russia and Janos Bolyaiin Hungary. Both men describedtheir resultsin strikingly similar terms, although they worked completely independently, and both men met with such dismal receptionsthat they died believing they had failed. Both men could have had much more support from Carl Friedrich Gauss, who wasin a position to influence

others and understood and accepted their work, but that too was insufficiently given. . Bolyai and Lobachevskii took the view that it was necessary to describe a three-dimensional geometryif it was to have any chance

of convincing people that it could bea logically valid alternative description ofphysical space, a status Euclidean geometry had attained after the work of Newton. Both took the same novel definition ofthe parallel line to a given line in a given plane and passing through a given pointin the plane: it is the line that separatesall the lines that meetthe given line fromall the lines

that never meetthe givenline. There are always twosuchlines, one in each direction, and these twoparallel lines get arbitrarily close

to the given line but never meet it. Both men found a surfacein their three-dimensional space upon which Euclidean geometry wastrue,just as in Euclidean geometry we have spheres on which spherical geometryis true. Both were able to play off these various geometries until they could express their findings in the language of trigonometry. Both concluded tharit was henceforth an empirical question as to whether geometry wascorrectly described by

Euclidean or non-Euclidean geometry.



Manyreasons have been put forward to expl ain why the new geometry was not immediately accepted, amo ng thema supposed stranglehold of Kantian philosophy, andit is almost certainly the case that Kant believed that space had to be conside red as Euclid’s Elem ents presented it. But the circumstances

of the original publications, the lack of suppor t from Gauss, and

the profound challenge it presented to cent

uries ofbelief and

successful practice were also factors, Wh at changed opinions was the posthumous discover y of Gauss’s sympathy for the new work and, paradoxically, the disco very by Riemannin Germany and Beltrami in Italy tha t Gauss’s ownideas of geometry could be deepened to provide a much more rigorous and persuasive account than Bolyai an d Lobachev skii had themselves been able to present. Thecrucial idea that these me n seized on indepe

ndently was that a geometry is a systematic acc ountof such conceptsas length s andan

gles, straight and curved lines, and lines

of shor

test length between their points, Gauss had showed that these questions can be posed abou

ta surface without regard for any thre e-dimensional space in which it may exist. Rie mann and Beltrami, in different ways, showedthat this approach could be extended to geometries in higher dimensions. In Particular, the non-Euclidean geometries of Bolyai and Lobachevskii cou ld b € given coherent accounts in this framework, and emerged as ge ometries in spaces ofwhatis called constant negative curvat ure, What Riemann,Beltrami, and after them Poincaré did was to describe a geometryby giving amapof it, much as mapsof the earth are depicted in an atlas.

Theconsistency of that description was enoughto establish existence of the new geometry,


It is often, and correctly, remarked that the emer gence ofa new geometry, as plausible as Euclid’s but different in its theorems,

spelled the end for the idea that Euclidean geometry wasthe only true geometry and thatit could be knowna priori. Henceforth,any



claim about the nature of space would have to contain a modicum of empiricalfindings.It is also true that the new geometry marked the end, for a time,of the idea of geometry as an axiomatic subject, and Euclid’s Elements was subject to a renewedlevel of

criticism that found many gapsandinconsistencies. However, by 1900 Hilbert, as well as several Italian mathematicians, had done muchtorestore axiomatic systems to geometry. In Hilbert’s case, consistency of these systems wasprovided by giving them arithmetic (coordinate) models and led to deeper questions about the consistency of arithmetic itself, Nonetheless, the framework for geometry provided by Gauss, Riemann,and their successors, knownas differential geometry because it grew out of sophisticated applicationsof the calculus to geometry, changed the way people regarded geometry.Fifty years later the most successful theory of physical space (or rather of space-time), Einstein’s general theory ofrelativity, was written precisely in the language ofdifferential

geometry that Riemann had done so much to promote. Einstein’s ideas show, of course, that we are no longerrestricted

to a straight choice between two accountsof the geometryof space. Norindeed,in the late nineteenth century, was there ever much doubt that of the two, Euclidean geometry would bethe useful one.

But the new geometry did dislodge the old onefromits unique place and open newquestions about whatit is for an account of space to be true. Riemann himselfwasvery clear that it had become the business of mathematics to propose many accounts ofgeometry, andby implication of other topics, to the scientific community, and

no longer could mathematics be regarded as a passive description of simple truths. Gradually, and not just because of these geometrical discoveries, the gap widened between true mathematics and proven mathematics, and the view prevailed thatall one could ask of a piece of mathematics was thatit be self-consistent. This was, for example, Hilbert’s view in his disputes with Frege.


What had begun as an inquiry into a possible weakness in

Euclid’s Elements became the source of two ofthe fundamental ideas of modern mathematics: that there are geometries of spaces other than the one imagined in ancient and even ever yday

geometry, and that many mathemati cal theories, not only in geometry butin algebra and analysis, can be fully axiomatized.




transcribes musical ideas from a musical space ofhis or her


George Hart


imagination to a score which embeds the music in the space of some particular instrumentation. In an anal.

ogous way, I transcribe ideas from a mathem atical space to scul pture in physical space. Figure R7.1 shows a wood sculpt ure, SuperFrabjous, which

originated from a relationship I notic ed between the compound offive tetrahedra and the orderl y tangle ofsix pentagons, These exist in ordinary Euclidean thre €-space, so it is not difficult to make a physical transcription. My art istic choices include the curves I chose, the wood material, and the use of color to distinguish planarsurfaces from darker edges, But Euclidean three-spaceis just one source ofinspiration,

Geometrical notions of space provide me, a mathematiciansculptor, with a frameofrJeference for embeddedobjects. I work

with both the physical space that my body and my sculpture inhabit and also the variousidealized spaces which are landscapes for the mathematical imagination. My work starts out as visualizations in one of many possible idealized spaces, perhaps a hyperbolic space, a projective space, or a four-dimensionalspace. Oneof my methodsis to choose objectsI feel are worthy of physical existence and transcribe them in some wayfrom a space " my mind’s eye to a piece of matter that can survive in the space o the physical world. . [use the word “transcribe” as a musician might, thinking here

of one of the many commonalities between mathematics and music. A musicalpiece originally written for one instrument may

be transcribed for another instrument or ensemble of instruments. A faithful transcription preserves someessential character of the original while adapting to the tonal and dynamic space of the new instrument. In a deepersense,I feel the original composer i by:: AndrewJaniak, i Oxfor ford George Hart, f MathemraticalSculptor’s Perspective onSpace. n: Space, Edited University Press (2020). © Oxford University Press, DOI: 10.1093/080/9780199914104.003.0014

FIGURE R71, SuperFrabjous, wood, 30 cm.





A majorrevolution in the history of mathematics developed from the nineteenth century discoveries of non-Euclidean and higher-dimensional geometries. Mathematics is no longer focused on quantitative and geometric properties of the physical world. Mathematicians nowfeel their job is to explore any set of assumptionsthey wish and develop their logical consequences into a consistentstructure.If the resulting systemofideasis rich, insightful, and elegant (according to a mathematical aesthetic

which onelearnsto appreciate in the process of becoming a mathematician), then the resulting structure is “good” mathematics.

Hyperbolic space is one of the notions which developed in this way and hasentered the repertoire of mathematical culture. Mysculpture Echinodermania, shownin Figure R7.2, derives from uniform tessellations in two-dimensional hyperbolic space, ie. the hyperbolic plane, which I transcribed into physical form. Simply put, hyperbolic space can be intuitively described as having .

“alot more room”than Euclidean space. Four squares exactlyfit together around a pointin the Euclidean plane,as oneinfers by lookingattiled floors orceilings. But in the hyperbolic plane, one can fit five squares around each point, or more. Muchmore. Ina

geometer’s imagination, this provides a rich canvasfor fantastic

FIGURE R7.2, Echinodermania, nylon, 3D printed, dye d, 11 cm,

Figure R73 shows a sculpt ure based on a four- dimensional object called the 120-cell. In threedimensio


con wlio vex polyhe indra, called rh the Platonic a solids. lideEuclid’s ee Elements

The Poincaré disk is a standard technique for mappingthe hyperbolic plane to an ordinary,flat circle. But as a sculptor, my goal in creating Echinodermania was to make something nonplanar, something fundamentally three dimensional. I ended up using a

ion of triangles in which the four classical elements and the universe are associated with the five solids. But in 1852, Ludwig Schlafli discovered that

nonlinear helical mapping of the Poincaré disk to create a 3D form

that I considervisually engaging as a sculpture. The pattern ofholes in Figure Rz.2 is subtle, but a mathematicaleye will see a regular

pattern to how theholes meet. Holes which are congruentin hyperbolic space appearas differentsizes in this transcription—an unavoidable consequenceofthe differences between hyperbolic and Euclidean space.

ene proof that there are only five, Plato used themin a proto-chemistry

theory based on conservat

in four-dimensional space there are six analogousobjects. Subsequent mathematicians have explored many properties ofstructures in higher-dimensional spaces. With Practice, onecan visualize

a great variety of beautiful higher-dimensional forms. Ofthe regular 4D polytopes, myfavorite is the 120-cell , comprising 120 regular dodecahedra, three around each edge. Transcribing the 120-cell to our physical three-dimensional space might be done in many ways.






FIGURE R73. 120-Cell (detail), nylon, 3D printed, dyed, 9 cm.

Forthe sculpture in Figure R73, I used a Schlegel projection of the

edges, adjusted the proportions by eye, and smoothed thesurface to give an organic sensibility.These artistic choices are just one possibility; I have revisited this form manytimes. Asa final example, Figure R7.4 shows Mermaid’s Delight. Oneaspect ofits internal coherenceis thatall of its surfaces are spherical. I designed it using a process called central inversion. This operation effectively turns space inside out, swappingpoints

near the origin with points near infinity. This technique is not commonly used by sculptors, but is natural to mathematicians accustomed to thinking aboutprojective space, in which a welldefined “pointat infinity”is available to be swapped with the

origin. Again, the mathematician’s comfortable habitation of a variety of abstract spaces provides a foundationfor transcribing conceptual objects into physical forms. A separate aspect of my work process is the struggle between coordinate-based and coordinate-free spaces. The space in Euclid’s Elements has no origin and nopreferred axial directions. An object in Euclidean spaceis not at any specifiable place in that space, and

FIGURE R7.4. Mermaid’s Del ight, color 3D printe d, 20 cm,

it has no orientationrelat ive to emptyspace. Th is is an elegant way to think about objects in space because it minimi ze Contrast this with Descar tes’s notion of a coordi te-based space. na Each point has coordina tes which placeit at a specific location in Cartes ian space. The axes dete rmine speci al directions; e.g., a cube randomlyplacedin Cartes ian space is mo re difficult to visualize than a cubealigned with th e axes. That m ental baggageis not

present when one visualiz es a cube in Euc lidean space.I generally think of my sculpture as liv ing in anisotr.opic space with no origin and no preferred direction

s, but as a practical matter the methods

Tuse to represent objects in co mpu ter software are coordinatebased ased. So while computer softwareis an essential toolin my

work,it also imposes notions ofposition and orientation that are

ultimately artificial from the artistic perspective.



To be more precise, the idea of sculpture living in a space with nopreferred directions applies strongly to handheld sculpture that one mightturn about to explore fromall directions. But larger works are mounted or suspended in some permanent manner, which meansthey inhabit a space with one special direction. Gravity defines a special up direction which the designer (and


often the viewer) must understandascentralto issues of tension,

compression,and support. One should walk aroundanylarge sculpture, because a successful work should be interesting from all sides, Ona scale of familiar human-size objects, such as sculptures

at the surface of the earth, the ambientspaceis neitherisotropic, like Euclidean space, norfully directionalized, like Cartesian space.I inhabit a special kind of sculptural space, with no standard

mathematical equivalent, in which gravity defines a special Z direction but X and Yarenot distinguishable. Mysculpting handsare prisoners of this physical space, while my

mindis free to explore an infinity of abstract spaces. Enriched by a long history of mathematical creations, I command aninventory of mathematical spaces in whichtovisualize lovely objects. Via sensitive transcription, I hope to create worthy sculpture in which ° anyonecan see somethingofthe beautyI see in the spaces of my mind’s eye.


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For the benefit ofdigit! al user s, indexe,d terms that Spa v0 n tr pages (e.g., 52—53) may, on occa appear on only one of those Page sion, s. $ Figures are indicated byJ . Ffol lowing the page number

absolute power of God, 1416

absolute reality of space andtim

absolute space

Anaximander, 21~24, 22n.22,

e, 285n.8

2.4.27, 2.4.28

astronomy in, 188-89

Kant and, 286n.9, 299 ~300n.27

atomists andnotion of

late medieval conceptions, 105-6 Leibniz’s criticism of, 263~68 , 288n.12

void, 28~31, 30n.39, 30n.42,


Newton's notion of, 260 ~65

carly cosmology, 21-24 Hesiod, 18-21

Abu-Lughod,L., 55

accommodation, 212-13, 219 -20

overview, 1-13 Plato, 32-38

Aguilén,F, de, 2101.28

ait PUMP, 244-47, 274-95

possible criteria for space, 14~15

Akmerkez shopping mall, Tur key, 58-59 Anaximander, 21-24, 22n.22,

spatial termsin, 15~18 Zeno, 25-28

24.27, 24,28

angels, place of, 149~34

ancient Greece, See also Aristotl e

anisotropy, 15n.4, 30N.41, 30-31 49




annihilation of matter within moon, 115-16,


space,religion, and, 52-54 - veiling and, 55-57

LIGHT, 122-25

convergence,triangle of, 208-10, 200f corpuscular theory oflight, 211-12

ant colonies, 99~103 Antoine-Laurent Lavoisier and Marie-Anne-

Bolyai, J., 308-9 Boyle, R., 244-47, 2451.20, 274-75

cosmology, in ancient Greece, 2I-24.

Pterrette Paulze (David), 277-79, 279f apeiron, 22-24, 290.26, 2.4.27 aphantastos, 89.89, 89

Bradwardine, T., 1111.6, 130.36,

Critique ofPure Reason (Kant)

151-§2, 156-58

Aristarchus, 188-89

Brahe, T., 270-71, 272f° brain, neuronalactivity in, 223-29, 2256

Aristotle. See also Physics (Aristotle)

Brunelleschi, E, 179-80, 180f, 180.9

anisotropy, 1511.45 30N.41, 30-31, 49

Buridan, J., 115-17, 135-39) 152

atomists, criticisin of, 30-31 conception ofplace, 109-16

Burnyeat, M., 44n.69

definition oftopos, 42-44

Cartesian space, 316~17

development of account, 40-42

Cartesian theory of motion criticism by Leibniz, 251-55

intelligible matter, 74.42, 74~75 mathematical space, 282n.4 natural places and universe, 48-49

criticism by Newton, 249-52 categorematic terms, 113-14

notionof topos, 38~40, 39N.55, 46n.72,

471.74, 48N.76 on optics and geometry, 188-89, 189n.4 overview, 38


Philoponus,criticism from,125-31 revised understanding, 45-48 spatial and temporal magnitudes, combining, 50-51 Zeno’s runner paradox, 26-28

Zeno’s fopos paradox, 25 astronomy in ancient Greece, 188-89 atomists, 28-31, 30.39, 30n.42,

31N.43) 31.44 auditory meter, 226-28, 22.7/°

Averlino, A., 177-79 Averroes, 111

Categories (Aristotle), 38-39 catoptrics, 189 centralinversion, 316 chasm, notion of, 18-19n.15, 18-21 Chaucer, G., 155

chemicallaboratories Chemical Revolution and, 277-79 in English Restoration context, 274-76

origin of, 270-71 semipublic, 273-74 chéra as place in society, 16n.10

Plato’s receptacle, featuresof, 34~37 Plato’s receptacle, functions of 32~34 Plato's receptacle, time distinguished from, 38

Bacon, FE, 273-74

topos and, 16-17 city space,veiling and, 57-61

Basilica of San Lorenzoin Florence,Italy,

Clarke, S., 234-35, 263-67, 264.39, 266n.42

179-80, 180f

Cleomedes, 195.16

Critique ofPractical Reason (Kant), 303 analysis of Newtonianview, 284-87 critique of Leibnizian-Wolffian philosophy, 287-91, 2881.13 differences with Leibnizian conception of space, 295-96 form and matter of intuition, 282n.4

infinite divisibilicy of matter in Space, 288~91 space as intuition, 281~89

cylindrical helix, 67~70, 68n.18 d'Alembert,J. le R,, 277-79. 278f Darwin, C,, 98-99

David, J-L., 277~79, 270f Davies, C., 135.43 da Vinci, L., 176 De Causa Dei (Bradwardine ), Igi=5§2 De Gravitatione (Newton),

235~36, 2.48,

rejection of substance/ property framework, 257~59 Democritus, 29~30, 31 De Risi, V,, 2.32N.3,,.290,


Descartes, R.

laws of motion, 236-40 Leibniz’s criticism of, 251~55 Newton's criticism of, 248-49 physiology ofvision, 206f 211~ 16, 2120.31, 2130.34, 21.41.35 Dialogus Physicus (Hobbes), 245 -49

Bellini, G., 180-81, 182f

grounded, 223-24

Beltrami, E., 309

hierarchy oftypes of, 82-83

dianoia, 70-73, 73n39 diastéma, 17, 42 Diderot, D., 277-79, 278f

Berkeley, G., 216-21 Beyazit Square, Turkey, 60-61 body cognitive processing oflanguagerelated

neuronalactivity, 223-29, 225/°

differential geometry, 309-10

practical, 2961.23, 3031.33, 30375 color perception, 196-97; 200

dimensionsofsoul, 82-83 dioptrics, 189

constant negative curvature, 309 convergence, 219-20

Dioptrics (Descartes), 211, 213, 214 distance, spatiunz used to describe, 163-66

Being, 28-29, 750.47

to, 228


distance perception Berkeley and, 218-20 Descartes and, 212-13

Keplerand, 208-10 divine omnipresence, 286-87, 287n.10, 287n.11, 303-5

Echinodermania (Hart), 314, 315 Einstein, A., 309-10

Elements (Euclid), 306-7, 30910, BIS—17

Spatiune in, 155~56, 1§6n.72 Elements ofPhilosophy, The (Hob bes), 240—

47, 241—42n02

viewof Leibniz’s idealism, 292— DS» 294N.20

249-50, 255


emanation, 82n.69, 259. 35, 259

embodied cognition, 223-24 empty space, 240-41, 241n41 Encyclopedie (Diderot and dAlember t),

277-79, 278f

Epicureans, 30~31 equilateral triangle, kinematic const ruction of, 68-69 Eratosthenes, 188~89

Erebos, 19,19, 19-20 eternity of God, 14548, 150, 2871.11, 287 Euclid, 306~7, 309~10, 31S-17 optics, 189~92, 192f Spatinmand,155-56, 156n.72 extramission theories, 189-97 cye, optical system of, 204-5, 206f. See

also optics false possibles, 161-62 Fatih district, Turkey, 59-6 0 Forms, 260.33, 32-34

frame ofreference, topos as, 45~48

Gaia, 18-19, 197.18 Galileo Galilei, 247-48 Gauss, C. F, 308, 309-10

geography, 52-54 Geometria Speculativa (Bradwardine), 156-58 geometry : geometrical proofs, 67 Kant and scienceof, 282-84 kinematic constructions, 67-70



geometry (cont.) non-Euclidean, 306-11, 314 plane,67, 76

ofsize perception, 184-86, 211-16 ofspatial perception, 221-22 useofspatitenin, 155-58

geometry ofsight Berkeley's rejection of, 216-21 Ibn al-Haycham and,198-204, 199f Kepler and, 207 global megacolonyofants, 101-2 God

absolute powerof, 114-16 annihilation of matter within moon, 115— 16, WIGHT, 122-25


Hesiod, 18-21

Islam, veiling in, 55-61

Hobbes, T., 2.41-4211.12, 2-44.17

Istanbul, Turkey, veiling in, 57~61

analysis of motionby, 240-47 Boyle’s criticismof, 245-47 emptyspace, 241.11, 2.43/° rejection of possibility ofvacuum,

142-44. 243f

hyperbolic space, 314 hypodoché, 71.31, 71 Iamblichus, 80n.64, 85 Ibn al-Haytham, 198-204, 199f, 199— 2000.21, 2120.31

eternity of, 145-48, 150, 287n.11, 287

imaginary space, 15051, 153-54, 240~42

in infinite space, discussion by

imagination analogies for, 75-78

Bradwardine, 150-52

Kant’s pure practical reason, 303~5 omnipresence of, 286-87, 287n.10,

discursive thought, projection, and, 70-73

infinite space and, 83~84

placementofearth in space, 266n.42, 266

intelligible macter and dimensionality, 73-83

place of angels, 149-54

kinematic constructions, 67-70

presence outside cosmos, 142-48 vacuumscreatedby, discussion by Buridan, 135~39 vacuumscreatedby, discussion by

overview, 63-66

2870.11, 303-5

Oresine, 132~-36

vehicle of, 84-86

imprints in wax, imagination likened

to, 77-78 Inaugural Dissertation (Kant), 294n.19, 302

Grant, E., 107

inertia, principle Of, 233-345 235-37) 2360.7

groundedcognition, 223~24.

infinite circle, 158-59

infinite divisibilicy ofmatter in Harari, O., 68-69 Hart, G., 312-18

120-Cell, 315-16, 316f Echinodermania, 314, 315f Mermaid’s Delight, 316, 317°

sculptural space, 316~18 SuperFrabjous, 313f, 313 headscarves - body and, 55-57 city space and, 57-61 heavens, motionof, 170 helix constructions, 67-70, 68n.18

Helmig, C,, 73n.39 hemicircle, 7.4.43

Koyré, A,, 105-7, 230-31

Italian Renaissanceart, use of perspect ive in, 176-83

latitude, concept of, 166-67

Jones, J. P, 53-54.

space, 288-92 infinite space, 83-84

Oresme and, 150-52 shift in thinking in seventeenth century, 2310.2, 231-32 intelligible matter, 72-83, 74N.42 intromission theory, 198-204, 199f° intuition

distinction between form and matter of, 282.4 divine omnipresence, 287n.10, 2871.11

schematism, 29.4n.19, 296 " science ofgeometry, 282-84 space as, 281-82

Kant, I,

absolutereality of space andt ime, 285n.8 absolute space, 299-300n .27 analysis of Newtonian view , 284-87 appearances, 289, 295-96

community ofsubstances, 300n.28 » 300-1 critique of Leibnizian-Wolff ian philosophy, 287-94, 288n.13

differences with Leibni SPace, 295-96

divine omnipresence, 287.11, 303~5

zian conception of



human sensibility, 306n1

Inaugural Dissertation

infinite divisibility of Space, 288~92

, 294.19

n hatter in

matter and material sub

stance, 297-

99, 298n.24, 298n.25, 300N.28, 300-1, 30In,29 momentum, 298n.2 5,


monadology, criticism

of, 287~91,

296n.22, 297-~98, 298n.24.

neglected alternative

objection, 283~84 and Newtonian phy sics, 299.26, 299 -301

practical cognition,

3031.33, 303-5


schematism, 29.411.19, 296 science of geometry, 282 -84

Spaceas intuition, 281 -82 transcendental ideali sm, 280-81, 299~305 universal Bravitation, 303-4. viewof Leibniz’s ideali sm, 292~ DSs

29.4N.20 kenon, 15~16, 17

Kepler, J, 20410, 206f, 209 0.27

kinematic constructions, 67~ 70 hindsis, 4.4

Kuhn, T,, 270n1 Landino, C,, 177 Langton, R., 302-3n.32

Jammer, M., 30.42, 2824

Holloway,J., 5-4 Hooke, R., 244-46


Lavoisier, A.-L., 277~79, 279f Lavoisier, M.-A, P., 277-79 laws of motion, 235-40, 2.47~51, 300-1 Leibniz, G. W,, 233-35

absolute space, 2881.12

Cartesian theory of motion, criticism

of, 251—55 conception ofspace, 252-54 correspondence with Clarke, 263~67, 264n.39, 266n.42

Kant’s viewof, 292-96, 294N.20

Newton's absolute space, criticism of, 263~65 points in common with Newton, 251-52

substance/property framework, 256-57

Leibnizian-Wolffian philosophy, 288n.13, 290-94, 296N.22

Leucippus, 29-30, 31 Libavius, A., 270-91 lighe corpuscular theory of, 211-12 nature of, 921.103, 92-94, 93.111

refraction of, 207

linear Perspective in Renaissance art and architecture, 176-83 lines, 160-61

lines of the ray, 198-99, 199f, 199~200N.21 Livre du ciel et du monde (Oresme), 170 place of angels, 149-54 presence of Godoutside cosmos, 142-48 Lobachevskii, N. L, 308-9 local motion, 163-64 locus, 109-10, 19-20 logica moderna, 1131-4, 115, 153-544

logos, 70.28, 86, 88-89

luminous compactness, 192-93 Manetti, A, r80n.9 Marston, S., 53-54




Massey, D,, 52-33

mathematical sculpture, 312-18 a0-Cell, 315-16, 316f Echinodermania, 314,315

Mermaid’s Delight, 316, 317f sculptural space, 316-18 SuperFrabjous, 313f, 313 mathematics. See also geometry separation between natural philosophy and, 111 use ofspatium, 155-59 use ofspatinm, in Oresme’s commentaries, 160-62

use ofspatium, in Oresme’s treatises, 163-70

matter annihilation of, by God, 115-16, HIGH, 122—25

infinite divisibility in space, 288-92 intelligible, 74n.42 Kant’s conception of, 297-99, 298n.2.4, 298n.25

Melissus, 28-29

Mermaid’s Delight (Hart), 316, 317 Metaphysical Foundations (Kant), 298Nn.25, 299

Metaphysics (Aristotle), 30-31; 74 meter codes, 226~28, 227f

Middle Ages Aristotle’s conceptofplace, modifications of, 11016 overview, 104-9

use ofspatium, 154-59 use ofspatium, in Oresme’s commentaries, 160-62

use of spatium, in Oresme’s

true motion, 250-51, 261


ofangels, discussion by Oresme, 149-50 Aristotle and, combining space


time, 50-51

Aristotle and, notion oftopos, 38-42 Descartes and, 236~37 170 of heavens, discussion by Oresme, laws of, 235-40 24775 30071

theories of, in seventeenth century,

233-34) 235-37

moonillusion, 196=97, 197.18

nominalistic notionsof place, 11213 Nonbeing, 28-29

commentaries, 160-62

use ofspatium, in treatises, 163-70

non-Euclidean geometry, 306~11, 314.

Ouranus, 19-20, 21

numbers, void between, 2.4

Oxford Calculatores, 165-66, 167~68

287N.11, 303~5

(Boyle), 244-46

New ‘Theory ofVision (NTV) (Berkeley), 216, 217-19

Newton,L., 233-35 absolute space, 260-65

Cartesian theory of motion,criticism

of, 249-51

50, De Gravitatione, 235-36, 248, 249255) 257-59

distinction betweenspace and body, 249-50 divine omnipresence, 286-87, 287.11 Kant‘and, 284-87, 299n.26, 299-301

laws of motion,235-36, 2.47—$l 300—1 Leibniz-Clarke correspondence, 263-67,

points in common with

296N.22, 297-98, 298n.24

syncategoreimatic use ofplace, 14n.10 use ofspatinm, in

nominalistic mathematicians, 173

120-Cell (Hart), 315-16, 316f On the Causes ofMiracles (Oresme), 140-41 On the Configurations ofQualities and Motions (Oresme), 163~6.4, 165~6 6 Onthe Proportions of.Proportions (Oresine), 164

moderni, 111, 1.9-30, 167-68, 172~73, See

monadology,criticism by Kantof, 287-91,

Nikulin, D., 73n.39

nativism, 221-22

26.41.39, 266.42 ontologicalstatus of space, 257-60

Molyneux, W, 218~19n.39

corruptione, 166-67 Questions on Euclia’s Geometry, 158~5 9

Nicholson, M., 231n.2

obligations, in medieval education, 16 9-70 Ockham, W, of, 108—9, 112-14, 119 Odo,G.of, 123-24. omnipresence, divine, 286-84, 287n,10,

treatises, 163-70 mirror analogy for imagination, 75-77

also Oresme, N.

xuaestiones super De Generatione et

use ofchemistry, 274~76

Zeno’s paradoxes, 25 224~28 motor processing, neural basis of 142748 of, discussion multiple worlds, Muslim women, veiling of, 55-61 natural philosophy, separation between mathematics and, 111 natural places, topes and, 48-49 neuroscience, 223-29, 225/ New Experiments Physico -Mechanicall

Leibniz, 251-52 Principia mathematica, 247-48, 259-60

structure of space, 261-65 substance/property framework, rejection

of, 255, 257-59

optics, 188-89

Berkeley and, 216~21 Descartes and, 211~16 Euclid and, 189-92, 192f geometry ofsight, 198-204, 199f, 216-21 geometry of size perception, 211-16

geometry ofspatial perception, 221-22 Tbnal-Haythamand, 198—204 Kepler and, 204-10 Ptolemy and, 192-97, 194, 196f

visual pyramid, 204—16 Optics: Paralipomena to Witelg and Optical Part ofAstronomy (Kepler), 20.4~5, 26 Oresme, N.. See also Questi ones super



Physicam (Orasme) On the Causes ofMiracles, 140-4 1 Onthe Configurations ofQualities and Motions, 163~64, 165-66

ondifferent senses of impossible . 132-36 overview, 107-9 place of angels, 149~54 presence of God outside cosmos, 142-48

Onthe Proportions ofProportions, 164

Panofsky,E., 176-77

parallel postulate, 191-92, 192f, 204, 233-34, 306-7 Parasole, L., 181-83, 183f “Parliamentof Fowls, The” (Chaucer), 155 Parmenides, 24

Perspectiva (Vredeman de Vries), 178f Perspective as Symbolic Form

(Panofsky), 176-77

perspective in Renaissance art and architecture, 176-83 petitio principii, 129-30

Phantasia discursive thought, imagination, and Projection, 70-73, 73n.39 infinite space and, 83-84

intelligible matter and dimensionality, 73-83 kinematic constructions, 67-70 Overview, 63-66 vehicle of imagination, 84-86 phenomenology, 189~90n.7

Berkeley and, 216~21 Descartes and, 211-16 Euclid and, 189-92, 192° Tbnal-Haytham and, 198-204 Kepler and, 204~10 , Prolemyand, 192—97, 193.13, 194f, 196f° Philolaos, 24n.29 Philoponus, J., 125-31, 1281.34

Physics (Aristotle), 25. See also Questiones super Physicam(Orasme) conception ofplace, 109~10 optics and geometry, 188-89, 189n.4




Physics (Aristotle) (cont.) toposin, defined, 42~48 toposin, tasks of, 38-42

overview, 63-66

pictorial perspective, 176-77

Lypot, 77S 4 vehicle of imagination, 84-86

place. See also Questiones super Physicam (Oresme) , Aristotle's conception of, 109-10, 125—31 kinds of, 80+82

medieval modifications ofconcept, 110-16 Oresme’s conceptionof, place of angels, 149-54

Oresme’s conception of, presence of God

receptacle and, 710.31 space, 91-95

World Soul projection, 88-91 projection in imagination, 78-80 propositions, analysis of truth of, 113-14

proton, 43-445 45 Prussian Army analysis example, 256-57 Prolemy, C., 192-97 194f, 195-96n.17, 196f

Pythagoreans, 24

outside cosmos, 142-48

Philoponus’s conception of, 125-31 syncategorematic useof, 1148.10 plane geometry, 67, 76

Plato. See also Timaeus (Plato) receptacle,32 topos paradox, 26 use ofchéra, 16n.10 World Soul, 87

Platonic solids, 315-16

Quaestiones super De Generatione et corruptione (Oreste), 166-67 Questiones super Physicam (Orasme), 16~25,

139~42,159 answersto, 119-25 existence ofvacuuuns, 132-36 listofuz-i9 Questions on Euclid’s Geometry (Oresme), 158-59

Plotinus, 75 radiallines, 198-99, 199f 199-200n.21

pneumatic vehicle, 84-86

real space, 240-41

Poincaré disk, 314

receptacle, 341.49, 3411.50, 3711.52

points, 160-61

features of, 34-37

Porphyry, 85

functions of, 32-34 Proclus and, 71n.31

Principia mathematica (Newton), 247-48, 259-60

Principia philosophiae (Descartes), 236-37, 238-39 probolé, 78~80

Proclus, 63-97 Being, 75n.47

cognition hierarchy, 82-83 discursive thought, imagination, and projection, 70-73 -

Rohault,J., 216.36 runner paradox, 26-28, 27.35

Sacrobosco,J. de, 160-62 San Giobbealtarpiece (Bellini), 18081, 182 f° schematisin, 294N,19, 296 Schlafli, L., 315-16

science of geometry, a priori statu s of, 282-84 SDIH.See size-distance inva riance hypothesis Secor, A., 57~58

sensibility (human), 306n1 Sensory processing, neural

Proclus’s imagination and, 71

timedistinguished from, 38 rectilinearity of vision, 184-85

refraction oflight, 207 relative space, 260-65 religion, See also God city space, veiling and, 57—61 space, body, and, 52-54

basis of, 224-28

seventeenth century, 236 n.7. See also Leibniz G. W; Newton, L

Descartes, analysis of mot

ion by, 236-40 Hobbes, analysis of mot ionby, 240-47 shift in thinking during , 230-35,

veiling and, 55-57

Religion within the Limits ofReason Alone (Kant), 303~4

impossible and infinite space, 83-84 kinematic constructions, 67-70

Renaissance art and architecture,linear

matter and dimensionality, 73-83 nature and World Soul, 86-88

retinal images, 204~$, 206f 207, 208,

perspectivein, 176-83 210, 216-17

theories of motion in, 235 =40 shape perception Berkeley and, 216-17, 220~21 Descartes and, 215

shopping malls, use ofve ils in, 58-59 " sight. See optics sight, geometry of Berkeley's rejection of, 216-21 Ibn al-Haytham and, 198 “204, 199f° Simplicius, 63-64, 91-92 size constancy, 200-1, 202, 204

size- distance invariance hyp

(SDI), 194-95

Kepler and, 210 Prolemy and, 192-97, 194f, 196f solid geometry, 1351.43, 135-36 spatial and temporal magnitudes, combining, 50-51 spatial aspects ofvisual perception, 188~89 Berkeley and, 216-21 Descartes and, 211~16 Euclid and, 189-92, 192f geometry of spatial perception, 221-22

Ibnal-Haythamand, 198-204, 199f Kepler and, 20.4~10 overview, 184-87

Prolemy and, 192-97, 194f, 19 6f

231N.2, 247—48

pneumatical engine, 244-47, 274-75

practical cognition, 296n.23, 3031.33, 303-5 Presocratics, 2.4

Riemann, B., 309-10 river example, in Physics, 45-48



Berkeley and, 220~21 Descartes and, 213 Ibn al-Haythamand, 201-3

size perception, geometry of, 184-86, 186~89n,2, 211~16 Berkeley and, 216-17 Descartes and, 2140.35, 21.4—15

Ibn al-Haytham and, 200-4

spatial language terms, cognitive processing of, 228 ,

spatial maps, 22.4~28, 227/"

spatial perception, geometry of, 221-22 spatial scales, 99-103 spatiotemporal schemata, 296 Spatium, 104—5, 140—42

evolution in meaning of, 154-59) 1561.72 in Oresme’s commentaries, 160-62 in Oresme’s treatises, 163-70 Sphere (Sacrobosco), 160-62 straight-line vision, 184-85 structure of space Leibniz's criticism of Newton, 263-68 Newton's notion of, 261~65, 267 substance, Kant’s conception of, 297-99

substance/property framework, 255-57, 267~68 substances, 112 SuperFrabjous (Hart), 3136, 33 Syila, E., z90n14

syncategorematic terms, 113-14) 11.41.10, 167+68

Syrianus, 80-82, 89n.89, 94-95: 95117 Tartarus, 20-21 Thales, 21-23

Theogony (Hesiod), 18-21

Theory ofVision Vindicated andExplained (TVV) (Berkeley), 217-18





351 .

Timaens (Plato), 49n.80

universal gravitation, 299-301, 303-4

receptacle in, 231.23, 32-38

universe, topos and, 48~49

topos paradoxin, 26 use ofchérain, 161.10 World Soulin, 87

vacuums Boyle’s pneumatical engine, 244-47


| | |

Philoponus and, 125—31

description of, 86—88 projecrio, , we

VredemandeVries, J., 178f


Pythagoreannotion of, 24

pace water, as basic principle, 22-23

conception by Philoponus, 127

Wiens, J., 99

combining space and, 50-51

creation by God,115-16

Woodward, K,, 33-54

spacedistinguished from, in runner

discussion of impossibility by

World Soul, 87n.84, 88n.86

Oresme, 132-36

paradox, 26-28

questions on Physics, by Oresme, 12225 rejection by Hobbes, 242~44, 2.43

spacedistinguished from,in Timaeus, 32-38

essential features of, 40n.61

natural places and universe, 48~49 revised understanding of, 45-48 Zeno’s paradox, 25-26 transcendentalidealism, 299-305, 302n.31, 3023.32 absolutereality of space and time, 285n.8 _ differences with Leibnizian conception of space, 295-96

infinite divisibility of matter in space, 288-92

Leibniz’s idealism versus, 292-95 a priori status ofscience of

geometry, 282~84 space as intuition, 281-82 theses of, 280~81


Berkeley and, 217-19


Descartes and, 212-13 Ibnal-Haytham and, 198-204, 199f

visual space, 188-97 Berkeley and, 216-21

Descartes and, 211-16 Euclid and, 189~92, 192f° geometry ofsight, 198-204, 199f, 216-21

Ibn al-Haytham and, 198-204, 199f° Kepler and, 204-10

Tuan, Y.-F, 54



Treatise on the Commensurability or Incommensurability ofthe Motionsofthe

true motion, 250~51, 261 truth ofpropositions, analysis of, 113-14

funner paradox, 26, 271.35

topos paradox,25

vestibule, 7011.28 visual angle,size perception and Descartes and,214 Ibnal-Haytham and, 200-3 Kepler and, 208 visual maps, 224-28, 227/° visual pyramid, 204-10

geometry ofsize perception, 211-16 geometry ofspatial perception, 221-22

Heaven (Oresme), 16.4~65

eno, 25-28 metrical paradox, 289~go, 290n14

veiling, 52-62 body and, 55~57 city space and, 57-61

Treatise on Architecture (Averlino), 177-79 Treatise on Man (Descartes), 211, 212f°

triangle of convergence, 208-10, 209f


vehicle of imagination, 84-86

topos, 38-40, 390.55, 460.72, 471.74, 48.76

chéra and, 16-17 definition of, 42-44 developmentofaccount, 40-42

hic f. t ea as vehicle of, 91-95

overview, 184-87 Prolemy and,192-97, 194f; 196f

void, 28-31 atomists and,28~31, 30n..42 to kenon, 15-16

typol, 770.54, 77-78

Oresme and,as place of angels, 149~54

ultima lectura (Buridan), 116-17

Oresmeand, presence of God outside cosmos, 142-48

unicolonial ant populations, 1o0~1

Philolaos and, 24n.29


ae i attechee. to philosophy’s Oxford Philosophical Concepts offers an exciting ene past and its relation to otherdisciplines. Each vo sie ei ae international scholars to excavate the sources of ne ps ee aTele andexplore their histories. OPC aims to provoke 4 See, rete AtieecoutitanellemaneOe TIT al ca be ee

past and to makethatpastavailable to anyoneinteres 3

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ECURRENT questions about space have ee aoaians

ea times. Can an ordinary person Sia < icne cede to say what space is? Can geometry characterize t sat volume chronicles

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ae early antiquity

the development ofphilosophical conceptions of spWea eda central figthrough the medieval period to the early modern a ophy are discussed, ures from

the history of mathematics, science, and p eke tsicole Oresme, including Euclid, Plato, Aristotle, Proclus, Ibn al-Hayt ah other books Kepler, Descartes, Newton, Leibniz, Berkeley, and Kant. L ae ee

in theseries, shorter essays enrich the volume by a aie cates space foundin various disciplines including ecology, ie em oe neuroscience, cultural geography, art history, and the history o



is Professor and Chair of Philosophy at Duke University.


Coverdesign: Rachel Perk ins