Being and Number in Heidegger’s Thought 9781472546166, 9781847060303

Being and Number in Heidegger’s Thought examines the relationship between mathematics and ontology in Heidegger’s though

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Being and Number in Heidegger’s Thought
 9781472546166, 9781847060303

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Acknowledgements

This book began as a dissertation written at the Hebrew University of Jerusalem under the direction of Gabriel Motzkin. Motzkin's advice was always insightful, but more importantly, he suggested new ways of looking at the problems I was tackling, encouraging me to expand my intellectual horizons. Carl Posy also accompanied the project from its inception, and I have bene¢ted immeasurably from his wisdom and philosophical profundity. Other colleagues who contributed in one way or another to my understanding of the subject include (in alphabetical order) Ori Dasberg, Eli Friedlander, Avraham Mansbach, Christoph Schmidt and Dror Yinon. During a stay at the Archives Husserl at the E¨cole Normale Supe¨rieure, I gained further insight from discussions with Jean Franc°ois Courtine, Franc°oise Dastur, Jean Toussaint Desanti, Claude Imbert and Jean-Michel Salanskis. The manuscript was written in Hebrew, and translated into English by Nessa Olshansky-Ashtar, who not only did a superb job of the translation itself, but also helped considerably in clarifying my ideas. While working on the book, I received ¢nancial support from the following sources: the Edelstein Center for the History and Philosophy of Science, the Lady Davis Fellowship Trust, the SFB (Sonder Forschung Bereich) research programme, and the Israel Science Foundation (grant 858/03). I am grateful to all these institutions for their assistance. An early version of parts of Chapter 2 appeared in Angelaki 10 (2005), 181^6. My thanks to Routledge for permission to reproduce this material. I would like to close by thanking my family. My sister Daniella encouraged me through more than one creative impasse. My wife

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Acknowledgements

Sharon's un£agging support and con¢dence in my ability to bring this project to fruition sustained me throughout. Finally, I would like to thank my children David, Sarah and Ariel, who are a source of both energy and inspiration. I dedicate this book to them.

Introduction

The following admonition was inscribed above the entrance to Plato's Academy: `Let no one ignorant of geometry enter here.' The history of western philosophy can be neatly circumscribed by classifying philosophers into those who share this outlook, and those who reject it, denying that geometry and mathematics are the epitome of rational understanding. Martin Heidegger unquestionably falls into the latter category. Throughout his writings, Heidegger seeks to sever the connection between mathematics and the fundamental questions of philosophy. But it is important to distinguish two very di¡erent kinds of rejection: some philosophers, for example, Nietzsche and Kierkegaard, see mathematics as marginal, as irrelevant to the questions that engage them. For others, however ^ Bergson, for instance ^ the repudiation of Plato's admonition shapes their philosophies in a very basic way. Now Heidegger is generally classed with the former philosophers, mathematics being thought to play but a marginal role in the development of his thought. Indeed, most Heidegger scholars pay no attention whatsoever to Heidegger's understanding of mathematics. The main argument I will be putting forward in this book, however, is that at some of its most central junctures, Heidegger's thought crystallizes around the distinction between ontology and mathematics. To expose this aspect of his thought, a new method of reading Heidegger's works is called for. It is not possible to start from a preunderstanding of ontology, and on that basis evaluate the place of mathematics, for inasmuch as this pre-understanding will be formulated without regard to mathematics, the centrality of mathematics is certain to be overlooked. Another pitfall to be avoided is the tendency to rely on a particular conception of mathematics as the model in terms of which Heidegger's stance is to be evaluated. If it is assumed

2

Being and Number in Heidegger's Thought

that mathematics is ¢rst and foremost a science akin to the natural sciences, the attempt to grasp the nature of the relationship between Heidegger's thought and mathematics will focus on his discussions of science, and the link between mathematics and the most basic strata of ontology will be obscured. Hence, the approach I have adopted in this study seeks to reveal the mutual unfolding of the two spheres. This task is made easier by the fact that the reciprocal formative impact of ontology and mathematics can be detected early in Heidegger's thought: in his writings from 1912 to 1916. These works include his doctoral dissertation, his qualifying dissertation, or Habilitationsschrift, and a number of short articles.1 The ¢rst chapter examines these early writings. The focal point for exploring the ontology^mathematics nexus is the distinction between the transcendental one, which characterizes all that exists, and one as a number, which is a particular thing. Analysis of this distinction and its implications brings us face to face with yet another distinction, a distinction Heidegger sees as the central link between Being (Sein) and mathematics: the distinction between general ontology and particular realms of reality (Seiende). Heidegger's thought undergoes a radical shift between the early writings and Being and Time (1927). The discussion of ontology ceases to be a search for the most general characteristics of entities, and becomes a quest for the meaning of Being (Sinn von Sein). This transformation constitutes a radicalization of the di¡erentiation of distinct ontological regions, since it countenances variation, not only between di¡erent kinds of entities, but also in the meaning of Being itself. Consequently, on the approach taken in Being and Time, it is no longer possible to identify universal characteristics of entities. This change in Heidegger's thought makes possible an entirely new conception of ontology. Heidegger moves from the question `what is Being?' to that of `how is Being given?'. Since, for Heidegger, meanings are always relations, this transition transforms the ontological inquiry into an investigation of the relation between the meaning of the question `how is Being given?' and the party inquiring into that meaning: Dasein. The change has other aspects as well. For example, it reveals that the meaning of Being is an outcome of a hermeneutic

Introduction

3

process, and that the analysis of the di¡erent meanings of Being shows them to be di¡erent interpretations of time. Against the background of these developments, during this period there is also a change with respect to the reciprocal characterization of ontological regions and the framework that makes this di¡erentiation possible. Alongside the ontological regions of Being and Time, such as nature and history, we ¢nd modes of interpretation of Being, such as `presence-at-hand' (Vorhandenheit) and `readiness-to-hand' (Zuhandenheit). Universal ontology is replaced by fundamental ontology, the condition grounding all interpretations of Being. No longer a universal framework, this condition now becomes Dasein, the ontological region that grounds all the other modes of interpreting Being. In Being and Time we encounter numbers at the juncture of fundamental ontology and the di¡erent interpretations of Being, the juncture superseding that at which general ontology and ontological regions formerly converged. The emergence of the interdependence of fundamental ontology and the modes of interpretation of Being is profoundly connected to number. The second chapter of my study examines this interdependence and the role played by number in its elaboration in Being and Time. Around 1930, Heidegger abandons his plan to answer the question of the meaning of Being in terms of a fundamental ontology rooted in the privileged status of Dasein. But the question nonetheless continues to engage him. Throughout the decade, he approaches it in terms of the history of Being. Being is revealed di¡erently in di¡erent epochs. Historical epochs thus replace the di¡erential interpretations of Being in Being and Time, assuming the role previously played by ontological regions. In What is a Thing?, a series of lectures delivered in 1935, Heidegger argues that the modern period, the essence and implications of which were made manifest by Descartes, is a mathematical epoch, that is, an epoch in which the existence of entities is interpreted mathematically. In the third chapter of my study, I o¡er an interpretation of What is a Thing? and of the connection between ontology and mathematics on which it rests. Once again, Heidegger is addressing the problems of both general ontology and regional ontology, but now his approach is very di¡erent. Basing himself on an interpretation

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Being and Number in Heidegger's Thought

of Kant, Heidegger attempts to make the argument that the possibility of a universal/mathematical ontology hinges on its being applicable to a single region, the region of possible experience. At this point, we may well wonder why it is that Heidegger injects mathematics into the whole issue of the reciprocal characterization of universal ontology and ontological regions. Mathematics has always been a framework that lends itself to the development of an ontology, but the connection between mathematics and the question of the relation between universal ontology and ontological regions emerges in the wake of the resurgence of the ontological debate at the close of the nineteenth century and the beginning of the twentieth. In the post-Kantian era, it had seemed as if the general ontological question of the being of entities had been eliminated from the philosophical agenda. The restriction of the ontological question to the realm of possible experience, as set out in the Critique of Pure Reason, shattered the very basis for this question.2 The return of the ontological discussion to the philosophical spotlight is, I believe, directly related to developments in mathematics. First, one response to the critique of psychologism in mathematics and logic voiced by Gottlob Frege, and later, Edmund Husserl, was the unambiguous separation of mathematical entities from entities of other sorts, such as mental entities and physical entities. Such separation upsets the Kantian framework, within which the same categories apply to all objects. Complete separation between di¡erent kinds of entities leads to a separation of categories: the disparate types of entities are grasped by means of equally disparate categories. In his Logical Investigations, Husserl moves in this direction. Second, developments in mathematics, especially the emergence of set theory, enabled the formulation of a general ontology based on the concept of being in general, being that does not fall into any particular region or category. Mathematics thus provided a potentially useful framework for addressing the question of being in general, a question that could not be overlooked now that entities were construed as so entirely distinct from each other. Husserl, Heidegger's teacher and one of the greatest in£uences on the crystallization of his thought, adopted this approach, seeking to create a formal ontology that encompassed the mathesis universalis.3 Not only is the mathesis universalis

Introduction

5

a central element in the quest for the most general characteristics of being ^ being that has no speci¢c content ^ but it is a framework in terms of which a general theory of multiplicity can be constructed. Such a theory would cover all possible multiplicities. Given that ontological regions are themselves multiplicities of entities with a shared essence, the mathesis universalis has the potential to apply to every conceivable ontological region. Hence, mathematics plays a central role both in the emergence of completely discrete ontological regions, and in suggesting a general characterization of Being and of the ontological regions. The primary question engaging Heidegger was that of the meaning of Being. Given the many meanings of Being, which meaning is most basic? Heidegger reports that this question began to haunt him as early as his teenage years, when he read Franz Brentano's On the Several Senses of Being in Aristotle.4 As we noted, during the period in question, mathematics provides a framework in terms of which this question can be answered, and indeed Husserl, whose Logical Investigations greatly in£uenced the direction Heidegger took in dealing with the question of Being, chose this course.5 Hence Heidegger, who embraces the notion of ontological regions completely distinct from one another, again and again returns, in grappling with the problem of the relation between ontological regions and general ontology, to the relation between ontology and mathematics. This study, then, by analyzing the connection between mathematics and ontology in Heidegger's thought, furthers and refocuses our understanding of a number of seminal interpretive questions, each chapter suggesting perspectives for new exegesis of central elements in Heidegger's philosophy. Let me recapitulate. The ¢rst chapter presents a comprehensive characterization of the young Heidegger's ontology. This characterization allows us to situate the young Heidegger relative to various intellectual in£uences with far greater precision than would be possible were the role of mathematics left unexamined. We can better situate him vis-a©-vis neo-Kantian thinkers such as Heinrich Rickert and Paul Natorp, and we are a¡orded a heightened understanding of the relationship between his thought and Husserl's Logical Investigations and Ideas for a Pure Phenomenology and a Phenomenological Philosophy.

6

Being and Number in Heidegger's Thought

This investigation also reveals the as yet unrecognized impact on Heidegger of Georg Cantor's work in set theory and, less directly, Richard Dedekind's work on the notion of number and the foundations of mathematics, as well as the impact of the emerging trend toward formal logic, particularly in the work of Frege. The second chapter puts forward a new interpretation of Heideggerian ¢nitude and how it relates to various temporal concepts, solving one of the most recalcitrant problems in the interpretation of Being and Time. The third chapter suggests construing historical epochs as ontological regions, shedding new light on the changes Heidegger's thinking underwent around 1930. Moreover, it facilitates comparison of Being and Time and Heidegger's writings from the thirties, particularly with respect to the pivotal question of the relation between the Cartesian subject and modern ontology. Other issues addressed in the chapter include analysis of Heidegger's response to Rudolph Carnap's attack on him, and responses to Heidegger's critique of modernity and its underlying premises. Awareness of the ontology^mathematics nexus invites us to reassess both speci¢c issues in Heidegger's thought and the larger question of Heidegger's place in twentieth-century philosophy. In particular, it o¡ers new possibilities for looking at the divide between analytic and continental philosophy. There is widespread agreement that an important aspect of this divide has to do with the fact that continental philosophers tend to be drawn to the human sciences, while analytic philosophers are drawn to mathematics and the natural sciences. This study will demonstrate that another important aspect of the divide can be traced to two very di¡erent approaches to mathematics and its foundations.

Notes 1. A signi¢cant biographical detail is that in 1911, after being forced by ill health to give up his theological studies, undertaken with the goal of entering the priesthood, Heidegger decided to forgo theology and philosophy, and turn instead to mathematics. In 1912, however, after a year of studying mathematics, physics and chemistry, he returned to philosophy, writing a dissertation on `The Doctrine of Judgment in Psychologism'. In the introduction to his

Introduction

2.

3.

4. 5.

7

dissertation, Heidegger refers to himself as a mathematician. For a detailed description of these years, see Ott 1988, 67^105. In the Critique of Pure Reason, Kant generally rejects formal characterizations of things, and in particular, the characterization of the transcendental one as true of all things (B113^14). Here Husserl adopts a Cartesian/Leibnizian concept. But while, for Descartes, the purpose of the mathesis universalis is to extract method from mathematics and extend it to all realms of knowledge ^ principally, to extract the relations of order and magnitude, for Husserl, the mathesis universalis is appropriate only for those disciplines amenable to full formalization. These disciplines can be characterized by means of the formal attributes of being in general. Heidegger 1963, 74. It is important to stress that this is not the sole option available in seeking an answer to the question of Being. Another possibility that the Logical Investigations leaves open makes use of categorial intuition. Whenever Heidegger refers to the Logical Investigations, he emphasizes the importance of categorial intuition in clarifying the question of Being. In contrast, formal ontology is rarely mentioned. Tension between the `mathematical' approach to the question of Being and the approach based on categorial intuition, which we can call the `phenomenological' approach, is thus already palpable in Heidegger's reading of the Logical Investigations. I discuss this in Chapters 1 and 3.

1

One as Transcendental and One as Number

In Heidegger's earliest writings, gathered together in the ¢rst volume of his collected works, there is but a single discussion of the question of being. It is found in the qualifying dissertation, which explores Duns Scotus' theories of categories and meaning.1 Aside from this discussion, Heidegger's early writings consist, in the main, of surveys and analysis of contemporary views on logic, and in particular, on the theory of judgment; there is also one paper on the concept of time in history. On the surface, these disparate subjects do not appear to re£ect a uni¢ed focus, but rather, Heidegger seems to be making exploratory forays in various directions not necessarily related to each other. Nevertheless, it is possible to discern in these writings topics and themes that recur in the works yet to come, though at this stage their treatment lacks the intensity of the later writings. In this chapter, I examine these early writings through the prism of what I consider to be a crucial distinction: that between one taken as a transcendental attribute, and one taken as a number. Analysis of this distinction and its signi¢cance will bring to light a central yet still unrecognized facet of Heidegger's earliest work, providing a new perspective on the whole philosophic thrust of this period. The distinction brings to the fore the place of mathematics in Heidegger's early thought, and its importance in the crystallization of his ontology. Why has the interpretative literature, even the few studies devoted to his early thought, failed to address the question of the place of mathematics in Heidegger's thought? There are a number of reasons. First, it is markedly easier to link other themes and topics from

One as Transcendental and One as Number

9

Heidegger's early writings to his mature thought. From the 1920s on, time, meaning and the question of Being manifestly play a central role in his thought, while passages linking the question of being to numbers and mathematics are, at least in terms of the amount of space devoted to them, marginal. Second, in the preface he wrote in 1972 for the volume of early works in his collected writings, Heidegger downplayed the role of mathematics in the crystallization of his thought.2 In this essay, Heidegger portrays his earliest writings as a starting point for his explorations of two central themes that occupied him subsequently: the question of Being, formulated in his early writings in terms of the problem of categories, and that of language, expressed, in the early works, in terms of the problem of meaning. He cites two works as having had considerable in£uence on his early thought: Franz Brentano's On the Several Senses of Being in Aristotle, and Husserl's Logical Investigations. Aside from these works, he notes the in£uence of Lask, Hegel, Schelling, Nietzsche and Dilthey, and the poetry of Trakl and Rilke. Mathematics is not even mentioned in this essay. The few commentaries that have focused on Heidegger's earliest work have, to a great extent, taken their cue from Heidegger's remarks in this retrospective introductory essay.3 However, in striking contrast to the picture painted by the introductory essay, in his original introduction to his doctoral dissertation of 1913, Heidegger de¢nes himself as an ahistorical mathematician.4 Mathematics, the later-repudiated dimension of his early writings, is here presented in a manner that invites its use as an important interpretive route through the Heideggerian corpus. Reclaiming this route allows us to elucidate the otherwise veiled connection between mathematics and the manner in which Heidegger formulates the question of being in general. My analysis integrates both Heideggerian perspectives on the early work. The chapter opens with a look at how Heidegger formulates the question of being in general in the early writings; goes on to consider a speci¢c element of the question of being ^ the distinction between one as a transcendental and one as a number; and reexamines the question of being in general in light of this distinction.

10

Being and Number in Heidegger's Thought

The analysis of being in general in the framework of a theory of categories We can summarize the thrust of the young Heidegger's approach to ontology by citing the requirement that di¡erent categories remain utterly distinct. Contrary to Aristotle and Kant, Heidegger maintained that di¡erent categories determine di¡erent, and nonoverlapping, realms of entities. This view is based on three considerations. First, the categorial distinction between psychology and logic; second, the interpretation of this distinction as applying not only to the categories themselves, but also to the entities falling under them. Logical entities are entirely di¡erent from mental entities. Third, a distinction must be made between realms of inquiry where mathematics is applicable, such as the modern natural sciences, and those, such as history, where mathematics is inapplicable as a matter of principle. Heidegger distinguishes between the psychological categories and the logical categories in discussing the question of psychologism in logic, that is, in discussing the view that it is possible to reduce the fundamental laws of thought to the contingent laws of human thought. Psychologism had been roundly attacked by Frege, and later, by Husserl, partly in response to Frege's criticism of his views.5 The Prolegomena to the Logical Investigations attempt to refute psychologism by means of arguments designed to show that any version of psychologism is relativistic, and relativistic positions are selfdefeating. In his 1913 doctoral dissertation, `The Doctrine of Judgment in Psychologism', however, Heidegger denies the possibility of a decisive argument against psychologism. The whole argument against psychologism, he maintains, is premised on the very hypothesis it seeks to refute.6 Heidegger does not state his reasoning explicitly here, but given the context of his remarks, it is plausible to interpret him as relying on the following argument. We have to assume the distinction between psychological and logical categories in order for the proof that logic is not reducible to psychology to be valid, that is, we need a concept of proof that is not formulated with reference to the psychological categories. But the assumption that there is such a concept is itself an essential part of the debate.7

One as Transcendental and One as Number

11

Heidegger lays bare the distinctions between the various categories by pointing to a category mistake at the heart of the alleged applicability of psychological categories to the realm of logic. Asking psychological questions about logical objects is, indeed, the crux of the problem with the psychologistic stance. Heidegger sees such queries as analogous to questions like, `how many grams does a second-order curve weigh?'8 Such con£ations reveal to us the impossibility of grasping logical objects by means of psychological categories. The ultimate signi¢cance of the absolute separation between the mental realm and the logical is that any category by means of which we grasp the activity of judging is ipso facto irrelevant to our understanding of the judgment itself. Despite this, Heidegger maintains, the judgment, as a logical object, is revealed via the activity of judging that is grasped by means of psychological categories.9 The judgment as a logical object constitutes a ¢xed element in a manifold of activities involving judging. Heidegger brings the example, `The cover is yellow.' This sentence can be asserted in a variety of contexts, for instance, when comparing book jackets of volumes on a shelf in a library, or upon seeing a yellow object on the street and being reminded of the yellow book cover we saw earlier in the day. What persists through these activities is the judgment itself. To identify this persistent element, however, we need the manifold of activities in which it is manifested.10 The logical object that is revealed to us belongs neither to the realm of the physical nor to that of the psychological, but rather, to the realm of the logical ^ the realm of meaning (Sinn). The form of reality pertaining to meaning is validity (Geltung). According to Heidegger, what meaning is cannot be de¢ned, but only illustrated. We can, for example, speak of a meaningful work of art, or of a meaningless (Sinnlos) business plan, namely, a plan that cannot be carried out, a plan that has not taken into consideration all the potential problems. And we can speak of cases where meaning is absent, for example, the lack of meaning (Unsinnig) of a series of words such as `green is or'.11 While meaning cannot be characterized directly, it can be characterized indirectly by analyzing the categories that are relevant to the activity of judging. These categories will not be applicable in the domain of logic. The foremost characteristic of the activity of judging

12

Being and Number in Heidegger's Thought

is that it is subject to the temporal categories. Hence, given the correlation between the categories and speci¢c areas of applicability, it follows that the temporal categories do not constitute a fundamental backdrop against which judgments acquire meaning, and thus, that meanings must be understood a-temporally.12 The distinction between the mental and the logical realms is retained in the later writings, as is another distinction from Heidegger's early work: that between history and the mathematical natural sciences. Heidegger maintains that there are limits to the applicability of mathematics to the sciences. There are, he believes, sciences to which quantitative categories are inapplicable. But it is only after he has worked out the distinction between one as a transcendental and one as a number that Heidegger is in a position to expand on the substance of the distinction between disciplines to which mathematics is applicable, and those unencumbered by the constraints of mathematics. This distinction, which plays an important role in Heidegger's theory of science,13 is developed in his paper on the concept of time in history.14

Theory of science Heidegger's theory of science is premised on two central dichotomies: that between the logical and the psychological, and that between the disciplines where mathematics is applicable and those where, as a matter of principle, it is not. In turn, his theory constitutes a framework for the ontological distinctions drawn in his study of Duns Scotus. The theory of science that Heidegger is seeking to formulate is not a snapshot or survey of the various realms of science at a given moment in time, but rather, a search for the scienti¢c essence of these realms, an essence independent of any phase of growth that they might exhibit. Heidegger approaches the search for the scienti¢c ontologically, interpreting the sciences as realms of objects (Gegenstandsgebiete). The search for a theory of science is, therefore, reduced to the search for the most general conditions that a region of objects, considered without regard to their content, must ful¢l. Thus the starting point of Heidegger's discussion of science is not by any means an

One as Transcendental and One as Number

13

epistemological analysis of the sciences. For any epistemological analysis of the sciences, it seems to me, focuses ¢rst and foremost on the distinction between science, on the one hand, and a collection of baseless beliefs, on the other. It seeks to institute this distinction by suggesting a method that, if followed meticulously, will ensure that only true beliefs are generated. On Heidegger's ontological approach, by contrast, the distinction between knowledge and opinion is secondary.15 A theory of science tries to identify the most general categories applicable to objects and regions of objects, and addresses whatever can be conceived (All des Denkbaren), without invoking any such preemptive epistemic discrimination. This ontological orientation in formulating a theory of science does not rule out the possibility that the di¡erent sciences have di¡erent methodologies. It is therefore quite compatible with the positions of Wilhelm Dilthey, Heinrich Rickert and others, who claim that the methods used in the humanities are completely di¡erent from those used in the natural sciences. When the essence of science is unhitched from the concept of method, it becomes possible to engage in an inclusive discussion of the sciences in general without thereby reducing one scienti¢c discipline to another. Moreover, since throughout history there have been numerous changes in scienti¢c method, it could be argued that only ontological analysis in the Heideggerian vein can transcend the historical context and reveal the scienti¢c in its most general form. The implications of the intimate connection Heidegger envisages between theory of science and ontology are not limited to the characterization of the scienti¢c nature of the sciences, but extend, in great measure, to the very nature of the ontological enterprise itself. If the fundamental question of the being of beings is raised within the context of the analysis of the sciences, then the primary ontological concept will be that of the ontological region, since the various sciences are ontologically de¢ned by delineating speci¢c ontological regions. In light of this, we might expect Heidegger's ontological analysis to involve compilation of an exhaustive list of categories specifying all possible ontological regions. But Heidegger does not take this approach. On the other hand, he rejects the idea that an a priori framework determines the categories. Thus, for example, he would

14

Being and Number in Heidegger's Thought

reject the Kantian approach, on which the categories are derived from the forms of judgment. Despite the link between the categories and the forms of judgment, the former cannot be derived from the latter. That is, Heidegger rejects the metaphysical deduction of the categories as the basis for uncovering the di¡erent categories of being. Indeed, Heidegger rejects, as a matter of principle, the notion that the categories can be arrived at deductively.16 This conclusion is based on the radical divide between logic and the other disciplines. The deduction of the categories would subordinate them to deductive logic, whereas Heidegger seeks a comprehensive framework constituted by something other than a set of rules of inference. Hence, in order to identify the di¡erent categories, we need to show (Aufweisen) what they are.17 The problem now confronting Heidegger is how the categories are indeed to be shown. Showing requires an unmediated connection to that which we show. According to Heidegger, it can be said of two areas that they are given to us directly: the immediate environment (Umwelt) and consciousness. Although inclined to begin with the immediate environment, ultimately Heidegger decides against approaching the issue of mapping out the categories of the various ontological regions from the starting point of the immediate environment. Nor does he begin with consciousness. Rather, he starts o¡ with an investigation into the concept of being in general. Now in Heidegger's very early writings, he seems to be jumping from one notion, that of the immediate environment, to another, the notion of being. But in his writings from the 1920s, we see that he takes these notions to be closely related. On the surface, this seems to contradict his declarations as to how he intends to tackle the question. I believe that Heidegger felt that through this inquiry he could generate an a priori categorial framework in a non-deductive way. Were Heidegger to opt for the method of enumerating the categories associated with the di¡erent ontological regions, it would be unclear how his approach di¡ered from the approaches to science that engaged in compiling inventories of the various sciences. Since, according to Heidegger, the aim of a theory of science is the delineation of all the possible ontological regions, and not construction of an empirical science that surveys the various

One as Transcendental and One as Number

15

disciplines already familiar to us, a survey of the sciences would pose a circularity problem.

Being: one as transcendental and one as number Heidegger attempts, therefore, to integrate two seemingly contradictory mandates: on the one hand, to provide an a priori framework for all the realms of being, and on the other, to arrive at the categories by means of directly showing these realms. The solution he proposes in his early writings is based on a reciprocal characterization of being in general and the ontological regions. In his book on Scotus, his starting point is an investigation into the notion of being (ens).18 According to Heidegger, the notion of being in general is indispensable for understanding any ontological region. Being is prior to all categorial determinations and prior to the question of the existence of entities. Hence, the general concept of being is prior to every ontological region characterized by a speci¢c category. The transcendentals ^ the one, truth and the good ^ are endowed with a status similar to that of being. The transcendentals are the most general determinations of being, in that they cannot be classi¢ed under any more general categories.19 Their generality is equivalent to that of being. The transition from the discussion of being to that of one as a transcendental occurs when Heidegger attempts to ascertain what can be said about being. Yet statements about being are not ordinary assertions of subject^predicate form. On the contrary, being and the one are assumed whenever we have subject and predicate. Hence, being cannot be construed as a subject. Every statement about being is already circular.20 Nevertheless, Heidegger begins his discussion of the transcendental one with an analysis of the judgment. The idea underlying this approach is that the judgment reveals the characteristics of the transcendental one. The generality of being is best captured in the proposition `each thing is some one thing' (`Das Etwas ist ein Etwas'). This proposition (Satz), though banal, reveals a central aspect of being ^ the relation (Beziehung) of being to itself. This self-relation assumes the oneness of being. Something is a being precisely because it is one thing

16

Being and Number in Heidegger's Thought

and not anything else. Here a pivotal meaning of the transcendental one is revealed ^ the one is not the other (das Andere). What is the relation between the two? The one is not the negation of the other, but its absence. This relation is best understood in terms of the transcendental many ^ the transcendental one is the absence of the many. The transcendental many, conversely, is the absence of the one.21 Thus, the one and the many are reciprocally de¢ned. This reciprocal de¢nition expresses the di¡erences or variance between them, but does not construe them as negations of each other. Negation of the transcendental one yields, not the many, but that which is not an entity. The signi¢cance of the one, accordingly, is ¢rst and foremost the delineation of being relative to the many that it excludes ^ the other. And the transcendental one has another aspect as well: it is a condition for self-identity.22 The transcendental dimension is thus revealed through a certain circularity. Being, the one and the self-identical all belong to this stratum, and, as I noted, are of equal generality. Exposure of these di¡erent aspects of the transcendental requires us to go around in circles, to use Heidegger's metaphor. There is no primary element from which the others are derived, but rather, a starting point is chosen (in this case, the assertion that something is some one thing) and the various meanings of being in general are then revealed. But when the transition from the transcendental dimension to speci¢c ontological regions is contemplated, we cannot proceed in the same manner. Here, Heidegger must acknowledge the disparity between the two realms. This follows from the fact that, as we saw above, Heidegger rejects the possibility of deriving the ontological regions from the transcendental dimension. In particular, the sphere of numbers cannot be deduced from the transcendental one. At the same time, Heidegger does not want to posit a total break between the two, since, having established that the one is a transcendental, it must also be true of numbers and thus, true of the number one. The manner in which he decides to proceed is to demonstrate that one, construed as number, must be characterized independently of the transcendental dimension. For Heidegger, one ^ as a number ^ is pre-eminently a certain object. That is, whereas the transcendental

One as Transcendental and One as Number

17

one is a determination true of objects but not itself an object, one as a number is an object. A deep chasm thus separates the two senses of `one'. As an object, one plays a special role relative to the other numbers. The number one is the basis for the other numbers.23 Heidegger should not be understood as construing the number one as a unit, so that the number two is two units, and so on. Rather, the number one should be conceived as ¢rst in the number series. The difference between the numbers one and two is essentially the di¡erence in their respective places in the number series, determined by the law of numerical ordering.24 Thus, in his early writings, Heidegger sees the ordering role of numbers as prior to the counting role. Heidegger maintains that this link between numbers and position in a series mandates a homogeneous space within which the numbers can be ordered. This space is the pure continuum.

Ontology without sets Even at ¢rst glance it is evident that the distinction between one as a transcendental and one as a number satis¢es some of Heidegger's desiderata for a theory of categories. First, the transcendental one functions as the framework for the categories, yet the categories are not revealed by their deduction from this framework, but by being di¡erentiated from it. Second, the delineation of the mathematical allows Heidegger to `liberate' the realm of reality (Wirklichkeit) from number's grip. The realm of reality need not be grasped in terms of quantitative categories, in terms of number.25 Thirdly, as I noted, the structure of the transcendental one as the absence of multiplicity is analogous to the relationship Heidegger envisages between the judgment as a logical entity and the act of judging. But the most important aspect of the characterization of the transcendental one is that it cannot be interpreted as unifying a multiplicity. This feature of the ontological framework Heidegger proposes is, in my opinion, intended as a solution, ¢rst and foremost, to the problem of the relation between the transcendental one and the di¡erent categories. This is the deep motivation for the framework. The reason Heidegger rejects the notion of a uni¢ed multiplicity

18

Being and Number in Heidegger's Thought

characterizing the ontological framework is that this sort of characterization would undermine the framework's universality. For if the framework is understood as a unity, it cannot comprehend the di¡erences between the categories. This critique dates back to Aristotle, but Heidegger presents a new version of the argument, based on the mathematical concept of the set. If sets are the only possible means of conceiving a uni¢ed multiplicity, and if sets are part of the mathematical realm, then a uni¢ed multiplicity cannot function as the framework of a theory of categories. The discussions of the concept of the set and its philosophical signi¢cance that were so central in the early years of the twentieth century, are, then, also found at the heart of the young Heidegger's ontology. Although the notion of set plays this crucial role, it is mentioned only once in Heidegger's discussion of the distinction between the transcendental one and the number one. In the course of his characterization of the transcendental manifold, Heidegger compares it to a pile of stones that cannot be associated with one speci¢c number. Heidegger draws the following conclusion: Hence, the manifold remains outside the realm of mathematics. The fact that there are, today, foundational areas in mathematics that deal with multiplicities or sets, or more precisely, `powers,' only appears to be a counterexample to what was stated above. For when we calculate with sets, as well as with `in¢nite' sets ^ proving, for instance, that the power of the totality of the rational numbers is not equal to that of the real numbers ^ already at that point a quantitative factor creeps into the discussion, making possible this calculation in terms of sets (Menge) and classes (Klassen). This is, indeed, the basis for the justi¢ed reservations about the derivation of the cardinals from the seemingly simpler concept of the set.26 Heidegger's main claim in this passage is that every set is premised on the possibility of its countability. Heidegger uses two di¡erent terms here, `set' and `class'. The former he takes from the work of Cantor, the latter, from the work of Frege and Russell. In so doing, Heidegger

One as Transcendental and One as Number

19

seeks to target both of the main approaches to the concept of set with his critique. To understand Heidegger's claim, therefore, it will be necessary to examine the di¡erent interpretations of the set that were current during this period. This will enable us to appreciate how Heidegger con¢gures his ontology, and particularly, the realm of numbers, to achieve his objective of an ontology without sets.

The notion of set The concept of the set has been variously understood in both mathematics and philosophy. Even Georg Cantor, the founder of set theory, presents a number of de¢nitions. Common to all the de¢nitions is the idea that the set uni¢es a multiplicity. On the question of how this unity is to be understood, however, there are two di¡erent views. The Frege^Russell camp attributes the unity to the concept (Frege) or propositional function (Russell27); on this approach, every set is the extension of a concept. Cantor and Husserl belong to the other camp. Cantor puts forward three main de¢nitions, which re£ect the stages in the development of set theory.28 The ¢rst de¢nition appears in his 1883 Foundations of a General Theory of Manifolds. `By a ``manifold'' (Mannigfaltigkeit) or ``set'' (Menge) I understand in general any many (Viele) that can be thought a one, that is, every totality (Inbegri¡ ) of de¢nite elements that can be united to a whole through a law. By this I believe I have de¢ned something interchangeable with the Platonic Idea.'29 The lawfulness that lies at the heart of this conception of the set is closely related to the possibility of arranging a collection into a well-ordered set. A well-ordered set is a set that can be ordered so that it has a ¢rst element and every element has a successor.30 At this point in Cantor's work, the set is still not grasped as the fundamental building block that allows for the de¢nition of numbers. Cantor characterizes sets as, ¢rst and foremost, sets of numbers, and thus is uninterested in reducing numbers to sets. In 1895, in his Contributions to the Founding of Trans¢nite Set Theory, Cantor o¡ers a di¡erent de¢nition of set: `By a ``set'' we understand any assembly (Zusammenfassung) into a whole M of de¢nite and welldi¡erentiated objects m of our intuition (Anschaung) or thought.'31

20

Being and Number in Heidegger's Thought

On this conception, there is no longer any need for lawfulness as a means of transforming the multiplicity into a unity. Using the concept of set, Cantor de¢nes the power of a set, or the cardinal number (Kardinalzahl) associated with it. In this work, then, the set is unquestionably presented as the basis for de¢ning numbers.32 The third de¢nition, found in an 1899 letter from Cantor to Dedekind, is based on the distinction between a multiplicity and a set. A set is a consistent multiplicity.33 That is, we cannot attribute unity to every multiplicity, but only to those multiplicities that do not generate inconsistency. This distinction is motivated by the fact that the conception on which the absolute is a set leads to contradiction. In Husserl's Philosophy of Arithmetic, the notions of set (Menge), multiplicity (Mannigfaltigkeit) and aggregate (Inbegri¡ ) are all used. However, in his description of the creation of the concept of number, Husserl prefers to employ the concept of aggregate. An aggregate, he explains, is generated by a unifying interest that coincides with an act of attending to di¡erences in content.34 Husserl's conception is fairly close to that expressed in Cantor's ¢rst de¢nition, but the source of the unity is not simply lawfulness or order, but some sort of content of the act of uni¢cation.35 Even the term Husserl selects ^ aggregate (Inbegri¡ ) rather than set (Menge) ^ points to the idea that for a collection of things to constitute a unity, there must be a speci¢c content through which the unity is grasped. Thus Frege, Russell, Cantor and Husserl all used the concept of set to characterize the notion of number. This motivates their stipulation that sets are de¢nite multiplicities in the sense that the number of their elements cannot change, since otherwise it would be unclear how they could be associated with numbers. Each felt he had succeeded in satisfying this requirement, but insisted that some or all of the others had failed. For example, Cantor argued that de¢ning sets by means of concepts was problematic, given that concepts do not determine ranges. Frege and Russell, on the other hand, maintained that it was impossible to refer to in¢nite classes without using concepts. Frege's Foundations of Arithmetic deepened the gap between the different views. Frege argued against any attempt to derive numbers from mental associations. Since the validity of arithmetic is independent of any particular perspective, it is impossible to reduce numbers

One as Transcendental and One as Number

21

to acts of abstraction or uni¢cation. Such a reduction would undermine the absolute validity of the laws of arithmetic. In view of this, Frege criticized the attempts by Cantor and Husserl to reduce cardinal numbers to sets conceived as unities of multiplicities. Frege thus critiqued the concept of set (Menge), replacing it with that of the extension of a concept (Klasse), which is entirely independent of any mental act.36 Although these disputes point to signi¢cant di¡erences between the two camps on the question of how sets are to be understood, they also reveal much common ground. Speci¢cally, they reveal parallel pictures of what concepts and categories are. Concepts and categories are modes of unifying multiplicities so that the uni¢ed multiplicity is a de¢nite multiplicity, that is, a multiplicity with ¢xed elements. The only question on which there is disagreement is that of how the unity is to be explained ^ psychologically or logically/conceptually? But both approaches, I must stress, share the same notion of unity. Heidegger utilizes the broad applicability of the concept of set to map out his early views on ontology.37 His fundamental premise is that not all realms of being are countable. Realms of beings that are not countable cannot be construed as de¢nite multiplicities. That is, these realms cannot be amenable to description as sets. And since the notion of an object in general must apply both to objects belonging to countable realms and to those belonging to non-countable realms, the formal characterization of an object in general, that is, the characterization that precedes the application of the categories, should exclude unity. Another conclusion Heidegger draws from the concept of the set is that the various kinds of unity that bind multiplicities must be kept distinct. Were all notions of unity identical, that is, were the concept of unity uniform, every unity would de¢ne a set. Therefore, to ensure total heterogeneity among the various realms of being, the unity that is at the heart of each realm must have a unique meaning. Hence, unity cannot be a part of being in general. This interpretation of Heidegger's very early thought as centred on the notion of unity and its relation to that of set sharpens our understanding of the views held by Heidegger at this point in his career, and provides a new perspective from which to evaluate his work relative to

22

Being and Number in Heidegger's Thought

the theories that in£uenced him and constituted his frame of reference. First, this new perspective allows us to identify more precisely the di¡erences between Heidegger's views and the prevailing neoKantianism of his time. In particular, it allows us to re-evaluate his views vis-a©-vis those of Heinrich Rickert, who served as advisor for his dissertation on Duns Scotus, and Paul Natorp. Second, it reveals important aspects of the Husserl^Heidegger relationship. Third, it allows us to situate Heidegger's views on number vis-a©-vis the views current at the time. But before we commence this comparative analysis, let us look at another aspect of Cantor's set theory, one with important links to both the scholastic thought in terms of which Heidegger formulated his early views, and those views themselves. The connection between Cantor's set theory and Heidegger's ontology emerges out of Cantor's revolutionary treatment of the in¢nite.

In¢nity as a de¢nite multiplicity Cantor's work had great signi¢cance for Heidegger. First, it served as a link between his interest in medieval philosophy, especially the medieval discussions of being, and the mathematics and philosophy of his time. For there is an unmistakable a¤nity between the scholastic discussions of being, the one and the many, on the one hand, and set theory, on the other. Indeed, Felix Klein, one of the foremost mathematicians of the nineteenth century, argued that when scholastic speculation is stripped of its mystical-metaphysical trappings, it can be seen as a forerunner of set theory.38 But even more importantly, Cantor's work provided Heidegger with signi¢cant support for the conceptions he was developing. Until Cantor, philosophical thought had unanimously rejected the possibility that in¢nity could be a de¢nite multiplicity, that is, a multiplicity amenable to being uni¢ed and counted. This rejection was based on the premise that the fundamental assumptions governing ¢nite quantities should apply to all quantities, including the in¢nite. Were an in¢nite multiplicity a given quantity, it would satisfy the condition that the whole is not greater than its real parts. Since for ¢nite quantities the whole is always greater than its real parts, it

One as Transcendental and One as Number

23

was concluded that the in¢nite is not a quantity. Hence, on the preCantorian conception, there are no in¢nite quantities. Set theory changed this conception dramatically, demonstrating that there are in¢nite multiplicities that are sets, yet the laws that apply to ¢nite multiplicities do not all apply to them. For example, the set of the natural numbers is not bigger than the set of even numbers. Nevertheless, as Cantor's third de¢nition of set makes clear, there are multiplicities that are not sets. Cantor's `Absolute', the ultimate totality, is such a multiplicity. Thus, according to set theory, being as a whole is not a set. Hence set theory, despite the changes it entails for in¢nite multiplicities, nonetheless supports the scholastic conception of in¢nity, in that the Absolute is not an in¢nite quantity, and thus, not a uni¢ed multiplicity. Cantor's characterization of the Absolute can be linked to Heidegger's treatment of being. Since the characterizations `being' and `one' apply to every entity, it could be argued that the domain of `being' and `one' is the Absolute. This would make the Absolute a set, the set of all the `ones'. But Heidegger's characterization of the one as the absence of multiplicity avoids this outcome. Since the domain of `one' does not include multiplicities, it cannot be equal to the Absolute. The domain of `one' is limited. Hence, Heidegger's account of the transcendental one, which can be traced, via Duns Scotus, to Aristotle's claim that being and oneness are not general, is in perfect agreement with Cantor's claims about the Absolute. Cantor's limitation of the applicability of the notion of set reinforces Aristotle's claim. Hence, set theory and its limits can be seen as providing a new foundation for the question of being.

Rickert on the one, unity and the number one Heinrich Rickert served as the advisor for Heidegger's qualifying dissertation. There is no doubt that the chapter in that work on the transcendental one and the number one was inspired by Rickert's 1911 article, `The One, Unity and the Number One'. At the time Heidegger wrote his qualifying dissertation, there were two main neo-Kantian schools: the Southwest school, whose leading ¢gures

24

Being and Number in Heidegger's Thought

were Rickert and Lask, and the Marburg school, beginning with Hermann Cohen and then led by Paul Natorp and Ernst Cassirer. Whereas the Southwest school downplayed the importance of the natural sciences in the Kantian worldview, the Marburg school not only acknowledged the salience of mathematics and the natural sciences, but also introduced into the Kantian framework the changes needed to bring the Critique of Pure Reason in line with early twentiethcentury mathematics. The young Heidegger is generally considered closer to the Southwest school,39 but as I will show, several central aspects of his thought are actually more in keeping with Natorp's ideas. Indeed, Heidegger embraces these ideas for the very purpose of critiquing Rickert's approach. At the heart of Rickert's article, `The One, Unity and the Number One', is the question of whether it is possible to di¡erentiate between logic and mathematics. His answer is that mathematics presupposes an extra-logical element. On Rickert's view, logic is the theory of thought, and at its purest, the logical is whatever is thought by a subject, completely independently of any extra-subjective element.40 That is, the purely logical is also the purely subjective. The purely logical comprises both the activity of thought and its object. Rickert characterizes the purely logical object as that which is one. This characterization is contrasted with the characterization of that which is `other' ^ that which cannot be a purely logical object ^ as multiplicity. However, Rickert proceeds to integrate the `other' into the purely logical object. That which is one and that which is other are both, he asserts, aspects of the purely logical object: its form is one; its content, multiplicity. Hence the logical object is a unity of form (one) and content (other), that is, it is a uni¢ed multiplicity. Rickert then turns to analysis of the number one, asking whether it can be generated from the concepts `one' and `unity'. On the basis of his analysis of propositions such as 1 ˆ 1, he answers in the negative. In this type of proposition, ` ˆ ' expresses equality, not identity. That is, what we have here is not a proposition expressing the identity of the number one with itself as an instantiation of the principle of identity, but rather, the claim expressed by this equation is that some particular 1 is equal to some other 1. In general, identity is not a relation, for relations presuppose some sort of di¡erentiation. It is, of course, true

One as Transcendental and One as Number

25

that we use propositions such as A ˆ A to express the principle of identity, but this formulation is misleading, as it contains an element of a relation where no such relation exists.41 The ` ˆ ' sign in A ˆ A, and in 1 ˆ 1, expresses equality, and therefore implies some degree of di¡erentiation. For this di¡erentiation to be possible, it is necessary for there to be a mediating common ground relative to which this di¡erentiation can occur. The common ground that is required is quantity. Quantity is that which enables 1 ‡ 1 to become 2. It is the extralogical element that is necessary for mathematics. As the medium of numbers in this sense, quantity has no spatio-temporal characteristics. Hence, numbers, and the equality that may exist between them, are independent of space and time, requiring only the generic homogeneous medium of quantity. At ¢rst glance, there appears to be considerable similarity between these ideas and Heidegger's distinction between the transcendental one and one as a number. Indeed, the manner in which Rickert's distinction between numbers, on the one hand, and the more basic stratum, on the other, is constructed, seems almost identical to its Heideggerian counterpart. Yet despite this apparent a¤nity, there are profound disparities between the two. Whereas for Rickert, unity is part of the basic stratum, the stratum of the basic categories, it plays absolutely no role in Heidegger's theory of the two distinct types of oneness. This lack of any role for unity goes to the heart of Heidegger's disagreement with Rickert. According to Rickert, equality between numbers calls for something that is not present at the level of the basic logical stratum, for some additional element. That is, the realm of numbers presupposes the logical stratum, but adds something further to it. In contrast, Heidegger sees a gap between the two realms. The realm of numbers is not constructed out of the fundamental ontological building-blocks, but in opposition to them. In fact, the entire model of construction and successive layers being added to an ontological skeleton is quite foreign to Heidegger's ontology. Di¡erences between Rickert and Heidegger are also evident in their conceptions of number. Rickert focuses on understanding arithmetical propositions and the fact that they involve something beyond the purely logical. Here Rickert's position calls to mind the Kantian conception of number. Heidegger, however, focuses on numbers as

26

Being and Number in Heidegger's Thought

objects, and on the manner in which they are determined by a system of axioms. On this point, Heidegger is closer than Rickert to the work on the philosophical and mathematical foundations of arithmetic of his day ^ for example, Peano's axioms of arithmetic. Rickert and Heidegger do concur in asserting that numbers presuppose a homogeneous medium, but whereas for Rickert this homogeneous medium is necessary as a foundation for equality, the notion of a homogeneous medium does not play an important role in Heidegger's account of number. Order can be characterized without assuming a homogeneous medium, and hence, on Heidegger's conception of number, there is no real need for this assumption. As we will see in the following chapter, Heidegger's account of number in Being and Time dispenses with it altogether.

Natorp and Heidegger Heidegger's relationship with Natorp is more complicated. At least on the surface, it would seem that Heidegger is closer to Rickert's position than to Natorp's. I believe, however, that Heidegger rejects both Rickert's thesis of the primacy of unity, and Rickert's concept of number, and that he does so on the basis of the concept of number put forward in Natorp's Logical Foundations of the Exact Sciences. That is, Heidegger uses Natorp's concept of number to critique those aspects of Rickert's thought that are incompatible with his theory of categories. The central question addressed in The Logical Foundations of the Exact Sciences is that of how, on the basis of the Kantian outlook, the foundations of the exact sciences, and in particular mathematics, are to be secured. Natorp endeavours to make whatever modi¢cations to the Kantian programme are needed if it is to serve as a basis for mathematics. The essence of such modi¢cation is the exclusion of intuition, both empirical and pure, from any role in grounding the sciences. Natorp accepts the Kantian idea that the basic act of thought is the uni¢cation of a manifold, but, contrary to Kant, maintains that the source of the manifold need not be sensation. That which is

One as Transcendental and One as Number

27

given as the material for thought lies completely outside the scope of thought and hence cannot be spoken of. The given, as the object of thought, is totally undetermined (unbestimmt).42 Identifying its source is a way of speaking about it, and therefore also rejected by Natorp, and even its characterization as manifold constitutes thinking about it. Yet in assimilating the given's characterization as manifold to the realm of thought, and not limiting the given to sense data, Natorp's stance is not so very far from that of Rickert. Rickert and Natorp di¡er mainly with respect to their accounts of number. Natorp construes numbers as a purely logical construction presupposing no extra-logical component. This conception eliminates the boundary between the logical and the mathematical. The quantitative synthesis, for example, is seen as inherently connected to the qualitative, and both serve as the foundation for the mathematical object.43 The links between all these elements are brought out in Natorp's discussion of the fundamental series (Grundreihe). The fundamental series is the form of a series in general. All other series (series of numbers, series of events in time) are possible contents of this formal series. The fundamental series is generated by the connection between the one (das Eine) and the other (das Andere) via uni¢cation. It is constructed through the basic act of thinking, positing (setzung). In thinking, something ^ namely, the one ^ is posited. This positing of the one always takes place in relation to something else ^ namely, the other.44 The relation is ¢rst a separation between the one and the other, then their uni¢cation. The uni¢cation in turn serves as the one for a new act of positing, generating the fundamental series.45 The fundamental series underlines the signi¢cance of Natorp's work for Heidegger, in that it points to the di¤culties inherent in the distinction Rickert attempts to draw between numbers and the basic logical stratum. It follows from the argument of The Logical Foundations of the Exact Sciences that every uni¢cation of a multiplicity is ipso facto related to number. Therefore, in order for there to be a distinction between one as a transcendental and the number one, the transcendental stratum cannot be characterized as the uni¢cation of a manifold. The transcendental stratum is not countable since it cannot be interpreted as a de¢nite multiplicity.

28

Being and Number in Heidegger's Thought

Another point on which Heidegger is in£uenced by Natorp is the claim that sets presuppose the concept of number. This argument, which appears in The Logical Foundations of the Exact Sciences, is directed mainly against Frege. Natorp's criticism is that Frege's de¢nition of the numbers already presupposes the concept of number, since it is based on the relation of an individual's belonging to a class. According to Natorp, this relation in turn presupposes that the individual is one.46 Heidegger saw in this critique, I believe, an opportunity to formulate a more general critique of Frege and Russell. Frege and Russell, he charged, equate the domain of the thinkable (logic) with the domain of the countable. Heidegger hopes to avoid this outcome in the following way: if the approaches of both Frege and Russell presuppose the concept of number, as Natorp argues, then should it be possible to provide a general characterization of objecthood in terms of categories that are not linked to the realm of numbers, as Heidegger hopes to do, their work would remain con¢ned to the realm of mathematics. The new formal logic would be but a branch of mathematics, and not, as Frege and Russell claimed, a new framework for ontology and logic in its more basic sense.

Husserl's notion of formal ontology The problem of the relation between logic, ontology and mathematics plays a central role in understanding the young Heidegger's encounter with Husserl's phenomenology as well. Later, in the 1920s, Heidegger develops his own path in phenomenology. But even then, as I will argue below in Chapter 3, the interrelations between logic, ontology and mathematics continue to play an important role in his critical discussions of Husserl's phenomenology. In Husserl's thought, the interface between the three realms emerges in the notion of formal ontology. My main thesis here is that the young Heidegger wants to preserve this notion, but rejects the link between formal ontology and mathematics posited by Husserl. According to Husserl, formal ontology has several di¡erent but interrelated aspects. First, there are formal objective categories such as object, state of a¡airs, unity, plurality, number, and so on.47 The

One as Transcendental and One as Number

29

characterization of these categories is completely determined by analytical laws. Second, Husserl adopts this conception of the analytical determination of formal categories as a model for a theory of all possible theories. In this expanded sense, formal ontology is extracted from the mathesis universalis, understood as a pure theory of manifolds. In the last sections of the Prolegomena to the Logical Investigations, we ¢nd the ¢rst comprehensive elaboration of a pure theory of manifolds in the context of a theory of all possible theories.48 The aim of this theory of theories is to generate every possible form a theory can take, all actual theories being but particularizations of these forms. Husserl takes the idea for this theory of all possible theories from the notion of manifold in mathematics. The manifold is a ¢eld of objects that are determined solely and uniquely by the laws governing the manifold.49 Examples of manifolds include the cardinal and ordinal numbers and n-dimensional Euclidean geometry.50 In these manifolds, the objects (numbers or points) are completely determined by their relations to other objects. For Husserl, the mathematical manifold is the ideal theory. Achieving this ideal would yield all the possible realms of being and provide a general formal ontology. Husserl is, of course, well aware that it is not achieved in every realm (for example, it is not achievable in the empirical sciences), but ideally, all theories would meet this standard. Thus, Husserl's formal ontology plays a role similar to that played by Heidegger's theory of science. But the di¡erences between the two are considerable. Heidegger denies the possibility of deriving all possible realms of being, since the di¡erent meanings of unity are intra-regional and not determined in advance by an a priori framework. For Heidegger, the only ontological region that exempli¢es the structure of a mathematical manifold is that of the numbers, the place of each number being determined by a system of rules. Husserl's formal ontology is based on a mathematical notion, but for Heidegger, mathematics is only one region out of many; the others cannot be characterized mathematically. Heidegger is, essentially, relegating Husserl's formal ontology to a particular ontological region.51 The motivation for Heidegger's argument that the concept of the set presupposes number is now comprehensible. Heidegger is trying to demonstrate that since every de¢nite mathematical manifold is

30

Being and Number in Heidegger's Thought

a set, even if it is not always possible to characterize its members in a completely formal manner, every de¢nite mathematical manifold is already part of the realm of number and thus cannot serve as a general framework for all possible regions. In evaluating the relationship between Husserl's notion of formal ontology and Heidegger's very early work, we should not limit ourselves to the connections between formal ontology and mathematics, but analyze the overall role of formal ontology in Husserl's thought. This entails consideration of two central questions: the relation between formal ontology and formal logic, and the role of formal ontology in Husserl's phenomenology. Husserl distinguishes between two kinds of categories: formal ontological categories, and categories of meaning, such as concept, proposition and truth.52 According to Husserl, the ontological categories and the categories of meaning are mutually dependent. On the one hand, to every proposition there corresponds a state of a¡airs, and we can even say that the notion of an object or a state of a¡airs should be understood as a corollary of the categories of meaning. On the other hand, inasmuch as every proposition or concept is itself an object, the ontological categories are primary, the categories of meaning being but a kind of object.53 The young Heidegger endorses the distinction between ontological categories and categories of meaning. But in contrast to Husserl, for Heidegger the ontological categories are clearly prior to the categories of meaning. The notion of an object is not understood via its role as a corollary of a concept. Nor is this the only di¡erence between Heidegger and Husserl. They also di¡er over the judgment qua logical object ^ that is, as distinguished from the judgment as a mental act. I maintain that the special link between the one and the many (as ontological categories) enables Heidegger to achieve a thorough separation between mental act and logical object. If the judgment is interpreted as a logical object, as a unit that is the absence of a multiplicity of other acts of judging (statements), then, of course, it will not be possible to arrive at this judgment by abstracting some shared essence from the many other statements. We have no direct evidence that this was Heidegger's original intention, but in the 1925 lectures, in which he discusses Husserl at length, he declares that Husserl's

One as Transcendental and One as Number

31

fundamental mistake was his failure to establish a distinction between the logical, eternal and ¢xed, and the mental, temporary and changing. He detects Husserl's mistake as far back as the Logical Investigations, where Husserl makes the following assertion: `Each truth stands as an ideal unit over against an endless, unbounded possibility of correct statements which have its form and its matter in common.'54 Heidegger considers this construal of the true judgment or logical object as an abstraction from a manifold of acts of judging, which makes the judgment an `anthropological unity', to be the source of Husserl's error.55 It seems to me that, although he does not state this explicitly in the early writings, one of Heidegger's motivations for interpreting the one as the absence of multiplicity is that it allows him to create a clear distinction between logical object and mental act. The connection between the transcendental one and truth reinforces this conjecture. Since truth is also a transcendental notion, and the scope of truth should be identical to the scope of the one, and since the attribution of truth is the basic property of the judgment as a logical object, the linkage I am suggesting between Heidegger's account of the judgment and the structure of the transcendental one follows straightforwardly from his theory of the transcendentals.56 Another important aspect of the notion of formal ontology is its intimate connection to Husserl's main philosophical project, namely, phenomenology. At ¢rst glance, it might seem as if the two disciplines have nothing in common. Phenomenology is intrinsically linked to things themselves; modes of appearance are uncovered in a process of `go[ing] back to the ``things themselves'' '.57 That is, in contradistinction to formal ontology, where things themselves, being completely determined by the region's system of axioms, play absolutely no role, in phenomenological investigation things themselves take centre stage. Upon examination, it becomes clear that there is a signi¢cant gap between the theory of science advanced in the Prolegomena to the Logical Investigations, and the theory of categorial intuition advanced in Investigation VI. The import of this intuition is that the categories by means of which we grasp things are themselves part of those things. The di¡erence between the categories and the sensory material is a di¡erence between two types of

32

Being and Number in Heidegger's Thought

intuition, a distinction that is by no means sharp.58 Perceiving things is a single activity with two facets. In the Ideas, the relation between formal ontology and phenomenology changes signi¢cantly. To understand this change, we must ¢rst consider the new relation between formal and material ontology. In the opening sections of the Ideas, Husserl seems to maintain the distinction between the formal and the material categories, that is, between categories that apply to things in general, and those that apply to speci¢c empirical domains. But the change becomes evident as the Ideas progresses. Formal ontology is not only the ideal form of the material domains, but also relevant to their constitution, since the notion of something in general, with no characteristics other than formal ones, as the pole of the determinable thing, is not limited to the formal sphere, but crucial for the material ontological regions as well. The formal categories play a role in constituting every region. This development is a consequence of changes in the methodology of Husserl's phenomenology. Whereas in the Logical Investigations there was a divide between phenomenology, which is oriented to things themselves, and formal ontology, with its notion of something in general, in the Ideas this dichotomy is no longer present. Not found in the Logical Investigations, the phenomenological reduction, which brackets out the existence of the thing, constitutes the very heart and soul of the phenomenological method as presented in the Ideas.59 Thus disclosure of a thing's modes of appearance takes place apart from the assumption of its existence, which essentially contributes nothing to the manner in which it is given. The eradication of the tension between formal ontology and phenomenology can be interpreted in two ways. First, the phenomenological method can be interpreted as an extension of formal ontology to all realms of being. The only di¡erence between formal ontology and material/regional ontology is the application of the phenomenological reduction, which is not needed in the formal disciplines. On this new approach, the meaning of phenomenology changes. Instead of seeking out things themselves, it proposes a mapping of ontological regions, a mapping in which the categories that apply to the various regions are not determined by the entities of which the regions are comprised.

One as Transcendental and One as Number

33

A second interpretation of the transition re£ected in the Ideas is that formal ontology is transformed into a speci¢c region. Phenomenology now assumes the role of formal ontology and mathesis universalis. Hence, any discussion of being in general has no place within the phenomenological framework, leaving only the regions constituted by the transcendental ego. From this perspective, the constitutive role of formal ontology disappears altogether. This understanding accords well with Husserl's statement, in the third volume of the Ideas, that phenomenology is not ontology.60 In this context, where are Heidegger's early writings to be positioned? It would seem that during this period Heidegger vacillated between the approach of the Ideas and that of the Logical Investigations. Heidegger's theory of science diverges signi¢cantly from that put forward by Husserl in the Logical Investigations, since the formal characterizations of Husserl's theory of science are applied by Heidegger to a single region and do not serve as a general framework for the sciences. But Heidegger does not reject the approach of the Logical Investigations in its entirety. On the contrary, his account of how the categories for each region are extracted ^ that is, by re£ecting on things themselves, as they disclose to us the categories through which they are given ^ closely recalls Husserl's use of categorial intuition in Investigation VI. For example, he reformulates Scotus' conception of empirical reality as a `givenness' that already manifests a categorial determination.61 Yet Heidegger's views show some important a¤nities with Husserl's Ideas as well. Heidegger's notion of the transcendental one as a framework for the regional ontologies plays a role virtually identical to that played by Husserl's formal ontology. The principal element of the Ideas that is adopted by Heidegger is, I believe, the notion of `form'. In section 13 of the Ideas, Husserl distinguishes between `formalization' and `generalization' in order to make eidetic singularity possible. `Generalization' being inapplicable to the relation between the categories and the transcendental one, Heidegger's early ontology is formal in the sense delineated by Husserl. Heidegger does not use the term `formalization' during this period. However, the notion of the formal plays an important role in his lectures delivered in the years 1919^21. In his 1920 lectures, entitled

34

Being and Number in Heidegger's Thought

`The Phenomenology of Religious Life', Heidegger distinguishes between Husserl's notion of `formalization' and his own notion of `formal indication' ( formale Anzeige).62 Heidegger seeks to reject what he regards as the negative aspects of Husserl's notion of `formalization', while retaining the positive. The former are those aspects that formalization shares with generalization. Both are universal and theoretical notions. Heidegger regards as positive the fact that the notion implies a desire to approach the question of the being of entities not in terms of the content ^ the `whatness' ^ of being, but in terms of the way in which entities are given. Kisiel argues that Heidegger's `formal indication' already ¢gures in the book on Duns Scotus,63 but as I have shown, Heidegger's early ontology is much closer to the ideas set out in Husserl's formal ontology, including its understanding of formal mathematics, than to those associated with the later `formal indication' doctrine. But while Heidegger adopts the notion of formal ontology, he develops it very di¡erently. Heidegger's account departs from Husserl's on two points: ¢rst, Heidegger's formal ontology can accommodate regions made up of uncountable entities; and second, the special link Heidegger postulates between the one and the many enables him to achieve a thorough separation between mental act and logical object.

Heidegger and the mathematics of his time Having situated the views of the young Heidegger relative to both the neo-Kantians and Husserl, let me turn to an assessment of their place relative to developments in the foundations of mathematics and mathematical logic at the close of the nineteenth century and in the ¢rst decade of the twentieth. Two principal currents, corresponding to the two approaches to sets discussed above, can be distinguished. One, associated with Frege and Russell, derives the concept of object from the logical analysis of propositions. The other, associated with Dedekind, Hilbert and Cantor ^ and Husserl belongs here too ^ starts with the concept of object in general, or thing in general,64 and from it proceeds to construct the numbers, and all of mathematics.

One as Transcendental and One as Number

35

This approach envisages a direct connection between the object in general and mathematics, unmediated by any logical analysis of propositions.65 Heidegger's position is closer to the second of these approaches, for two reasons. First of all, he does not derive the object in general by means of the logical analysis of judgments. Indeed, he does not endorse the idea that logic is prior to ontology, and construes ontological categories as independent of logical categories. The sole connection between the two realms is that the transcendental one, as the absence of multiplicity, is true of every entity, and hence, true of judgments, and judgments are logical objects. Moreover, Heidegger endorses the centrality of the link between numbers and the object in general, a link that is particularly striking in the de¢nition of number. His characterization of numbers as ordered also re£ects an a¤nity with a fundamental aspect of Dedekind's conception.66 Heidegger, like Hilbert, rejects the reduction of arithmetic to logic. Indeed, while Heidegger's critique of the reduction of number to set is apparently based on Natorp, Natorp himself based his arguments on Hilbert's 1904 article `On the Foundations of Logic and Arithmetic'.67 In this article, Hilbert uses the general concept of the thing as the starting point from which he goes on to construct arithmetic.68 The various in£uences that have been outlined above come together to paint a complex picture of where Heidegger's very early thought should be situated relative to the revolutionary developments that had transformed logic and the foundations of mathematics in the preceding years. The young Heidegger was by no means hostile to these developments, and in fact, his own approach places him squarely in one of the two main camps, in the framework of which he seeks a treatment of the thing in general that transcends the mathematical context and can serve as a foundation for all the sciences. We can now resolve the interpretative dilemma raised at the beginning of this chapter. The reader will recall that, in interpreting Heidegger's very early work, two directions are open to us. One is suggested by Heidegger's 1972 preface to the volume of his early writings in his collected works, in which he describes his early writings as posing the question of the meaning of Being, and as inspired by the

36

Being and Number in Heidegger's Thought

work of Brentano, Husserl, Hegel, Schelling and others. The other takes its cue from Heidegger's 1913 characterization of himself as a mathematician. We are now in a position to assess which of these interpretative directions is in fact correct. Heidegger does indeed seek answers to ontological questions, but the questions are posed from within a conceptual framework that encompasses the foundations of mathematics and Husserl's Logical Investigations. The concept of the thing in general, and the attempt to characterize it, are directly and intimately related to the mathematics of the period. Heidegger's objective is to create a formal ontology embracing even areas where mathematics and numbers have no applicability. To this end, he utilizes research on the foundations of mathematics, recasting the entire realm of mathematics as an ontological region. In that they preserve the goal that underlies the mathesis universalis, namely, a formal characterization of being in general, the very early writings are mathematical. But they nonetheless deny mathematics as we know it the status of formal ontology. For Heidegger, mathematics as we know it is but peripheral to a general theory of categories. The con£icting sentiments expressed in the two introductions thus re£ect a profound dichotomy in the young Heidegger's thought: from one perspective, his very early writings are indeed mathematical, yet from another, they are utterly estranged from mathematics. Arriving at his ontology from within mathematics, and focusing, as he does, on the concept of set, Heidegger is well placed to draw philosophical/ontological conclusions from the developments that were occurring in logic and mathematics. The main notion Heidegger drops from the programme of general ontology is that of the uni¢ed multiplicity. He thereby drives a wedge between his position and the views of the neo-Kantians, on the one hand, and those of Frege and Russell, on the other.

The transition to Being and Time The cardinal challenge with which the young Heidegger grappled was that of ¢nding a general framework for all the categories.

One as Transcendental and One as Number

37

The main constraint on this framework is that it must maintain a total distinction between the di¡erent categories. Since deduction, for example, is a logical category, it cannot be applied to non-logical categories. Deriving the various categories, therefore, is not an option for this framework. Heidegger must come up with a framework that neither employs any categories of the ontological regions, nor allows their derivation in any way. The solution he o¡ers in his very early writings utilizes the transcendental one, in two distinct ways: ¢rst, as a formal framework for the categories, and second, as a starting point from which to di¡erentiate the categories by distinguishing between the transcendental one and the number one and then between the realm of numbers and the realm of real things. In Being and Time, the ontological discussion takes a signi¢cant turn. While Heidegger's starting point is indeed the general question of the meaning of Being, the way this question is formulated is indicative of signi¢cant change. Heidegger no longer inquires into the formal characteristics of entities, but rather, inquires into the meaning of Being. This question links being with the way it appears and what it means to people. Consequently, Heidegger does not stay with the general question of the meaning of Being, but moves to a discussion of the individual inquiring into the meaning of Being: Dasein. Dasein is not simply a speci¢c ontological region, but is, rather, the condition that makes all the other regions possible. That is, in Being and Time there is no trans-regional framework, no framework over and above the regions themselves. What is the import of this change? The conception of the thing in general as free of all content is already a certain understanding of the being of entities. Essentially, the book on Scotus o¡ers an interpretation of being that excludes certain realms from the scope of ontology. For example, the primary characteristic of being in general is amenability to being thought. This already presupposes a preliminary concept of thought as the starting point for ontology. For an ontology to be comprehensive, it must concern itself with the relationship between the interpretation of the being of entities and the interpreter. The interpreter is, on the one hand, encompassed by the interpretation, and on the other, the condition enabling this interpretative process.69 We can say, therefore, that in Being and Time Heidegger holds the view

38

Being and Number in Heidegger's Thought

that the apprehension of being `in general', as presented in his previous writings, is basically a reduction of being to a speci¢c region rather than a truly general examination of being. Heidegger's critique of his earlier treatments of being in terms of the general/formal properties of entities is also found in his discussion of metaphysics in the late 1920s.70 As Heidegger sees it, the central problem burdening metaphysics is the duality it harbours. On the one hand, it seeks formal characteristics of being as being; on the other, it seeks a primordial entity that is the foundation of all other entities. This duality, which Heidegger terms onto-theology, is present as far back as Aristotle. It might be thought that this criticism is equally applicable to Being and Time, where Heidegger sets out to answer the question of the meaning of being in general, and ultimately, puts forward an answer in terms of one speci¢c entity, Dasein. But this supposition is quite mistaken. To identify the error, let us recall the concepts that were central to Heidegger's earlier ontology: the one and the many. To read Being and Time as exemplifying the mistake that a¥icts metaphysics is to read it in a manner that construes Dasein as unifying the manifold ontological regions. But the relationship between Dasein and the other regions is more complicated. It is more accurately construed as one of reciprocal inclusion: Dasein is part of the world and in this sense is part of the whole that encompasses all the regions, yet this whole is, essentially, encompassed by Dasein. That is to say, the relationship between Dasein and the world is one of mutual inclusion. It would even be correct to describe this relationship as basically taking over the central place occupied by the notion of the one as the absence of multiplicity in the earlier writings. Heidegger, however, is unable to say this explicitly, as doing so would constitute commission of the very error he deplores ^ attempting to provide a general treatment of being. But nevertheless, there is no doubt that this characterization does capture the underlying theme of Being and Time. A direct discussion of this idea appears in 1957, years after Heidegger abandoned the use of Dasein as the framework for the question of being in general. In his paper, `The Principle of Identity', Heidegger explores the concept of identity, and its relation to the unity of a multiplicity.71 He begins by examining the view that identity should

One as Transcendental and One as Number

39

be understood as the unity of a multiplicity, a view that originated with Kant. Heidegger rejects this interpretation of identity. He then considers in its stead the conception of identity as mutual belonging, which he derives from Parmenides' identi¢cation of thought and being. As I noted above, the idea of mutual inclusion is, I believe, already present in Being and Time, but within the framework of that work's ontological approach, Heidegger is unable to reveal this structure without undermining Being and Time as a whole. The important di¡erences between Heidegger's early ontology and the ontology of Being and Time raise the question of whether the unique linkage between ontology and mathematics is preserved in Being and Time. The issue is complex. In Being and Time there are two contexts in which number plays a key role. Counted time is one such context (though it should be kept in mind that there are other interpretations of time). The other context in which number plays a key role in Being and Time is its discussions of the interpretations of the being of entities implicit in contemporary work in mathematical physics. As I show in the coming chapters, these contexts, though seemingly peripheral, are actually of profound signi¢cance, and in general, the linkage between ontology and mathematics is not abandoned in Being and Time, but preserved. Indeed, the connection between counted time and Heideggerian ¢nitude, treated in the next chapter, can be seen as a direct outgrowth of the ideas presented in the early writings. Whereas, in the early writings, the categories were revealed by di¡erentiating them from the transcendental one, Heidegger's discussion of the relation between ¢nitude and counted time in Being and Time can be interpreted as demonstrating that for Heidegger, any such di¡erentiation presupposes ¢nitude. Hence, while the number^ontology nexus in Being and Time di¡ers in many respects from that found in the earlier writings, it nonetheless plays a similar role, and is a crucial aspect of the architectonics of Being and Time.

Notes 1. Not all the texts that Heidegger used were in fact written by Scotus. Today, it is widely believed that the two main works on which he based his interpretation should in fact be attributed to other philosophers: De rerum principio to

40

2. 3.

4. 5. 6. 7.

8. 9.

10.

11. 12.

13.

14. 15.

Being and Number in Heidegger's Thought Vital de Faur, and Grammatica speculativa to Thomas of Erfurt. See Motzkin 1982, 1022^23, and Frede 1993, 68 n. 9. Heidegger, GA 1, 59. Van Buren (1994, 58^9) mentions Heidegger's intense interest in mathematics, but does not pursue it as a possible path for interpreting Heidegger's early writings. Kisiel (1993) ignores this path as well. Heidegger 1913, 61. Frege 1894. Heidegger 1913, 165. Heidegger is addressing the arguments put forward by Husserl in the Prolegomena to the Logical Investigations, arguments that had, in the decade since the book's publication in 1900, been critiqued on a wide range of grounds. Notable among the criticisms was the claim that Husserl's position was itself guilty of begging the question. For an overview of the various critiques of Husserl's attack on psychologism, see Kusch 1994. It is interesting that Frege was of the opinion that not every version of psychologism was subject to criticism, but only inconsistent versions (Frege 1893, xvii). A decisive refutation is impossible, Frege argued, since the truth of the laws of logic cannot be proven. On the fundamental problems in Husserl's sweeping denunciation of psychologism, see FÖllesdal 1958. Heidegger 1913, 160^1. In Heidegger's 1912 article reviewing recent work in logic, he declares: `It is one thing if psychology is the fundamental basis for logic and for its validity; it is something else if psychology's role is to serve as a preliminary ¢eld of activity.' (Heidegger 1912, 29) Heidegger 1913, 167. Heidegger makes use of Husserl's `Philosophy as a Rigorous Science' in formulating his view of the relation between the logical and the mental, expressing it in terms of Husserlian ideas or essences. According to Husserl, the essence is that which remains invariant through possible changes in the object. This example is taken from Husserl's Logical Investigations; see Hua XIX, Investigation I, ½15. This view is altogether di¡erent from that put forward in Being and Time, where Heidegger identi¢es meaningfulness with the modes of time and temporality. Meaning is that which endows entities with temporal interpretative frameworks. Heidegger argues that the distinction between one as a transcendental attribute and one as a number demonstrates that `pure number lies outside empirical reality and cannot capture the historical dimension of its individuality' (Heidegger 1915, 262, emphasis in original). Heidegger 1916. The secondary nature of the distinction is retained in Being and Time and the later writings. But in these writings Heidegger develops a critique of

One as Transcendental and One as Number

16. 17.

18.

19. 20.

21. 22.

23. 24. 25.

41

epistemology based on revealing its ontological presuppositions. On Heidegger's critique of epistemology, see Richardson 1986. Heidegger 1915, 213. There is a signi¢cant break with the Kantian approach in Heidegger's positing of completely distinct ontological regions. For Kant, something that exists is an object of possible experience. On the Kantian scheme, there are no `logical' objects. This is why Heidegger must distance himself from the Kantian mode of deriving the categories. In so doing, that is, in keeping the logical realm ^ the realm of validity ^ distinct from the realm of possible experience, he is following in the footsteps of Lotze, Husserl and Lask. On Heidegger's place, vis-a©-vis this distinction, in nineteenth-century and early twentieth-century philosophy, see Brach 1996. It is interesting to note that from this perspective, Frege, in deriving the objectuality of objects from the logical analysis of sentences, is much closer to Kant than is the young Heidegger. Although the examination is carried out within the context of interpreting Scotus, the analysis is not solely historical, but seeks to show that Scotus' concerns remain highly relevant to the contemporary philosophical agenda. In view of this, and also in view of the fact that the writings Heidegger analyzes were not written by Duns Scotus, I will relate to the positions put forward in the book on Scotus as Heidegger's own. Heidegger 1915, 216. An argument against the possibility of de¢ning either being or the one was already suggested by Aristotle (Metaphysics III3, 998b21^7). But there is a crucial di¡erence between the approaches of Aristotle and Heidegger: for Aristotle, oneness characterizes being, but it is not clear whether it can also be a number, since number is primarily associated with multiplicity; see Physics IV 12, 220a26. Heidegger 1915, 225. The one^many connection Heidegger lays out here echoes the relation between a judgment and di¡erent acts of making that judgment proposed in the dissertation on psychologism, as noted above. The judgment is de¢ned as a logical object that is ¢xed relative to di¡erent acts of judging, that is, di¡erent statements (Heidegger 1913, 168). The judgment can thus be conceived as the absence of the many. Heidegger 1915, 231. Heidegger 1915, 248. Heidegger's distinction between one as a transcendental and one as a number is without precedent in the history of philosophy. Traditionally, the subject has been approached in two di¡erent ways. It has been claimed that one is not a bona ¢de number at all, but rather, numbers are premised on one in the sense of unit. This approach, advanced by Aristotle, is adopted by Husserl in his Philosophy of Arithmetic. The other position that has been taken is that one is a number, pure and simple, and has no non-numeric meaning. Frege is an

42

26. 27.

28. 29. 30. 31. 32.

33. 34. 35.

36.

37. 38. 39. 40. 41. 42. 43. 44.

Being and Number in Heidegger's Thought ardent proponent of this view at its most radical. Heidegger rejects both these approaches, as they negate the distinction between ontology and mathematics. If one is just a number, there is no distinction between ontology and mathematics: everything is countable. If, on the other hand, it is that which numbers are premised on and extrapolated from, then the one engenders homogeneity, and again every multiplicity is necessarily countable. Heidegger 1915, 234. There are, of course, di¡erences between Frege's approach and Russell's. Frege focuses on the extensions of concepts; for Russell, a class is a collection of the expressions that satisfy a certain propositional function. For more on these di¡erences see Lavine 1994, 63^5. On the various de¢nitions of set found in Cantor's writings, see Hallett 1984, 33^4. Cantor 1883, 204. This translation is adapted from that given in Hallett 1984, 33^4. Cantor 1883, 168. Cantor 1895, 282. The translation is adapted from that given in Hallet 1984, 33^4. This change in Cantor's perception of the importance of sets is attested to by the location of the de¢nition of set in the respective works. In the 1883 volume, the de¢nition appears in a note at the end of the book, whereas in 1895, it is the opening sentence. Cantor 1932, 443. Hua XII, 74. Husserl was a friend of Cantor's and a colleague at Halle from 1886 to 1900. On the a¤nities between their theories, see Ortiz-Hill 1994, 346^7. On their personal relationship, see Schuhmann 1977, 19, 22. Ortiz-Hill reads Frege's attack on Husserl as a response to Cantor's criticism of his Foundations of Arithmetic, see Ortiz-Hill 1994. For a comprehensive discussion of the two approaches to de¢ning sets, see Lavine 1994. In light of his analysis of the dispute, Lavine suggests a new construal of the crisis in the foundations of mathematics, arguing that it was generated by the Frege^Russell interpretation of set, not Cantor's interpretation. His awareness of the notion's far-ranging potential is re£ected in his comment that Principia Mathematica assumes set theory; see Heidegger 1913, 174n. See Klein 1926, 52. For an interpretation of Heidegger's work that sees it as continuing the approach of the Southwest school, see Friedman, 1996. Rickert 1911, 30. Rickert 1911, 43. Natorp 1910, 41. Natorp 1910, 53. Natorp 1910, 99^100.

One as Transcendental and One as Number

43

45. The fundamental series is the basis for both the ordinal and the cardinal numbers. It is, moreover, the basis for space and time. The link between the fundamental series and time is made through Aristotle's de¢nition of time as that which separates the earlier from the later (Natorp 1910, 107). The fundamental series is both the dividing line between the earlier and the later, and that which unites them. Since the fundamental series does not necessarily require a ¢rst element, and can be extended at both ends (by adding a predecessor to the one or a successor to the other), it is suitable for the temporal series, which does not have a ¢rst moment. Moreover, in the realm of numbers, the fundamental series can serve as a basis for both the natural and the whole numbers. We will see in the next chapter that in Being and Time, far more explicitly than in his early writings, Heidegger utilizes the fundamental series as a foundation both for numbers and for a speci¢c interpretation of time. 46. Natorp 1910, 114^15. Natorp's claim rests largely on the fact that in German, `individual' (einzeln) derives from the root `one'. A similar argument is o¡ered by Poincare¨ (1905, 823). Couturat, replying to Poincare¨, emphasized that the circularity here is only apparent, since the number one is not assumed in its de¢nition, but only the notion of unity (Couturat, 1906, 224^6). For an excellent analysis of Poincare¨'s argument and its relevance to the programme of reducing mathematics to logic, see Goldfarb, 1988. 47. Husserl, Hua XVII, Prolegomena, ½67. 48. Husserl, Hua XVII, Prolegomena, ½69. 49. In his 1901 lecture, `The Imaginary in Mathematics', Husserl provides a criterion for a manifold's being de¢nite (bestimmt): a de¢nite manifold is a manifold that is completely determined formally without any of the objects of which the manifold is comprised playing any role whatever in this determination. The criterion Husserl o¡ers for a completely determined manifold is based on the notion of the completeness of the axiomatic system that describes the manifold: every assertion about the manifold follows from or contradicts the axioms (Hua XII, 441). 50. Husserl, Hua XVII, Prolegomena, ½70. 51. In the 1921 lectures, in which he presents a phenomenological interpretation of Aristotle, Heidegger argues that formal logic is ultimately a speci¢c ontological region (GA 61, 20). I am suggesting that this claim be interpreted as follows. Formal logic presupposes the concept of set, which in turn presupposes the concept of number. Given that numbers constitute a speci¢c ontological region, formal logic is itself an ontological region. 52. Husserl, Hua XVII, Prolegomena, ½62. 53. Husserl's distinction between ontological categories and categories of meaning is one of the elements that di¡erentiate his philosophy from the Fregean analytic tradition. Frege's philosophy is usually interpreted as claiming that the ontological categories are extracted through logical analysis of propositions (Dummett 1981, 241; Ricketts 1986), although there are also di¡erent

44

54. 55. 56.

57. 58.

59. 60. 61. 62.

63. 64. 65.

66.

Being and Number in Heidegger's Thought interpretations of Frege (Bar-Elli 1996, 45^54). Ernst Tugendhat has put forward an argument to the same e¡ect, recommending that ontology be replaced by formal semantics. The argument clearly has much to do with his `conversion' to analytic philosophy; see Tugendhat 1982, lecture 3. For a recent defence of Husserl's position against Tugendhat, see Soldati 1999. Husserl, Hua XVII, Prolegomena, ½50 (Findlay translation). Heidegger GA 21, 58^9. In the 1925 lectures, Heidegger links Husserl's error to the views of Lotze, and in particular, to the connection Lotze makes, in the third volume of his Logic, between the general and the logical. In support of this contention, Heidegger adduces a 1903 article written by Husserl as a self-commentary on the Logical Investigations, in which Husserl states that his treatment of logical objects was in£uenced by Lotze's interpretation of Plato's Ideas (Husserl Hua XII, 156). A similar association between sets and the Ideas is made in Cantor's ¢rst de¢nition of set; it is likely that Cantor's position in£uenced Husserl. Husserl, Hua XVII, introduction to vol. 2, ½ 2. This position is quite di¡erent from that held by Kant, according to which the categories belong to the logical realm, and intuition to the sensory. Whereas for Husserl the divide is fuzzy, Kant construes it as absolute. On the constitution of regions and things in general in the Ideas, see Husserl Hua III, ½9, ½150. This will be discussed in Chapter 3 below. Heidegger 1915, 318. Kisiel supports this interpretation of Heidegger's early writings, see Kisiel 1993, 27. GA 60, 55^65 (Eng. trans. 38^45). Kisiel (1993) regards `formal indication' as a key notion for understanding Heidegger's agenda from 1919 up to the publication of Being and Time. On the role of the notion of `formal indication' in Heidegger's early thought, see Dahlstrom 2001, especially chapters 4.0 and 5.5. Kisiel, 1993, 19^20. Dedekind 1888, 44; Hilbert 1904, 131. In the book on Duns Scotus, Heidegger distinguishes between logic and mathematics in the following way: the province of mathematics is quantity, that of logic, intentionality. But there is an asymmetry implicit in Heidegger's distinction ^ mathematics is coextensive with quantity; but intentionality is not limited to the realm of logic. See Heidegger 1915, 283. Heidegger is closer to Dedekind than to Cantor inasmuch as Dedekind holds that the primary characteristic of numbers is that they are ordered, whereas Cantor maintains that for trans¢nite numbers, the ordinal and cardinal aspects are on a par. For a comparison of the views of Cantor and Dedekind on the ordinal aspects of numbers; see Ferreiros 1995. It is quite probable that Heidegger's a¤nity with Dedekind grew out of Natorp's stance in The Logical

One as Transcendental and One as Number

67. 68.

69.

70. 71.

45

Foundations of the Exact Sciences. Natorp upholds the primacy of the ordinal aspect, and thus favours Dedekind's conception of number; see Natorp 1910, 124^8. Natorp 1910, 4. Hilbert 1904, 131. In the article, Hilbert proceeds from things in general to the number one, from which, using addition, he constructs the rest of the numbers `by combining the object 1 with itself '. The change in Heidegger's position can already be discerned in his 1919 lectures. Its three principal aspects are the following: 1. Heidegger considers the question of philosophy as primordial science in relation to the notion of a worldview. He does so in order to reveal the being of the investigator as a basic part of the project of a primordial science. This was not the case with respect to his prior ontological framework. 2. In attempting to grasp the meaning of the thing in general, we always grasp the meaning of a speci¢c thing that has concrete content (GA 56/57, 68). Hence, any discussion of the thing in general will ultimately reduce the question of being to that of the being of a certain ontological region, without any awareness that the question has been quali¢ed in this manner. 3. The question, `Is there something?' (Gibt es etwas?), is always raised with respect to the self (ich) (GA 56/57, 68). For example, the discussion in Fundamental Concepts of Metaphysics, GA 29/30, 65. Heidegger 1957.

2

Number and Time in Being and Time

The transition from the early writings to Being and Time marks a change in the relationship between ontology and mathematics. Whereas, in the early writings, the role played by mathematics in the formation of the ontological concepts is manifest, in Being and Time it is not conspicuous. The ontological question is no longer a question of the characteristics of being as being, but rather, that of the meaning of Being (Sinn von Sein). This question is inextricably linked to the question of the being of its formulator ^ Dasein.1 In Being and Time, the formal general question of unity and multiplicity is no longer the focus. Every multiplicity is a speci¢c multiplicity given in a speci¢c way.2 The modes in which multiplicities are given are the modes in which the question of the meaning of Being can be interpreted, and are, ultimately, derived from di¡erent interpretations of time. These interpretations must satisfy the constraints of a general framework that determines the di¡erent possibilities for interpreting time ^ fundamental ontology ^ which is not an empty formal framework, but determined by Dasein. The temporal correlate of fundamental ontology is primordial temporality (ursprÏnglichen Zeitlichkeit). These di¡erences reduce the role of mathematics in ontology. In the early writings, the principal connection between mathematics and ontology had to do with the relation between mathematics and formal ontology, and the question of how to construct a formal ontology without limiting ourselves to currently known mathematics. But this question is no longer of any relevance in Being and Time. Nevertheless, as I will argue in this chapter, mathematics, and particularly the interpretation of numbers, is by no means peripheral to Being and Time, and indeed, constitutes an important interface between the various modes of interpreting time. That is, the role of number is to connect primordial temporality to the concepts of time it encompasses.

Number and Time in Being and Time

47

The role of mathematics, then, and especially number, is to link fundamental ontology, as understood in Being and Time, and the ontological regions demarcated by the di¡erent interpretations of time.3 Thus, focusing on the question of the time^number nexus can resolve one of the central problems in Being and Time: the interconnectedness and interdependence of the di¡erent strata of time.4 Furthermore, the interpretation that will be presented in this chapter makes it possible to reassess another important question, namely, the relationship between Being and Time and developments in the sciences of the period. Our examination of the time^number nexus in Being and Time will be prefaced with a look at Heidegger's treatment of the subject in the early writings, as this will provide a useful perspective from which to approach the interpretation of time in Being and Time.

The distinction between number and time in the early writings In his 1916 article on the concept of time in the science of history, Heidegger distinguishes between physical time and historical time.5 He arrives at this distinction by observing the way in which physics and history function as sciences in practice. His discussion of the two concepts of time starts with physics, as Heidegger seeks to develop the concept of time underlying the science of history by contrasting it to that underlying physics. Heidegger maintains that the aim of physics is to reduce all the phenomena to fundamental laws of general dynamics that can be determined mathematically.6 The concept of time that makes it possible for physics to achieve this objective is that of measurable time, which Heidegger characterizes thus: As soon as time is measured ^ and only as time that is measurable and to be measured does it have a meaningful function in physics ^ we determine a `so many'. The registering of the `so many' gathers into one the points of time that have until then £owed by. We as it were make a cut in the time scale, thereby destroying authentic time in its £ow and allowing it to harden. The £ow freezes, becomes a £at surface, and only as a £at surface

48

Being and Number in Heidegger's Thought

can it be measured. Time becomes a homogeneous arrangement of places, a scale, a parameter.7 The concept of time that is a fundamental premise of physics, therefore, itself presupposes that time is a homogeneous medium amenable to being measured, that is, that we can count homogeneous units of time. Heidegger contrasts time conceived of as measurable to the time that serves as the basis for the science of history. The concept of historical time is not, as is that of physical time, homogeneous. Historical time cannot be expressed by a mathematical series, for there is no general rule establishing how times succeed each other. Historical time has, of course, succession, but lacks location in a series.8 That is, we speak of the end of an era and the beginning of a new era, but we have no quantitative measure of `an era'. It is certainly possible to describe historical events within a framework of quantitative time, but what makes these events historical is not some kind of homogeneous time. I would like to suggest that the distinction between these two concepts of time is an application of the distinction between the transcendental one and one as a number: a historical event is one in the sense of the absence of multiplicity, whereas time as conceived by physics is the projection of the number series onto time. It is important to realize that in moving from the number series to physical time, the numbers serve both to order and to count time. The distinction between the time of nature and historical time is, in many respects, quite similar to Bergson's distinction between time and duration.9 These two temporal concepts correspond to two di¡erent kinds of multiplicity, quantitative and qualitative. The former, quantitative multiplicity, is associated with time as counted, which is space; the latter, qualitative time, is associated with identity and difference, but is not amenable to quanti¢cation.10 This a¤nity with Bergson is interesting in light of the critique of Bergson in Being and Time, which will be discussed below. It will be recalled from the last chapter that Heidegger di¡ered with the neo-Kantian thinker Heinrich Rickert over the notion of `one'. Heidegger's stance here reveals another issue on which he

Number and Time in Being and Time

49

di¡ers with Rickert. The di¡erence stems from the fact that, according to Heidegger, time can be interpreted in several ways. Interpreted one way, time is related to the stratum of the transcendental one; interpreted another way, it is related to the stratum of number. Heidegger sees physical time as derived from numbers, and not the other way around, because physical time is the outcome of quanti¢cation or counting, and all counting presupposes numbers as objects. Rickert, on the other hand, upholds but a single notion of time. Time presupposes a homogeneous medium, he maintains, hence the notion of number. For Rickert, only numbered time is time. This disagreement nonetheless highlights a basic harmony between their conceptions: for both Rickert and Heidegger, number is prior to time (in Heidegger's case, to physical time). Natorp, too, embraces the position that time presupposes number. The di¡erence between his view and the accounts of Rickert and Heidegger is that he sees the connection between number and time as based, ¢rst and foremost, on ordinal numbers and the fundamental series. The measure of time, which is also dependent on number, is secondary, and dependent on the fundamental series and the successive ordering of events.11 Despite the di¡erences between them, in the very early writings, Heidegger is closer to Rickert and Natorp than he is to Kant. For on the Kantian outlook, there is an intimate connection between number and time. Kant asserts: `Number is therefore simply the unity of the synthesis of the manifold of a homogeneous intuition in general, a unity due to my generating time itself in the apprehension of the intuition'.12 In taking this position, Kant is linking number and time inextricably, in a manner whereby number is not prior to time. In fact, Kant's discussion of time in the `Transcendental Aesthetic' makes no reference to number whatsoever. Number enters the discussion only in the context of discussing the form of time, but not in discussing time as a form of intuition.13 This position on the relationship between number and time is very di¡erent from that taken by Heidegger in his very early writings. Another question on which their views di¡er is the connection between numbers and counting. While Kant upholds an intimate

50

Being and Number in Heidegger's Thought

connection between number and the act of counting, for Heidegger numbers are not fundamentally associated with counting. Counting is merely the result of projecting a series of numbers onto an essentially heterogeneous multiplicity. The ontology proposed by the young Heidegger, therefore, is that of a hierarchy of interdependent strata, the primary stratum being the ontological stratum of the transcendental one, a subordinate stratum being that of the category of quantity and the numbers, and the lowest stratum being the concept of physical time. It is di¤cult to tell, from this picture, whether there is indeed a general concept of time that is prior to the distinction between physical and historical time. That is, it seems unclear whether time is prior to the application of mathematics to it, or whether physical time is entirely a product of projecting the number series onto a homogeneous medium, and has no independent existence. It appears that Heidegger's position is that time has a status akin to that of objects in the world. Time, like the empirical stratum, is a realm to which the category of quantity is applicable, but which is independent of it. Heidegger is committed to this premise, since time is not necessarily related to projection of the number series. The existence of two di¡erent notions of time, historical time and physical time, entails a distinction between these notions and a general concept of time in which they are grounded. But in the very early writings, there is no discussion at all of a concept of time that is independent of the ontological framework within which it is interpreted. This shortcoming in Heidegger's very early thought is corrected in Being and Time, which does postulate a stratum of time that is prior to the distinction between historical time and time in nature: But still more elemental than the circumstance that the `time factor' is one that occurs in the sciences of history and nature, is the fact that before Dasein does any thematical research, it `reckons with time' and regulates itself according to it. And here again what remains decisive is Dasein's way of `reckoning with its time' ^ a way of reckoning which precedes any use of measuring equipment by which time can be determined. The reckoning is prior to such

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51

equipment, and is what makes anything like the use of clocks possible at all.14 Heidegger's view, then, is that there is some sort of reckoning taking place with respect to this incipient time that precedes the division into the time of the historical sciences and that of the physical sciences. But what exactly is this reckoning, and how is it connected to number? Moreover, what is the connection between time and number in Being and Time? To answer these questions, we must ¢rst describe the di¡erent strata of time in Being and Time. Three basic strata of time are distinguished in Being and Time: ordinary time, world time and primordial temporality. The ordinary conception of time is its interpretation in terms of a sequence of `now's. Ordinary time is also the basis for the measurement of time. World time is time as dateable, spanned and public. One of its aspects is a use of a natural clock, or what Heidegger calls `time reckoning', as a means of orientation in time that does not necessarily involve measurement. Primordial temporality is ¢nite time. It is based on Dasein's being-towards-death, and its basic temporal dimension is the future. The three layers of time are linked by dependency relations: ordinary time presupposes world time, and world time presupposes primordial temporality. In the context of the present chapter, the question that arises is where number is situated. Is it germane only to the ordinary interpretation of time, or does it also play a role in world time? Answering these questions will suggest a new argument in support of Heidegger's claim that ordinary time presupposes primordial temporality. My analysis of these questions will unfold in three stages. First, counting and reckoning will be distinguished from measuring. Second, the notion of counting, having been demarcated, will be explicated in terms of Heidegger's interpretation of Aristotle. Lastly, the role of number in the world time^primordial temporality relationship will be examined. My interpretation of this relationship will be based on a comparison between Heidegger's position on the relationship between number and time, and some issues at the heart of Brouwer's intuitionism.

52

Being and Number in Heidegger's Thought

Counting, measuring and reckoning According to Heidegger, `measuring is constituted temporally when a standard which has presence is made present in a stretch which has presence'.15 Measured time is conceived as a present-at-hand multiplicity of `now's. Hence measured time presupposes the ordinary notion of time mentioned above. A clock is Heidegger's main example of a standard for measuring time. Use of a clock is connected to number, since using a clock involves counting the positions of the pointer. But not every use of a clock constitutes measuring. The clock is also something ready-to-hand (zuhanden), use of which is similar to the dating of events by invoking particular points of reference. For example, the statement `when we last met for lunch, I was tired' uses the meeting at the restaurant to date the tiredness event.16 Such dating does not necessarily entail measurement of time. A clock can be used for dating in the same way. I can say, `At 3 in the afternoon on August 1, when we last met for lunch, I was tired.' This use of the clock does not necessarily involve measurement, since it refers to a certain location of an event and not to its distance from the present. In Being and Time there are thus two aspects of time that involve numbers: measuring and reckoning. Measuring necessarily involves numbers, while the connection between number and reckoning is much looser. At one point Heidegger claims that reckoning is distinct from the application of numbers to dating;17 elsewhere he maintains that counted time is an aspect of world time, and not only of the ordinary notion of time.18 Hence, according to Heidegger, the relationship between time and number is the following. In time as ordinarily conceived, the role of number is to measure. In world time, however, reckoning and orientation in time do not necessarily involve numbers, though it is certainly possible to use a clock with numerals in order to orient oneself in time. The use of number in the temporal context, then, is not limited to measurement. How is this other use of time to be characterized? Heidegger does not o¡er an explanation, but I think an explanation can be extracted from his interpretation of Aristotle, which will be analyzed in the next section.

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53

Thus the approach to numbers taken in Being and Time is altogether di¡erent from that taken by Heidegger in the early writings. Whereas in the early writings counting is only carried out within the framework of the category of quantity, in Being and Time a new option comes to light: the possibility of counting time in the context of our reckoning with time, which is not derived from time as a homogeneous measurable medium. This development can be attributed to the revelation of `readiness-to-hand' in Being and Time. Readiness-to-hand, as an ontological stratum, is associated with world time and hence with reckoning. What is Heidegger's notion of number as it emerges from the distinction between measurement and counting? While in the earlier writings number was characterized as a certain region of entities, the conception put forward in Being and Time is more complex. In Being and Time there are not only di¡erent regions of entities, such as nature and history, but di¡erent meanings of being as well (present-at-hand, readyto-hand, etc.). It follows that number can be interpreted in several ways: in measurement, as present-at-hand, in counting, as ready-tohand, and so on. But is there a privileged or basic interpretation of number? This question is critical. Consider the case of Dasein. Dasein can be interpreted as present-at-hand, and has been so interpreted by the philosophical tradition. But although this interpretation has indeed been o¡ered, there is nonetheless a more basic interpretation of the meaning of Dasein's being, namely, that proposed in Being and Time. The question that arises in our context is analogous: does number also have such a privileged interpretation? Granted, there is no explicit mention of a privileged interpretation in Being and Time, but not long after the publication of Being and Time, Heidegger does indeed invoke such an interpretation, and as I will show, it is in fact alluded to in Being and Time. In a series of lectures entitled `Introduction to Philosophy', delivered in 1928/9, that is, soon after the publication of Being and Time, Heidegger explicitly characterizes the mode of being of number as subsistence (Bestand).19 As I will show, Heidegger endorses this idea, and claims that there is a meaning of being that is particularly appropriate to mathematical entities. He does so in the context of his discussion of Descartes' notion

54

Being and Number in Heidegger's Thought

of the world. Heidegger interprets Descartes' notion of res extensa as re£ecting Dasein's approach to mathematical entities, which are interpreted as that which `enduringly remains' (stÌndigen Verbleibs).20 Later, in comments he added in the margins of his copy of Being and Time, Heidegger writes that Descartes' position was `oriented to mathematics as such'.21 Another hint of the existence of a privileged interpretation of numerical being can be found in the introduction to Being and Time, where Heidegger claims that numerical and spatial relations are `non-temporal'.22 That is, these relations have a special, non-temporal mode of being. It would appear, then, that there is a privileged sense of numerical being, namely, Heideggerian subsistence. Nevertheless, because of the dependency of numbers and all `non-temporal' entities on primordial temporality, numbers must undergo `temporalization' via the notion of counting as described above.23 In the thirties, when Heidegger abandons the claim that the framework of Being and Time su¤ces for all conceivable examinations of the question of being, he returns to the interpretation of mathematical entities as enduringly remaining. As I will show in the next chapter, Heidegger then proposes a new framework from within which to interpret and delimit the meaning of numbers. This suggests that in Being and Time he saw primordial temporality as providing such a framework.

Aristotle and Heidegger on the relationship between number and time The question now facing us is how to characterize the notion of counting in Being and Time. Reference to the Aristotelian concept of time, and Heidegger's interpretation of it, can assist us in answering this question. For Heidegger's characterization of counted time has a pronounced a¤nity with the Aristotelian de¢nition of time. Heidegger characterizes counted time as follows: `This time is that which is counted and which shows itself when one follows the traveling pointer, counting and making present in such a way that this makingpresent temporalizes itself in an ecstatic unity with the retaining and

Number and Time in Being and Time

55

awaiting that are horizontally open according to the ``earlier'' and ``later''.'24 This position is put forward as an existential-ontological interpretation of Aristotle's de¢nition of time. To clarify Heidegger's position, we must ¢rst examine Aristotle's notion as he himself presents it, and only after doing so proceed to an examination of Heidegger's interpretation of the Aristotelian concept. We will then turn to the question of how to interpret the Heideggerian concept of counting that is not measuring. According to Aristotle, time is just this ^ `number of motion in respect of ``before'' and ``after'' '.25 This raises the question of the role of number in de¢ning time. The ¢rst answer that suggests itself is that the role of number is to enumerate the `now's. This view is presented by Aristotle in the paragraph preceding his de¢nition of time. Aristotle there states that time is said to exist when the mind pronounces the `now's to be two ^ one before, and one after. These remarks, together with Aristotle's views on number, lead to the conclusion that the `now's are the units of time. But if the `now's are the units of time, then time is made up of them. But this idea is inconsistent with another of Aristotle's positions, namely, that the `now' is not a part of time, and time is not made up of `now's.26 Thus, de¢ning time in terms of counting `now's is problematic. A solution to this tension in Aristotle's position is to interpret it as maintaining that time is number in the sense of the ordering of before and after.27 Construal of the ordinal aspect of number as primary allows for the emergence of the use of number for measuring. The ordinal aspect of number is that which motivates Aristotle's linking number and time directly, in contrast to the indirect linkage between number and movement, or size and number. The very distinction between the categories of place, time and quantity points to the fact that the link between time and number is not solely a quantitative connection. That is, time is not merely the quantity of movement, since if it were, there would be no explanation for the distinction between the category of time and the category of quantity. Aristotle embeds the irreversible order of time in number in order to avoid a circular de¢nition that itself employs the concept of time. Such a de¢nition would be inevitable were we to interpret `before' and `after' relative to time itself.28

56

Being and Number in Heidegger's Thought

This interpretation of the basic role of number in Aristotle's de¢nition of time helps to clarify some of the tensions it embodies. As we saw, the `now' is both the boundary between before and after, and that which number counts. This poses a problem, as the two cannot be the same, since the `now' that serves as a boundary has no quantity: however many `now's ^ in the boundary sense ^ we concatenate, they will not yield a quantity of time. My suggestion is that the primary role of number is to mark the limits of a movement (its beginning and end) by marking the `now' of its beginning and the `now' of its end. At this preliminary stage there are as yet no homogeneous units by means of which counting is carried out. Through this process of marking the beginning and the end of movements, this process of numbering, units can be established and counting can function as measurement.29 The main problem with this interpretation is that it appears to be incompatible with Aristotle's account of number. In the Metaphysics, Aristotle contends that all number presupposes units.30 Applying this position to the de¢nition of time, we see that in attributing number to time, units are presupposed. Hence, counted time turns out to be a homogeneous domain merely by virtue of being counted. However, a solution to this problem can be found in Aristotle's distinction between the number of things that are counted, and the number with which we count them.31 Aristotle argues that time is the former, that is, the number of things counted, and not the number used in counting them. The distinction can be interpreted as follows: number that counts things calls for a ¢xed unit of measurement equally applicable to all counted things, whereas number in the sense of the number of things counted is not constant, and therefore does not call for such a ¢xed measure.32 In the case of number as counted, number functions as a sign of the ever-di¡erent now. To every `now' there corresponds a number that marks its di¡erence from the other `now's.33 The interpretation of Aristotle that has been presented here corresponds to that o¡ered by Heidegger himself. (In terms of its presentation, it is not identical to Heidegger's interpretation; for example, Heidegger does not use the notion of `ordinal number'. But I think it captures Heidegger's intent.) In the interpretation of Aristotle he presents in The Basic Problems of Phenomenology, Heidegger

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clearly distinguishes between counting and measuring. Counting, far from being identical to measuring, is a precondition for measuring.34 Heidegger does not make explicit use of the distinction between ordinal and cardinal numbers, but in my opinion, this distinction is helpful in clarifying his position. We now have a sense of Heidegger's distinction between counting and measuring. This distinction arises not only with reference to his interpretation of Aristotle, but also in his critique of Bergson. As was noted above, in his early writings Heidegger's stance was generally close to that taken by Bergson. In Being and Time and other writings from this period, however, Heidegger is critical of Bergson. As I will show below, Heidegger's new distinction between counting and measuring plays a crucial role in the critique. More generally, Heidegger's new understanding of the relation between number and time plays an important role in the transition from Heidegger's early position on time to his mature view in Being and Time. Bergson claimed that counted time was space, basing this claim on his analysis of number. Every number necessitates a sum of units, and this sum necessitates spatiality.35 Let me explain Bergson's reasoning. In time, there is only succession, and succession does not su¤ce to constitute number. This is so because we need to somehow save each passing moment so that the multiplicity of moments can be tallied up. By virtue of being so `saved', traces of the £eeting moments are left behind. That is, on Bergson's view, the counted units must have some sort of continuing presence, a presence that cannot exist in pure time. It might be thought that only material objects can be counted, but Bergson shows that counting is relevant in the case of conscious states as well ^ that conscious states can indeed be counted, though this is by no means a fundamental aspect of consciousness. In order for conscious states to be counted, they must be symbolically represented in some way. Because representation calls for abstract units that do not merge with one another, which is precisely the spatial quality of impenetrability, such symbolic representation must necessarily involve spatiality.36 Indeed, Bergson maintains that spatiality and number are mutually interdependent. Bergson understands impenetrability to be a logical necessity that follows, ultimately, from the nature of number.37 That is, when we count

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Being and Number in Heidegger's Thought

conscious states, we relate to them as if they were a material manifold, and thus we assume impenetrability. Hence, Bergson reasons, time, as the domain within which we count and distinguish, is none other than space. This stance is based on a conception of space as a homogeneous medium. Indeed, on his view every homogeneous medium is modelled on space. Since Bergson's main argument is based on his interpretation of number and on the connection between time and number, Heidegger construes Bergson as critiquing Aristotle's de¢nition of time. But according to Heidegger, Bergson does not really understand Aristotle, and as a result, his notion of duration does not succeed in providing an alternative to Aristotle's notion of time.38 Heidegger does not explain exactly where he thinks Bergson went wrong, but I would argue, based on the interpretation of Aristotle proposed above, that Heidegger is referring to Bergson's misunderstanding of the relation between number and time.39 This relation is not that of measurement, hence there is no need to save any element of the numbered `now's. Counting does not in and of itself necessitate a yardstick that `freezes' time.40 Having established that Heidegger has a distinctive notion of counting, the question that arises is that of its role in Being and Time. I will approach this question by ¢rst examining the degree to which the number^time relation, as presented by Aristotle, also applies to Heidegger's own notion of time. Although, as we saw above, Heidegger, like Aristotle, distinguishes between counting and measuring time, there is an important di¡erence between their positions. Heidegger does use the ordinal characterization to distinguish between measurement and counting, but the two thinkers di¡er with regard to how they conceive the role played by number. According to Aristotle, number plays two roles in the de¢nition of time: First, as we saw above, number functions as a means of distinguishing between earlier and later. Second, number is the soul's contribution to the de¢nition of time.41 This follows from Aristotle's understanding of the connection between number and time as a connection between an element in the soul ^ number ^ and an element outside the soul ^ change. In Heidegger's own framework, however, number does not play these two roles. In Being and Time numbered time presupposes world time and primordial temporality. Moreover, Heidegger claims

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that Aristotle's notions of earlier and later, in addition to connoting `order', necessarily have a temporal connotation.42 This is part of Heidegger's claim that Aristotle's notion of time presupposes primordial temporality. In Heidegger's own account of temporality, number does not serve to connect an element of the soul to the physical world. With respect to the relation between the soul and time, Heidegger's notion of time is not a synthesis between a subjective and an objective component. Indeed, Heideggerian temporality is neither subjective nor objective. Time is more objective than any object, since it is a precondition for the appearance of objects; and it is more subjective than the subject, inasmuch as it is intimately linked to care (Sorge).43 Given that in Being and Time number does not play the role it did in Aristotle's de¢nition of time, there are two possibilities for explicating the role of number in Heidegger's account of time: 1. Number is a marginal element, even in world time. Hence we cannot speak of counted time as a stratum that is additional to world time and primordial time. On this interpretation, Being and Time must be understood as positing an absolute disparity between the sciences and mathematics, on the one hand, and the basic ontological framework, on the other. 2. Heidegger sees number as playing a role in the framework of the temporal strata, but quite a di¡erent role from that envisaged by Aristotle. Heidegger scholars have generally taken the ¢rst interpretive route. Thus, for example, BÎhme maintains that for Heidegger, in contrast to Aristotle, number is alien to time,44 and number does not play any signi¢cant role in Blattner's meticulous reconstruction of Heidegger's notion of time.45 The interpretation that I am about to put forward, however, takes the second route. That is, in my estimation, in Being and Time number ¢lls several important roles, roles quite di¡erent from those it ¢lled in earlier theories of time, such as Aristotle's. What, then, is the role of counting in Being and Time? First, number, which in the introduction to Being and Time was characterized as non-temporal, can now, via the notion of counting, be integrated into the framework of primordial temporality. But it is the second role of number that is the most central: number links the three strata of

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time: primordial temporality, world time and ordinary time. That is, counting connects primordial temporality, which is essentially ¢nite, to measured time, which is in¢nite. By virtue of this connection, in¢nite time is dependent on ¢nite primordial time. In the following sections, I will present an argument in support of this claim.

Finitude and in¢nity Heidegger's interpretation of Aristotle and Bergson makes it clear that Heidegger sees counting and measurement as fundamentally different from each other. The di¡erence is rooted in the di¡ering characteristics of the various strata of time. Measured time is in¢nite in the sense that it is unbounded; primordial time, on the other hand, is ¢nite. Heidegger declares that `Only because primordial time is ¢nite can the ``derived'' time temporalize itself as in¢nite.'46 He must thus provide an account of how it is that in¢nite time presupposes the primordial ¢nitude of time. A ¢rst step in interpreting this claim is to understand Heidegger's conceptions of ¢nitude and in¢nity. Two basic conceptions of in¢nity recur throughout the history of western thought. One is that of the incomplete in¢nite, that is, the potentially in¢nite. When time is said to be in¢nite, this generally refers to the potentially in¢nite. Time is in¢nite in this sense in that it cannot be exhausted; to use the Aristotelian formulation, the in¢nite is that which `always has something outside it'.47 On this conception, the ¢nite is the complete, the actual, whereas the in¢nite is the partial, lacking an end or boundary, and existing only within the realm of the possible. The other conception of in¢nity is that of the perfect in¢nite, that is, that which cannot be added to. This is, then, a notion of the actually in¢nite. On this conception, the ¢nite is the imperfect. Regarding the relation between the ¢nite as imperfect and in¢nity as actual, Descartes argued that the imperfect presupposes the perfect. That is, to understand ourselves as imperfect and non-omniscient, we must have the concept of God's in¢nity. Hegel too upheld this dependence: the ¢nite, in that it is bounded, presupposes the in¢nite as perfect. The relation between the ¢nite and the in¢nite as perfect is

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not, however, that of opposites, but rather, that of part to whole. Hegel makes an additional claim, namely, that the two conceptions of in¢nity are themselves linked, in that the potentially in¢nite, which is the `bad' or negative in¢nite ^ the concept of the in¢nite used in mathematics ^ presupposes the in¢nite as perfect. According to Hegel, the in¢nite as perfect is not to be characterized in quantitative terms.48 Given these two conceptions of ¢nitude and in¢nity, where is Heidegger's position situated? Heidegger's position is based on a unique conception of ¢nitude that does not characterize it in terms of its contrast with in¢nity. On the contrary, every conception of in¢nity presupposes this radical ¢nitude. Heidegger's ¢nitude is best understood as a boundary or demarcation that is not de¢ned in relation to what lies beyond it. `The ¢nitude of Dasein can be shown and radically clari¢ed only out of Dasein itself.'49 In Being and Time, the elucidation of ¢nitude is carried out by invoking death. Death is not understood from a vantage point beyond death, but rather, from within its meaning for Dasein. `Death is certain (Gewissheit), and yet indeterminate (unbestimmt).'50 It is certain that we will die some day, but we do not know when this will transpire. The ¢nitude that is linked to death is incomplete; it is being-towards Being-a-whole without being complete. Death, as the end of Dasein, is the uttermost possibility (Ìusserste MÎglichkeit). Death is not the end of a road, nor is it the most mature stage of development, but rather an impossible possibility. That is, from Dasein's perspective, death is always a possibility. The realization of this possibility, however, marks the end of Dasein, and hence Dasein cannot realize it. In this sense, it is impossible. Heidegger's conception of ¢nitude as incompleteness is closer to Descartes' understanding of it than to Aristotle's. But contrary to Descartes, Heidegger does not endorse the view that this conception of ¢nitude presupposes in¢nitude as perfection. Furthermore, the notion of in¢nity in Being and Time is that of potential in¢nity, which Heidegger interprets in temporal terms: If one directs one's glance towards Being-present-at-hand and notBeing-present-at-hand, and thus `thinks' the sequence of `nows'

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through `to the end,' then an end can never be found. In this way of thinking time through to the end, one must always think more time; from this one infers that time is in¢nite.51 But what about Descartes' conception of the in¢nite? Heidegger does not address this point directly, but his position can be inferred from other contexts.52 Heidegger reads Descartes as upholding the idea of the perfect in¢nite: an in¢nite understood as presence, presence being understood as `a de¢nite mode of time ^ the ``Present '' '.53 Since we can understand the temporal present only against the background of the past and future, it follows that the Cartesian^Hegelian conception of perfection presupposes incompleteness. This demonstrates the dependence of the in¢nite as perfection on the potentially in¢nite. But it does not yet demonstrate the dependency of potential in¢nity on Heideggerian ¢nitude. How can this dependency be proved? There are two options: to show how the potentially in¢nite can be arrived at from Heidegger's notion of ¢nitude, or to prove that the notion of the potentially in¢nite necessarily presupposes Heideggerian ¢nitude. In Being and Time, only the former approach is taken. Heidegger claims that in¢nite time results from Dasein's £ight (Flucht) from death.54 Since this £ight is a mode of relating to death, the conception of time as in¢nite can be arrived at from ¢nitude and death. That in¢nite time is not to be accorded primacy is demonstrated by the fact that it is an interpretation of death, but does not itself contain the entire spectrum of interpretations of death. It follows that in¢nite time is but secondary.55 This line of argument only shows how, given the primacy of ¢nite time, an interpretation of time as in¢nite can emerge, but Heidegger's claim as quoted above requires the second approach, namely, establishing that in¢nite time necessarily presupposes Heidegger's conception of ¢nitude.56 Unless such an argument is provided, one of the key points of Being and Time, namely, the dependence of all the possible interpretations of time on Dasein's primordial temporality, is left without any convincing support. In the coming sections, I present an argument for Heidegger's claim that in¢nite time presupposes Heideggerian ¢nitude. Though not put forward by Heidegger himself, my argument is based on Heidegger's conception of counting

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and ¢nitude, but also draws on Brouwer's intuitionistic approach to the foundations of mathematics.

Brouwer's intuitionism The writings of L. E. J. Brouwer set out a thoroughgoing reformation of mathematics and its foundations. The details of his position changed over the years, but some theses always remained central to his approach: 1. Mathematics is a pre-linguistic activity; language is a later stage of human intellectual development. 2. The source of mathematical activity is the basic intuition of time, namely, the intuition of two-oneness ^ the separation and uni¢cation of moments of life. 3. Logic does not lay down preconditions for mathematics.57 Hence logic must accommodate mathematical activity. It follows that the law of the excluded middle is not a valid logical principle, as it is not valid for all mathematics, but only for speci¢c areas of mathematics ^ those that are ¢nite. 4. Hence mathematics must undergo a fundamental revision so as to be in conformity with principles (1)^(3). Brouwer reconstructs mathematics on the basis of the intuition of twooneness. This intuition is generated out of `the falling apart of moments of life into qualitatively-di¡erent parts, to be reunited only while remaining separated by time'.58 Brouwer generates all the ordinal numbers on the basis of this fundamental intuition. It is also the basis for the linear continuum as a `between' that cannot be interpreted as a collection of units. The series that is generated from the primordial intuition is that of the natural numbers. This is a series with a speci¢c lawfulness. According to Brouwer, every series that is generated by a law is legitimate. At the outset, Brouwer rejects the possibility of an arbitrarily generated series. But he is then confronted by a problem: if we allow mathematics to rely only on series generated by means of

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lawful regularities, the mathematics we have at our disposal is highly impoverished. In particular, it lacks the concept of the mathematical continuum, which, at the start of the twentieth century, was at the eye of the stormy debate over the foundations of mathematics. Following Cantor's set theory, the continuum was conceived as a set of points each of which corresponds to a particular real number. One of the problems Brouwer had to contend with was that these numbers cannot be constructed by means of a lawful regularity. To know these numbers requires knowledge of an in¢nite series. Brouwer did not want to forgo the notion of the mathematical continuum, and therefore sought an element that would on the one hand function like the mathematical continuum, but on the other, would not violate the epistemic constraint that it is not always possible to know every individual member of an in¢nite series. The answer he came up with was to introduce the notion of the choice sequence or freely generated series, and distinguish it from that of the rulegoverned sequence. An example of a rule-governed sequence is the series of the squares of the natural numbers (1, 4, 9, 16 . . . ). A choice sequence is not rule-governed, and each member is chosen independently of the others; an example would be the series (8, 1, 92, 6 . . . ) with no rule for continuation. Using such series, Brouwer developed an entire mathematics, including set theory and a new notion of the continuum. One of the best-known results of this new intuitionistic mathematics is the uniform continuity theorem, which states that every function de¢ned on the real numbers (i.e., every total function) is continuous. This result is incompatible with classical mathematics, in which a total function is not necessarily continuous.

Heidegger and Brouwer In the introduction to Being and Time, Heidegger links the crisis in the foundations of mathematics to the need to return to the question of Being. According to Heidegger, the debate between formalism and intuitionism is about `obtaining and securing the primary way of access to what are supposedly the objects of this science'.59 Heidegger thus creates a direct link between the situation in mathematics (and in

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other sciences as well), and the need to go back to the question of Being to provide a framework for the debate. In this sense Being and Time is indeed distant from both formalism and intuitionism. Yet there is nonetheless a certain a¤nity between intuitionism and Heidegger's approach: Brouwer's thesis that time is the framework for mathematics in its entirety is re£ected in Heidegger's approach to the question of the meaning of Being. An additional point of agreement is Heidegger's denial that the scope of the law of excluded middle is unlimited. In a well-known passage in Being and Time, he claims that Newton's laws had no truth value prior to their discovery. They were, he says, neither true nor false.60 Nevertheless, Heidegger and Brouwer are explicitly at odds over some important issues.61 Divergence between their views is most evident in that whereas Brouwer's position is foundationalist in the sense that it seeks to establish a ¢rm foundation upon which mathematics in its entirety can be constructed, Heidegger rejects the primacy of such a foundation. Brouwer conceives of number as emerging from the basic intuition. His model of the foundational enterprise is thus a creationist model. Heidegger, on the other hand, understands Being as a necessary horizon. Entities are not created by the meaning of Being, but this meaning enables them to show themselves as entities. This disparity between the outlooks of Brouwer and Heidegger leads to a signi¢cant di¡erence between them with respect to how they conceive time. Brouwer stresses the £owing character of time. For Heidegger, primordial time is not an intuition we utilize in order to construct the world's entities, but rather a framework for all possible interpretations of time, including the interpretation of time as £owing. For Brouwer, time functions as the basic building block of human knowledge, whereas for Heidegger, time is a horizon. But beyond the aforementioned di¡erences between them, their positions attest to a shared intellectual orientation, an orientation that is re£ected in a basic tension underlying the intuitionistic position. This tension is most readily discernible by examining one of the proofs Brouwer gives for the uniform continuity theorem, which states that every function de¢ned on the real numbers (i.e., every total function) is continuous. One version of this theorem states that `Any discontinuous real-valued function is not total.'62 Let us assume a

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discontinuity in the following way: f(x) ˆ 2 for any x < 1 and f(x) ˆ 3 for any x  1. Then the value of the choice sequence 0.99999 . . . with the rule of free choice is not always determined. If at some point we choose a digit other than 9, then the value the function assigns to the number will be 2, because no future choice will change the fact that the number is de¢nitely less than 1. If we choose 9, the value of the function is not necessarily 3, since there is always the possibility that at a subsequent stage a di¡erent digit will be chosen, in which case the value of the function will be 2. To prove his theorem, Brouwer requires that the value of the choice sequence be indeterminate, as this will prove that if a function is not continuous, it is not completely de¢ned. But this requirement can be met only if the choice made continues to be 9, though the option of choosing otherwise remains open. This result is paradoxical, since if 9 is indeed always chosen, then the value of the choice sequence is determined. Carl Posy describes this situation as `forced indeterminacy'.63 In situations of forced indeterminacy, we must continue to choose a certain digit in order to maintain a series' indeterminacy. According to Posy, the state of forced indeterminacy captures a basic tension in the intuitionistic position. It characterizes Brouwer's position with respect to the notion of in¢nity,64 and also lies at the heart of his approach to the continuum. For it is precisely the indeterminacy of the choice sequence that makes it possible for Brouwer to successfully steer clear of two outcomes he seeks to avoid: reducing the continuum to a set of determinate points, on the one hand, and restricting the arithmetization of the continuum to points that can be de¢ned by rule-governed series and ¢nite choice sequences, on the other. What is the connection between the notion of forced indeterminacy and Heidegger's thought? My claim is that the relation between the notion of in¢nite time, viz., time composed of an in¢nite series of `now's, and primordial temporality, should be interpreted in terms of the relation between the arithmetization of the continuum and forced indeterminacy. An important element of this thesis is the idea that Heidegger's notion of death exempli¢es the situation of forced indeterminacy. But to establish the link between Brouwer and Heidegger, we must ¢rst identify what is common to the problems confronting them. And

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indeed, both are preoccupied by the notion of the continuum, more speci¢cally, the question of reducing the continuum to a set of points. For Heidegger, this question arises in the context of his discussion of time. Heidegger explicitly acknowledges the importance of the notion of the continuum, inasmuch as it underlies the ordinary notion of time.65 Although Heidegger mentions some of the modern literature on the notion of the continuum (Georg Cantor, Hermann Weyl and Bertrand Russell), his own approach is in£uenced chie£y by Aristotle. The relation between the point and the continuous line plays an important role in Aristotle's rejection of the conception that time is a continuum of `now's.66 For example, on the basis of the analogy between `now's and points, Aristotle rejects the conception that a `now' has an immediate successor. One point cannot immediately succeed another on a continuous medium, since `things are called continuous when the touching limits of each become one and the same'.67 But in the case of points, the limit is identical to the point itself. Therefore on a continuous medium there are no successive points. Brouwer also rejects the idea that the continuum can be reduced to a set of points, although his argument is a little di¡erent. Brouwer's main concern is individuating points in terms of their numerical characterization. Since only points that can be constructed are acceptable, and these points cannot generate the notion of the mathematical continuum, where every point is determinate, Brouwer maintains that the continuum is irreducible to a set of determinate points. More recently, Michael Dummett has put forward a Brouwerian approach to the question of the reducibility of time to a set of instants.68 Dummett begins by rejecting the possibility of discontinuous change. Instead of discontinuous change, he proposes a conception of time on which instants cannot be separated from their surrounding interval, a conception he calls `fuzzy realism'. Dummett then o¡ers an intuitionistic model of this conception of time that is based on the notion of choice sequences. By recourse to choice sequences, we are able to narrow the intervals, but we cannot eliminate them completely. Now it seems to me that Dummett's approach to the question of the reducibility of continuous time to a set of instants can be applied to the interpretation of Heidegger's stance on the dependency of

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Being and Number in Heidegger's Thought

in¢nite time on primordial temporality, as it provides grounds for the position that time cannot be construed as composed of instants without previously assuming the structures of Dasein's ¢nite temporality. The argument can be taken a step further. For Brouwer, as we saw, the characterization of an in¢nite choice sequence mandated the notion of forced indeterminacy. My contention is that this is also true of Heidegger's account. Heidegger claims that in¢nite time presupposes Dasein's ¢nite time. And, I contend, Heidegger's notion of ¢nite time likewise exempli¢es forced indeterminacy. In this sense, Being-towards-death enables freedom.69 For Heidegger, it is death that opens the space of possibilities, and without which there are no possibilities at all. In Brouwer's case, the possibility of choosing the sequence and maintaining its indeterminacy is created by the forced choice. The indeterminacy of the series' limit preserves the possibility of a further free choice. The two notions are not, of course, identical. Heidegger conceives of death as ¢nite, while forced indeterminacy is not necessarily ¢nite. As we saw above, it applies to in¢nite series as well. But this di¡erence is not crucial, since Heidegger's notion of ¢nitude is not ¢nitude in the sense of a ¢nite quantity, but in the sense of having a limit. The profound a¤nity between the two notions, namely, Heidegger's notion of death and the notion of forced indeterminacy that emerges from Brouwer's approach, is rooted, in my opinion, in the almost identical roles they play in the respective philosophies. Both Heidegger and Brouwer are seeking to delineate a limit where there is no possibility of characterizing it from the outside, that is, both require an immanent notion of limit. Let me recapitulate my argument. Given the reasons for rejecting the conception of time as a continuum of instants, as argued by Dummett, among others, and the fact that Brouwer's choice sequences are invoked in describing the process by which instants of time are determined, the notion of forced indeterminacy plays a crucial role in explaining why the very possibility of formulating the notion of an in¢nite time composed of instants is conditioned on Heidegger's notion of ¢nite temporality. In Heideggerian terms, this argument shows the dependency of world time on primordial temporality.70 This last argument completes the task of delineating the role of number in providing an account of the relations between the di¡erent

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strata of time in Being and Time. In the ¢nal sections of this chapter, I examine the more immediate relationship between Being and Time and actual developments in the ¢eld of mathematics. As mentioned above, Being and Time promises to provide a new foundation for the sciences. How, then, should the relationship between Heidegger's fundamental ontology and real-world mathematics be conceived? I open my examination of this question with a discussion of Oscar Becker's interpretation of the Being and Time^intuitionism nexus; I then compare Becker's interpretation with the interpretation I o¡ered above. Against this background, I attempt a more general evaluation of the role played by contemporary developments in the foundations of mathematics in Heidegger's thought during the Being and Time period.

Oscar Becker on Being and Time and intuitionism Oscar Becker studied with both Husserl and Heidegger. Most of his work explored issues related to mathematics and physics. His ¢rst publication dealt with the question of the foundations of geometry and the application of geometry to the physical world.71 His second, Mathematische Existenz, originally published in the same volume of the Jahrbuch fÏr Philosophie und phÌnomenologische Forschung as Being and Time in 1927, deals with the debate between intuitionism and formalism in mathematics.72 One of its aims is to examine this debate through the prism of Heidegger's ontology.73 Heidegger's philosophy plays two major roles in Mathematische Existenz.74 First, it provides a general framework for the book, in the sense that Becker interprets the debate over the foundations of mathematics in terms of the di¡erent meanings of `existence'. Second, Heidegger's ontology provides a means of deciding which of the two positions in question, namely, intuitionism and formalism, best captures the notion of existence in mathematics. Formalism considers a theory's consistency to be a su¤cient condition for the existence of the entities of which it speaks; intuitionism deems it insu¤cient. Entities do not exist unless the means for constructing them can be provided.75 Becker maintains that, from a Heideggerian point of view, intuitionism is the correct position. In the ¢nal analysis, he has two reasons

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for this. For one thing, the centrality of the concrete lives of individuals in Heidegger's ontology is paralleled by the intuitionistic requirement of constructability: according to the intuitionist, the very possibility of intuiting mathematical objects rests on our ability to realize them concretely in the world. Concrete reality restricts the possibilities for mathematical construction.76 For another, intuitionism endorses a distinction between sequences generated by a rule and choice sequences. Becker contends that this distinction is meaningful only on the assumption of a ¢nite perspective. From a God'seye point of view, there is no essential distinction between these two types of sequence.77 Becker interprets the distinction between sequences generated by a rule and choice sequences in terms of the Heideggerian distinction between natural time and historical time. Natural time is based on the possibility of iteration; it is time as measured. Historical time, on the other hand, is best grasped by looking at the phenomenon of the passage from one generation ^ not in the biological but in the cultural sense, as in `the generation of 1914' or `Gen X' ^ to another. According to Becker, the emergence of a new generation is not determined by the previous generations. That is, in historical time the transition from one period to another is not rule-governed. Applying this distinction to the mathematical realm, it is easy to see that a rule-governed series such as the natural numbers exempli¢es natural time, while a non rule-governed choice sequence exempli¢es historical time. Hence, fundamental elements of Brouwer's intuitionistic mathematics are clearly echoed in Heidegger's notions of time as interpreted by Becker. But parting with Heidegger, Becker does not endorse the aforementioned view that both the notion of time in the natural sciences and that of time in the historical sciences presuppose a more basic stratum of time. Becker claims that there is an unresolved tension between nature and history that does not arise out of any more basic framework. This tension, he contends, is mirrored in the debate between formalism and intuitionism, both of which are rooted in classic philosophical approaches to mathematics. He situates Brouwer's position within the critical tradition that includes Aristotle and Kant,78 contrasting this to the approach taken by Plato and Leibniz, which he characterizes as the `symbolic' approach. In the ¢nal chapter, Becker

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goes even further, concluding that the meaning of being in mathematics is not only the ontology of concrete life, but the overcoming of death.79 Mathematics evolves in the living world, and in this sense, he argues, the intuitionistic approach is correct, but eventually mathematics advances from the realm of history to that of nature. This conclusion leads Becker further away from Heidegger. Becker's basic approach to the link between Heidegger's philosophy and the foundations of mathematics is, in my opinion, correct. Heidegger's thought has a greater a¤nity to the views of Brouwer than to those of Hilbert, and choice sequences are the appropriate place to demonstrate this a¤nity. Nevertheless, as it stands, Becker's approach is problematic. The chief di¤culty is that it is far from clear from his analysis why choice sequences presuppose Heidegger's notion of ¢nitude and death, and not some other notion, or even the notion of potential in¢nity. On Becker's interpretation, the intuitionistic approach to mathematics presupposes a notion of time that has certain characteristics in common with Heidegger's notion of historical time, but it is not clear that it requires Heidegger's notion of primordial ¢nite temporality. To establish the necessity of Heidegger's primordial temporality, the dependency of historical time on primordial temporality must be demonstrated. But as we saw above, in Being and Time this turns out to be a serious problem. Moreover, my proposed solution to the problem makes use of certain features of Brouwer's intuitionism. Hence the relation between intuitionism and Heidegger's fundamental ontology is considerably more complex than Becker seems to realize. Becker maintains, in line with the Heideggerian view, that mathematics can be grounded in fundamental ontology, but, as I have argued, in some sense the reverse is true: it is thanks to the mathematics of Brouwer's intuitionism that we can better understand the claims of Being and Time.

Being and Time and the foundations of mathematics In Heidegger's early writings, the relationship between mathematics and the fundamental concepts of ontology was premised on their separation from each other. In Being and Time, on the other hand, the

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relationship is one of dependence: the dependence of mathematics on fundamental ontology. Thus far we have focused mainly on the dependence of mathematical objects, such as number and space, on fundamental ontology, and, more speci¢cally, on Dasein's primordial temporality. But there are two additional aspects of mathematics that are important in Being and Time. First, mathematics, which comprises arithmetic and geometry, is a positive science that deals with a certain domain of entities: arithmetic studies numbers, and geometry studies space.80 Science in this sense has its source in Husserl's notion of a positive science, which studies a particular region. Heidegger's remark about the crisis in the foundations of mathematics refers to this aspect of mathematics within the Heideggerian worldview. Mathematics takes its place alongside the other sciences, and just as a foundational crisis can be discerned in these other sciences, so it can be detected in the foundations of mathematics. Second, from the beginning of the modern era, mathematics has served as the paradigm of a science and of the scienti¢c orientation. A discipline must be mathematical if it is to be deemed a science. This aspect of mathematics endows it with a privileged status relative to the other sciences. For example, in the `Introduction to Philosophy' lectures from 1928, Heidegger makes a direct connection between mathematics and the question of whether philosophy is a science. According to Heidegger, the Cartesian view of philosophy as the primordial science was in£uenced by the conception that mathematics was the loftiest and most rigorous science.81 This second additional aspect of mathematics will be at the forefront of our discussion in the coming chapter. Here, in this concluding section of the present chapter, I will focus on the ¢rst of the additional aspects of mathematics just enumerated, and relate it to the issues raised above. Heidegger sees the positive sciences as inherently limited. This limitedness is manifested in various ways. For one thing, every science is directed toward the speci¢c region of entities it studies. Biology studies living things, physics studies the material world, and geometry studies space. Moreover, the sciences cannot account for themselves. That is, they cannot grasp the relevant region in its entirety. To accomplish this, the borders of the region must be transcended. For although each science is directed toward the speci¢c region of

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entities it studies, it cannot grasp the Being of these entities.82 This distinction, namely, the distinction between Being and entities, points to an additional limitation of the sciences: scienti¢c investigation is ontic, and thus it presupposes ontological investigation, which reveals the givenness of that which is.83 Therefore, every science necessarily presupposes fundamental ontology, since it is the condition for any access to entities. But this dependency on ontological investigation does not constitute the basis for the sciences in the sense of providing a foundational principle, an elemental axiom from which the positive sciences can be deduced. Heidegger seeks to institute a clear distinction between ontic and ontological investigation. An axiomatic foundation would have created a link between the positive sciences and fundamental ontology that would have undermined this distinction. Although Heidegger maintains that the positive sciences are dependent on fundamental ontology, certain aspects of his position in Being and Time are incompatible with a clear separation between the two levels. In some cases, results in the positive sciences help to shape fundamental ontology as well. In this respect Heidegger continues along a path already evident in his early writings as interpreted in Chapter 1 above. Recall that in `Duns Scotus' Doctrine of Categories and of Meaning', Heidegger formulates his ontological concepts in light of developments in mathematics, particularly the foundations of mathematics. The close connection with mathematics continues in Being and Time, but its thrust is di¡erent. Whereas in the very early writings, the dual meaning of `one' was at the heart of the ontology^mathematics nexus, in Being and Time a similar duality can be found in the notions of number and space. Heidegger uses the various possibilities for interpreting these notions to establish his conception of fundamental ontology. A similar approach can be detected in remarks on the theory of relativity he makes in his 1924 lecture on the concept of time. After discussing the theory of relativity brie£y, he turns to his own investigation of time, which he characterizes as `pre-scienti¢c' (Vorwissenschaft), analyzing use of the clock, which also, of course, plays a pivotal role in the theory of relativity.84 The analysis is very similar to the treatment of world time in Being and Time. Here too, then, there is a close link between a positive science and fundamental

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ontology.85 Indeed, there is a marked similarity between the role played by the clock in world time and its role in the theory of relativity. Time reckoning in world time does not presuppose a pre-given notion of absolute measurable time, paralleling the use of clocks in relativity. All these points of contact make it plausible to interpret Being and Time as an attempt to justify certain currents in the mathematics and physics of Heidegger's day. Heidegger is seeking, it appears, to put forward a vision of an alternative science that would be in line with Aristotelian physics and the theory of relativity, while parting ways with the `modern' science of Galileo and Newton. I therefore agree with Caputo that there is an existential conception of science in Being and Time.86 But I must add that this conception is related to certain elements of the scienti¢c thought of the 1910s and 1920s, and not derived solely from Heidegger's fundamental ontology. Yet two years after Being and Time came out, this existential approach to science can no longer be detected in Heidegger's writings. In the 1928 `Introduction to Philosophy,' Heidegger is careful to distinguish between philosophy and ontology, on the one hand, and the speci¢c sciences, on the other. He continues to uphold this dichotomous approach in the years that follow, and from the 1930s onward, it even intensi¢es. How is the change to be accounted for? It seems to me that the considerable emphasis Heidegger places on the division in his writings in the 1930s can largely be attributed to the connection between the methodological framework of Being and Time and the modern conception of mathematical physics, and the problems generated by this connection. This nexus is, in fact, the principal reason for Heidegger's abandonment of the framework of Being and Time; I discuss it in detail in the next chapter. Here, let me just say that the problems arise out of the vicious circle that pervades Heidegger's ontology, namely, the fact that certain ontological characterizations of mathematical physics turn out to condition the possibility of articulating the notion of Dasein itself ! The upshot of these methodological problems was that Heidegger sought to reestablish a clear distinction between philosophy and the sciences, and hence repudiated any a¤nity between his notion of philosophy, or thought in general, and his interpretation of modern science.87

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To summarize, while in his early writings Heidegger maintained that ontology and mathematics were separate realms of discourse, in Being and Time the distinction between them is blurred. In the 1930s, Heidegger ¢nds his way back to the views espoused in the early writings. In the works of the 1930s, to which the next chapter is devoted, Heidegger once again attempts to establish a distinction between ontology and mathematics, but now, in contrast to the strategy adopted in his early writings, he assumes that there is a total and uncompromising divide between the two.

Notes 1. As we saw at the end of the previous chapter, this aspect of Being and Time is related to the manner in which Heidegger links a speci¢c ontological region and the formal question of the being of entities. 2. The transition from consideration of multiplicity in general to consideration of speci¢c multiplicities is evident in the 1926 lecture series entitled `Logic: the Question of Time'. The lectures in the second half of the series deal with Kant. According to Heidegger, one of the peculiarities of the Critique of Pure Reason is that it has no pure multiplicity, in the Scholastic sense, but rather, multiplicities are always given in speci¢c ways: side by side, one after another, etc., see Heidegger GA 21, 299. 3. It is important to note that in Being and Time there is a distinction between ontological regions in the strict sense, such as nature, history, life, space, Dasein, etc., and di¡erent modes of Being, such as `present-at-hand' and `ready-to-hand'. The modes of Being are a precondition for interpreting the di¡erent ontological regions. In Being and Time there is no strict correspondence between ontological regions and modes of Being: several modes can apply to a single region. 4. Ricoeur sees this as the cardinal problem confronting Heidegger's analysis of time, see Ricoeur 1985, 131. 5. Heidegger 1916. 6. Heidegger 1916, 421 (Eng. trans. 53). 7. Heidegger 1916, 424 (Eng. trans. 54^5). 8. Heidegger 1916, 431 (Eng. trans. 50). 9. In the early writings, Heidegger quotes a number of times from Bergson's Time and Free Will: An Essay on the Immediate Data of Consciousness, see, e.g., Heidegger GA 1, 168, 306. In these references, there is a manifest a¤nity between Heidegger's position and that of Bergson. For example, in Heidegger's doctoral dissertation, `The Doctrine of Judgment in Psychologism',

76

10. 11.

12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24. 25. 26. 27.

Being and Number in Heidegger's Thought the ¢rst place he refers to Bergson, there is a clear parallel between Bergson's discussion of the distinction between the mental and the extra-mental, and Heidegger's discussion of the distinction between the activity of judging and the judgment as a logical object. On Bergson's concept of multiplicity, see Deleuze 1966. Natorp 1910, 284. In the remainder of the chapter, I will argue that in Being and Time, Heidegger adopts Natorp's stance. That is, he construes the connection between number and time as based primarily on number as orderimparting. Critique of Pure Reason, A143/B102. Here too we observe a blurring of the di¡erences between the various streams of neo-Kantianism. As noted, there is indeed a signi¢cant di¡erence between Natorp and Rickert, namely, that while for Natorp numbers have no extra-logical component, Rickert holds that the existence of such an extra-logical component is essential if there is to be equality between numbers. However, in opposition to the Kantian view, both thinkers see number as prior to time. Heidegger SZ, 404. Heidegger SZ, 417. On this interpretation of datability in Heidegger, see Blattner 1999, 168^9. Heidegger SZ, 412. Heidegger SZ, 421. GA 27, 71^2. Heidegger SZ, 96. Heidegger's remark appears in the edition of Being and Time included in his collected works (GA 2, 128). Heidegger SZ, 18. Examination of Heidegger's treatment of space in Being and Time will support this interpretation. In the introduction, space too is characterized as `nontemporal', and toward the end of the book (in ½70) Heidegger brings an argument in favour of the dependence of space on time, an argument that seeks to interpret space in temporal terms, speci¢cally, in terms of directionality and de-severance (Ent-fernung). Heidegger SZ , ch. 11, 421. Physics IV, ch. 10, 219b1. Physics IV, 218a7. This interpretation is put forward by several interpreters of Aristotle. The idea that the role played by number in de¢ning time is that of number as ordering ¢rst appears in Simplicius: `But since there are two sorts of number, the one enumerating or enumerated in quantity, as when we say ``one, two, three,'' the other ordinal, as when we say ``¢rst, second, third,'' time is the latter sort of number' (Simplicius 1992, 125). A contemporary version of this interpretation can be found in BÎhme 1974.

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28. On the problem of circularity in Aristotle's notions of `before' and `after', see Corish 1976. Corish maintains that the function of number is to count the `now's, which must be inherently temporal and not merely positions in space or states of motion. As I see it, the function of number is to create order. 29. On this point, I concur with the analysis of Waterlow (1984), who sees the `now' as the terminus of an interval. But unlike Waterlow, I retain the distinction between counting and measuring. 30. Metaphysics X, ch. 1, 1053a31. 31. Physics IV, ch. 12, 220b8. 32. On a similar approach to the meaning of number in the context of the de¢nition of time, see Brague 1982, 134^44. 33. On the tensions in Aristotle's thinking on time, and particularly the tension between his account of time in the Physics and his position on number in the Metaphysics, see Annas 1975. The interpretation I have suggested reconciles the tensions identi¢ed by Annas. 34. Heidegger, GA 24, 353^5 (Eng. trans. 250^1). 35. Bergson 1921, 58^9. 36. Bergson 1921, 65. 37. Bergson 1921, 66. 38. Heidegger, GA 24, 328^9 (Eng. trans. 232). 39. Heidegger's analysis of space in Being and Time suggests an additional point on which he and Bergson diverge. Heidegger distinguishes between space in the primordial sense, which is grounded in time, and measured space. Space per se is not the space that is measured. See Heidegger SZ, 112. 40. On Heidegger's critique of Bergson, see Crocker, 1997. Crocker and I both understand Heidegger to be alluding to Bergson's notion of number as the problematic element in his account of time. Crocker argues that number is not bound to the thing that is counted, and hence does not eliminate the transitional aspect of the `now'. But he does not provide a precise analysis of why numbered time is not necessarily `spatialized' time. 41. Physics IV, ch. 14, 223a 22^8. 42. Heidegger GA 24, 349 (Eng. trans. 247). 43. Heidegger GA 24, 359 (Eng. trans. 254^5). 44. BÎhme 1974, 4. 45. See Blattner 1999. 46. Heidegger SZ, 331. 47. Physics III, ch. 6, 207a1. 48. On the two conceptions of the in¢nite and how they have been interpreted by philosophers throughout history, see Moore 1990. On Hegel's two notions of the in¢nite, see Hegel 1975, ½94^5. 49. Heidegger GA 25, 155 (Eng. trans. 107). 50. Heidegger SZ, 259. 51. Heidegger SZ, 424.

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52. Especially relevant is Heidegger's critique of Parmenides, since Parmenides' notion of Being has close a¤nities with Descartes' conception of the in¢nite; see Heidegger SZ, 26. 53. Heidegger SZ, 25. There is a debate as to whether Heidegger should be interpreted as equating presence (Anwesenheit) and the present (Gegenwart). Olafson reads Heidegger as equating presence and Being, and construing the present, past and future as dependent upon presence; see Olafson 1994. In contrast, Carman understands presence as the metaphysical interpretation of Being, such interpretation being a mode of temporal interpretation; see Carman 1995. On this point, I concur with Carman. 54. Heidegger SZ, 424. 55. According to Heidegger, in¢nite time is a de¢cient mode of ¢nite time. For a profound analysis of the notion of the de¢cient mode in Being and Time, see Hartmann 1974. 56. As Blattner puts it, Heidegger does not show that his explanation of in¢nite time is the only explanation, see Blattner 1999, 226. 57. For a clear exposition of this point, see Detlefsen 2002. 58. Brouwer 1912, 127. 59. Heidegger SZ, 9. Heidegger mentions the intuitionism^formalism debate in the context of his general discussion of the crises that occur in the sciences, including physics, biology and theology. His survey of these crises seeks to emphasize the need to illuminate the foundations of these sciences. Ultimately, these sciences rest on a pre-understanding of the areas in question, which is in essence a pre-understanding of the Being of the entities they deal with. Thus crisis in the sciences reveals the relevance ^ and necessity ^ of an investigation into the question of the meaning of Being. 60. Heidegger SZ, 226 61. There is another interesting connection between Heidegger and Brouwer, having to do with their respective notions of number. It will be recalled that Heidegger's very early stance was based on Natorp's fundamental series. Brouwer's student Heyting claimed that Brouwer's position on the creation of the natural number series was the same as Natorp's (Heyting 1931, 106). 62. The proof appears in Brouwer 1927. Both my reconstruction and the conclusions drawn from it are based on Posy 2000. 63. Posy 2000, 202. Posy uses an analysis of the Hangman paradox to help articulate this notion. 64. Posy 2000, 206^7. 65. Heidegger GA 22, 75^6. 66. Physics IV, ch. 10, 218a 10^21. 67. Physics V, ch. 3, 227a 10^15. Heidegger discusses this de¢nition in GA 19, 112^21 (Eng. trans. 77^83). On the importance of this text for understanding Heidegger's conception of geometry, see Elden 2001.

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68. Dummett 2000. 69. On the connection between freedom and ¢nitude, see Heidegger 1929, 175 (Eng. trans. 135). 70. The interpretation being put forward here as to the role of number in Being and Time solves an interpretive problem raised by Ricoeur, who argues that Heidegger did not succeed in establishing the dependence of the ordinary notion of time on primordial temporality and on Dasein. He concludes that ascribing autonomy to the ordinary notion of time is thus unavoidable. This leads Ricoeur to reintroduce a bifurcated account of time that posits two totally separate temporal realms: objective time, and the subject's internal time; see Ricoeur 1985, 131. 71. Becker 1923. Heidegger cites this work in Being and Time, SZ 112. 72. All my page references to Mathematische Existenz refer to the stand-alone edition. 73. Becker's discussion is based largely on Heidegger's 1924 lecture on the concept of time and on Heidegger's Freiburg courses, which Becker attended, but not on the text of Being and Time, which came out together with Mathematische Existenz. 74. For a detailed examination of the role of Heidegger's thought in Mathematische Existenz, see Gethmann 2003. 75. Becker 1927, 29. 76. Becker 1927, 196. 77. Becker 1927, 230. It could be argued that there is a distinction between a lawful and a non-lawful in¢nite series, since there can be an algorithm for the lawful series but not for the non-lawful series. If so, Becker's claim is mistaken. 78. Becker 1927, 311. 79. Becker 1927, 321. 80. This characterization of mathematics appears in Heidegger GA 24, 17 (Eng. trans. 13). 81. Heidegger GA 27, 18. 82. Heidegger GA 25, 35. 83. Heidegger SZ, 11. 84. Heidegger 1924, 3^4. On the relation between the theory of relativity and Heidegger's thought, see Keller 1999, 197^201. 85. Heidegger comments on a parallel similarity between Hermann Weyl's approach in Space, Time, and Matter and the Aristotelian conception of the continuum, which he takes to support his view that the continuum should not be conceived primarily as a set of points. See Heidegger GA 19, 117. 86. Caputo 1986. 87. Questions pertaining to number and mathematics also play another role in the fact that the approach taken in Being and Time was abandoned. In his 1962 lecture, `On Time and Being', Heidegger asserts that, in retrospect, the problem that stymied Being and Time was how to reduce space to time (Heidegger

80

Being and Number in Heidegger's Thought 1962, 24). As noted above, `subsistence' (Bestand ) is the mode of being of space (and number). Hence it may well be that Heidegger could not ¢nd an adequate way to incorporate `subsistence' in the framework of primordial temporality. Franck takes a di¡erent approach to the problem of space in Heidegger's thought. Franck contends that the problem of space is related to that of Dasein's corporeality (1986, 55^6). While it is true that the body is certainly a problem in Being and Time, Heidegger was nonetheless also preoccupied by the meaning of the being of number and space.

3

The Mathematical Epoch

In the preceding chapter, we discussed two contexts in Being and Time where there is a connection between ontology and mathematics. The principal such context, we saw, is the role of number in fundamental ontology, the connection in question being the link between primordial temporality and ordinary time. The second such context is that of the relation between fundamental ontology and the possibility of authentic approaches to mathematics that is hinted at by Being and Time. An additional salient context where there is a signi¢cant interface between ontology and mathematics is Heidegger's characterization of the modern attitude toward nature. On the modern understanding, nature is determined by mathematical physics. The implications of Heidegger's discussion are not limited to the understanding of nature that arises from mathematical physics, but extend to our understanding of modernity as a whole. Heidegger's engagement with mathematical physics is, in fact, an attempt to characterize modern ontology from Descartes on, as well as to articulate his own critique of modernity and of the Cartesian/Husserlian subject. The main goal of my exposition in this chapter is to uncover the various meanings latent in Heidegger's discussion of the mathematical, focusing on the two primary texts where the concept appears: Being and Time, and What is a Thing?1 Most of the chapter is devoted to a careful examination of the argument Heidegger presents in What is a Thing? concerning the self-limitation of modern ontology. At the end of the chapter, I take issue with a number of the argument's premises.

Mathematical physics in Being and Time Heidegger discusses mathematical physics in the framework of his general account of the existential characterization of theory. According

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to Heidegger, a theory is not ¢rst and foremost a theory in the sense of a collection of interrelated true propositions, but rather, is theoretical by virtue of its existential character. This character is rooted in circumspection (Umsicht), which is the mode of examining the ready-tohand.2 Deliberation (Ueberlegung), which plays an important role in circumspection, has an `if-then' structure. For example: if I use a hammer, then I'll be able to drive the nail in deeper. The `if-then' structure already presupposes that something is understood as some thing. Though pre-predicative, this understanding is the existential root of the copula (something is something).3 Heidegger sees such circumspective deliberation as exemplifying the basic meaning of `is'. Heidegger's characterization of theory, here, is very close to Husserl's conception of phenomenology. Heidegger seeks to situate Husserl's notion of phenomenology, and the centrality of intention to phenomenology, in the practical sphere. If Husserlian phenomenology is, as Husserl claims in his `Philosophy as a Rigorous Science', the basis for all the other sciences, as well as the primary theoretical science itself, Heidegger sees himself as uncovering the existential roots of this theoretical framework. It seems, therefore, that in Being and Time Heidegger adopts Husserl's stance with respect to the nature of the theoretical.4 The central feature of mathematical physics in Being and Time is not simply the application of mathematics to nature, but rather, as Heidegger puts it, the way in which nature is `a mathematical projection' (Entwurf ).5 According to Heidegger, projection is the disclosure of Dasein's possibilities as possibilities.6 Mathematical projection is characterized as an a priori disclosure ^ in most cases, disclosure of a particular ontological region. The disclosure generates a foreconception of the subject under discussion, that is, of the aggregate associated with that projection. The projection releases the entities from the contexts of our encounter with them and `counter-throws' (entgegenwerfen) them, transforming them into objects.7 In mathematical physics, therefore, the entity is transformed into an object, and devoid of any ties to the ready-to-hand. Furthermore, mathematical physics is not only an exempli¢cation of the theoretical, but also has unique characteristics. That is, not every theory manifests the characteristics found in mathematical physics. One of the speci¢c

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characteristics of mathematical physics is thematization (thematisieren), which is generated by the pre-demarcation of a given region that is mandated by the mathematical approach. Thematization endows mathematical physics with a signi¢cance that goes beyond its role as exemplifying the paradigmatic science, for the notion of `theme' is very central to Heidegger's entire method in Being and Time. Indeed, the ¢rst section of the ¢rst part of Being and Time, in which Heidegger discusses Dasein's characteristic features, is entitled `The Theme of the Analytic of Dasein'. It appears, then, that adoption of the notion of science that is unique to mathematical physics can be viewed as the starting point for Heidegger's discussion of Dasein. Now the conclusion that would seem to follow from this is that Being and Time rests on the notion of science assumed by modern mathematical physics.8 But the rest of the volume does not bear this out. Being and Time unfolds as follows: mathematical physics is interpreted by way of existential analysis of the theoretical approach. We arrive at the characteristic features of the theoretical approach, and situate it within a wider interpretive framework, by way of, among other things, transforming Dasein into a theme, but in the process of doing so, gain insight into the concept of the `theme', and thereby overcome our dependence on a particular theoretical starting point.9 The process of ongoing self-understanding that takes place in Being and Time teaches us that on Heidegger's interpretation, mathematical physics is not simply a particular case of theory, but Heidegger's own starting point. It follows that for Heidegger the status of mathematical physics is of the utmost importance. This being so, the changes his philosophy underwent during the 1930s, particularly the abandoning of fundamental ontology, gave rise to a need to return to mathematical physics, the most signi¢cant aspect of this return being Heidegger's examination of his own premises in Being and Time. Heidegger undertakes his renewed discussion of the status of mathematics in What is a Thing?, which is based on lectures Heidegger delivered in 1935. In this work, as in Being and Time, most of Heidegger's attention is focused on the question of how a conception based on mathematics determines a thing's thingness, the most salient example of this determination being, once again, modern physics. But already

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at the opening of Heidegger's discussion, we ¢nd a signi¢cant di¡erence between What is a Thing? and Being and Time. In What is a Thing?, principles and deductive structure are pivotal to our understanding of mathematics and the ontological role it plays. While the characterization of mathematics as projection remains in place, it is now anchored in the deductive structure of mathematical theory. Another major difference is that the mathematical project is now clearly perceived as a historical event that takes place in the modern period, an event at the heart of which lies the thought of Descartes. Whereas in Being and Time there was a certain separation between the discussion of mathematics and the discussion of Descartes' interpretation of being,10 in What is a Thing? the two are one and the same.

What is a Thing? on the mathematical Heidegger's discussion of the mathematical approach to ontology begins with an examination of the meaning of the concept in Greek thought. `Mathematics' is derived from mathesis, meaning learning. More precisely, it is the element that must be presupposed if learning is to be possible. For example, in order for it to be possible to learn something about a gun, it must be recognized as a gun. Such recognition requires that some preliminary criteria for the gun's being identi¢ed as a gun have been met. These criteria, or more generally, all the necessary conditions for the possibility of recognizing a thing as what it is, are what Heidegger sees as the essence of mathematicity. The connection between this basic sense of the mathematical, and the realms of number and geometric form, is that these realms manifest the essential character of the mathematical in the most unequivocal way. We cannot count unless we already have a concept of what a number is.11 The characterization of counting as presupposing the concept of number only applies, however, to the number three and the numbers that follow. This is because the characterization of three necessarily makes reference to the number series and to a place in that series: three is the number that occupies the third place in the number series. One and two are not necessarily characterized in this manner. One, as a characterization of an entity, does not yet make reference to

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a place in the number series, and two can be characterized as a pair, which necessitates neither reference to the number series, nor reference to an order within the pair. That is, Heidegger distinguishes between the determination that something is one or two, and the determination that it is three. Three, therefore, is what establishes the number series. After it has been established, we go back and characterize the second and ¢rst places in the series, rendering two and one numeric characterizations. The chief objective of this interpretive process vis-a©-vis numbers is to demonstrate that if number is understood as mathematical, that is, if it is maintained that the concept of number is already implicit in every instance of counting, and this concept itself necessarily encompasses relations to places in a series, it follows that this characterization applies, initially, to the number three, and only retrospectively to two and one. But there is another objective as well. On the basis of its characterization as mathematical, number is not autonomous relative to its foundations. It is not self-grounding. Rather, number presupposes the pre-numeric, namely, the concepts of: place in a series, one and pair. It is against the backdrop of this understanding of the mathematical that Heidegger attempts to clarify why modern science is characterized primarily by its mathematical nature. Heidegger extracts the mathematical nature of modern science from the law of inertia as presented in Newton's Principia. This law determines a thing's motion in advance. We cannot perceive a body as described by the law of inertia. Hence, there is no way this law can be interpreted as derived from our experience. The determination of a thing's motion in advance, prior to any sensory perception of it, is the mathematical component of modern physics.12 This mathematization of nature, as construed by Heidegger, does not allow for any distinction between di¡erent natures or essences of bodies. The di¡erences between things ^ for example, the di¡erence, on the Aristotelian worldview, between celestial and earthly bodies ^ must therefore be grounded in the realm of observation. In this sense, modern mathematical physics uses observation in contexts where it was considered irrelevant from the standpoint of the Aristotelian conception of the world. For Aristotle, the di¡erence between the moon and a terrestrial object such as a stone,

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being one of nature or essence, cannot be arrived at by observation. But modernity's use of observation is only meaningful in the context of modern physics. Hence, the modern conception of nature, as expressed in the law of inertia, mandates a pre-determination of natural being in general. In the determination of the motion of being, there can be no preliminary di¡erentiation between locations or ontological regions. Modern science, then, puts forward a new approach to the question of what a thing is, the salient characteristic of this new approach being the pre-determination of the `thing'-ness of things. Heidegger calls this pre-determination, which involves `jumping past' things, `projection'. In so doing, Heidegger is returning to the characterizations presented in Being and Time. However, it is here that we also discover the principal di¡erence between the expositions of modern mathematical physics in What is a Thing? and Being and Time. In What is a Thing?, the mathematical projection does not receive an existential interpretation. It is not the connection between the mathematical projection and fundamental ontology that is emphasized, but rather, Heidegger argues that the essence of the mathematical projection is axiomatic structure. The axioms are the foundational propositions (GrundsÌtze) that determine in advance the region of physical things. That is, the range of applicability of the concept of a body is determined by these foundational propositions. Indeed, these basic axioms also determine the range of applicability of the concept of the experiment, with which the concept of the mathematical is so intimately linked. The centrality of the experiment in modern physics, Heidegger argues, is a product of its mathematical nature. The mathematical orientation of modern science even determines the substance of mathematics itself. Heidegger sees Descartes' analytic geometry, Newton's £uxions and Leibniz's in¢nitesimals as determined by the mathematical in the sense that they are determined by the fundamental principles of the `thing'-ness of things.13 Heidegger does not justify this claim directly, but an argument for it can be provided on the basis of what he says about the mathematical projection. The mathematical projection must generate a uniformity of all bodies with respect to space, time and motion. Hence this uniformity, which is necessitated by the need for the mathematical projection to

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pre-determine the characterizations of bodies without reference to their particular natures, in turn necessitates the uniformity of the ¢eld of mathematics itself. This uniformity is generated by means of Descartes' concept of magnitude, and Leibniz's and Newton's concept of the in¢nitesimal. These concepts apply to all magnitudes, whether spatial or temporal. The Cartesian concept of magnitude, which is the basic notion underlying algebra and analytic geometry, is not a concept of some particular kind of magnitude, but magnitude in general. By contrast, in Greek mathematics, the meaning of number varies according to the object quanti¢ed. For example, the number four has one meaning when it refers to the area of a quadrangle, and a di¡erent meaning when it refers to the length of a line. It follows that the speci¢c manner in which mathematical theories developed in the modern period should be seen as arising from the mathematical projection.14 A ¢nal point Heidegger makes in What is a Thing? concerning mathematics has to do with the debate between formalism and intuitionism. The controversy, it will be recalled, is also referred to in Being and Time. But it is in What is a Thing? that Heidegger makes his ¢rst attempt to suggest a direction for addressing this debate. His initial suggestion follows from his claim that in the modern epoch, the mathematical, as the determination of the thingness of the thing from principles or axioms, and experience, or intuition, are reciprocally de¢ned. It is impossible to speak of experience or intuition without reference to the axiomatic aspect of mathematical ontology. Hence the tension between formalism and intuitionism arises from the two aspects, namely, the axiomatic and the intuitive, of the mathematical (understood in the Heideggerian sense). The main direction in which the clari¢cation of the intuitionism^formalism debate should proceed, then, is via clari¢cation of the notion of the mathematical and how it relates to sensory, viz., experienced, intuition. Such clari¢cation will also supply an answer to the principal question with regard to the mathematical, namely, its foundations and limits. Heidegger provides the answer in the context of the core discussion in What is a Thing?, which analyzes the Kantian answer to the question of what thingness is. For if we are to examine the Critique of Pure Reason from the standpoint of the mathematical in its Heideggerian sense, we

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must ¢rst consider how the mathematical projection determines speci¢c systems of metaphysics.

Mathematical metaphysics The mathematical projection determines not only the characterization of modern science and modern mathematics, but also, and more importantly, the characterization of modern metaphysics. Metaphysics inquires about what is, about the thingness of the thing, including things that are not part of nature. Whereas in Being and Time the mathematical projection was limited solely to the realm of nature, and inapplicable to other regions ^ history, for instance ^ in What is a Thing? the mathematical projection serves as the general ontological framework for modernity itself. As metaphysics, the mathematical projection is an attempt to characterize being in general in terms of the mathematical. Heidegger locates the fundamental change with respect to metaphysics and its mathematization in Descartes' thought. The change in question was not that Descartes put epistemology before theories of the world, but rather, the revolution he engendered within metaphysics itself. Now what, precisely, did this revolution consist in? It consisted in understanding metaphysics as modeled on mathematics, that is, as determining the being of that which is on the basis of foundational principles that serve as axioms. And Descartes' conception of the arche¨, the foundation in the metaphysical sense, Heidegger argues, di¡ers profoundly from both Aristotle's conception, and the conceptions advanced in the Middle Ages.15 The signi¢cance of this claim is far-reaching. Heidegger seeks to ascribe all the changes that resulted from Descartes' philosophy to this revised understanding of the notion of the foundational principle. The Cartesian foundational principle functions as a basic proposition or premise (Grundsatz).16 Hence, the foundation is, ¢rst and foremost, an axiom in the sense of a claim from which all other claims are derived. Thus, the primary characteristic of a mathematical principle is that it is located within an ordered system of propositions.17 Introduction of the notion of order here points, it would seem, to a certain

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discontinuity between the numerical realm ^ as Heidegger interprets it ^ and metaphysics under the mathematical interpretation. In the former, order is not de¢ned on the basis of the numbers one and two, but only beginning with the number three. That is, in the numerical realm we have no internal foundation from which the complete order of the numbers can be derived. The numbers themselves are not autonomous relative to their foundation. In contrast, in the framework of the mathematical as metaphysical, the foundational principle is autonomous. This di¡erence between the realm of number and the general characterization of the mathematical is, in my opinion, one of the justi¢cations for Heidegger's claim that the mathematical approach, in its modern sense, impacts the very content of mathematics and not just its methodological framework. The question that arises at this point is, therefore, how we are to understand this concept of the foundation in modern metaphysics. Being a ¢rst premise, the foundational proposition (satz) must ground itself. Hence, it cannot presuppose any prior logical or ontological structure. According to Heidegger, these constraints on the foundational premise necessitate a changed conception of logic. The subject^predicate sentence cannot be adopted as the basis upon which a system of deductive logic is to be constructed, because the subject^predicate structure presupposes that the subject is given as such beforehand. Hence the subject cannot ground itself ^ the givenness of the subject can only be generated by the axioms. An axiomatic system must therefore be in place before we can establish the logical structure of the sentence; this axiomatic system can then serve as the basis for the logical analysis of the sentence. In this respect Heidegger's interpretation of the role of the axiomatic system in the characterization of mathematical metaphysics is akin to the approach of modern mathematical logic, which sees the logical analysis of a sentence as largely dependent on the roles it can play in valid arguments. In light of this understanding of the function of the foundational premise in modern metaphysics, let us now return to the thought of Descartes. Descartes' inquiry into how we are to understand the role of the foundational principle in modern metaphysics is itself mathematical. What, he inquires, is the pre-condition for a foundational

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principle that is itself the foundation for every sentence? His answer is simple: the subject (subiectum), which must be given, that is, which must be presupposed in the ¢rst premise. If we add to this condition the condition that the foundational principle must be a sentence that every other sentence always assumes, we arrive at the cogito ^ I think. It follows that the subject of the ¢rst foundational principle is the `I', and so too we arrive at the transition from a world-based theory to a subject-based theory. This transition follows from the nature of the mathematical, in its Heideggerian sense. To the cogito as a foundational principle of any metaphysics we must add two other fundamental premises ^ the principle of su¤cient reason, and the law of contradiction.18 These three principles form the framework within which the question of the Being of beings can be answered. This framework arises from pure reason, since nothing can be given outside the realm of reason. Reason is absolutely autonomous in determining the thingness of that which is.

Epochs as ontological regions Comparing Heidegger's treatment of mathematical physics in Being and Time to that of What is a Thing?, we ¢nd that although there are similarities ^ mainly with respect to the characterization of the mathematical projection ^ there are several important di¡erences. In Being and Time, mathematical physics does not characterize an entire epoch, but is at most a speci¢c instance, albeit a signi¢cant one, of the theoretical approach. In What is a Thing?, on the other hand, the mathematical comes to de¢ne an epoch as a whole, determining both its logic and its ontology, and hence, its speci¢c sciences. Even history, which in Being and Time was distinct from mathematical physics, is now conceived in terms of the mathematical approach. In order to better grasp this change, we must return to some of the basic distinctions set out in Being and Time. There is, in Being and Time, a clear distinction between two types of realms, a distinction that parallels the ontological di¡erence between Being and entities. One type of realm is that of regions (Gebiet). Examples of regions include history, nature, space, life, Dasein and language.19 These regions are

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the subjects investigated by the speci¢c sciences. Scienti¢c research determines the boundaries of each discipline; these boundaries determine which entities are studied by the science in question. At the same time, there are also modes of interpreting the Being of entities, such as presence-at-hand, readiness-to-hand, and Dasein, who is both a mode of Being and an entity. Each of these modes is divided into sub-modes.20 Mathematical physics is a sub-interpretation of the theoretical mode, which is itself a sub-interpretation of the mode of presence-at-hand. All these interpretations of Being are constrained by primordial temporality, which is the horizon of every interpretation of Being. The di¡erence between the two types of realms in Being and Time opens up the possibility of di¡erent interpretations of the same region. Nature can be interpreted as present-at-hand but also as ready-to-hand. In What is a Thing?, such di¡erential interpretation is no longer a possibility.21 The modern epoch is characteristically mathematical, and the mathematical approach determines the Being of the entities in every region. History, in the modern epoch, can only be interpreted from the mathematical vantage point. Mathematics is universalized, becoming the sole mode through which it is possible to gain understanding of the Being of entities in the modern epoch. In light of these developments we can ask: is Heidegger ruling out any other possible way of interpreting Being? Are there no other possibilities for interpreting Being? It would certainly seem that the universality of the mathematical approach leaves no room for non-mathematical interpretation of the Being of entities. Nevertheless, Heidegger attempts to draw the boundaries of the mathematical without detracting from its universality. The boundary he comes up with is conceived through the limits of the epoch. We can even say that in Heidegger's thinking in the 1930s, the epoch plays a role similar to that played by the ontological region in the early writings. Heidegger struggled with the problem of demarcating the modern epoch well into the 1950s. But before we turn to the question of how the mathematical epoch is to be demarcated, let us ¢rst try to understand why Heidegger views mathematics as the universal ontology of modernity. To grasp this, we must return to the question of Husserl's position on the status of ontology and its relation to mathematics,

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since in my view it was in the course of critiquing Husserl's approach that Heidegger's stance on the mathematical crystallized.22

Universal ontology, mathesis and the transcendental ego In Husserl's writings, universal ontology is discussed in two distinct contexts. The central context in which it features is that of the connection between universal ontology and formal ontology/mathesis universalis. A second context is that of the connection between being in general and the given-ness of being to transcendental consciousness. I discussed the former in Chapter 1 above. Husserl links formal ontology, which studies things in general, and formal mathematics. Starting with the Ideas, formal ontology is distinguished from regional ontology, which is also referred to as `material ontology'.23 The less-central context where universal ontology plays a role is implied by the foundational thesis of phenomenology: `Every originary presentive intuition is a legitimizing source of cognition.'24 This principle expresses the intimate link Husserl posits between something's being a thing and its being given to intuition. On this conception, transcendental consciousness is not itself a thing, and everything that exists, exists in relation to it. It is within the framework of this transcendental phenomenological inquiry that the essences of di¡erent regions are revealed. The relation between the two approaches to universal ontology is never really worked out adequately in Husserl's thought, and remains unclear. Phenomenological study focuses on the second context alone, seeking the modes of the given-ness of things. Nevertheless, there are references to formal ontology throughout the entire Husserlian corpus. We can even say that an unresolved tension between the formal dimension and the discussion that takes the transcendental ego as its point of departure pervades Husserl's phenomenology.25 There is no doubt that in its formative years, aspects of phenomenology ^ for instance, the critique of psychologism, the pure theory of manifolds, and the method of variation for extracting essences ^ were greatly in£uenced by developments in mathematics. But as a basis for philosophy, it would appear that formal ontology, like mathesis universalis, mathematics in the broadest sense of the term, is incompatible with

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the transcendental approach. If thinghood has a formal meaning, then grasping this meaning is a precondition for phenomenological inquiry itself, hence phenomenology cannot be the primary science. If, on the other hand, it is phenomenology that is a precondition for any other inquiry, then formal ontology is just one of several areas of investigation, others being time, space and so on, and loses its primacy. Husserl does not deal directly with this tension in his writings. But in the 1920s, he stresses the distinction between phenomenology and ontology, asserting, e.g., that the transcendental ego lacks any ontological characterization, but is, rather, the condition for every investigation, including ontological investigation.26 This stance is already found in the earlier writings, but the emphasis now placed on it is to some extent a reaction to Heidegger's identi¢cation of ontology with phenomenology. The two contexts within which universal ontology ¢gures in Husserl's phenomenology have parallels in Being and Time. Heidegger uses the concept of `formal indication' ( formale Anzeige) to characterize his method; this is, essentially, an appropriation and critique of Husserl's notion of formal ontology. But since Heidegger poses the question of the meaning of Being through Dasein, his principal concern is the transcendental ego^ontology nexus, an issue he addresses via a discussion of the meaning of the `sum' in Descartes' `Cogito ergo sum'. But here too, although he is speaking of Descartes, the real target of this discussion is, I believe, Husserl's transcendental phenomenology. In Being and Time, the main problem Heidegger ascribes to Descartes' ontology in the Meditations is that the sense in which the cogito itself is said to exist remains uninterpreted, and as a result Descartes merges the being of the cogito into the being of extended bodies.27 Descartes interprets the being of extended bodies as constant presence. Returning to Husserl, we can see that one of Heidegger's principal arguments against Husserl is that the transcendental ego does not, in fact, lack any ontological characterization, but on the contrary, presupposes the meaning of constant presence. Hence Husserl's phenomenology has not succeeded in eliminating all ontological presuppositions. Furthermore, because he makes these implicit ontological assumptions, Husserl fails to uncover the di¡erence between the meaning of Dasein's being and the being of the other

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ontological regions. Hence the question of the meaning of being could not be raised within his phenomenology. In this sense, Being and Time can be interpreted as a radicalization of the phenomenological inquiry. Heidegger does indeed mention the centrality of mathematics for Husserl and Descartes, but it is far from clear to what extent he sees this centrality as shaping their ontological frameworks.28 In Being and Time and the other writings of the Being and Time period, Heidegger does not claim that these frameworks presuppose a meaning of being that is derived from mathematics. Rather, his claim is that by way of mathematics it is possible to arrive at constant presence as the meaning of being. In fact, in Heidegger's early writings, his references to formal ontology are not negative. In his very early writings, formal ontology is referred to in connection with Heidegger's own notion of `formal indication'.29 In Being and Time, Heidegger mentions the notion of `formal being-something'. He rejects the uni¢cation of formal ontology and logic, but not totally.30 Formal ontology is also mentioned in lectures, given in 1926, entitled The Phenomenological Interpretation of Kant's Critique of Pure Reason. In the course of clarifying what formal logic meant to Kant, Heidegger claims that the important concept of formal ontology, which is the counterpart of formal logic, is not found in the Critique of Pure Reason.31 Clearly, then, it is not the case that Heidegger completely rejects the notion of formal ontology. In the period up to and including Being and Time, Heidegger criticizes the transcendental approach to phenomenology, but recognizes the value of the idea of formal ontology.

Dasein and formal ontology To summarize our discussion thus far, we can say that whereas Being and Time does not set out any clear connection between Heidegger's critique of Descrates' and Husserl's respective approaches to the meaning of the transcendental ego's being and his analysis of the place of mathematics in their philosophies, such a connection is indeed suggested in What is a Thing?. This change in Heidegger's thinking on this issue is best explained by considering it from a di¡erent perspective,

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and inquiring into the place of formal ontology in Being and Time. To articulate the link between this Husserlian notion and the ontological approach of Being and Time, it will be helpful to examine the marginal notes made by Husserl as he read Being and Time. I will then proceed to my account of the role of formal ontology in Being and Time and in What is a Thing?. In the marginal notes on Being and Time, Husserl seeks to situate Heidegger's discussion of ontology with respect to two possibilities. First, the possibility that the goal of Being and Time is to put forward a formal ontology. This is suggested by the fact that Heidegger declares that the question he is asking is the question of the Being of entities. Now the answer to this question given by Husserl is formal ontology, that is, mathesis universalis. However, Husserl maintains that ontology can also be understood di¡erently, namely, as referring to the world as a framework or horizon within which entities appear. That is, according to Husserl, the world is a general condition for the regional ontologies. For Husserl, these two meanings of ontology are completely distinct from each other. This distinction rests on the distinction between generalization and formalization.32 It is with these distinctions in mind that Husserl sets out to read Being and Time. Husserl's perplexity as to how to understand the ontological discussion in Being and Time arises from the fact that from Husserl's perspective, Heidegger's question has to do with the formal conditions for the being of entities, but the answer provided is framed in terms of the world, that is, in terms of material ontology. Ultimately, Husserl indeed interprets Heidegger as engaged in a discussion of material ontology.33 He therefore reads Being and Time as seeking to reduce all the ontological regions, as well as the most general region, the world, to one particular region, namely, Dasein. Husserl sees this approach as anthropological, and subjects it to a scathing attack in `Phenomenology and Anthropology'.34 His main argument in this article is that any position that takes the concrete individual as its starting point will be unable to make the radical move of establishing ontology as the primary science.35 It can be made only by way of a transcendental reduction that discloses the transcendental ego, which is the foundation for any positing and any ontology, including that of the concrete individual.

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We can now proceed to examine Being and Time against the background of these ontological concepts. The question to be answered is clear: in Being and Time, what sort of enterprise is Heidegger engaged in: formal ontology or material ontology? As I see it, this question is not only central to the attempt to formulate a Husserlian understanding of Being and Time, but one of the keys to grasping the thrust of Being and Time in and of itself. To explore the question adequately, we must focus on the meaning of Dasein given the notions of regional ontology and formal ontology. Dasein serves both as a region and as a mode of being. In the terminology of Being and Time, the ¢rst characterization is `ontic', the second `ontological'. Dasein's two roles are connected, since `Dasein is ontically distinctive in that it is ontological.'36 But what is the relation between these characterizations and the question of Being? Dasein has primacy, vis-a©-vis the question of Being, in three respects: ontically, ontologically and onticontologically.37 Dasein's ontic primacy follows from the fact that it is that which is closest to us. Dasein has ontological primacy by virtue of understanding its being as part of its ontic constitution. And it has ontic-ontological primacy because it understands not only its own existence, but the being of all other entities as well. This third primacy follows from the second, since an aspect of understanding its own being is grasping the distinction between it and the being of other entities. All three respects in which Dasein is primary, particularly the third, seem to con¢rm Husserl's judgment that the philosophical approach of Being and Time is anthropology, since the modes of Being are all reduced to Dasein's understanding, which is a part of Dasein's ontic constitution. In response to the charge that Being and Time is an exercise in anthropology, Heidegger stresses the importance of the role played by the notion of transcendence in Being and Time. I would argue that this means that in taking Dasein to be transcendence, we cannot interpret Dasein as simply the ontic region occupied by mankind.38 The unique role of Dasein should, therefore, be explained in the following way: every ontological region is de¢ned not only with respect to the entities it comprises, but also with respect to that which is beyond it. Hence every region necessitates going beyond ^ transcendence. As an ontic-ontological condition, however,

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Dasein is not transcended, but transcendence itself. Ontically and ontologically, Dasein is a region, but ontically-ontologically, Dasein is the condition for every other region. There is, in my opinion a further justi¢cation for the special role of Dasein and its priority vis-a©-vis the question of Being, and the limitations of any formal ontology. If formal ontology supplies the characterizations of all entities, it becomes a general or universal ontology. But characterization of the Being of entities in general terms is problematic. Paradoxically, if Being (or its meaning) is characterized as general, it loses its universality, because the particular di¡erences between beings lie beyond it. It is, indeed, this very problem concerning the question of Being that is the basis for Aristotle's claim that Being is not a genus. Heidegger's conception of the interdependence of the ontic and the ontological levels is, I believe, an attempt to solve the problem. If universality is to be retained as a characteristic of Being, there cannot be a total separation between Being and entities, though the distinction between them must be preserved. In light of all this, it thus seems that Heidegger's treatment of fundamental ontology does not remain solely at the level of ontological regions, and it is incorrect to characterize it as exclusively a material ontology. On the other hand, Heidegger rejects formal ontology in the sense of a treatment of Being that is outside the framework of a speci¢c region. If my account is correct, Being and Time can be seen as resolving the deep-seated tension between phenomenology and formal ontology that pervades Husserl's thought.39 From the phenomenological perspective, there are only ontic regions whose essence is constituted by the transcendental ego, but there is no room for a transregional formal ontology. Yet it is not clear how it is possible to speak of ontical regions without a formal concept of being. No solution to this dilemma can be found in Husserl's writings. It may even be the case that Husserl's sensitivity to the tension between phenomenology and formal ontology was due largely to the impact of Being and Time. Indeed, this may explain why Husserl distinguishes, in The Crisis of European Science, between the formal/mathematical realm, on the one hand, and the realm of phenomenology and the real world, on the other.40 He thereby links phenomenology with ontology, the latter

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being directly anchored in the concept of the world, without any invocation of the formal-mathematical aspects of the characterization of being.41 In this section, we have attempted to clarify the web of connections between Dasein, the transcendental ego, and formal and material ontology. Formal ontology, so inextricably linked to mathematics, was a central factor in the crystallization of the role of Dasein in the ontology of Being and Time. And as we are about to see, the change Heidegger's thought undergoes in the 1930s mandates a renewed discussion of formal ontology, which, I will argue, is the underlying theme of What is a Thing?.

The change marked by What is a Thing? That Heidegger's philosophical position has changed radically by the time he writes What is a Thing? is most clearly re£ected in the fact that Dasein does not provide a demarcation of an ontology on the basis of mathematics, but rather, is de¢ned or situated by the mathematical qua ontology.42 As a result, Heidegger weaves together two elements that Husserl separated: the transcendental ego and its role in grounding all ontology, on the one hand, and formal ontology, on the other. Whereas in Being and Time Heidegger's critique of the transcendental ego and his discussions of mathematics and formal ontology remained distinct, in What is a Thing? Heidegger argues that the role of the transcendental ego is a consequence of formal-mathematical ontology. This change in Heidegger's approach can be discerned if we examine the principal focus of his interpretation of Descartes' philosophy. Whereas in Being and Time he focused primarily on the Cartesian `I', in What is a Thing? the `I', and its primacy, follow from the role played by the concept of the foundational principle in the ontology of the modern epoch. This change is just one aspect of the broader turn toward the history of Being in Heidegger's thought. In light of the interpretation of Being and Time I have presented above, this change should be understood as Heideggerian self-critique. As we saw, the characterizations

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of Dasein in Being and Time are arrived at, in large measure, in seeking to articulate the connections between regional and formal ontology, and to invoke this connectedness in attempting to establish how, from a phenomenological perspective, formal ontology is possible at all. Heidegger's characterization of Dasein is thus arrived at by way of the constraints set down by the Cartesian tradition, despite the fact that there are signi¢cant di¡erences between his, Descartes' and Husserl's respective philosophical orientations. Thus in Being and Time the break between Heidegger and the preceding philosophical tradition is only partial.43 Heidegger's writings from the 1930s, and in particular, the turn toward the history of Being, can be seen as a radicalization of a process that was nascent in Being and Time, namely, an attempt to answer the question of Being without de¢ning it in terms of the philosophic tradition. This claim might seem somewhat perplexing, since these writings appear to be almost entirely devoted, at least on the surface, to the philosophical tradition. But as I will try to show, in the writings from the 1930s the tradition is interpreted via a lexicon that includes such terms as `history of Being' and `epoch', which although not de¢ned by way of the concepts that guided the discussion in Being and Time, have the same objective ^ to set out a framework for dealing with the question of Being. Another factor also contributed to Heidegger's concentration on mathesis and mathematics: the trenchant criticism of his philosophy that was being voiced in the early 1930s. Two main arguments were adduced against Heidegger. The ¢rst is articulated in Rudolph Carnap's `The Elimination of Metaphysics through the Logical Analysis of Language', which takes issue with the thrust of Heidegger's 1929 article, `What is Metaphysics?'. Carnap's basic claim is that, like all metaphysical statements, the statements that make up `What is Metaphysics?' are meaningless. It might seem that Carnap's argument addresses a question pertaining to the theory of meaning. But in an `Afterword' to the article, published in 1943, Heidegger interprets Carnap's position in `The Elimination of Metaphysics' as reducing all thought to calculation, and as construing being as amenability to being counted. In other words, he reads Carnap as ascribing being only to that which can be counted. That is, Heidegger takes Carnap's attack to be premised on this notion of being. In response, he argues

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that all counting and calculation presuppose an uncountable whole.44 In making this claim, Heidegger appears to be reverting to the position he had upheld in the very early writings.45 But as I will establish in the next section, Heidegger's understanding of mathematics has changed signi¢cantly since the very early writings. His basic critique of formal logic or `logistics', as it was sometimes called, remains in place, but what he takes to be the meaning of mathematics has evolved considerably, enabling him to address positivistic views such as Carnap's.46 The second criticism directed at Heidegger's philosophy in the early 1930s was raised by Oscar Becker.47 Becker argued that the notion of Dasein does not su¤ce for ontology, and must be supplemented by that of Dawesen, understood as essence or ground; Becker links it to both mathematics and nature. Dawesen is also related to the possibility of artistic creation. When the Nazi party came to power, this critique of Heidegger's philosophy broadened. Becker saw Heidegger's approach as a continuation of Husserlian idealism. The notion of Dawesen now grew to encompass, on Becker's view, not just art and mathematics, but also race and blood.48 According to Becker, there are primordial essences that Dasein cannot alter. It may be that Heidegger's discussion of mathematics in his course lectures from 1935 is an attempt to counter Becker's interpretation of the Nazi ideology. This might be another reason why mathematics came to occupy such a central place in Heidegger's thinking in the second half of the 1930s. What is a Thing? can also be related to the Davos debate between Heidegger and Cassirer in 1929. In the debate Heidegger takes the position that ¢nitude is the framework for all of Kant's philosophy, while Cassirer claims that Kant's account of mathematics requires the transcending of human ¢nitude. Furthermore, mathematics is much more central to Cassirer's philosophy, and in particular, his interpretation of Kant, than it is for Heidegger. Although mathematics plays a far more important role in What is a Thing? than in Kant and the Problem of Metaphysics, I believe, as will become clear below in the section on the Critique of Pure Reason, that this is due to internal developments in Heidegger's thought, and not to his having adopted Cassirer's interpretation of Kant.49

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Mathesis, ontology and logic The challenge Heidegger faces is how to demarcate mathematics from ontology without recourse to fundamental ontology. He approaches this challenge from two directions. First, he attempts, as we saw above, to pinpoint the essence of mathematics as ontology; second, he attempts to ¢x its boundary by delineating the bounds of an era. Granted, Descartes ushered in the mathematical epoch, but this determination does not su¤ce. Unless boundaries can be set for mathematics, it will be the only ontology possible, since qua ontology, mathematics is universal. On the other hand, demarcation of the realm of mathematics by adducing another ontological framework is out of the question, since that would constitute a return to the approach taken in Being and Time. The sole solution available to Heidegger is to make mathematics self-delineating. This self-delineation is carried out in What is a Thing? with the help of the Critique of Pure Reason. The process of self-delineation presupposes that the realm of the mathematical has an essence. This essence was described above, in `What is a Thing? on the mathematical', but now, having completed our comparison of Husserl's and Heidegger's respective conceptions of ontology, it will be instructive for us to undertake a similar comparison of their notions of mathematics. My argument has been that in several important respects, Heidegger has actually embraced the same Husserlian position he is trying to undermine. That is, although Heidegger attempts to liberate himself from the Husserlian ontological framework, when it comes to delineating the boundaries of the mathematical he adopts, to a considerable degree, Husserl's understanding of mathematics. First of all, it appears that Heidegger regards the axioms that establish the thingness of a thing as axioms establishing the domain of things. In Formal and Transcendental Logic, Husserl claims that formal mathematics, taken at its broadest, mandates a universal extension that is de¢ned as the extension of the concept `thing in general'.50 An additional aspect of Husserl's conception of formal mathematics is characterization of the notion of a system of axioms as de¢nite. A system is de¢nite if: `Any proposition which can

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be constructed out of the distinctive axiomatic concepts, regardless of its logical form, is either a pure formal-logical consequence of the axioms or else a pure formal-logical anti-consequence ^ that is to say, a proposition formally contradicting the axioms.'51 Combining these two characterizations ^ the characterization of the extension of `thing in general', and the characterization of de¢nitude ^ gives us Heidegger's conception of the mathematical as ontological. The mathematical as ontological is the determination of the realm of the thing in general by way of axioms. The key di¡erence between the positions of Husserl and Heidegger is that Husserl maintains that there is a distinction, within the realm of logic, between apophantic logic and formal ontology, whereas Heidegger does not. Apophantic logic deals with the structure of propositions, linguistic categories and deductive relations, in contrast to formal ontology, which deals with being in general and the categories relevant to it. There are reciprocal inclusion relations between these two aspects of logic. Apophantic logic subsumes formal ontology, since ontological determinations are propositions that describe being in general. On the other hand, every proposition presupposes that which it is about. Moreover, every proposition is a kind of thing and therefore apophantic logic is a part of formal ontology. Both these characterizations are linked to mathematics: formal ontology is linked to mathesis universalis and apophantic logic to the characterization of a mathematical theory as a deductive system. As I said, the distinction plays no role in What is a Thing?. For Heidegger, the deductive system has primacy in apophantic logic, since it determines the structure of sentences, but also in formal ontology, since it determines the being of entities. Heidegger's premise of the primacy of the deductive system, and his rejection of the distinction between apophantic logic and formal ontology, can be seen as part of his response to Carnap's article. In the introductory remarks that preface his critique of Heidegger, Carnap presents a method for determining the meaningfulness of a word. The ¢rst stage in determining meaningfulness is to transform the word into a sentence. For example, if we want to discuss the word `stone', we examine the sentence `x is a stone', which we can call s. For s to be meaningful, we have to answer the following question, which

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has several equivalent formulations: from what sentences can s be deductively derived, and what sentences follow from s?; what are the truth conditions of s?; how do we verify s?; what is the meaning of s? According to Carnap, the ¢rst formulation is the correct one.52 Deductive relations thus play a decisive role in determining a sentence's meaning. Moreover, on the Carnapian conception, ontological analysis cannot be prior to logical analysis. Carnap can thus be interpreted as claiming that it is not just meaning that deductive relations determine, but ontology too. Frege, many elements of whose thought Carnap adopted, had also maintained the primacy of logic over ontology. One question that had been left open to a certain extent by Frege was that of which element of logic was primary, the proposition or the deductive system. In the preface to the Begri¡sschrift, the inferential sequence is prior, that is, what Frege calls `conceptual content' is extracted by focusing on deductive considerations only.53 In The Foundations of Arithmetic, on the other hand, it seems that the proposition has priority. If we accept, as a fundamental principle of the logical analysis of language, the contextuality principle, according to which we can inquire about the meaning of a word only in the context of a proposition, then, clearly, the ontological categories depend primarily on the logical structure of propositions.54 Heidegger's understanding of mathematical ontology, and the di¡erences between this understanding and Husserl's formal ontology, yield a general framework for mathematics that re£ects the positions of both Husserl and Carnap. Heidegger's understanding of mathematical ontology, it should be noted, re£ects the changes in Heidegger's position after the Being and Time period. In Being and Time mathematics is interpreted via the existential characterizations of Dasein, whereas in What is a Thing? it is characterized as a system of propositions.55 Heidegger's position with regard to the status of the foundational principles of ontology in the mathematical epoch is in certain signi¢cant respects quite close to Hilbert's characterization of an axiomatic system. The central feature of an axiomatic system is that no distinction is made between de¢nitions and axioms, but rather, the system as a whole de¢nes the objects of which it speaks.56 Hence the theorems of geometry, like those of any other formal system, are not about objects

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given beforehand. The axiomatic system in its entirety determines its ontology, and nothing is given prior to that determination.57 Thus the subject of a sentence in a formal system has no primacy over the sentence as a whole, but rather, it is de¢ned by the entire axiomatic system. As we have seen, if this stance is adopted, it is very similar to Heidegger's understanding of the role of foundational principles in the mathematical epoch.58 Heidegger's move can thus be viewed as an attempt to capture the insights of the Cartesian, Kantian and Hilbertian understandings of what an axiom is. This synthesis might seem strange, since Descartes takes the cogito to be certain and absolute; its status qua axiom is not established by virtue of the realm it is taken to ground. Kant, on the other hand, does indeed posit a link between axioms and the realm of possible experience; for him, therefore, axioms, being linked to a speci¢c realm, lose their absolute status. Lastly, Hilbert takes axioms to be but a tool for the generation of propositions. In themselves, they have no autonomy whatsoever, and are merely instrumental. Despite these di¡erences, Heidegger is essentially taking the position Hilbert took with regard to the question of how to establish the thinghood of the thing in that Heidegger establishes the thinghood of the thing on the basis of fundamental principles that do not themselves presuppose anything.

The limits of the mathematical in the Critique of Pure Reason Given his characterization of the essence of mathematics, Heidegger must now confront the question of how to demarcate the mathematical. In What is a Thing? this demarcation is carried out by means of the Critique of Pure Reason. The mathematicity of the Critique of Pure Reason arises from the fact that the discussion of the thinghood of a thing rests on fundamental principles and is prior to any discussion of the various ontological regions. In Kant's time, the regions in question were God, nature and man. This division into regions re£ects the medieval approach to ontology, the basic premise of which is that entities were created.

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The question, then, is how the mathematical conception of the being of entities ¢ts with the Christian conception of being, at the centre of which lies the notion of Creation. The mathematical approach mandates the primacy of the clari¢cation of the being of entities over the discussion of the being of the various ontological regions ^ the very approach taken by Kant in the Critique of Pure Reason. Hence Heidegger's claim that the Critique of Pure Reason is the unfolding (Entfaltung) of the mathematical approach. But Heidegger goes further, arguing that in the Critique of Pure Reason the mathematical approach is elevated to the point where it reaches its own limits and supersedes itself.59 In light of the connection Heidegger makes between the limits of pure reason and the limits of the mathematical approach, his interpretation might be expected to focus on the dichotomy between mathematics, which is restricted to the realm of possible experience, and the realm that is beyond any possible experience. In other words, as the mathematical cannot overstep the limits of possible experience, they constitute its own limits as well. But I would argue that for Heidegger, additional limits are, of necessity, interwoven into the boundaries of reason. With respect to the move that Heidegger is making, the decisive boundaries are the limits of thought, and the dichotomy between thought and intuition. Heidegger sees in this dichotomy the principal limitation of the mathematical approach. This might be surprising given that, as is well known, for Kant intuition is thoroughly imbued with mathematics. Intuition has a form, a form that is intimately linked with geometry and arithmetic. But we must not lose sight of the fact that here Heidegger is not interested in mathematics as a science. Mathematics is, like any other science, anchored in intuition. This is precisely the import of the Critique of Pure Reason. But the mathematical approach itself cannot be anchored in intuition, since its very essence is that the foundational principles must entail the being of entities on their own, nothing else being presupposed. The need for intuition limits both the foundational principles and the autonomy of the determination of the existence of the being of entities on the basis of the foundational principles, and in this sense, intuition constitutes a limitation on the mathematical

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perspective. That is, on the Kantian position, intuition re£ects the need for givenness that is prior to the foundational principles that establish the thinghood of things. How does reason delineate its boundaries? To understand this, we must go back to the idea of establishing ontology on the basis of foundational principles. This project mandates the primacy of ontology over logic, in the following sense: not only must the being of entities be established on the basis of foundational principles before it is possible to analyze the structure of a judgment, but the structure of the judgment is subject to the same principles. The linkage between the structure of a judgment and the being of entities is already present ^ by way of the categories ^ in Aristotle's thought, but according to Heidegger, Kant makes a crucial change in the nature of this connection. Heidegger interprets this fundamental change in the relationship between logic and ontology as a change in the essence of logic itself. That is, according to Heidegger, making reason subject to intuition radically changes the very nature of logic. It mandates not merely supplementing the existing logic of the period, but rather, completely re-conceiving it.60 This new conception is supplied by the mathematical approach, that is, by establishing the being of entities on the basis of foundational principles. To understand the change wrought by Kant, it will be helpful for us to go back to Aristotle's notion of the assertion (Aussage). According to Aristotle, an assertion says something about something. That is, it is the predication of an attribute to a subject. Though this characterization is found in the Critique of Pure Reason (B10), elsewhere in the Critique Kant takes a step forward, whereby he does not merely add to this characterization, but penetrates its core. This move is related to the notion of the judgment presented in the writings of Christian Wolfe. Wolfe characterizes the act of judging as the connection or separation of concepts. Kant is frequently interpreted as having adopted Wolfe's position, but, as Heidegger emphasizes, he rejected it in the `Transcendental deduction of the pure concepts of understanding': `I have never been able to accept the interpretation which logicians give of judgment in general. It is, they declare, the representation of a relation between two concepts' (B140). In place of the position upheld by Wolfe, Kant argues that `a judgment is nothing but

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the manner in which given modes of knowledge are brought to the objective unity of apperception' (B141). For Heidegger, the import of this change is the following: `Judging, as an action of understanding, is not only related to intuition and object, but its essence is de¢ned from its relation and even as this relation.'61 The meaning of the change, therefore, is that the subject^object axis is the foundation for the subject^predicate axis. It is now possible to understand why Heidegger calls the Critique of Pure Reason the unfolding (Entfaltung) of the mathematical approach. As explained above, in the mathematical epoch, logic and ontology are established on the basis of foundational principles. We saw that Heidegger interprets Kant as having transformed logic by grounding it in the subject^object nexus. The principles that determine the objecthood of the object are synthetic a priori principles. This determination is also a determination of the ground from which the structure of judgment is derived, hence these foundational principles are the foundation for both the objecthood of the object and for logic in the sense of grasping the relation between subject and predicate. This interpretation of Kant is very di¡erent from the standard interpretation, on which Kant takes general logic as a premise from which he proceeds to deduce the categories. It also di¡ers markedly from Heidegger's earlier reading of Kant, which took the power of imagination to be the basis for Kant's position in the Critique of Pure Reason. This interpretation also serves as Heidegger's starting point for his critique of mathematical logic, or as he calls it, `logistics'. Mathematical logic is not logic in the very basic sense of the term, nor is it the logic governing mathematics. It does not govern the fundamentals of mathematics, but can at most be characterized as the mathematics of logic. The primary explanation for this limited scope, according to Heidegger, is that logistics, being based on Aristotle's conception of the judgment, is unable to give an account of mathematics in the deepest sense. Hence one of Heidegger's objectives in o¡ering a reading of Kant is to attack mathematical logic. In contrast to logistics, the logic Kant arrives at does reach the bedrock that allows for the construction of true mathematical logic. This profound change in the way logic is understood is also the basis for the distinction between analytic judgments and synthetic

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judgments. The distinction is not found in the traditional theory of the judgment, namely, the theory that a judgment is a unity of subject and predicate. Every judgment requires a divide between the subject and the predicate, but also their uni¢cation. Thus, every judgment is a synthesis but also an analysis.62 Heidegger's claim is that the distinction between analytic judgments and synthetic judgments is only tenable on the basis of a completely new conception of the structure of the judgment, at the centre of which is the subject^object axis. The distinction between an analytic judgment and a synthetic judgment is that the former requires only the concept of an object, whereas in the latter, the object itself is referred to. In light of this, we can describe Heidegger's objection to Kant's position as follows: in the ¢nal analysis, the Critique of Pure Reason, in stipulating boundaries of thought, makes formal logic necessarily subordinate to transcendental logic. This dependence is illustrated in Heidegger's discussion of what Kant calls the `principle of analytic judgments'. On Heidegger's reading of Kant, the law of contradiction only serves as a constraint on judgments, but does not constitute the highest formative principle of analytic judgments. Rather, it is the law of identity that is the governing principle of analytic judgments. The novelty of this conception is that principles traditionally interpreted as constitutive for all judgments are now conceived by Kant as constitutive of only some judgments. Therefore the principles of all synthetic judgments and the relation to intuition that they require should, on Heidegger's reading of Kant, be interpreted as necessary for the completeness of formal logic. Hence, restricting the law of indentity to analytic judgments alone is an important aspect of the delineation of the mathematical ontology. Since transcendental logic applies to only one ontological region, limiting formal logic to a single ontological region proves that the ontology determined by foundational principles, that is, mathematical ontology, cannot be universal but is only regional. We can, therefore, formulate Heidegger's argument as follows: mathematical ontology can only ful¢l itself if there is a new logic; this new logic mandates that thought be subject to intuition, a conclusion that is arrived at in two steps. First, the concept of object is necessary for every judgment. Second, the concept of an object is determined fully only by synthetic

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a priori principles, necessitating intuition. Therefore, the determination of the thingness of the thing on the basis of principles, which is the hallmark of mathematics as ontology, can be realized only by means of its relegation to the region of possible experience. This move is to a great extent the maturation of Heidegger's earlier writings on Kant, especially the 1927 lectures on the Critique of Pure Reason.63 In these lectures, Heidegger tries to show, among other things, that in the ¢nal analysis Kantian formal logic presupposes transcendental logic: since formal logic requires the concept of the object, and since any discussion of objects falls under transcendental logic, formal logic presupposes transcendental logic. That is, formal logic presupposes ontology, which, ultimately, is reduced to the region of nature that is interpreted by Heidegger as `presence'. In these lectures, human ¢nitude is a central element in this linking of formal logic with transcendental logic, since it is the reason why, in human thought, relating to an object requires intuition. One of the most signi¢cant di¡erences between The Phenomenological Interpretation of Kant's Critique of Pure Reason and What is a Thing? is that in the former, Heidegger distinguishes between formal ontology and formal logic,64 a distinction that will serve as the basis for the remaining steps in the broader project of linking Kant's transcendental logic with fundamental ontology. In fact, I would argue that there is an implicit correspondence between formal ontology and fundamental ontology, inasmuch as, according to Heidegger, Kant's disavowal of formal ontology is the reason for his restricting existents to natural objects. Hence fundamental ontology, which exposes the limitations of interpreting the Being of entities as presence, ful¢ls the function of formal ontology, namely, preventing the reduction of the meaning of Being to a single ontological region. In What is a Thing?, on the other hand, the distinction between formal logic and formal ontology is not made, and the goal of Heidegger's reading is to generate a self-delineation of the mathematical/formal approach to ontology.65 Heidegger's objective, in 1927, was to make formal logic subject to transcendental logic, interpreted as ontology, with formal ontology serving as a possible escape route from the limitations of what Kant took to be the meaning of being. In What is a Thing?, on the other hand, Heidegger reads the Critique of Pure Reason as an attempt

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to ground formal ontology, or mathesis universalis, an attempt that highlights the self-delineative nature of formal ontology, which also characterizes formal logic. This self-delineation is found on a number of levels. First, the general determination of the being of entities on the basis of foundational principles excludes ontological regions such as the world, the soul and God from ontology. Second, exclusion of these regions is created by subjecting thought to intuition. Third, the need for pre-determination of the thingness of things, and the subjection of thought to intuition, create a clear distinction between analytic and synthetic judgments. The logical principles serve as foundational principles only in the sphere of analytic judgments. That is, the analytic/formal sphere is delineated at this level too. On the traditional conception of logic, there was no clear distinction between analytic and synthetic judgments, and the logical principles applied to every judgment. The interpretation of the role played by the Critique of Pure Reason in the general framework of What is a Thing? suggests a reevaluation of Heidegger's relation to Kant's thought. On the surface, it might seem that during the Being and Time period Heidegger felt close to Kant, and clearly distinguished between his views and those of Descartes,66 but in the 1930s distances himself from Kant, considering him Descartes' direct successor. Focusing on formal ontology reveals a di¡erent relationship between Heidegger and Kant. In the 1927 lectures, Heidegger highlights the limitations of the Kantian conception of being, whereas in 1935 these limitations have come to be seen as an advantage, since they allow for the self-delineation of mathematics.

The signi¢cance of Heidegger's argument This reading of Kant is, in my opinion, Heidegger's response to Carnap.67 The distinction between analytic judgments and synthetic sentences, the cornerstone of Carnap's philosophy, and of his notorious paper attacking Heidegger, presupposes transcendental logic and therefore ontology as well: it presupposes an ontology that is prior to formal logic. If Carnap has to assume an ontology in order to implement his logical distinctions, what he has to presuppose is

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nothing less than a pre-logical ontology, where logic is understood in the Carnapian sense. This exposure of his presuppositions radically undermines the basis for Carnap's critique of Heidegger.68 But Carnap's theory is not Heidegger's chief target. Heidegger is seeking to demarcate formal/mathematical ontology without relying on fundamental ontology. He tries to demonstrate that when an ontology based on predetermination of the being of entities via foundational principles attempts to be universal, that is, attempts to be a general ontology, it is inevitably transformed into a merely regional ontology ^ the ontology of the realm of possible experience. And indeed, Heidegger believed he had succeeded in creating a selfdemarcating mathematical epoch as an ontological region with clear boundaries. Furthermore, What is a Thing? can be read as collapsing the distinction between universal ontology and regional ontology. That is, the overcoming of the mathematical approach is not the transcending of the possibility of establishing a universal ontology based on an axiomatic system, but the rejection of the entire stock of Husserlian ontological concepts, particularly those pertaining to formal ontology and regional ontology. And it was in relation to these concepts that the ontological concepts of both Heidegger's early writings and Being and Time crystallized. A hint that this is, indeed, part of the basic agenda of What is a Thing? can be found in Heidegger's `Science and Re£ection', written in the early 1950s. In this paper, Heidegger contends that the division into ontological regions is closely related to the notion of reckoning.69 That is, Heidegger connects the delineation of a region and the demarcation of its boundaries to mathematics in the ontological sense. The ontological regions are possible only on the basis of formal, trans-regional ontology.70 Since formal/mathematical ontology can realize itself only as a regional ontology, the very distinction between them collapses. More generally, given that Heidegger characterizes metaphysics as an inquiry into the essence of the thing, viz., its thinghood, What is a Thing? can be understood as an attempt to demarcate metaphysics, in its modern incarnation, so as to reveal the internal limits of the modern epoch in the history of Being, and in so doing, bring about its destruction (Destruktion).

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This critique of Husserl also opens up the option of re-appropriating Being and Time without its `mathematical' aspects. As we have seen throughout this chapter, there are a number of links between Heidegger's project in Being and Time and his characterization of mathematical ontology. But although Heidegger abandoned Being and Time as the framework for examining the question of Being, he did not want to reject it completely. From the 1930s on, Being and Time is interpreted as a partial answer to the question of Being. It explores Dasein's understanding of the question of Being, but does not put forward a comprehensive answer to the question of Being. As such it is very di¡erent from the project of mathematical ontology set out in What is a Thing?

Are logic and mathematics mathematical? The main premise of What is a Thing? is the continuity between mathematics, mathematical physics and the mathematical approach to the question of Being. That is, modern science and mathematics are the embodiment of the mathematical approach. In making this assumption, Heidegger is denying that there is a divide between the positive sciences and basic or general ontology. This divide, in many respects an adaptation of Husserl's distinction between the positive sciences and the foundational science, namely, phenomenology, was one of the central pillars of Heidegger's approach in Being and Time, where he develops it via the distinction between the ontic and the ontological. The sciences are directed toward speci¢c regions of entities (nature, the mind), and presuppose the existence of the entities they study. These presuppositions are not accessible within the positive sciences. To render them apparent we must engage in an ontological discussion that discloses the conditions that make possible the delineation of particular ontological regions. The ontological characterization of these regions presupposes an additional stratum, that of fundamental ontology.71 In What is a Thing?, however, these distinctions collapse. Both the distinction between a region and the underlying ontology of that region, and that between regional ontology and fundamental

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ontology, disappear. Heidegger is thus by default committed to a conception that does not di¡erentiate between the sciences and the ontology that serves as their foundation. Therefore, any di¡erentiation between mathematics and both the natural sciences and mathematical ontology in the Heideggerian sense, is ruled out. The import is that in the modern epoch, there are no ontological distinctions within a time period, and the only ontological distinctions that can be identi¢ed are those between the di¡erent epochs. Given the comprehensiveness of every epoch, on the one hand, and the multiplicity of epochs, on the other, the question arises, what would explain an epoch's coming to an end, and being followed by another? There is no clear answer to this question. Heidegger's metahistorical framework for the history of Being is based on the notion of the truth of Being. The truth of Being, with its essential element of concealment, is the main reason why in every epoch there is only a partial answer to the question of Being, but does not explain the passage from one partial answer to another. Heidegger did not uphold a general account of the transitions from one epoch to the next. There are no Hegelian explanations in this vein in his oeuvre, and he does not appear to have adopted such a stance with respect to the history of Being. Moreover, such a stance would have amounted to returning to a variant of the kind of fundamental ontology from which Heidegger struggled to free himself throughout the 1930s. Indeed, this accounts for Heidegger's ongoing engagement with modernity and modern science/technology from the 1930s through the 1950s. This engagement is Heidegger's attempt to ¢nd in modernity an internal crack or ¢ssure that will enable it to be overcome, despite being fully aware that even on his own stance, such a ¢ssure is hardly possible. What is a Thing?'s destruction of mathematical ontology does not put an end to the discussion of modernity, for it does not provide modernity with the sought-after internal ¢ssure or di¡erence. One possible solution to the dilemma of the Heideggerian stance is to situate the required internal di¡erence in mathematics itself. This possibility arises out of the introduction to Being and Time apropos Heidegger's description of the crisis besetting the various sciences. This crisis, which can only a¥ict a science that has matured, overturns (wanken) the connection between a positive science and its

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object. Heidegger mentions the debate between formalism and intuitionism as an example of such a breakdown in mathematics.72 But in What is a Thing? Heidegger interprets the formalism^intuitionism debate as part of mathematics qua ontology. That is, the debate cannot be interpreted as undermining the mathematical approach to ontology. It seems to me that Heidegger's position in What is a Thing? is based more on his interpretation of the views of the two sides with respect to the fundamental question of whether mathematics is based on some primordial intuition or is a content-less, empty formalism, than on the substantive positions actually espoused by Brouwer and Hilbert. According to the interpretation I ascribe to Heidegger, the Kantian approach, on which formalism and intuitionism imply each other, can be adduced to demonstrate that the debate does not constitute a threat to the conception of mathematics qua ontology but rather is implied by it, much as the entire Kantian view is.73 At the same time, it is important to point out that this sort of conception does not really match the level of profundity of Brouwer's intuitionism. If we evaluate it on the basis of Heidegger's own characterization of mathematics qua ontology, it is clear that Brouwer's position cannot be said to follow unequivocally from the mathematical approach to ontology. On the one hand, the manner in which Brouwer's intuitionism is articulated exhibits, to a great extent, the mathematics-as-ontology approach. Brouwer's starting point is the principle of the intuition of time, from which he derives the logic and the ontology of mathematics. But on the other hand, the ontology derived from intuitionism is incompatible with the requirement, laid down by both Husserl and Heidegger, that the objects of mathematics have the characteristics of a de¢nite manifold. The continuum is not a de¢nite manifold, since the question of whether a choice sequence belongs to an open segment of the continuum has no decisive answer. But focusing on the concept of the de¢nite manifold proves to be useful; because through it we come to see that the ontological di¡erence we have been seeking is present, as I suggested above, in mathematics itself. Above, I o¡ered an interpretation, based on Husserlian concepts, of Heidegger's claim that the mathematical as ontology determines the thingness of the thing by means of foundational premises. But the problem is that mathematics does not constitute a

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realization of Husserl's suggested characterization of the de¢nite manifold. GÎdel's incompleteness theorems are a serious blow to Husserl's criterion.74 Arithmetic cannot satisfy the condition for nomologicity, and mathematics and physics, which presuppose arithmetic, cannot be nomological either. At ¢rst glance, it seems that GÎdel's incompleteness theorems lead to a conclusion in mathematics that is, in essence, no di¡erent from Heidegger's conclusion with regard to the mathematical ontology in What is a Thing?: any attempt to provide a universal ontology based on an axiomatic system collapses, since no manifold that contains the natural numbers can be determined by an axiomatic system. But these theorems also reveal a gap between the mathematical ontology, as Heidegger understands it, and mathematics. Mathematics is not mathematical.75 It might seem as if GÎdel's incompleteness theorems ¢t in with Heidegger's critical stance on the mathematical ontology. The collapse of the mathematical ontology is found in mathematics itself. But on the other hand, GÎdel's theorems make Heidegger's discussions of the mathematical epoch super£uous. There can be no overarching mathematical epoch in terms of which ontology and all the sciences can be framed. If the ontological di¡erence is generated at the ontic level, there is no need for Heidegger's attempts to create an ontological gap by means of which the mathematical epoch can be delineated. The mathematical epoch is demarcated by mathematics itself. GÎdel's results are not in keeping with Heidegger's views on the relation between the sciences and the question of Being. Heidegger maintained that the sciences presuppose ontology or the question of Being, a position with regard to which he never wavered. In `Science and Re£ection', for example, he declares that `if we want to assert something about mathematics as theory, then we must leave behind the object-area of mathematics, together with mathematics' own way of representing. We can never discover through mathematical reckoning what mathematics itself is'.76 Heidegger continued to uphold this view even when he adopted the notion of the history of Being. Although there is no de¢nitive answer to the question of Being, the shift from one epoch to another cannot come from entities, but only through Being. In this sense Heidegger is still a captive of the Husserlian position.

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Heidegger's broader ontological outlook assumes that entities can only be disclosed from within a certain ontological framework. Given this outlook, it is impossible to ascribe to entities autonomy vis-a©-vis a particular ontological framework, or to maintain that there are nonontological ways of revealing something concerning entities. Nevertheless, I would argue that the notion of an `ontological di¡erence' can be interpreted more symmetrically than Heidegger suggests.77 Indeed, I would argue that it can arise from the ontic stratum, from the realm of entities, and not only from the ontological stratum, the realm of Being.78 To recapitulate, in this chapter I have identi¢ed the connections between the concepts of formal ontology, regional ontology and fundamental ontology, and Heidegger's discussion of the modern epoch, the mathematical sciences and the mathematical ontology. Central to this context are the links between the Husserlian transcendental ego and the Heideggerian Dasein. In light of my analysis, I interpreted Dasein as the Heideggerian solution to the problem of the possibility of formal ontology given the constitutive role of the transcendental ego. The links between the transcendental ego and formal ontology are the key signposts to understanding the Heideggerian connection between the mathematical approach to ontology and the Cartesian subject. I have interpreted What is a Thing? as an attempt to undermine the mathematical ontology. In order for general ontology, which establishes the thinghood of things a priori on the basis of foundational principles, to be possible at all, it must be restricted to the realm of possible experience. Heidegger's argument for this thesis is based on his interpretation of the Critique of Pure Reason. The thinghood of a thing cannot be established by means of analytic principles, since clearly, distinguishing between analytic judgments and synthetic judgments calls for transcendental logic and its predetermination of the concept of object. Hence the sole possibility for the pre-determination of the thingness of things is by way of synthetic a priori principles, that is, by limiting the realm of things to that of possible experience. Heidegger's argument is plagued by two main problems. The ¢rst concerns his interpretation of Kant, and the second, his understanding of mathematics. As regards his interpretation of Kant, it can be

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argued that according to Kant it is general logic that provides the notion of an object, whereas transcendental logic has to do with the relation between the general notion of an object and the conditions of sensibility. This explains why Kant continues to use the table of categories in the second Critique, viz., in the realm that lies beyond any possible experience. Thus although there is some justi¢cation for Heidegger's interpretation of Kant, overall it cannot be endorsed. Moreover, Heidegger's interpretation is problematic inasmuch as it rests solely on Kant's account of the distinction between analytical and synthetic judgments. But as is well known, the distinction can be made in ways that do not require synthetic a priori principles, as Frege and Carnap demonstrated. At the same time, I must point out that Quine's critique of the analytic^synthetic distinction can be interpreted as in line with, or at least in the same vein as, Heidegger's conclusion, insofar as it undermines any attempt to provide an a priori set of principles that exhaustively determine the thinghood of the thing. The second, and to my mind, the central problem with Heidegger's argument, lies in his assumptions regarding the mathematical ontology. Heidegger extracts its characterization through the Husserlian notion of the de¢nite manifold. A manifold is completely de¢nite if the axiomatic system that determines it is complete in the Husserlian sense. GÎdel's theorems rule out the possibility that arithmetic can be characterized as complete in the Husserlian sense. It follows that mathematics is not mathematical in the Heideggerian sense of the term. On the one hand, the theorems support the thrust of Heidegger's approach, since if such an axiomatic system does not exist in arithmetic, then a fortiori it cannot be used to develop a general ontology. On the other hand, contrary to Heidegger's approach to the question of Being in the 1930s, it is not Being but entities that delimit mathematical ontology. Here we can see the chief weakness of Heidegger's thought from the 1930s on ^ his abandoning the idea of a hermeneutic approach to the ontological di¡erence. The reciprocal conditioning of the ontic and the ontological on each other, which was the cornerstone of Being and Time, is rejected and replaced by the idea that entities do not play any role in the elaboration of the question of Being. Heidegger

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adopts this position because he wants to avoid making the question of Being dependent on Dasein's understanding, and seeks to increase the di¡erence between Being and entities. The interplay between entities and Being that is a feature of the hermeneutic approach allows for a possible blurring of the ontological di¡erence. Heidegger makes this move in the context of working out his understanding of the history of Being, particularly, the case of modern ontology, which is characterized as mathematical. But on this point I have to say that Heidegger erred. The limitations of modern mathematical ontology do not necessitate a leap or a God who will intervene, as Heidegger said in the 1960s, to save us, but originate from within and can be overcome with the help of science or mathematics themselves. The problematic nature of Heidegger's stance is not just a matter of the ontological import of its premises, but is also due to its implications for Heidegger's involvement in the Nazi regime. The selfunderstanding implicit in Heidegger's post-war writings is that Nazism is the continuation of the modern interpretation of Being.79 On this interpretation, the question of whether a particular entity ^ in this case, Heidegger himself ^ supports or distances itself from Nazism is unimportant. Hence rejection of the hermeneutic approach to the ontological di¡erence is in e¡ect rejection of personal responsibility vis-a©-vis espousal of political views, which take on the status of inevitable aspects of the history of Being.

Notes 1. Many features of the discussion in What is a Thing? are also in evidence in Heidegger's well-known article, `The Age of the World Picture'. My main reason for focusing on What is a Thing? is that only with the help of the book is it possible to extract the argument underlying the claims made in the paper. 2. Heidegger SZ, 69. 3. Heidegger SZ, 359^60. 4. On the relation between Heidegger's notion of science and Husserl's thought, see Kockelmans, 1985. 5. Heidegger SZ, 362. 6. Heidegger SZ, 145. 7. Heidegger interprets the word `object' (Gegenstand ) literally, and understands it as `[something] thrown against' (entgegenwerfen); see Heidegger SZ, 363.

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8. Glazebrook also maintains that Heidegger's model for science and philosophy in Being and Time is mathematical physics, but does not justify this claim; see Glazebrook 2000, 16. I believe it should be argued for on the basis of the role played by the notion of `theme' in Being and Time. 9. On the problematic role of thematization in Being and Time, see Dahlstrom 2001, 206^10. Dahlstrom interprets Heidegger's notion of `formal indication' ( formale Anzeige), discussed in `Husserl's notion of formal ontology' in Chapter 1 above, as an attempt to solve this problem, see 241^52. 10. See Chapter 2 for a discussion of the connection between Descartes and mathematics as set forth in Being and Time. 11. Heidegger GA 41, 74 (Eng. trans. 74). 12. Heidegger GA 41, 90 (Eng. trans. 89). 13. Heidegger GA 41, 94 (Eng. trans. 89). 14. See Klein 1966 on the emergence of the concept of `general magnitude' in the mathematics of the modern era, and for a comparison of this concept to that found in Greek arithmetic. Klein identi¢es a discontinuity between the various Greek approaches, and the modern approaches of Vieta, Descartes and Stevin, a ¢nding that supports Heidegger's claims. 15. On the di¡erences between the Greek, medieval and Cartesian conceptions of the arche¨, see SchÏrmann 1987, part 3. 16. Heidegger stresses the relation between the verb setzen, to place, and the noun for proposition, satz. 17. Heidegger GA 41, 103 (Eng. trans. 103). 18. According to Heidegger, in the transition to modernity, the meaning of the law of contradiction changes. Whereas in former times it applied to entities, in the modern epoch it applies to sentences; see Heidegger 1961a, I 602^16. Heidegger argues that the transformation of the principle of su¤cient reason into a foundational principle can be attributed to the mathematical approach. Although Heidegger addressed the law of contradiction and the principle of su¤cient reason in his 1929 article `Vom Wesen des Grundes' (On the Essence of Ground), the discussion in What is a Thing? adds to what was said there in arguing that the mathematical approach lies at the heart of the modern understanding of the two principles. 19. Heidegger, SZ 9. 20. On the duality of Dasein, see below. 21. Here Heidegger is, in the 1930s, essentially returning to the stance he took in the very early writings, viz., to the position that there is a precise correspondence between the characterization of a region and the entities it comprises. This return to his earlier position can even be detected in the `Introduction to Philosophy' lecture series delivered in 1928/29, in which he posits a correspondence between entities and modes of Being (Seinart). For instance, the mode of Being `subsistence' is associated with number and space. At this stage in the evolution of his thinking, however, Heidegger still retains the

120

22.

23. 24. 25.

26.

27. 28.

29. 30. 31. 32. 33.

34. 35. 36. 37. 38. 39. 40.

41.

Being and Number in Heidegger's Thought notion of the region as distinct from that of the mode of Being; see Heidegger GA 27, 71^2. Heidegger makes the connection between Husserl's notion of mathesis universalis and Descartes' philosophy, both of which highlight the mathematics^ ontology nexus, as early as his 1923 lectures; see Heidegger GA 17, 210^11. In Formal and Transcendental Logic, ½27, Husserl asserts that the idea of formal logic is already present in Logical Investigations. Husserl, Ideas, ½24. This tension has already been remarked upon earlier, in Chapter 1. But there, I focused on the meaning of formal ontology, whereas here, I will be emphasizing the transcendental ego^ontology nexus. In his Cartesian Meditations, Husserl claims that ontology carries out tasks preparatory to, and hence distinct from, phenomenology, see Husserl Hua I, ½59. In the third part of the Ideas, he claims that that phenomenology is not ontology, Husserl Hua V, 129 (Eng. trans. 117). Heidegger SZ, 98. On the centrality of mathematics to Descartes' thought, see Heidegger SZ, 96; on its signi¢cance for the development of Husserl's thought, see Heidegger GA 21, 31. See `Husserl's notion of formal ontology' in Chapter 1 above. Heidegger SZ, 160. Heidegger GA 25, 64, 205 (Eng. trans. 44, 140). This is discussed in `Husserl's notion of formal ontology' in Chapter 1 above. The move toward this interpretation of Being and Time can be traced in his marginal comments on the ¢rst pages of Being and Time. By the time he gets page 13, he has settled on interpreting its philosophical approach as anthropology. See Husserl 1994, 10^13 (Eng. trans. 275^85). Husserl 1931. Interestingly, Husserl does not mention Heidegger by name. But it is evident that the basic position he is attacking is Heidegger's. Heidegger SZ, 12. Heidegger SZ, 13^14. Heidegger 1929, 162. I thus concur with Courtine's claim that Husserl's phenomenology plays a central role in Being and Time, see Courtine 1990, 168^70. On the incompatibility between formal ontology and Husserl's phenomenology, see Marion 1989, Chapter 5. Marion reads the incompatibility as attesting to the primacy of epistemology over ontology in Husserl's phenomenology. I am not, of course, suggesting that there is no such thing as a phenomenological account of mathematics. Not only is it possible to put forward s uch an account, but even today there are ongoing attempts to do so, e.g., Tieszen 1989. Rather, my argument is that it is not possible, from a phenomenological perspective, to arrive at an ontology that is based on purely

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42. 43.

44. 45. 46.

47. 48.

49.

50. 51. 52. 53.

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formal-mathematical considerations. For phenomenology, formal ontology must be a positive science. There is an essential gap between formal ontology and phenomenology. For a thorough discussion of the subject not limited to Husserl's phenomenology, see Imbert 1992. Heidegger GA 41, 96. In a note in the margins of `On the Essence of Ground', Heidegger describes the problem of the position taken in Being and Time as its being too transcendental-phenomenological, see Heidegger 1929, 159 (Eng. trans. 123). Heidegger 1943, 309 (Eng. trans. 235). Discussed in Chapter 1 above. It could be argued against my reading of this development that Heidegger paid little attention to Carnap, but it seems to me that, though responding to Carnap is by no means the central goal of What is a Thing?, Carnap does play a signi¢cant role in the crystallization of Heidegger's mature thinking on the mathematical outlook. His lectures from 1934 ^ two years after the publication of Carnap's article ^ are entitled `Logic as the Question concerning the Essence of Language', a formulation that clearly calls to mind Carnap's paper, and attests to the seriousness with which he relates to Carnap's critique. In 1951, Heidegger identi¢es these lectures as embodying the transition from the primacy of logic to the primacy of language in the inquiry concerning the question of Being. It seems to me that this change is related to Heidegger's new understanding of mathematical ontology, and in particular, the logic^ ontology nexus that follows from it, which he arrived at, among other things, in response to Carnap's attack. For a discussion of this transition, see Courtine 1997. We can also gain insight into the importance of Carnap's views for Heidegger from a 1964 letter in which he describes the di¡erences between Carnap and himself as `the most extreme counterpositions'; see Heidegger GA 9, 70. See Becker 1963. This can be traced back to Becker's Mathematical Existence, where Becker claims that mathematics is anchored in man's primitive `natural' stratum; see Becker 1927, 321. Schalow has o¡ered an interpretation of What is a Thing? that highlights its emphasis on the centrality of modern science. He maintains that the Davos debate in£uenced Heidegger's subsequent reading of Kant; see Schalow 1996. Husserl Hua XVII, ½24. Husserl Hua III, ½72. Carnap 1932, 62. In the preface, Frege claims that he is only expressing factors that have signi¢cance for an inferential sequence; see Frege 1879, 6. He then introduces the function^argument distinction; see Frege 1879, 7. It thus seems that during this period, Frege sees the function^argument distinction as based on deductive-inferential considerations.

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54. Frege, 1884, x. As I noted in Chapter 1 above, Bar-Elli argues that neither of these alternatives can be de¢nitively said to be Frege's position. The characteristics of the object do not arise out of either deductive or linguistic logical constraints, but rather depend on how the object is given. 55. In holding this view, Heidegger downplays what I consider to be one of the fundamental di¡erences between the Fregean and Husserlian stances on logic. Husserl distinguishes, within the general framework of logic, between apophantic logic, which addresses the realm of meaning, and formal ontology, which addresses the notion of being in general and that of de¢nite manifolds. According to Husserl, mathematics, though expressed, of course, by means of propositions, is, with respect to its subject matter, very close to formal ontology. Frege, on the other hand, saw no room for the distinction. Indeed, I would argue that the distinction was a salient factor in the emergence of the infamous divide between philosophy in the Fregean tradition, and philosophy in the Husserlian tradition. In What is a Thing?, however, Heidegger does not endorse the distinction, and rede¢nes the philosophy that preceded him in such a way that the principal demarcation line is that between his own view and the views of all those who construed mathematics as ontology. 56. Hilbert outlined this stance in an exchange of letters with Frege in 1900^1; see Frege 1971, 13. 57. Hilbert can be read as claiming that geometry is not a system of propositions. On this reading of Hilbert, and its connection to the broader change in the way logic was understood, including Carnap's position, see Co¡a 1991, 135^40. 58. Heidegger's a¤nity with Hilbert can be linked to Husserl, whose positions on formal ontology and the de¢ned manifold were close (though not identical) to Hilbert's. Heidegger's conception of the nature of mathematics can thus be viewed as an attempt to think through all the rami¢cations of Husserl's Hilbertian premises. On the Hilbertian themes in Husserl's thought, see Mahnke 1923. On Husserl's and Hilbert's respective notions of completeness, see Bachelard 1957, 118^23, Majer 1997 and Hill 2000. 59. Heidegger GA 41, 123 (Eng. trans. 121). Heidegger uses the distinctly Hegelian term Aufhebung. It seems very likely that his use of the term attests to vestiges of Hegelianism in Heidegger's conception of history in the 1930s. In my opinion, however, these vestigial links to Hegel are no longer present in his discussions from the 1950s. 60. Heidegger maintains that Kant himself vacillated over the question of whether making logic subject to intuition mandated a fundamental change in logic itself, but that ultimately, Kant reached the conclusion that a fundamental change was indeed mandated. Heidegger interprets Kant's famous declaration, in the preface to the second edition of the Critique of Pure Reason, that logic has not changed since Aristotle, in this vein, see Heidegger GA 41, 152^3 (Eng. trans. 148^50). 61. Heidegger GA 41, 160 (Eng. trans. 157).

The Mathematical Epoch 62. 63. 64. 65.

66. 67.

68.

69. 70.

71. 72. 73.

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Heidegger GA 41, 164 (Eng. trans. 160). See Heidegger GA 25, 182. Heidegger GA 25, 205. In his article `Heidegger, lecteur de Husserl', Pavlos Kontos (1994) links The Phenomenological Interpretation of Kant's Critique of Pure Reason to Husserl's Formal and Transcendental Logic. There is indeed a certain similarity between the two works, but Kontos ignores the profound di¡erences, which I discussed above, between the Husserlian and Heideggerian approaches to questions of ontology. But in view of the general similarity, What is a Thing? can be interpreted as an attempt, on Heidegger's part, to completely cut his ties to Husserl with regard to both logic and ontology. Heidegger GA 27, 20. Heidegger's treatment of logistics together with the traditional Aristotelian theory of the judgment indicates, in my opinion, that his reading of Kant was explicitly intended as an answer to Carnap's charges. The extent to which this constitutes a satisfactory answer to Carnap is unclear. Carnap's distinction between analytic sentences and synthetic sentences is by no means perfectly parallel to Heidegger's distinction between analytic and synthetic judgments. Moreover, even if Heidegger's interpretation of Kant's notion of judgment is correct, and the notion of a judgment is inseparable from that of a thing, this does not necessarily lead to the conclusion that general logic requires a prior ontology or a prior transcendental logic. Given Heidegger's interpretation it could be still maintained that the notion of a thing can be derived from the logical analysis of sentences, as Frege claims. For a comparison of the Kantian, Fregean and Carnapian notions of analyticity, see Proust 1986. Heidegger 1954, 54^5. Nevertheless, it is important to understand that the critique of ontological regions, or more precisely, their reframing as a consequence of modern ontology, does not constitute a rejection of the various modes of interpreting Being. These possibilities continue to play a role in Heideggerian thought. Heidegger, then, is intent on completely divorcing the ontological regions from the modes of interpreting Being. On the distinction between ontology and the positive sciences, see Heidegger SZ, 8^11; GA 26, 3. Heidegger, SZ, 9. In this context it is fascinating to compare Heidegger's position with that taken by Foucault in The Order of Things (Les mots et les choses). Foucault sees modernity and the concept of language on which it rests as generating a dichotomy between the formal approach to language, and the historical^ anthropological^archaeological approach that discloses the pre-linguistic basis for language (Foucault 1966, 307^13). The modern episteme is the framework for Kant's philosophy as well. Moreover, according to Foucault the

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74.

75.

76. 77.

78.

79.

Being and Number in Heidegger's Thought classical episteme is governed, to a large extent, by Descartes' notion of mathesis. But Foucault, unlike Heidegger, sees no continuity between the tradition of the classic episteme, and the modern episteme (361). As to the history of mathematics, Tiles (1991) argues for continuity between Descartes and Hilbert, whereas Mehrtens (1990), who explicitly embraces Foucault's account, highlights the discontinuity that emerged at the beginning of the nineteenth century. Cavaille©s (1947, 70^7) was the ¢rst to assert the relevance of GÎdel's theorems to Husserl's position, contending that they undermine all of Husserl's philosophy. Lohmar maintains that they are not relevant to phenomenology as a whole, but only to a limited technical aspect of Husserl's work (Lohmar 1989, 192^3). Scanlon (1991), on the other hand, argues that GÎdel's theorems are a constraint on any phenomenological discussion of formal systems. GÎdel's theorems have various philosophical interpretations. Some see them as strengthening the Platonistic approach to mathematics, others, as strengthening the case for intuitionism. It is interesting to note that toward the end of his life, GÎdel himself thought that Husserl's transcendental phenomenology could provide a philosophical framework for mathematics. On GÎdel's interpretation of Husserl, see Atten and Kennedy 2003. Heidegger 1954, 62 (Eng. trans. 177). In this sense my approach is similar to Salanskis's hermeneutic approach to mathematics. This approach is akin to that of Heidegger, but allows mathematics itself to reveal the truth about its entities; see Salanskis 1991. In `On Time and Being', Heidegger describes his thought as an attempt to think about Being without entities. He sees this feature of his thought as di¡erentiating it from metaphysics, see Heidegger 1962, 24. Heidegger's thought from the 1930s onward no longer takes the hermeneutic approach to the question of Being, which focuses on the relationship between Being and entities. Courtine interprets this change as integral to Heidegger's turning away from phenomenology, since Being is no longer understood in relation to entities, as would be the case if the question of Being was being approached phenomenologically, on the basis of intentionality; see Courtine 1990, 399. This understanding of the signi¢cance of Heidegger's turn to the history of Being has been put forward in Habermas 1985, Chapter 6, among others.

Conclusion

Toward a Continental Philosophy of Mathematics

The mathematical thread in Heidegger's thought that we have uncovered and traced both illuminates Heidegger's own philosophy, and provides some fruitful suggestions for re£ecting on mathematics in the continental tradition. With regard to the former, we explored the di¡erent ways in which Heidegger sought to demarcate mathematics so as to limit the application of mathematics to a certain ontological region. Heidegger believed that this restriction of the applicability of mathematics was necessary because of the role mathematics played in interpreting the question of Being. The strategies he adopted to carry out this demarcation changed in accordance with his overall philosophical stance. Nevertheless, we saw that underlying all the di¡erent strategies is the assumption that mathematics is not only a certain region, but also a certain conception of what a region is. In this respect Heidegger retained one of Husserl's basic teachings concerning the relationship between ontology and mathematics in its most general form ^ mathesis universalis. Husserl, we saw, had maintained that mathesis universalis provides a formal characterization of the notion of a region. Heidegger's delineation of mathematics thus rests on a speci¢c interpretation of the philosophical meaning of mathematics in its most general form. In bringing to light this aspect of Heidegger's thought, I have demonstrated that the connection between mathematics and the question of Being is by no means unidirectional. Mathematics and the various sciences are not only determined by the question of Being, but also, in an important sense, determine the way this question is conceived. Attending to Heidegger's discussions of mathematics thus a¡ords us a fuller understanding of all of Heidegger's thought.

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But these insights have relevance well beyond the context of Heideggerian philosophy, and shed light on the far broader, and more pressing, issue of the role of mathematics in the continental approach to philosophy. Generally speaking, the divide between the analytic and continental traditions in philosophy is characterized by their deeply divergent attitudes to logic, mathematics and the natural sciences. The continental tradition is taken to be critical of the rational approach on which the sciences are based, and to widen the gap between philosophy and formal logic. The analytic tradition, on the other hand, is seen as upholding mathematics as the paradigm of rational thought, and taking its development to be closely linked to developments in formal logic.1 But in recent years this picture has begun to erode, principally due to the growing interest in what is now referred to as `continental philosophy of science', an epithet that would have been regarded as an oxymoron in the half century following the logical positivist era.2 This emerging discipline seeks to reveal the complex and little-recognized network of relations between continental philosophy and the natural and formal sciences. It has two main focuses. One focus is exploration of the conceptions of science explicit or implicit in the thought of the seminal continental philosophers, among them Heidegger, Merleau-Ponty, Foucault and Deleuze. The second is to gain a better understanding of the continental thinkers, such as Bachelard, Cavaille©s and Becker, whose primary concern is science yet who are connected, in one way or another, with the central ¢gures of continental philosophy. Recently these two directions have converged in the philosophy of Alain Badiou, one of the leading contemporary continental philosophers; his main area of interest is the philosophy of mathematics. In the framework of this emerging continental philosophy of science, my study has, as I said, made two modest contributions. In addition to charting the unfamiliar territory of Heidegger's conception of mathematics, it has, I hope, suggested a broader perspective on the relationship between philosophy and mathematics in the continental tradition. Three principal conclusions have emerged from the analysis undertaken in the preceding chapters. First, the connection between philosophy and mathematics is not made via the discipline of formal logic, but rests chie£y on the mathematics^ontology nexus

Conclusion

127

These connections can be traced back to Husserl's work, and remain important in present-day continental philosophy: one of Badiou's core theses is the identi¢cation of ontology with mathematics. The second conclusion, which follows from the ¢rst, is that mathematics and ontology are linked primarily through the notion of multiplicity. The roots of this idea can, again, be found in Husserl, and here too, the thesis lives on not only in Heidegger's thought, but in that of Deleuze and Badiou as well. My third conclusion, namely, that Brouwer's thought is as signi¢cant for Heidegger's philosophy as it is for Husserl's, has particular relevance for the increasingly discussed question of the relationship between Brouwer and phenomenology.3 Interest in this question can be attributed, to some extent, to a growing interest in the phenomenological elements in the thought of Hermann Weyl and Oscar Becker. My analysis has, I hope, helped sort out some of the themes common to Heidegger and these pre-war philosophers of mathematics. Though the aforementioned three conclusions are hardly exhaustive, and there is still much work to be done on the role of the mathematical in continental philosophy, they indicate the directions in which a continental philosophy of mathematics can pro¢tably be developed. The divide between the analytic and the continental philosophical traditions has been a major preoccupation of philosophers and historians of philosophy in the past few decades. While in many respects the divide is not as relentless today as it was just thirty years ago, the hoped-for rapprochement has yet to emerge. The blurring of the divide, far from generating a rewarding dialogue between the two traditions, has instead produced a growing compartmentalization of philosophy. Although the two traditions ^ the analytic and the continental ^ are no longer at loggerheads to the point where the presence of one automatically excludes that of the other, and continental philosophy is widely embraced as a legitimate `branch' of philosophy, this cordial coexistence nonetheless still precludes any genuine dialogue. In writing this book, I have sought a way out of this impasse. One lesson I hope readers come away with is that instead of this compartmentalization and specialization, we should pursue an exchange of ideas based not on super¢cial inclusiveness, but on a real understanding of the assumptions underlying each tradition. The philosophy of

128

Being and Number in Heidegger's Thought

mathematics can only be enriched by an encounter with the alternative approaches that grew out of the continental tradition. I suspect that the direction in which philosophy develops in the twenty-¢rst century will to a great extent be determined by how directly it confronts the alternatives ¢rst proposed in the twentieth.

Notes 1. For a recent example of this view, see Friedman 2000, Chapter 9. 2. See, e.g., Gutting 2000 and the special issues of Angelaki 2005. 3. See, e.g., Van Atten 2007.

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Index

absolute, notion of 23 aggregate, see also multiplicity 20 analytic/synthetic 107^8, 110, 116^17 Aristotle 10, 18, 23, 51, 52, 54^8, 59, 60, 67, 70, 85, 88, 97, 106, 107 arithmetic 26, 72, 115 axiomatic system 26, 88, 89, 101^2, 103^4, 117 Badiou, Alain 127 Becker, Oscar 69^71, 100, 127 Being 90^1, 96, 105, 125 history of 99, 111, 113, 115, 118 meaning of 2, 3, 5, 9, 36, 37, 46, 65, 93, 94, 109 notion of 15, 99 vs. entities 73, 90, 97, 117^18 Bergson, Henri 48, 57^8 Brentano, Franz 5, 9, 36 Brouwer, L. E. J. 51, 63^71, 114, 127 Cantor, Georg 6, 18^23, 34, 64, 67 Carnap, Rudolf 6, 99^100, 102^3, 110^11 Cassirer, Ernst 24, 100 categories 9, 10^12, 13^14, 17^18, 21, 25, 28^30, 32, 36^7, 39, 55, 103, 107 cogito 90, 93, 104 consciousness 57 continental philosophy 6, 126^8 continuum 17, 64, 65^8, 114 contradiction, law of 90, 108 counting see also time, counted 84, 100

Dasein 2, 37, 38, 46, 50, 51, 53^4, 61, 82, 83, 90, 91, 93, 95^7, 98, 116 Davos debate 100 Dedekind, Richard 6, 34, 44n. Descartes, Rene¨ 53^4, 60, 61^2, 72, 81, 84, 86^7, 88^90, 93, 94, 98, 101, 104, 110 death 61^2 being-towards-death 51, 61, 68 Dilthey, Wilhelm 9, 13 Dummett, Michael 67^8 ¢nitude 22^3, 39, 51, 60^3, 68 formalism 64^5, 87, 113^14 formal indication 34, 93, 94 formalization 33^4, 95 Foucault, Michel 72n. Frege, Gottlob 4, 6, 10, 18, 19, 20^1, 28, 34, 36, 103 Galilei, Galileo 74 geometry 1, 29, 72, 87, 104 GÎdel, Kurt 114^15 Hegel, G. W. F. 9, 36, 60^1, 113 Heidegger, Martin: works `Age of the World Picture' 118n. The Basic Problems of Phenomenology 56^7 Being and Time 2, 3, 6, 26, 36^9, 46^7, 50^4, 57^62, 64^5, 67^9, 71^5, 81^4, 86, 87, 88, 93^9, 101, 103, 110, 111^12 `Doctrine of Judgment in Psychologism' 6n., 10

138

Index

Heidegger, Martin: works (continued ) Duns Scotus' Doctrine of Categories and of Meaning 8, 12^18, 73 Fundamental Concepts of Metaphysics 45n. Introduction to Philosophy 53, 72, 74 Logic as the Question concerning the Essence of Language 121n. `On the Essence of Ground' 121n. `On Time and Being' 79n., 124n. `The Problem of Time in the Science of History' 14, 47^8 Phenomenological Interpretation of Kant's Critique of Pure Reason 94, 109 The Phenomenology of Religious Life 34 `The Principle of Identity' 38 `Science and Re£ection' 111, 115 What is a Thing? 3, 81, 83^92, 94, 98^118 Hilbert, David 34, 35, 103^4, 114 Husserl, Edmund 4, 5, 7, 9, 10, 20^2, 28^34, 36, 72, 91^7, 101^2, 111^12, 114^15, 117, 125 identity, principle of 25, 108 in¢nity 22^3, 60^2, 66, 68 intuition 26, 63, 65, 87, 105, 108, 110 categorial 31^3 intuitionism 63^71, 87, 113^14 judgment vs. act of judging 30^1, 106

11, 17,

Kant, Emmanuel 4, 7n., 10, 25, 26, 49, 70, 87, 94, 100, 104^10, 114, 116^17 Kisiel, Theodore 34 Klein, Felix 22 Lask, Emil 9, 24 Leibniz, G. W. 70, 86^7 logic 10, 24, 27, 89 apophantic 102 formal 28, 30, 100, 126 logistics 100, 107

mathematical 34, 89, 107, 112 transcendental 108^9, 116 manifold 29 de¢nite 29, 114, 117 mathesis universalis 4, 5, 29, 33, 36, 92, 95, 102, 109 mathematical physics 39, 74, 81^3, 85, 91 meaning 11^12, 30, 99, 102^3 modernity 72, 81, 84, 88, 98, 113 multiplicity, see also aggregate 127 absence of 16, 31, 35, 38 consistent 20 de¢nite 22 theory of 5 29, 92 uni¢ed 17, 27, 36, 38^9 vs. set 18 Natorp, Paul 5, 22, 24, 26^8, 35, 49 Nazism 100, 108 neo-Kantianism 5, 23 Newton, Isaac 74, 85, 86^7 numbers 3, 16^17, 19, 24^5, 28, 29, 39, 46^60, 68^9, 73, 84^5, 89 natural 23, 63, 115 ordinal 55 real 18, 65 series 48, 50, 64 object in general, see also formal ontology 21, 28, 34^5, 116 one 24, 27 as transcendental 2, 8, 12, 15^17, 27, 31, 35, 37, 38, 49, 50 as number 8, 12, 16^17, 27 ontological di¡erence see also being vs. entities 90, 116, 117^18 ontological region, see also regional ontology 86, 90, 91, 93, 97, 104, 105, 108, 109, 110, 112 ontology 28, 73^4, 81 formal 28^9, 31^4, 36, 92^ 8, 99, 102, 109 general 4, 5, 36, 111, 116

Index fundamental 3, 46, 71, 73, 83, 86, 111, 112 material 31, 95, 97^8 mathematical 87, 103, 108, 111, 115, 116, 117 regional 3, 5, 13, 96, 99, 111, 112 universal 3, 92^3, 97, 111 phenomenology 28, 31^3, 82, 92^3, 97^8, 112, 127 Plato 1, 70 Posy, Carl 66 presence 62, 93 presence-at-hand 3, 53, 61, 91 psychologism 10^11, 92 Quine, W. V. O.

117

ready-to-hand 3, 52, 53, 91 realms of being, see also ontological region 12, 15, 21, 29, 91 relativity theory 73^4 Rickert, Heinrich 5, 13, 22, 23^7, 48^9 Ricoeur, Paul 75n. Russell, Bertrand 18, 19, 20, 28, 34, 36, 67

139

science theory of 12^13 positive 72^3, 112, 113 set, notion of 4, 18^23, 36, 64 Simplicius 76n. space 17, 57, 73, 86, 90 subject, Cartesian 6, 81, 116 subject predicate 15, 89, 106^8 subsistence 53^4 thematization 83 time 67^8, 86 before and after 55 clocks 52, 73 counted 39, 53, 54, 56^8 historical vs. natural 47^52, 70 measurable 50, 51, 52, 53, 56^8, 60, 74 primordial 46, 51, 54, 58^60, 65, 68, 71, 81 vs. duration 48 ordinary 50, 52, 81 world 51, 52, 58^60, 68, 73 transcendental ego 97^8, 116 Weyl, Hermann Wol¡, Christian

67, 79n., 127 106