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Kant's Schematism and the Foundations of Mathematics KANT’S SCHEMATISM AND THE FOUNDATIONS OF MATHEMATICS · KLAUS FROVIN JØRGENSEN SECTION FOR PHILOSOPHY AND SCIENCE STUDIES ROSKILDE UNIVERSITY ROSKILDE 2005 Kant’s Schematism and the Foundations of Mathematics PhD Thesis by Klaus Frovin Jørgensen, M.Sc. November 4, 2005 This dissertation is submitted in partial fulfillment of the requirements for obtaining the degree Doctor of Philosophy at the Section for Philsophy and Science Studies, Roskilde University. The study has been conducted within the Ph.D. program Science Studies at the Sec- tion for Philosophy and Science Studies, Roskilde University, and was financed by a scholarship from Roskilde University. The study has been supervised by Professor Stig Andur Pedersen, Section for Philosophy and Science Studies, Roskilde Univer- sity. Roskilde, November 4, 2005 Klaus Frovin Jørgensen Abstract The theory of schematism was initiated by I. Kant, who, however, was never precise with respect to what he understood under this theory. I give—based on the theoreti- cal works of Kant—an interpretation of the most important aspects of Kant’s theory of schematism. In doing this I show how schematism can form a point of departure for a reinterpretation of Kant’s theory of knowledge. This can be done by letting the concept of schema be the central concept. I show how strange passages in, say, the first Critique are in fact understandable, when one takes schematism serious. Like- wise, I show how we—on the background of schematism—get a characterization of Kant’s concept of ‘object’. This takes me to an analysis of the ontology and episte- mology of mathematics. Kant understood himself as a philosopher in contact with science. It was science which he wanted to provide a foundation for. I show that, con- trary to Kant’s own intentions, he was not up-to-date on mathematics. And in fact, it was because of this that it was possible for him to formulate his rather rigid theory concerning the unique characterizations of intuition and understanding. I show how phenomena in the mathematics of the time of Kant should have had an effect on him. He should have remained more critical towards his formulation and demarcation of intuition, understanding and reason. Finally I show how D. Hilbert in fact gives the necessary generalization of Kant’s philosophy. This generalization provides us with a general frame work, which func- tions as a foundation for an understanding of the epistemology and ontology of math- ematics. Resume´ Denne afhandling handler om skematisme, som er en teori, der p˚abegyndes af I. Kant. Kant var aldrig helt præcis med hensyn til, hvad han forstod under denne teori. Baseret p˚a Kants teoretiske værker giver jeg en fortolkning af de vigtigste dele af Kants skematisme, og jeg viser, hvorledes skematisme giver anledning til en nyfortolkning af Kants erkendelsesteori ved at lade begrebet være det centrale begreb. S˚aledes viser jeg, hvorledes mærkværdige passager i f.eks. den første Kritik kan gøres forst˚aelige, n˚ar Kants teori om skemaerne tages alvorligt. Jeg viser ligeledes, hvordan vi p˚a baggrund af skemaerne f˚ar karakteriseret Kants begreb om ‘objekt’, hvilket leder frem til en analyse af matematikkens ontologi og erkendelsesteori. Kant s˚a sig selv som en filosof i kontakt med videnskaberne. Det var videnskaberne han søgte at fundere med sin erkendelsesteori. Jeg viser, hvorledes han desværre var tidsligt et stykke bagefter med hensyn til matematikken. Dette førte ham til at formulere sin noget rigide teori ang˚aende den entydige og endelige karakterisering af anskuelsen og forstanden. Jeg viser, at fænomener i matematikken, som den s˚audp˚a Kants tid, burde have f˚aet Kant til at forholde sig mere kritisk i forhold til netop sin formulering af anskuelse, forstand og fornuft. Endelig viser jeg, hvorledes D. Hilbert faktisk giver en generalisation af Kants filosofi. Denne generalisation giver anledning til formuleringen af en generel ramme, der kan fungere som grundlag for en forst˚aelse af matematikkens erkendelsesteori og ontologi. Prologue and Acknowledgments In contemporary philosophy of science there is a renewed interest in Kant’s theory of knowledge and his philosophy of science. One of the reasons for this is the compre- hensive work done by Michael Friedman which has prompted a variety of responses. The present thesis can also been seen as a reaction on Friedman’s work on Kant. I was introduced, through Friedman (1992), to Kant’s notion of schematism in geome- try. Now, by reading the Critique of Pure Reason I became acquainted with that fact that Kant claims to have a theory of schemata, not only for geometry, but for all the concepts of the understanding—pure as well as empirical. As it was clear to me, that the theory of schemata could provide an alternative understanding of Kant, it became a goal to get a deeper understanding of it. But also other interests motivated me. Today, within the philosophy of mathematics, there is also a growing interest in the use of diagrammatic reasoning. Euclid is paradigmatic in this respect—some find the reasoning style used by Euclid highly problematic, others do not. Kant belongs to the latter category. Euclid uses diagrams throughout Elements and Kant wants to give an account of and a foundation for this kind of reasoning. With Kant’s theory of schematism follows a notion of schematic construction in pure intuition—and this is precisely Kant’s device when providing an epistemological frame work for Eu- clid’s reasoning. Schematic reasoning is not exclusively about justification, nor ex- clusively about discovery in mathematics—it is generally about the whole reasoning process which the mathematician makes from discovery to justification. Mathemati- cal schematic reasoning is not in particular about mathematical proofs, it is generally about mathematical thinking. I have not written this thesis for the sake of writing history, rather it is for the sake of understanding what mathematics really is about. Understanding Kant’s schema- tism is on my path. Nevertheless, for an understanding of schematism I need to use and produce elements from history of science and history of philosophy. On the other hand I will also use theories and concepts from contemporary philosophy and math- ematics in order to get something meaningful out of Kant’s schematism. We need to interpret Kant’s theory of schemata, as Kant does not present a clear, nor a detailed theory; rather he outlines some remarkable and very fruitful ideas. But these ideas are much in need of elaboration. I am interested in understanding Kant’s theory of knowledge, not for the sake of history, but for the sake of truth. Thus this thesis should be read as a thesis on philosophy of mathematics, not as a thesis on history of philosophy. In consequence of this I allow myself to use notions and concepts which were not know at the time of Kant. Chapter 1 and 2 treat my interpretation of Kant’s schematism. The former of the ix x two chapters shows how the geometrical schemata and Euclid’s postulates go hand in hand. The geometrical schemata are the part of Kant’s schematism which is best explored in the literature. In this thesis, this chapter on the geometrical schemata functions mostly as an introduction to schematism and the central notions of schema- tism such as types, tokens and rules. Chapter 3 is on the schematism of the pure concepts of quantity. I show how Kant operates with both a concept of a particular number and a concept of a schema determining the properties of numbers. A particu- lar number, to Kant, signifies unity in a collection of objects falling under a concept. The unity that the collection can posses is, that by counting the elements of the col- lection we reach a finite number just in case we can judge the collection to be a unit. I furthermore show that numbers are not determined extensionally. Rather they are determined by a schema—an intensional element. Chapter 3 and 4 deal with the relations between space, schemata, geometry and the notion of object. It has always been difficult for me to understand the following words: “Space is represented as an infinite given magnitude” (B40). I also show in chapter 6 that also Hilbert and Bernays had difficulties here. As it turns out, my interpretation of the arithmetical and geometrical schemata as presented in chapter 1 and 2 actually provide a framework for an understanding of Kant’s concept of infinity. Geometrical schemata need a space to exercise in; a space which is unbounded. How unbounded is the space? Well, it is in-finite in the sense that no magnitude can be ascribed to space. In chapter 4 I elaborate on an observation due to Carl Posy. It is well-known that according to Kant all objects are completely determined, in the sense, that given any object x and any predicate P, then either P(x) is true or false. Posy (1995) observes that this is expressible by the following first order formula: ∃y(y = x) → P(x) ∨¬P(x). I note that it is not quite clear to which language this formula belongs. In elaborating on the formula I provide a class of modal models which (semi-)validates a modalized version of Posy’s formula. I chapter 5 I critically discuss Kant’s philosophy of mathematics, and I generalize some of his notions in order to incorporate some of the new elements which have been discussed and introduced in mathematics since the time of Kant. This leads me to: Chapter 6. In this chapter I outline a relation between Kant’s general theory of knowledge and Hilbert’s philosophy of mathematics.
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