Abstract
Though my title speaks of Kant’s mathematical realism, I want in this essay to explore Kant’s relation to a famous mathematical anti-realist. Specifically, I want to discuss Kant’s influence on L. E. J. Brouwer, the 20th-century Dutch mathematician who built a contemporary philosophy of mathematics on constructivist themes which were quite explicitly Kantian.1 Brouwer’s theory (called intuitionism) is perhaps most notable for its belief that constructivism (whatever that means) requires us to abandon the traditional (classical) logic of mathematical reasoning in favor of a different canon of reasoning, called intuitionistic logic. Brouwer thought that classical logic is intrinsically bound up with a nonconstructive (or “realistic”) view of mathematics. This means that, according to Brouwer, when we do mathematics we must give up bivalence (the principle that a given sentence either is true or is determinately false), we must no longer use such familiar logical laws as excluded middle, and we must sometimes forebear from the classic method of reductio ad absurdum. All of these are intuitionistically invalid classical principles.
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Notes
See for instance Brouwer’s remarks in Chapter 11 of his dissertation. Over de grondslagen der wiskunde. (Translated on pages 11–101 in L. E. J. Brouwer, Collected Works v. J. North Holland, Amsterdam, 1975
Hereafter I will make page references to this anthology when citing Brouwer’s works. I will abbreviate it CW.) See also Brouwer’s suggestion in “Intuitionism and Formalism” (1913, CW page 123) that his own position accepts Kant’s views on the a priority of time, and rejects Kant’s notions about space.
I have discussed these matters a bit more extensively in “Dancing to the Antinomy”, “Transcendental Idealism and Causality,” and “The Language of Appearances and Things in Themselves”. “Dancing to the Antinomy” American Philosophical Quarterly,vol. 20, (1983), pages 81–94.
J. Bennett, Kant’s Dialectic,Cambridge University Press, London, 1974, section 40.
B. Russell, Principles of Mathematics,Norton. See especially section 435.
It is tempting to confuse this notion of regulativity with Kant’s notion of continuation in indefinitum (A510—I 1/B538–9). I fell into this mistake in “Dancing to the Antinomy” as did Bennet in Kant’s Dialectic section 46. But the in infinitum/in indefinitum distinction concerns only the initial conditions on a given regressive series. Thus for instance the series of divisions discussed in the second antinomy is given regulatively, but nevertheless is continuable in infinitum.
See my “Transcendental Idealism and Causality” in Kant on Causality, Freedom and Objectivity,edited by R. Meerbote and W. Harper. University of Minnesota Press, Minneapolis, 1984.
I have discussed some of these notions in “Transcendental Idealism and Causality”. B This is especially prominent in the A-Deduction, A 103 ff.
To know anything in space (for instance a line) I must draw it, and thus synthetically bring into being a determinate combination of the given manifold….Actually this is a theme that goes back in Kant’s writing at least as far as the Prize Essay of 1763:
A cone may signify elsewhere what it will: in mathematics it originates from the arbitrary representation of a right angled triangle rotated on one of its sides. The explanation obviously originates here, and in all other cases through synthesis. (AK II, 296)
See C. Parsons “Mathematical Intuition”, Proceedings of the Aristotelian Society,vol. 80 (1979–80), pages 145–68.
there are iterative complexes of sensations whose elements are permutable in point of time. Some of them are completely estranged from the subject. They are called things. For instance individuals, i.e., human bodies, the home body of the subject included, are things,…
Mathematics comes into being when the two-ity created by a move of time is divested of all quality by the subject, and when the remaining empty form of the common sub-stratum of all two-ities, as basic intuition of mathematics, is left to an unlimited unfolding creating new mathematical entities…. (“Consciousness, Philosophy and Mathematics” 1948, CW,p. 480.)
That’s because some constructions may leave some properties of the constructed objects eternally undecided.
In general — under the minimal deviation from the truth tabular meanings — if A is truth-valueless so too are —A and thus -A. Now take A as (Pt V —Pr) with Pt undecided.
Thus from the perception of the attracted iron filings we know of the existence of a magnetic matter pervading all bodies, although the constitution of our organs cuts us off from all immediate perception of this medium…The grossness of our senses does not in any way decide the form of possible experience in general. (A226B273) (See also A52–2B550).
See Bxl(n), B70–71, B274 and the “Appendix” to the Prolegemena.
See D. Hilbert, “Mathematische Probleme”, Arch, d. Math. u. Phys. (3), 1901; and “Über Das Unendliche”, Math., Ann., (96), 1926.
Consider for instance Dummett’s argument from language learning (i.e., from the premise that assertability conditions are inevitably the novice’s first contact with meaningful complete sentences; see Truth and Other Enigmas,Harvard, University Press, Cambridge MA:, 1978, pages 217ff.)
This argument certainly encompasses the Peircian assertabilism. For the novice can’t escape the preponderance of deferred justifications in our everyday discourse. Quite similarly, Dummett’s Wittgensteinian argument (as found in Truth and Other Enigmas pages 216–27 and Elements of Intuitionism,Oxford, 1977, pages 360–89) can be adapted to the Peircian case as well. The Peircian, like the Dummettian, ties “understanding” a sentence to a grasp of its assertability conditions, and thus does link understanding with actual behavior.
It might seem strange that the optimist should reject classical logic, and even stranger that he should do this while advocating the long semantic view. But you must recall that logic per se is linked with asserting behavior rather than with beliefs about the future. Under the strict constructivist view we cannot use these guarantees to assert as yet unwitnessed existentials or undecided disjunctions.
See for instance the criticism of Hilbert towards the end of Chapter III in Brouwer’s dissertation, and the similar criticism almost fifty years later in “Historical Background, Principles and Methods of Intuitionism” (1952), CW pages 508–15.
See Über Definitionsbereiche von Funktionen“ (1927). CW pages 390–405, and the editor’s note (4) (CW page 603) to ”The non-equivalence of the constructive and the negative order relation on the continuum“ (1949) CW pages 495–96.
See for instance “Intuitionistiche Betrachtungen über den Formalismus” (1928) CW pages 409–14.
I have treated this in more detail in “The Language of Appearances and Things in Themselves, ” Synthese,vol. 47, (1981), pages 313–352.
For the natural appearances are objects which are given to us independently of our concepts, and the key to them lies not in us and our pure thinking, but outside us; and therefore in many cases, since the key is not to be found, an assured solution is not to be expected.“ (A480–18508–9).
This is how I understand the Appendix to the “Dialectic”, entitled “The Regulative Employment of the Ideas of Pure Reason.” See in particular A547–8TB674–6.
Compare the dissertation description of the “basic intuition” of continuity with the presentation of the continuum as a “spread” in subsequent writings, e.g., Begrundung der Mengenlehere unabhangig vom logischen Satz vom ausgeschlossenan Dritten, CW 150–221.
I would distinguish here between notions like this iterative conception of infinity and notions which depend intrinsically on multiple quantifier shifts (e.g., the arithmetical hierarchy). The latter, I think, are wholly language bound, and do not correspond to any specific Kantian ideas.
Since the first appearance of this paper I have published four additional pieces which are relevant to the topics I have covered here. “Autonomy, Omniscience and the Ethical Imagination” (in Y. Yovel, ed. Kant’s Practical Philosophy Reconsidered, Kluwer, 1989) explores the parallels in Kant’s practical philosophy to the view that I have here called his “mathematical realism.” “Where Have All the Objects Gone?” (The Southern Journal of Philosophy (1986) XXV, Supplement) touches on Kant’s views about the nature of mathematical objects. This ontological theme as well as some phenomenological and semantic issues are taken up in “Mathematical as a Transcendental Science.” (in D. F011esdal et. al. eds., Phenomenology and the Formal Sciences, University Presses of America, 1990 )
That paper also expands on the remarks in note (5) above concerning the “Second Antinomy.” Finally, “Kant and Conceptual Semantics” (Topoi,10, 1991) qualifies my attribution to Kant of a modem assertabilism and considers his anti-realism in the context of his Leibnizian background.
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Posy, C.J. (1992). Kant’s Mathematical Realism. In: Posy, C.J. (eds) Kant’s Philosophy of Mathematics. Synthese Library, vol 219. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8046-5_12
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