Abstract
The interest and influence of Kant’s philosophy as a whole have certainly been great enough so that this by itself would be enough to make Kant’s philosophy of arithmetic of interest to historical scholars. It is also possible to show the influence of Kant on a number of important later writers on the foundations of mathematics, so that Kant has importance specifically as a figure in the history of the philosophy of mathematics. However, my own interest in this subject has been animated by the conviction that even today what Kant has to say about mathematics, and arithmetic in particular, is of interest to the philosopher and not merely to the historian of philosophy. However, I do not know how much of an argument the following will be for this.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
An earlier version of this paper was written while the author was George Santayana Fellow in Philosophy, Harvard University, and presented in lectures in 1964 to the University of Amsterdam and the Netherlands Society for Logic and the Philosophy of Science. I am indebted to J. J. de Iongh, J. F. Staal, and G. A. van der Wal for helpful comments. I am also grateful to Jaakko Hintikka for sending me two unpublished papers on the subject of this paper.
I.e., 1st edition, p. 320, 2nd edition, pp. 376–377. All passages are quoted in the translation of Norman Kemp Smith (London, 1929) with slight modifications. Other translations from German are my own. Translations of Kant’s Inaugural Dissertation are by John Handyside, in Kant’s Inaugural Dissertation and Early Writings on Space, Chicago and London, 1929.
Kants Gesammelte Schriften, ed. by the Prussian Academy of Sciences, Berlin, 1902–1956, IX, 91. This edition will be referred to as “Ak.”
“Kant’s `new method of thought’ and his theory of mathematics”, p. 130. Hintikka argues in detail for this thesis in a paper, “On Kant’s notion of intuition (Anschauung) “ in Terence Penelhum and J. H. MacIntosh (eds), Kant’s First Critique (Bemont, Calif., 1969). The same idea seems to underlie the analysis of Kant’s theory of mathematical proof in E. W. Beth, ”Über Lockes ‘allgemeines Dreieck“ in Kant-Studien 48 (1956–1957), 361–380.
“It is a mere tautology to speak of general or common concepts” (Logic 1, Ak. IX 91).
One might attribute to Kant the view that there are no such representations. The classification Kant makes in A 320 = B 376 and Logic §1 is of Erkenntnisse, which Kemp Smith translates as “modes of knowledge” but which in many contexts would be more accurately though inelegantly translated as “pieces of knowledge.” Then the relation of a representation to its object is that through which one can know its object, and it might be held that intuition in the full sense is the only singular representation which can provide such knowledge. This view would have the perhaps embarrassing consequence that an object which is not in some way perceived is not really known as an individual.
Cf. the examples of “truths of reason” given by Leibniz, Nouveaux Essais, IV, ii, §1.
Arithmetik and Kombinatorik hei Kant (Diss. Freiburg 1934), enlarged ed. Berlin 1972; Kant’s Metaphysics and Theory of Science, Manchester, 1953, ch. i; Klassische Ontologie der Zahl, Kant-Studien Ergänzungsheft 70, Köln, 1956, §12.
Neither Leibniz nor Schultz seems to mention the fact that in order to prove formulae involving multiplication, such as ‘2.3 = 6’, one also needs instances of the distributive law.
Aus mehrern gegebenen gleichartigem Quantis durch ihre successive Verknüpfung den Begriff von einem Quanto zu erzeugen, d. i. sie in ein Ganzes zu verwandeln. 2. Ein jedes gegebenes Quantum, um so viel, als man will, d.i. sie ins Unendliche zu vergrössern, and zu vermindern (Prüfung, I, 221).
Arithmetik and Kombinatorik bei Kant, p. 64–5.
C. J. Gerhardt (ed.), Leibnizens mathematissche Schriften, Halle, 1849–1963, VII 78. Leibniz gives a definition of addition from which he claims commutativity follows immediately. One could read his argument as deriving the commutativity of addition from the commutativity of set-theoretic union.
“Kant’s ‘new method of thought’,” “On Kant’s concept of intuition,” also “Are logical truths analytic?” Philosophical Review 74 (1965), 178–203, “Kant on the mathematical method,” This volume, 21–42.
It ought to be remarked that while no doubt the distinction which Kant makes between axioms and postulates derives historically from that of “common notions” and postulates in Euclid, Kant’s distinction does not correspond exactly to Euclid’s. Euclid’s division is between more general principles and specifically geometrical ones. For Kant postulates are “immediately certain practical judgments,” the action involved is construction, and their purport is that a construction of a certain kind can be carried out. The role they play is thus that of existence axioms. Euclid’s common notions are all of a type which Kant asserted to be analytic propositions (A 164 = B 204, B 17), while axioms proper must be synthetic.
Cf. W. V. Quine, Methods of Logic, revised ed., New York, 1959, §28.
“Kant’s ‘new method of thought’,” p. 130, also “Kant on the mathematical method.” The texts are A 717—B 745, A 734—B 762.
In “Are logical truths analytic?” Hintikka develops a distinction between analytic and synthetic according to which some logical truths are synthetic. He suggests that the logical truths which are analytic according to this criterion are roughly those which Kant would have regarded as analytic. It follows, however, that in some of the arguments which according to Beth and Hintikka involve for Kant an appeal to intuition, the conditional of their premises and conclusion is analytic. In particular, this is true of the example that Beth works out in detail in Über Lockes ‘allgemeines Dreieck’.“ §7. In order to be applied to mathematical examples like Kant’s, Hintikka’s criterion would have to be extended to languages containing function symbols. The way of doing this which seems to me most in the spirit of Hintikka’s definition has some anomalous consequences. See also Logic,Language-Games, and Information, Oxford, 1973, chs. 6–9.
In fact, (1) is analytic according to the criterion of “Are logical truths analytic?” (see note 21 above). However, according to another criterion which might be more in the spirit of Kant, to consider as synthetic a conditional whose proof involves formulae of degree higher than its antecedent, (1) is synthetic. Hintikka takes account of this in “Are logical truths tautologies?” by making an additional distinction between analytic and synthetic arguments, such that in the relevant sense the argument from the conjuncts of the antecedent of (1) as premises to its consequent as conclusion is synthetic.
Cf. Hao Wang, “Process and existence in mathematics,” in Y. Bar-Hillel, E. I. J. Poznanski, M. O. Rabin, A. Robinson (eds.), Essays in the Foundations of Mathematics, dedicated to A. A. Fraenkel, Jerusalem, 1961, 328–351, p. 335.
“Frege’s theory of number” (1965), in Mathematics in Philosophy, N.Y. 1983.
Die Mathematik betrachtet in ihren Auflösungen, Beweisen, und Folgerungen das Allgemeine unter den Zeichen in concreto, die Weltweisheit das Allgemeine durch die Zeichen in abstracto (Erste Betrachtung, §2. heading, Ak. II 278).
Denn da die Zeichen der Mathematik sinliche Erkenntnismittel sind, so kann man mit derselben Zuversicht, wie man dessen, was man mit Augen sieht, versichert ist, auch wissen, dass man keinen Begriff aus der Acht gelassen, dass eine jede einzelne Vergleichung nach leichten Regeln geschehen sei u.s.w. Wobei die Aufmerk-samkeit dadurch sehr erleichtert wird, dass sie nicht die Sachen selbst in ihrer allgemeine Vorstellung, sondern die Zeichen in ihrer einzelnen Erkenntnis, die da sinnlich ist, zu gedenken hat. (Dritte Betrachtung, §1, Ak II 291).
But cf. the following: in der Geometrie, wo die Zeichen mit den bezeichneten Sachen überden eine Ähulichkeit haben, ist daher diese Evidenz noch grösser, obgleich in der Buchstabenrechnung die Gewissheit evenso zuwerlässig ist. (Ibid., 292).
[This “general accord” now seems to me quite tenuous, and Manley Thompson is probably right in saying that the synthesis required for analytic judgments is clearly distinguishable from that in mathematical judgments (“Singular Terms and Intuitions in Kant’s Epistemology,” p. 342, n. 23). Nonetheless, a reply to the main point, that logic is not entirely independent of intuitive construction, would demand a lot of Kant’s distinction between intuitive and discursive proofs, as is clear from Thompson’s interesting discussion of this distinction (ibid., pp. 340–342). His interpretation implies rather extreme limits on the role of logic in mathematics. This raises a doubt whether Kant’s distinction is in the end tenable.]
One might say that it is possible to construct tokens. The sense of possibility in which this is possible is, however, derivative from the mathematical possibility of constructing types (or mathematical existence of the types). For we declare that the tokens are possible either directly on the basis of the mathematical construction, or physically on the basis of a theory in which a mathematical space which is in some way infinite is an ingredient.
Cf. my “Infinity and Kant’s conception of the possibility of experience” in Mathematics in Philosophy (1964).
This does not imply that there is an upper limit on the numbers which can be individually represented, once we admit notations for faster-growing functions than the successor function. This happens already in Arabic numeral notation. The number 1,000,000,000,000, if written in 0 and S notation with four symbols per centimeter, would extend from the earth to the moon. That there is such an upper limit follows, of course, from the assumption that human history must come to an end after a finite time.
“Kantian Intuitions”, Inquiry 15 (1972) 341–345. In this Postscript this paper is cited merely by page number.
Kemp Smith translates einzeln in this passage as “single.” The translation “singular” fits its use in Logic, §1 (see above, p. 112), where it is paired with the Latin singularis. Thompson suggests that in the Critique passage it may mean that an intuition is a single occurrence. (“Singular Terms and Intuitions, in Kant’s Epistemology” this volume, p. 105, n. 13.) If intuitions are thus in effect events, that would rule out Hintikka’s interpretation (though not the view of Robert Howell discussed below). Though this seems to me to agree with Kant’s characteristic way of speaking about institutions, the point is not so clear as to be a serious argument in the present dispute.
“Institution, Synthesis, and Individuation in the Critique of Pure Reason,” Noüs 7 (1973), 207–232, p. 210.
In the discussion of “the black man” in a letter to J. S. Beck, July 3, 1792 (Ak., XI, 347); cf. Howell, “Intuition, Synthesis, and Individuation,” p. 210. Other examples that can be given, such as the Idea of God, involve either Ideas of Reason or mathematics and might therefore be regarded as exceptional.
“Intuition, Synthesis, and Individuation,” pp. 210–211. The distinction between what he takes to be the definition and his further interpretation is not explicit in the paper; here I rely on his clarification of his views in a recent letter.
Ibid., p. 215. A somewhat similar picture is presented in Thompson, “Singular Terms and Intuitions.” However, Thompson rejects Howell’s view that certain demonstratives can be the linguistic expression of intuitions; see Thompson, pp. 91–92, and Howell’s reply, p. 232. Thompson holds that a Kantian “canonical language” would be virtually without singular terms (ibid., p. 101).
In a discussion in March 1983, after this Postscript has been written, Hintikka stated that Howell’s interpretation of immediacy was the view Hintikka had maintained all along.
But obviously one needs to look carefully at how Kant and his contemporaries actually viewed the relation of concepts and their “marks.” This matter was explored by my student Alan Shamoon in his dissertation, “Kant’s Logic,” Columbia University 1979.
I owe this last observation to Manley Thompson. However, the direct-reference view might itself suggest an assumption such as I attribute to Kant; compare the connection between direct reference and sense-perception in the philosophy of Bertrand Russell.
Kant’s Theory of Science, p. 50, n. 15. Chapter 2 of this book is a very clear and instructive discussion of Kant’s philosophy of mathematics, with expositions both of Hintikka’s and my own views. Howell seems not to be free of the same misunderstanding; see “Intuition, Synthesis, and Individuation,” pp. 210–211.
Ak., IV, 281–282, translation from Beck’s edition.
“But in mathematical problems there is no question of this [the conditions under which the perception of a thing can belong to possible experience], nor indeed of existence at all, but only of the properties of the objects in themselves, solely in so far as these properties are connected with the concept of the objects” (A 719 = B 747). This passage is instructively discussed by Thompson, “Singular Terms and Intuitions,” pp. 98–101. It is largely through this paper that I became aware of the difficulties faced by my own views about mathematical objects and existence in Kant. Brittan comments on the same passage (Kant’s Theory of Science, p. 66), but he seems to me to misread it in saying that “mathematical problems” have to do with real possibility. That seems to me to neglect the clear statement of A 223–4 = B 271 that construction does not establish such possibility, and even the remark in the present passage that in mathematical problems there is “no question” of the conditions under which perception of a thing can belong to possible experience. The point is subtle because Kant holds that mathematics is about really possible objects and that this can be established. But it is not mathematics that establishes it.
A 218 = B 265. Cf. the whole discussion of possibility this statement introduces. Thompson (p. 106, n. 21) sees a difficulty with a modalist interpretation of how Kant might deal with reference to mathematical objects in Kant’s distinction between demonstrations and discursive proofs. I am not sure I understand what the difficulty is, but it is not evident that the attenuated version of modalism suggested for Kant in the text is more exempt from it than the direct version. But cf. note 30 above.
See my “Ontology and Mathematics” in Mathematics in Philosophy (Ithaca, Cornell University Press, 1983), section III; also “Mathematical Intuition, Proceedings of the Aristotelian Society N.S. (1979–1980).”
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Parsons, C. (1992). Kant’s Philosophy of Arithmetic. 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_3
Download citation
DOI: https://doi.org/10.1007/978-94-015-8046-5_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4105-0
Online ISBN: 978-94-015-8046-5
eBook Packages: Springer Book Archive