Kant on arithmetic, algebra, and the theory of proportions

Journal of the History of Philosophy

Volumn 44, Issue 4, 2006, Pages 533-558

  Sutherland, Daniel ^a  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 60950050018     PISSN: 00225053     EISSN: 15384586     Source Type: Journal    
DOI: 10.1353/hph.2006.0072     Document Type: Article

References (56)

1. Immanuel Kant, Prolegomena to Any Future Metaphysics, ed. and trans. Gary Hatfield (Cambridge: Cambridge University Press, 1997);
2. and Immanuel Kant, Lectures on Metaphysics, ed. and trans. Karl Ameriks and Steve Naragon (Cambridge: Cambridge University Press, 1997).
3. The contributors are too numerous for an exhaustive listing. The relatively recent interest was spurred by the work of Jaakko Hintikka and Charles Parsons; subsequent authors include Philip Kitcher, Gordon Brittan, Manley Thompson, J. Michael Young, Arthur Melnick, and Michael Friedman, among others. A helpful selection of these papers and references to others published before 1992 is found in Kant's Philosophy of Mathematics: Modern Essays [KPM], ed. Carl Posy (Dordrecht: Kluwer, 1992).
4. A few works not in that collection, as well as more recent work, include: J. Michael Young, "Kant on the Construction of Arithmetical Concepts" ["Kant on the Construction"], Kant-Studien 73 (1982): 17-4 6;
5. Michael Friedman, Kant and the Exact Sciences [KES], (Cambridge, Mass: Harvard University Press, 1992),
6. as well as "Geometry, Construction and Intuition in Kant and His Successors," in Between Logic and Intuition: Essays in Honor of Charles Parsons, ed. Gila Sher and Richard Tieszen (Cambridge: Cambridge University Press, 2000), 62-100;
7. Emily Carson, "Kant on Intuition in Geometry," Canadian Journal of Philosophy 27 (1997): 489-512,
8. as well as "Kant on the Method of Mathematics," Journal of the History of Philosophy 37 (1999): 629-52;
9. and Lisa Shabel, "Kant on the 'Symbolic Construction' of Mathematical Concepts" ["Symbolic Construction"], Studies in History and Philosophy of Science 29A (1998): 589-621.
10. I have argued for this interpretation in "Kant's Philosophy of Mathematics and the Greek Mathematical Tradition" ["Kant's Philosophy of Mathematics"], Philosophical Review, 113 (2004): 157-201.
11. It includes C. D. Broad, "Kant's Theory of Mathematical and Philosophical Reasoning," Proceedings of the Aristotelian Society 42 (1941): 1-24;
12. Jaakko Hintikka, "Kant's Mathematical Method," Monist 51 (1967): 352-75;
13. Charles Parsons, "Kant's Philosophy of Arithmetic" ["Kant's Arithmetic"], in Mathematics in Philosophy (Ithaca: Cornell University Press, 1983);
14. Charles Parsons, "Arithmetic and the Categories" ["Arithmetic and Categories"], Topoi 3 (1984): 109-22;
15. Manley Thompson, "Singular Terms and Intuitions in Kant's Epistemology," Review of Metaphysics 26 (1972): 168-89;
16. Philip Kitcher, "Kant and the Foundation of Mathematics," Philosophical Review 84 (1975): 23-50;
17. and Gordon Brittan, Jr., "Algebra and Intuition," in KPM, 315-39.
18. I set aside whether Plato thinks of these numbers as Forms or as "intermediates." For a detailed account of the Greek conception of number and its role in Greek mathematics, see Jacob Klein, Greek Mathematical Thought and the Origin of Algebra [GMT], trans. Eva Brann (New York: Dover Publications, 1968).
19. The following summary of the Greek conception of number relies on Klein's work. See Klein, GMT, 22 and 46-47 on pure numbers.
20. Klein GMT, 71, 60.
21. Plato, The Republic, ed. G. R. F. Ferrari, trans. Tom Griffith (Cambridge: Cambridge University Press, 2000), 526A.
22. Cf. Plato, Philebus, trans. Dorothea Frede (Indianapolis: Hackett Pub. Co., 1993), 56D-E.
23. Plato, The Republic, 525E.
24. For a lucid account of the Eudoxian theory of proportions, see Howard Stein, "Eudoxus and Dedekind: On the Ancient Greek Theory of Ratios and Its Relation to Modern Mathematics," Synthese 84 (1990): 163-211. My exposition draws from his account.
25. It is not entirely clear what kinds of things the Greeks counted as magnitudes. Although the kinds mentioned were clearly paradigmatic, angles, weights, and other things may have been counted as magnitudes as well. Aristotle, who followed Eudoxus and preceded Euclid, does not use the term magnitude when discussing ratios and proportions, and he may have wished to reserve the term for geometrical magnitudes. See Ian Mueller, "Homogeneity in Eudoxus's Theory of Proportion" ["Homogeneity"], Archive for the History of the Exact Sciences 7 (1970-71): 1-6. Euclid, on the other hand, thought of numbers as magnitudes, since magnitudes are characterized as what can stand in ratios, and numbers can stand in ratios.
26. The requirement of homogeneity for standing in ratios may not have been a part of Eudoxus' original theory, but it was a requirement by the time of Euclid. See Ian Mueller, "Homogeneity," 1-6.
27. For a fuller account, see Sutherland, "Kant's Philosophy of Mathematics."
28. I discuss the Critique's lack of a section devoted explicitly to mathematical cognition and argue for the role of the Axioms in Kant's philosophy of mathematics in "The Point of Kant's Axioms of Intuition," Pacific Philosophical Quarterly 86 (2005): 135-59.
29. For an explication of Kant's concept of magnitude and the arguments that rest on it in the Axioms of Intuition, see Sutherland, "The Role of Magnitude in Kant's Critical Philosophy" ["Role of Magnitude"], Canadian Journal of Philosophy 34 (2004): 411-42.
30. Friedman, KES, 121;
31. Parsons, "Arithmetic and Categories," 116.
32. Julia Annas, "Aristotle, Number and Time," The Philosophical Quarterly 25 (1975), 99.
33. Euclid means an "aliquot part," that is, a part that is a submultiple. See Heath, Elements, Vol. 2, 280.
34. Leonhard Euler, Elements of Algebra [EA] , trans. Rev. John Hewlett (New York: Springer Verlag, 1972.), xxxiii.
35. Christian Wolff, Anfangsgründe aller Mathematischen Wissenschaften [AMW] (1st ed., 1710; 7th ed., Franckfurt und Leipzig: Rengerschen Buchhandlung, 1750-77);
36. Kant singles him out as "the most subtle arithmetician" (29:52). Nichomachus of Gerasa (fl. 100 A.D.) authored an arithmetic that was significantly less developed than that of Diophantus. Michael Psellus of Constantinople (1018-1078) wrote a great number of compendia, including works on arithmetic. He used what may have been the only remaining Greek copy of Diophantus's Arithmetic. At the fall of Constantinople, that copy was taken to Italy, where it was rediscovered during the Renaissance. See entries for Psellus and Diophantus in Dictionary for Scientific Biography, ed. Charles Coulston (New York: Charles Scribner's and Sons, 1970-80).
37. T. L. Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra (New York: Dover Publications, 1964);
38. See Henk J. M. Bos, Redefining Geometrical Exactness: Descartes's Transformation of the Early Modern Concept of Construction [RGE] (New York: Springer Verlag, 2001), 129. There was nevertheless an early appreciation of the parallels between geometrical problems and algebraic equations, as I will discuss in what follows.
39. See B. L. van der Waerden, A History of Algebra: From Al-Khwarizmi to Emmy Noether [HA] (New York: Springer Verlag, 1985), 13. Nevertheless, in the Renaissance and early modern period it was thought that Diophantus was the source of Arabic algebra, making Diophantus' presumed influence twofold: indirectly through Arab sources and directly on Viète.
40. See Bos, RGE, for an extended and thorough treatment.
41. Wolff, ML, 35-37.
42. See, for example, Parsons "Kant's Arithmetic," 134-35,
43. Klein, GMT, 197, 209.
44. Descartes, Rules for the Direction of the Natural Intelligence, ed. and trans. George Heffernan (Atlanta: Rodopi, 1998), Rule XIV.
45. Wolff, Elementa Matheseos Universae [EMU], in Christian Wolff: Gesammelte Werke II. Abteilung, Band 29 (Hildesheim: Georg Olms, 1968), 92-98.
46. Wolff's German Elements (Anfangsgründe aller Mathematischen Wissenschaften) [AMW] first appeared in 1710;
47. See Klein, GMT for an extended discussion of the representation of indeterminate and unknown quantities.
48. Johannes Tropfke, Geschichte der Elementarmathematik, Band 1, 4. auflage (New York: Walter de Gruyter, 1980), 137;
49. Wallis, Mathesis Universalis, 56;
50. Wolff, EMU, 35.
51. Euler, ERK, 2.
52. See Wolff, KU, 8. The arithmetical books were omitted in many if not most editions of Euclid; they were nevertheless available in Kant's day.
53. Furthermore, Kant explicitly says in a letter to Schultz in 1788 that arithmetic has no axioms (10:555). See Parsons, "Kant's Arithmetic," 122,-23.
54. The view that arithmetical cognition rests on pure units was endorsed by nineteenth-century philosophers such as Johannes Thomae, Elementare Theorie der analytischen Functionen (Halle. a.S.: L Nebert, 1880);
55. Rudolf Lipschitz, Lehrbuch der Analysis (Bonn: M. Cohen & Sohn, 1877);
56. Kant's letter to Rehberg, September 25, 1790 (11:208, 14:57). See footnote 76 above.