‘With a Philosophical Eye’: the role of mathematical beauty in Kant's intellectual development

Abstract

This paper shows that Kant's investigation into mathematical purposiveness was central to the development of his understanding of synthetic a priori knowledge. Specifically, it provides a clear historical explanation as to why Kant points to mathematics as an exemplary case of the synthetic a priori, argues that his early analysis of mathematical purposiveness provides a clue to the metaphysical context and motives from which his understanding of synthetic a-priori knowledge emerged, and provides an analysis of the underlying structure of mathematical purposiveness itself, which can be described as unintentional, but also as objective and unlimited.

Keywords::

• mathematics
• beauty
• Kant
• Wolff
• essence
• purposiveness
• demonstration of God's existence

Acknowledgements

I thank the editors of this special issue, the members of the Fox Center at Emory University and ENAKS, as well as Oliver Thorndike, Dilek Huseyinzadegan, Rudolf Makkreel and Mark Risjord for providing helpful critical comments on earlier drafts of this paper.

Notes

^1. It must be kept in mind that in the KU, Kant does not want to apply the term ‘beauty’ in a strict sense to mathematical objects or propositions. However, because the rather technical considerations motivating this typology do not figure in Kant's earlier work, he does speak there of the ‘Schönheit’ of mathematical figures (e.g. BDG 02:94–95). I will use the term ‘mathematical beauty’ in this less strict sense, except in the context of the KU, where I will employ the more precise locution ‘formal objective purposiveness’. As we will see, this difference in terminology does not arise from a change in Kant's view of mathematical purposiveness, but from a refinement in his understanding of other forms of beauty. All translations from the Critique of the Power of Judgment are from Kant (2002) unless otherwise noted.

^2. This is how commentators uniformly treat the passage, when they discuss it at all. For some recent examples, see Nuzzo (2005, 330–332) and Wicks (2007, 189–193).

^3. This diagram was drawn using the software GeoGebra 4.4.41.0.

^4. This diagram is from Euclid (1908, vol. 2, 71).

^5. It is not entirely clear whether the sentence ‘The other curves … construction.’ (‘Die andern krummen Linien … gedacht war.’) introduces another example or is rather the conclusion to the second. Here I take the latter to be the case.

^6. Cf. Nuzzo 2005, 332.

^7. They are analytic in the sense that the perfection of, e.g. a circle, consists in its possessing precisely and only the features contained analytically in the general concept of a circle. Likewise, the perfection of something produced intentionally is judged purely by a logical comparison of it with the concept of what was to be produced. So, that a figure is or is not perfect in this sense is merely an analytic judgement. Kant speaks of such analytic perfection, e.g. at B113–B116.

^8. It is very likely that what Kant has in mind here is something like a purely mathematical analogue of a higher-order physical law, such as de Maupertuis' Principle of Least Action. What both de Maupertuis and the young Kant found so remarkable about such a principle was that it is unified, in a single universal law and in one principle, so many diverse individual laws, which until that point had been demonstrated from entirely independent grounds.

^9. Wolff offers a similar proof in his German Metaphysics (cf. Wolff 1747, §38), but as we will see below, it is this passage from the Latin Ontology in particular that Kant has in mind when later criticizing this doctrine.

^10. An internal determination not grounded in any of the remaining components of the essence would have itself to be a component of the essence for this very reason.

^11. Leibniz first announced his discovery of this series in the paper ‘De vera proportione circuli ad quadratum circumspriptum in numeris rationalibus expressa,’ which appeared in the Acta Eruditorum 15 September 1713. See Leibniz (1858, 118–122). Wolff later discusses the series in a letter to Leibniz, mentioning how some involved in the controversy over the invention of calculus were stating that the series had previously been discovered by the English mathematician James Gregory (1638–1675) (Leibniz and Wolff 1860, 152–153). The series is also discussed in Wolff (1738, 182–183).

^12. This has not been previously noted by commentators. Walford and Meerbote seem to suggest Kant is referring to the Secant Tangent Theorem itself, because their explanatory note gives only reference to it (Kant 1992, 431, n. 42). Kreimendahl and Oberhausen, 173, n. 117, explicitly follow Walford and Meerbote, and do not provide any further explanation. But the theorem Kant describes is clearly not the Secant Tangent Theorem, although it is related to it as described.

^13. The diagram is from Euclid (1908, vol. 2, 74).

^14. I have altered the Cambridge Translation here, which by using ‘straightforward’ obscures the significance of the term Kant uses, namely ‘Anstalten,’ and masks its relation to the later occurrence of this term in the same passage, as we will see below. This term occurs throughout the BDG, and always means a kind of purposive arrangement or design. The Cambridge Translation translates this word in other contexts variously as ‘design’ and ‘provisions.’ See BDG 02:127–137 passim.

^15. Again, no other commentator has noted the source of this example, which in this case is historically quite significant, because it suggests that Kant was familiar with Galileo's book. Indeed, I think that if Kant had simply learned it as a standard part of physics, it would be quite a coincidence for him to choose it as one of his examples. Most likely his choice is motivated by a desire to demonstrate his knowledge of Galileo and his sophistication at not needing to mention him by name. Walford and Meerbote provide a modern demonstration in Kant (1992, 431, n. 43). Kant (2011, 173, n. 118) again follows Kant (1992).

^16. No explanation of this is given in either Kant (1992) or Kant (2011). Of course, the mathematics in this case is elementary and of no particular historical significance. Yet the point Kant makes with it is important for our purposes. The diagram given here is in the public domain and copied from Wikipedia.

^17. This does not conflict with the fact that this section is supposed to provide an a-posteriori demonstration of God's existence, because Kant is using the term here in its traditional meaning of moving from the effects to the causes, not from causes to effects, which would be a priori in this way of speaking. This is essentially different from Kant's later use of the term to mean what is cognized independently of experience.

^18. Kant seems to be echoing the language of de Maupertuis, who he cites in the next section:

If it is true that the laws of motion and rest are indispensable consequences of the nature of body, that proves all the more the perfection of the supreme being: It shows that everything is ordered such that blind and necessary mathematics executes what is prescribed by the most enlightened and free intelligence. (1751, 65–66)

I think it is reasonable to assume that de Maupertuis' teleological interpretation of the Principle of Least Action, as well as his related criticisms of all supposed teleological proofs of God's existence from the contingent design of nature, had as much influence on Kant's views on mathematical purposiveness as did Hutcheson's writings. However, as discussion of this would take us rather deeply into the field of physics, it must be omitted here.

^19. I have again changed the translation of ‘Anstalten’ from ‘provisions’ as in the Cambridge Translation, to ‘design,’ so as to keep the link with the use of the term earlier in this passage.

^20. Fistioc (2002) seems to overlook this point, taking Kant's remarks as genuinely about the historical Plato. See esp. 36–43.

^21. Thus, although it might seem as if we could render any given mathematical truth analytic by redefining its subject so as to include its predicate (e.g. render ‘2 = 1+1’ analytic by defining ‘2’ as meaning ‘1+1’), Kant thinks it is true of most if not all mathematical concepts that the number of such truths would in fact prove to be inexhaustible. So the process of redefinition could never be completed. One familiar with the wealth of purposive relations in mathematics would thus at once realize mathematical truths to be non-analytic. Of course, the synthetic character of such truths is not dependent upon the failure of such a process, although its failure does seem to indicate that there is something in these truths that essentially exceeds what can be captured in an analytic judgement.

Additional information

Notes on contributors

Courtney David Fugate

Courtney D. Fugate received his PhD in philosophy from the Catholic University of Leuven, Belgium, in 2010 and is currently assistant professor of Civilization Studies at the American University of Beirut. His books include Alexander Baumgarten's Metaphysics: A Critical Translation with Kant's Elucidations, Selected Notes and Related Materials (Bloomsbury 2013) (with John Hymers) and The Teleology of Reason: A Study of the Structure of Kant's Critical Philosophy (de Gruyter 2014). He has interests in Kant's pre-critical and critical work, as well as early modern German philosophy and philosophical cosmology.