Transcendental and mathematical infinity in Kant's first antinomy

ABSTRACT

Kant's first antinomy uses a notion of infinity that is tied to the concept of (finitary) successive synthesis. It is commonly objected that (i) this notion is inadequate by modern mathematical standards, and that (ii) it is unable to establish the stark ontological assumption required for the thesis that an infinite series cannot exist. In this paper, I argue that Kant's notion of infinity is adequate for the set-up and the purpose of the antinomy. Regarding (i), I show that contrary to appearance, the Critique can even accommodate a modern notion of infinity without consequences for the antinomy; and regarding (ii), that the ‘ontological’ consequence indeed follows once its covertly epistemic character is adequately understood.

KEYWORDS:

• Kant
• first antinomy
• infinity

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Kant's works are cited by reference to the translation used (if available) and by reference to his Gesammelte Schriften (Kant 1900). The Critique of Pure Reason is quoted via reference to the A/B editions, translations are taken from Kant (1998).

2 The number of objections to the antinomy that can be allocated to these claims is considerable: Cantor (1932), Russell (2009), Bennett (1974), Guyer (1987), Priest (1995), Smith (2003), and Kreis (2015). From this selection we can see that they are prominent both amongst mathematically oriented philosophers as well as Kant scholars.

3 Of course, Kant will go on to specify this notion with respect to many different applications, such that later on it will become a task to determine which one he uses in the Antinomy.

4 A notable exception here is Zermelo in his function as editor of Cantor's collected works. Cantor's polemic agains Kant prompted him to add a footnote in which he comments with remarkable insight that the antinomy is not about the concept of the infinite as such, but its application to the ‘world whole (Weltganze)’ (Cantor 1932, 375). Moreover, at the end of Zermelo (1930) he acknowledges that Kant's antinomies have ‘[t]he two diametrically opposed tendencies of the thinking mind, the ideas of creative progress and summary completion’ (Zermelo 1930, 431) at their basis. These ideas are equally characteristic for his proposal of a growing hierarchy of domains and models for set theory which he proposes in that paper. For parallels between Zermelo's solution to the set theoretic paradoxes and Kant's resolution of the antinomy, see De Bianchi (2015).

5 For an in-depth discussion of these issues see Mancosu (2016, ch. 3.)

6 The Berliner Akademie der Wissenschaften, for instance, announced an essay competition (Preisfrage) in 1784 whose central aim was to eliminate the concept of the infinite in mathematics by reducing it to known and uncontroversial terms (cf. Schubring 1982). Use and requirement of the notion of the infinite was also a central theme in the debate between Kant and Abraham Kästner where Kästner accused Kant of not being restrictive enough with regards to the infinite (cf. Kästner (1790), Kästner (1790), Kant (2014) (AA XX: 418) and Onof and Schulting (2014)).

7 See, for instance, B17, A142/B182, A240f.

8 This is mentioned in multiple places in the Critique: A39/B56, A25/B41, A47/B65, B120, B147. The details are, of course, beyond the scope of this paper.

9 Finally, Schultz tries to apply his theory to infinite space and euclidean geometry where he committed several mistakes. Criticism of his contemporaries, however, was mainly focused on the metaphysical and epistemological part of his concept of the infinite such that his technical mistakes remained unknown to him and his contemporaries. These technical concerns, however, do not touch upon the conceptual issues that we are concerned with and will therefore not be discussed.

10 Apart from having the required mathematical expertise, Schultz was also a competent interpreter of Kant's work. His critical examination of the doctrines of the first Critique in Schultz (1789) includes a discussion of the antinomy, but, remarkably, there is no attempt at a refutation and no reference to any mathematical notion of the infinite to be found. Furthermore, Kant was aware of Schultz's work of reception and gave it his seal of approval. As mentioned in a letter from 29.7 1797 (AA XII: 367–368) he recommends Schultz as one of the few who have understood him well.

11 This is testified in a letter Kant wrote to Schultz, which, to my knowledge, was first mentioned in Büchel (1987, 185–220). In that letter, Kant refers to an additional piece of paper (‘sheet b’) sent to Schultz: ‘I believe that the included sheet b […] contains new material to reconcile your theory with what the Critique has to say in the Antinomy about the cognition (Ansehung) of the infinite in space ’ (AA XI: 2.8.1790, my trans.) This comment indicates that the first Critique does not yet fully accommodate Schultz's theory of the infinite, but at the same time that this doesn't suggest wide-ranging changes to the antinomy. Unfortunately, this key document which he refers to (‘included sheet b’) seems to be lost.

12 Kant occasionally uses the word ‘magnitude’ for quantities (i.e. numbers) as well. This is to some extend due to the fact that the german original ‘Größe’ is a much more general and colloquial word. More importantly, however, his whole concept of number is intimately connected to the notion of magnitude and measurement which additionally informs his wording (for this connection, see Sutherland 2004). Nonetheless, the distinction between quantity and magnitude by degree of abstraction remains contentful in both Kant and Schultz.

13 Perhaps it is interesting to compare Kant's notion of multiplicity and Schultz's infinite set to the Cantorian intuition behind the set concept. Cantor captures his intuition in a quote that comes closest to a definition: ‘By an aggregate (Menge) we are to understand any collection into a whole (Zusammenfassung eines Ganzen) M of definite and separate objects m of our intuition or our thought’ (Cantor 1952, 85). This is surprisingly similar to what has been discussed before. But Kant and Schultz's concept differs in one main aspect: it still adheres to the notion of a unit. This yields two distinct consequences: for one, Cantor's notion is broader in that it allows a more general type of object to be an element of a set, and second, the restriction to units (as conceived by Kant and Schultz) also retains the connection to the notion of measure and with it to space (and time).

14 This is a persistent thread in Kant's reflections during and after his work on the first Critique. Cf. Kant (2005, 192, 200, 228, 265, AA XVII: 725, AA XVIII: 37, 161, 277).

15 Another type of synthesis would be that of a manifold into a singular representation in the first place (which would be transcendental). This, too, lies under the conception of finitude, in that our cognition requires an external impulse and material to work (we are finite in that we don't create the objects of our thinking). This is emphasized by Kant in the very first sentence of the introduction (cf. B1). Moore (1988) argues that the operations based account characterized in this paragraph is a consequence of how this metaphysical dependence appears to us.

16 In this sense Kant notes that ‘no concept can be thought as if it contained an infinite set of representations within itself’ (B40). Cf. also Kant (1992, 632, AA IX: 142).

17 See also A430/B457 (esp. the footnote) and A518/B546 for the same claim. Moreover, Kant (2004, § 52c, AA IV: 342) and similarly (Kant 2005, 190–191, AA XXVII: 712f).

18 It is thus to be distinguished from certain accounts of finitism that feature an additional limiting account of feasibility (cf. Stenlund 2012).

19 Cf. also the discussion in Grier (2004, 186).

20 Cf. also Kant (2005, 182, 211, AA XXVII: 701, AA VIII: 85) and Kant (1997, 195, AA XXIX: 838).

21 Huby (1971) in her criticism of Russell talks of different types of numbers out of which the transfinite one is not a number type to apply to the structure of our cognitive faculties.

22 Further evidence for this epistemic based set-up is the reduction of the case of space to the case of time in the proof of the thesis. Kant argues (roughly) that infinite space cannot be an object of the understanding, because there wouldn't be enough time to synthesize it (A428/B456). Again this points to a representation of the totality with respect to our cognitive faculties. The fact that space is not considered according to its nature as an aggregate, but by our way of synthesizing it, again shows that epistemic concerns dominate in the antinomy (cf. A412/B439). Incidentally, Allison (2004) and Grier (2004), neither of which explicitly note the epistemic character of the antinomy, only mention this reduction of the case of space to the case of time, but they don't provide an explanation as to why. The explicitly epistemological nature of the antinomy was most extensively discussed by Falkenburg (2000). For a systematic development of this view the reader is referred to her book.

23 Cf. also Kant (2004, § 52b), Kant (2005, 315, 321, AA XIIX: 403,413), and Kant (1997, 211, AA XXIX: 854).

24 According to the explanation in Grier (2004) ‘the application of the principle of reason to the world is only undertaken, because appearances are taken to be things in themselves’ (187). If so, ‘we are conceptually bound that the conditions necessary for their existence have already been met’ (186). But it is left open why then a first beginning needs to be assumed in order to account for their existence. The epistemic origin of the claim is not made explicit. The ingenuity and suitability of Willascheck's interpretation for our purpose lies in the simplification of the explanation and its compatibility with the epistemic setting of the antinomy and its underlying notion of human finitude.

25 Kant's approach of replacing a completed infinity with a regress in indefinitum is, as De Bianchi (2015) argues, what Zermelo (1930) has called the ‘idea of creative progress’ (431), and what stands in relation to his understanding of the open-endedness of the height of the set theoretic universe. Unlike in the resolution of the first antinomy, however, the height of the set-theoretic hierarchy has relative stopping points (the so-called inaccessible cardinals). Zermelo equally links these to Kant by reference to the idea of ‘summary completion’ as the other ‘tendency of the thinking mind’ (ibid.). De Bianchi argues for a connection of this idea to the second antinomy and its resolution by reference to a regress ad infinitum, where the objects under consideration already exists and are successively decomposed. Each decomposition into smaller parts, however, can be further refined by choosing yet a smaller unit. As De Bianchi points out, these units of decomposition bear resemblance to the relative stopping points in the set theoretic hierarchy. The details of the argument, however, cannot be discussed here.

26 For a discussion of Russell's remarks in particular see Huby (1971).

27 Perhaps this is also one reason why Kant doesn't defend it more. It seems to be a very well established epistemological premise. Again, this is not an argument against any mathematical treatment of the infinite, it just reiterates the difference between the two that was developed in the first two sections.

28 The idea for this paper was first developed while I was part of the masters program Theoretical Philosophy at LMU Munich. I want to thank Axel Hutter, Christian Martin, Thomas Oehl, and David-Benjamin Berger for their help and encouragement. I am grateful to Colin Johnston and Johannes Nickl for their comments and discussions of previous drafts, and I appreciate the time and work of the anonymous reviewer which lead in particular to an improvement of section 3.