Abstract
Chapter5 analyzes the general structure of the antinomy of pure reason and the specific structures of its four versions, including a detailed logical reconstruction of all thesis and antithesis proofs. To understand the logical structure and epistemological significance of the antinomy, it is crucial to distinguish carefully between the pre-cretical views which correspond to the standpoint of transcendental realism on which the proofs are based, and the critical point of view, which is the key to Kant’s resolution of the antinomy. Kant was well aware that the proofs are defective. His critical diagnosis is that they seem conclusive from the point of view of transcendental realism, whereas transcendental idealism reveals that they derive from a self-contradictory cosmological concept. Our reconstruction shows that the proofs employ rationalist, empiricist, or verificationist arguments, including Kant’s own pre-critical conception of the infinite, but do not depend fatally on claims of transcendental idealism; and that the proof results are due to the logical fallacy of an ambiguous middle term in the proofs derive from semantic equivocations inherent to the cosmological concept of the spatio-temporal world, which Kant considered to be inevitable in particular in the case of the “mathematical” antinomy.
The second class of sophistical inference is applied in general to the transcendental concept of absolute totality in the series of conditions for a given appearance; and from the fact that I always have a self-contradictory concept of the unconditioned synthetic unity in the series on one side, I infer the correctness of the opposite unity, even though I also have no concept of it. (CPR, A 340/B 398)
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Notes
- 1.
- 2.
“Es dürfte kaum jemals […] mehr zur Diskreditierung der menschlichen Vernunft und ihrer Fähigkeiten geschehen sein, als mit diesem Abschnitt der ‘kritischen Transzendentalphilosophie’. Ich werde gelegentlich zeigen, daß es diesem Autor nur durch einen vagen, distinktionslosen Gebrauch des Unendlichkeitsbegriffs […] gelungen ist, seinen Antinomien Geltung zu verschaffen, und dies auch nur bei denen, die gleich ihm einer gründlichen mathematischen Behandlung solcher Fragen gern ausweichen.” My translation.
- 3.
“Nicht um eine Widerlegung oder Ablehnung des Unendlichkeitsbegriffes handelte es sich hier bei Kant, sondern um seine Anwendung auf das Weltganze, um die Tatsache, daß die menschliche Vernunft sich durch ihre innere Natur ebenso gedrängt findet, die Welt als begrenzt wie als unbegrenzt, als endlich wie als unendlich anzunehmen—eine Tatsache, die weder durch mathematische Theorien wie die Cantorsche Mengenlehre noch durch seine wohl nicht sehr tiefgreifende Polemik aus der Welt geschafft werden kann.” Zermelo’s note [1] to Cantor’s text; my translation.
- 4.
- 5.
See the Ethics lectures Brauer (Menzer 1924, 59), Collins (27:280), and Mongrovius (27:1431) from the mid 1770s.
- 6.
According to Malzkorn (1999, 38–53), Kant’s ideas of reason are defined as follows (my translations): they are concepts (I) “of the form of cognition” (ibid., 38), (II) “of the form of the relation between cognition of the understanding (judgments)” (ibid., 39), (III) “of the unity of all cognition of the understanding” (ibid., 45), and (IV) “of ‘unconditioned conditions’ of a uniform system of all empirical cognition” (ibid., 47). According to this determination, ideas are not concepts of empirical objects, but “meta-concepts of empirical science” (p. 45); the world as the “sum total of appearances […] is not an object that could fall under such a meta-concept” (ibid., 76). Accordingly, he states that the world as the sum total of appearances for Kant is no possible object of an idea of reason (i.e., it lacks real possibility): “It only appears as such if one conflates the realm of appearances with the realm of cognition about them” (ibid., 77). However, he disregards that Kant locates the origin of the cosmological antinomy exactly in this problem. For Kant, the sensible world lacks not only real possibility, but also logical possibility as an object of an idea of reason. Malzkorn (1999, 77), however, criticizes Kant as follows: “A description of the world should not be conflated with the world itself. Such mistaken identity would make the world appear to be the object of a concept of reason, but it represented an illegitimate conflation of two distinct realms. Indeed, traces of such conflation can be found […] in Kant’s doctrine of the antinomy”; and he concludes: “Kant’s theory of reason may […] be coherently reconstructable and be an important part of his epistemology; however, it cannot convincingly present the world as an object of natural and inevitable speculation of reason”. Indeed Kant himself attributes the inevitability of the antinomy precisely to an immanent tendency of reason to commit those logical fallacies of which today’s interpreters would accuse him. For a comprehensive interpretation of Kant’s theory of reason and the rationality of its natural disposition toward metaphysics, see Willaschek (2018).
- 7.
For the precise relationship between the amphiboly chapter and the antinomy, see Grier (2001).
- 8.
- 9.
Here, Kant differentiates as follows between ‘world’ and ‘nature’. The world is “the mathematical whole of all appearances and the totality of their synthesis in the great as well as in the small”, whereas nature is the world “insofar as it is considered as a dynamic whole and one does not look at the aggregation in space or time” (CPR, A 418–419/B 446–447). In a corresponding footnote, Kant explains that here he means the concept of nature “taken substantively (materialiter)”, in contradistinction to the concept of nature “taken adjectivally (formaliter)” (ibid.).
- 10.
- 11.
Al Azm (1972) correctly observes that the arguments do not exactly correspond to rationalist and empiricist positions. Against the background of the Leibniz–Clarke debate, he then associates the thesis positions with Newton’s views and the antithesis positions with Leibniz’s views, overlooking that both Newton (or Clarke) as well as Leibniz themselves put forth rationalist as well as empiricist (or phenomenological) arguments, despite all their differences. See also the references to Bayle and Crusius in Heimsoeth (1960, 260), and the detailed discussion in Grier (2001, 182–229).
- 12.
- 13.
See for example Kreimendahl (1990, 424–430). Schmucker (1990, 116–117), too, claims that Kant already takes the critical position when he relates the cosmological ideas to sensory perception. In my opinion, he conflates Kant’s critical conception of the phenomena with the common sensualistic concept of experience of the seventeenth/eighteenth-century schools. Guyer (1987, 386–387) accuses Kant of either reducing truth to empirical confirmability, or rendering the argument in favour of transcendental idealism circular. His analysis of the proofs (ibid., 405–412) suffers from not distinguishing between the point of view of transcendental realism, which underlies the thesis and antithesis proofs, and the critical point of view, which gives rise to the resolution of the antinomy.
- 14.
Malzkorn (1999) reconstructs the semantic aspects of the antinomy as far as possible by syntactic tools, including temporal logic and existential statements. Guided by a “principle of charity” (ibid., 118–119) he overlooks the fact that each kind of antinomy emerges from a crucial semantic equivocation which Kant wants to reveal.
- 15.
A thoroughly syntactic reconstruction of the semantic claim cannot capture these features. It has to violate the proof principles of consistency. Not all of Kant’s proof premises can be consistent with one another. As regards Kant’s concerns for a “dialectical” logic of transcendental illusion it is sufficient that the premises seem to be satisfied from the point of view of transcendental realism. Uncovering the transcendental illusion shows that indeed they are not. The syntactic reconstructions of Malzkorn admit of this insight only for the dynamical antinomy; see Malzkorn (1999, 199–200 and 221).
- 16.
Malzkorn (1999, 128) uses a predicate ‘non-F’ in order to preserve the contrary relation of thesis and antithesis.
- 17.
According to Allison (2004, 367–371), this is an assumption of transcendental realism, which presumes that the spatio-temporal world is a totum syntheticum and contradicts the definition (I) of infinity. My reconstruction of the thesis proof essentially agrees with Allison’s interpretation; see also his refutation of the objections raised by Russell (1903), Moore (1953), Strawson (1966), and Bennett (1974).
- 18.
Malzkorn (1999, 257) concedes this interpretation as admittable, in contrast to Strawson (1966, 176), Mittelstaedt and Strohmeyer (1990, 156), and Schmucker (1990, 115–116). However, he thinks that the proof of the thesis in this case fatally depends on a transcendental philosophical premise: namely on (4a) “When a series has elapsed, then it can be completely synthesized successively” (Malzkorn 1999, 257; my translation). According to my interpretation, this premise is not needed, given that “elapsed” for an infinite time series would mean that this time series was actually or cardinally infinite, in contradiction to the very concept of a series. The thesis proof confounds two concepts of the infinite which are incompatible regarding their real possibility; see below.
- 19.
In contrast, van Benthem (1983, 33) assumes that both concepts are logically rather than semantically incompatible. In Kant’s view, in the face of the resolution of the antinomy, this is just a consequence of the logical fallacy of taking the middle term of the cosmological syllogism in different meanings, which is what gives rise to the antinomy.
- 20.
Malzkorn (1999, 118–119 and 130–141) neglects this crucial point. To avoid the diagnosis of a non sequitur, he employs a syntactic “principle of charity” and reconstructs a logically valid proof.
- 21.
Allison (1983, 49) also emphasizes that the proof result depends decisively on Leibniz’s principle of indiscernibles, and interprets this in the sense of a verificationist position. In the revised edition, Allison (2004, 373) no longer accepts this view, but argues that “the argument takes an epistemological turn, which is not to be confused with verificationism”; in the corresponding note he claims that it “is a confusion because it (falsely) implies that the assumption of an absolute beginning is meaningless rather than simply false” (ibid., 504, n. 33). I am puzzled about this claim; it fits in with the critical resolution of the antinomy, but not with the point of view of transcendental realism taken in the proof. Against the background of Kant’s 1755 and 1758 arguments (see Sect. 3.3.1), the verificationist interpretation seems plausible to me; whereas I do not share the view that Kant employs verificationist arguments in all four antinomies (Allison 1983, 61 and 312): he only does so in the first antinomy, as far as I can see. In his revision, Allison (2004) seems to take into account the criticism of Grier (2001, 190); she suggests an interpretation of the proof in terms of real relations, according to which “an empty time would lack any ‘distinguishing condition of existence’ (A 428/B 456)”, i.e., any real (ontic) ground of the world. However, the claim in step (4) that “the relation of the world to empty space would be a relation of the world to no object” may well be understood as a claim about a meaningless concept (given that relations are usually understood as two-place predicates), and hence as a verificationist argument.
- 22.
For a much more comprehensive analysis of the second antinomy, its historical background, its genesis, and its relation to Kant’s theory of cognition, see Engelhard (2005).
- 23.
For the axioms which the relation < of mereology obeys, see e.g., Simons (1987). The following reconstruction does not depend on specific mereological axioms.
- 24.
- 25.
According to (A 440–442/B 468–470), the thesis does not directly refer to Leibniz’s monads, but to real monads, see Engelhard (2005, 176–177). Kant’s own explanation of the thesis proposition in terms of “transcendental atomistic” or “the dialectical principle of monadology” (A 442/B 470) does not really contribute to clarifying this question. However, see also below.
- 26.
- 27.
In Falkenburg (1995, 15), I still did so.
- 28.
However, the issue is not completely clear; see n. 25 above.
- 29.
Malzkorn (1999, 279, n. 96), in contrast, neglects the traditional metaphysical concept of a substance and defends the view that Kant does not employ an analytical argument.
- 30.
See Russell (1903, 460): “It is indeed obvious that the proposition, true or false, is concerned purely with whole and part, and has no special relation to space and time. Instead of a complex substance, we might consider the numbers between 1 and 2, or any other definable collection. And with this extension, the proof of the proposition must, I think, be admitted; only that terms or concepts should be substituted for substances, and that, instead of the argument that relations between substances are accidental (zufällig), we should content ourselves with saying that relations imply terms, and complexity implies relations.” Vogel (1975, 299; my translation) charges the proof with circularity: “That the argument of the removal of all composition can indeed only apply to an object which already presupposes what is to be proved, we have already commented upon as against the proof in the Monadologia physica […].’
- 31.
Engelhard (2005, 175) challenges this claim, pointing to Kant’s third and fourth arguments against the discursive character of space in the Transcendental Aesthetic (B 39–40). However, Kant’s theory of mathematics does not admit of the abstract concept of a set, or a logical multiplicity, given that from 1770 on he considers the objects of mathematics to be generated in concreto in pure intuition.
- 32.
The extent of the multiplicity, or the number of individuals that form a concrete composite, remains open. The proof does not claim that a composite has finitely or countably infinitely many simple parts. The thesis here does not imply a position of finitism; in contrast to the Prolegomena (§52c, 4:342), where Kant states the thesis and antithesis claims as follows: “[…] that bodies in themselves consist of infinitely many parts or of a finite number of simple parts.”
- 33.
My reconstruction sets aside all details concerning the cosmological concept of freedom.
- 34.
One could also express this claim by a predicate N with the meaning ‘obeys a law of nature’, or within second-order logic by a proposition of the form ∀x∃N Nx, where N stands for the law of nature. Malzkorn (1999, 194) defines two predicates ‘causality according to the laws of nature’ and ‘causality through freedom’, which he formalizes as three-place relations of two events x, y at a time t. This approach makes it possible to distinguish the absence of a cause (indeterminism or contingency) explicitly from causality out of freedom. In his approach, contradictory claims are also only obtained if the middle term ‘world’ of the cosmological syllogism is used equivocally (Malzkorn 1999, 199–200).
- 35.
In contrast, in his reconstruction Malzkorn (1999, 231–234) uses a modal operator of contingency, which is defined by ♢Px ∨♢¬Px.
- 36.
The cosmological antinomy has this aspect in common with the paralogism of pure reason. The paralogism is based on a fallacious categorical syllogism in which the middle term ‘I’ is used equivocally in an empirical or intelligible sense. The antinomy, on the other hand, is based on a fallacious hypothetical syllogism in which the complete series of empirical conditions is conflated with an intelligible unconditioned; see the distinction between paralogism and antinomy (A 407/B 433–434) as well as the diagnosis that the antinomy is based on a sophisma figurae dictionis (A 499/B 527–528, and Logic §90, 9:135). See also Malzkorn (1999, 110). Seifert (1989) concludes from this equivocation that Kant’s argument is inconclusive, similar to Malzkorn (1999), who does so for the third and fourth antinomy. Kant, however, only has the burden of showing that the proof seems conclusive from a dogmatic metaphysical point of view. He himself was convinced that they are not.
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Falkenburg, B. (2020). The Antinomy of Pure Reason. In: Kant’s Cosmology . European Studies in Philosophy of Science, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-030-52290-2_5
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