Was Kant anticipating Gödel's incompleteness in his antinomies?

Question

Kant's attempts to prove that there's a limit to pure reason based on the existence of antinomies, i.e. pairs of propositions where each one is rational, but the propositions contradict each other. One example is:

Thesis: The world has a beginning in time, and is also limited as regards space. Anti-thesis: The world has no beginning, and no limits in space; it is infinite as regards both time and space.

Kant's concludes from this that there is a limit to the knowledge that can be attained by pure reason.

But isn't this just a very non-rigorous informal version of Gödel's proof that any theory capable of expressing elementary arithmetic cannot be both consistent and complete? The implication here is that any complete system will contain inconsistencies, i.e. contradictory statements. But isn't this just the same things as Kant, who says pure reason is limited because it can lead to contradictions?

Answer 1

This reminds me of the older question Was Wittgenstein anticipating Gödel? There is more to it in the case of Kant than there was in the case of Wittgenstein though, at least in spirit. One could say that Kant pioneered in epistemology the stratification into levels of discourse, which Gödel later applied to formal semantics.

When the Gödel theorem first appeared many mistook it for a paradox, like Russell's, a contradiction within a system of mathematics. Many included Russell himself, Wittgenstein and Zermelo, at least according to the traditional view, see however What sources discuss Russell's response to Gödel's incompleteness theorems? for a different view. The issue was that the paradox only arises if one mixes the levels of language. Gödel sentence is unprovable in the object language, the proof that it is nonetheless true is done in the meta language, if one properly distinguishes between the two the paradox disappears, and we uncover an interesting property of the object language. Russell, Wittgenstein and Zermelo were presumably thinking universalistically, within an all-encompassing logical system.

What does this have to do with Kant? Kant also has a two level distinction, not in the language but in ontology, appearances and things in themselves. Like Gödel's, Kant's predecessors were in the habit of instinctively identifying the two, and antinomic reasoning was a direct result of taking this identification to its logical conclusion. What prevents Gödel sentence from being a paradox is a subtle rephrasing of "I am false" into "I am unprovable [in a language]". Kant similarly resolves the antinomies by relating them to appearances, our 'language of mind'. What creates the Liar is the language trying to handle unrestricted truth within itself, what creates the antinomies is mind trying to handle unrestricted reality within itself.

Both paradoxes result from disregarding self-limitations, and are resolved by explicitly reinstating them. As long as we do not regard the "world" as both an appearance and a thing-in-itself there is no antinomy of it having and not having a beginning in time, as long as we do not regard true and provable as a single item there is no Liar. Taking the analogy further one could say that Kant would have regarded Frege's universalist logicism program as a logical case of transcendental illusion, reasoning about appearances as if they were things-in-themselves.

Answer 2

1) The first incompleteness theorem of Gödel:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.

This theorem restricts the scope of provability, in short: Not all theorems of the theory are provable within the theory.

2) The second incompleteness theorem of Gödel:

For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.

In short: This theorem hinders a theory to prove its own consistency.

Both theorems are statements from the field of mathematical logic. Both have been proved and both do not refer to the physical world. They show that Hilbert’s program of formalization cannot be kept up.

3) Kant’s antinomies refer to the boundaries of our rational argumentation and experience. Kant argues in Critic of Pure Reason that statements about all embracing concepts like the world create antinomies. These concepts are only applicable within the domain of experience, which is the domain of the physical world.

Was Kant anticipating Gödel's incompleteness with his antinomies?

Both, Kant and Gödel prove the impossibility of certain rational approaches. That’s what both have in common.

But both refer to different fields: Mathematical logic (Gödel), epistemology (Kant).

I also have never heard that the idea has ever been discussed in Kant’s time, that an axiomatized system may have principal boundaries of its scope. On the opposite, Euclid’s concept of axiomatization was held by all as the paradigm of scientific reliability.

Hence my answer to the question of the post is „No“.

Answer 3

I'd be surprised; the intuitive way of looking at Gödel's theorems is to consider the following statement:

What is proveable is true

Given the intuitive understanding of what is understood by proof, this appears to be trivially true, and not a statement that one would object to; interestingly, this means in Kants terminology, that it is an analytic proposition.

but consider now it's converse:

What is true is proveable

This is much more problematic: that God exists may be true, but unprovable.

Now, Gödel showed, via formal languages, that there are true statements that are unproveable; so the above is actually false.

Can Kantian antinomies show the same thing? If so, one can largely say they, if not anticipating Gödel's exact result, have a family resemblence to them.

But this doesn't appear to be possible, since both statements in a Kantian antimony are plausibly true, in Kants own hands this means there is a limit to reason, whereas in Hegels and Priests readings they are more akin to dialethism.

Answer 4

on the contrary, Kant thought that he can prove the unprovability of God, while Goedel's second theorem stand in contradiction to Kant https://link.springer.com/article/10.1007/s11787-019-00235-z,

Answer 5

There is a connection, through Leibniz's distinction between infinitary and finitary proofs, which is directly related to issue of completeness of arithmetic (Gentzen's 1936 infinitary proof of completeness of arithmetic based on transfinite induction, Gödel's 1931 incompleteness theorem, Hilbert's formalist programme etc.). The root of Kant's antinomies, as everyone knows, is extending ad infinitum certain structures, whether taken from the Tr. Aesthetic or the Tr. Analytic, that are perfectly valid in their finite employment by the understanding, but lead to "paradoxes" when applied infinitely, by pure reason. Kant utilizes here his Leibnizian background and utilizes Leibniz's logico-mathematical work.

This means that all of Kantian antinomies are structured in a way where finitary and infinitary solutions to certain paradoxes are juxtaposed. The solution that Kant proposes is always finitary - which is more suited, he argues, for the aims of science, of its completeness. Since Kant, when introducing the cosmological ideas, speaks of various modes of 'completeness'. His sense of completeness, I believe, is basically the same as one employed by modern logicians: provability (syntactic derivability), and thus knowability in principle, of all true statements. In this sense Gödel's theorem challenges some of Kant's considerations by demonstrating that many (indeed, almost all important) domains of knowledge fail to meet this criterion.

Gödel, who was very enthusiastic about Kant (and Husserl) by the way, himself acknowledged that Leibniz anticipated some of his ideas, when he begun studying his work in the 1940s. Is this also true of Kant, as your question suggests? That's arguable, since Kant's work in this regard seems too narrowly philosophical, although Kant definitely was aware of the early developments in logic which anticipated Hilbert, Gödel and other mathematicians and philosophers working in this tradition.

My answer thus is: maybe, depending on how you look at it.