9

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?

  • The whole point of the antinomies is showing that they are only antinomies if you're not a transcendental idealist and can with the help of it shown to be not contradicting each other. I think there is a misunderstanding regarding the position. Kant does not actually endorse them as antinomous. – Philip Klöcking Jan 19 '16 at 21:54
  • I suppose you could make that comparison, but I don't think there's much to it. Any "redundancy" would be relatively tiny for either work. – Nelson Alexander Jan 19 '16 at 22:26
  • I'm not really up to speed on the "foreshadow" relation. Is there a different way to articulate that? – virmaior Jan 19 '16 at 22:32
  • 1
    @virmaior any other term I can think of "Anticipating","Predicting" pretty much conveys the same meaning. What I am trying to convey is that Kant was giving an intuitive version of an idea that Godel later expresses in a formal way. – Alexander S King Jan 20 '16 at 0:11
  • Q: may i please request an authentic and accessible reference to Goedel's work and that of Bertrand Russell possibly by Russell himself ? is there an accessible reference to Wittgenstein as well, please ? i hesitate to share my untutored thoughts on these serious topics for fear that it might cost me my audience and perhaps my membership as well --- a sad development it would be ! i am an admirer of Kahlil Gibran's poetry and Sri Ramakrishna Paramhansa's religious philosophy and i like to read Russell and Goedel and Wittgenstein from a humanistic point of view --- please help ! --- Krishna Rao – Krishna Rao PUTCHA Oct 15 '16 at 11:54
7

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.

  • I would want to disagree here, Conifold. Not to argue but to give an alternative view. If we dispose of the 'thing-in-itself' then phenomena become paradoxical. The only way to say that the world does and does-not have a beginning is to see this as referring to two aspects of reality, one which is superficial and time-bound, the other of which is changeless. Kant's mistake ( I would say) is not to see that by his definition there can only be one 'thing-in-itself'), Pardon me if this a little off-topic. I just feel that Kant did not get his logic right. – PeterJ Sep 8 '17 at 17:23
  • @PeterJ I think on this Kant's view is compatible with yours. He does not postulate one thing-in-itself, or many, he makes it clear that applying any categories of experience to it/them, including oneness, objecthood, temporality, etc., is an abuse of those categories. Thing-in-itself is just unknown X of which nothing can be legitimately predicated, like God of negative theology. Of course, he also denied us intellectus archetypus, which is supposed to give insight into thing-in-itself that goes beyond pure reason and its categories, that's where you two might disagree. – Conifold Sep 10 '17 at 18:14
  • @Confold -Interesting. I hope you're right. I thought Kant got it wrong but perhaps I've misread him. I would agree with him that the intellect cannot know his 'thing-in-itself' but would say that it is knowable all the same. It is the possibility of direct knowledge hat Kant seems to deny, But according the his theory the 'thing-in-itself' is what we are, so by following the Oracle's advice to 'know thyself' we can know the thing-in-itself. I think he calls this 'non-intuitive immediate knowledge', but I'm not well acquainted with his epistemology. – PeterJ Sep 11 '17 at 13:12
  • @PeterJ Thing-in-itself is what reason posits as behind the appearances, and its impenetrability applies even to us. Kant distinguishes empirical and noumenal self, and the latter is beyond knowing ("non-intuitive immediate knowledge" is Fries's idea rather than Kant's, I think). To the extent that we "make contact" with the noumenal self at all it is through practical reason and action, but that to him is not knowledge (we might call it knowledge-how today). He calls the moral law "sole fact of reason" and implies that noumenal self somehow exercises its freedom through maxims of behavior. – Conifold Sep 11 '17 at 19:28
  • I see this, and it is where I'm sure that Kant went wrong. It seems relevant that the sages who actually go in search of what lies behind the phenomenal world say that the Ultimate is a phenomenon that has no noumenon. Thus we do not have to know the noumenon, there would not be one. We would have to become the 'thing-in-itself' to know it at all. Thus the Biblical 'I Am'. I like Kant but I wish he'd been able to read Nagarjuna, whose view is more sophisticated and systematic. Interesting chat! – PeterJ Sep 12 '17 at 9:30
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“.

  • You say "Kant’s antinomies refer to the boundaries of our rational argumentation", exactly. My point is, isn't "rational argumentation" just an informal way of saying logic? What is the difference between a limit on rational argumentation and a limit on logic? – Alexander S King Jan 20 '16 at 18:57
  • @Alexander S King The content of argumentation can be quite arbitray: E.g., I can use arguments referring to natural science. On the opposite, logic has no "material" content, it is a purely formal science. – Jo Wehler Jan 20 '16 at 19:00
  • Kant antinomies do not refer beyond the boundaries of rational argument. They show that we are wrong to reify the distinction on which they depend, and thus show us how to rationally proceed to a fundamental theory. It is just that Western thinkers do not want to go there so they treat the antinomies as if they are a problem and a road-block. They're not a problem if we treat them as telling us something about the world, but many philosophers do not want to hear it. . – PeterJ Sep 19 '17 at 13:03
1

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.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.