电影“幽灵场”不能继续游荡

Question

I once watched a lecture on YouTube about Wittgenstein’s philosophy of mathematics.

The speaker said something very interesting: Wittgenstein did not simply dismiss G?del’s incompleteness theorems as nonsense (though he famously said that the liar’s paradox does not interest him). Rather, he thought that G?del himself did not fully understand the meaning of his own discovery.

The issue, according to this interpretation, is not that a formal system is incomplete per se. Instead, the point is that there are things within a system of integers that are not, strictly speaking, integers. For example, the famous self-referential G?del sentence (“I cannot be proven”) can be encoded numerically using G?del numbering (e.g., via primes raised to powers), yet what it expresses is not purely numerical. This suggests that something non-numerical is embedded within the formal system of arithmetic.

If this interpretation is correct, what could that mysterious non-numerical thing be?

Answer 1

The relevant primary passage is this:

I imagine someone asking my advice; he says: "I have constructed a proposition (I will use 'P' to designate it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: 'P is not provable in Russell's system'. Must I not say that this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable."

Just as we can ask, " 'Provable' in what system?", so we must also ask, "'True' in what system?" "True in Russell's system" means, as was said, proved in Russell's system, and "false" in Russell's system means the opposite has been proved in Russell's system. —Now, what does your "suppose it is false" mean? In the Russell sense it means, "suppose the opposite is proved in Russell's system"; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by "this interpretation" I understand the translation into this English sentence. —If you assume that the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is not provable" again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called "losing" in chess may constitute winning in another game.)

-- Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, Blackwell, 1956. Part I, Appendix III, #8. Translated from the original German by G.E.M. Anscombe.

In a letter to Karl Menger, G?del comments:

As far as my theorem about undecidable propositions is concerned it is indeed clear from the passage that you cite that Wittgenstein did not understand my Theorem (or that he pretended not to understand it). He interprets it as a kind of logical paradox, while in fact it is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics).

-- Kurt G?del, Letter to Karl Menger, 1972. Collected Works. Volume V. Correspondence H–Z. S. Feferman, J. W. Dawson, W. Goldfarb, C. Parsons and W. Sieg (eds). Oxford University Press, 2003.

Many early commentators on the passage agreed with G?del that Wittgenstein had simply misunderstood the first incompleteness theorem. However, in subsequent years others have come to regard Wittgenstein as saying something more subtle. The fact that Wittgenstein summarises the theorem succintly suggests that he is not challenging its correctness. Some commentators have interpreted him as disagreeing with G?del’s platonistic interpretation of the philosophical significance of the theorem. Others have claimed that Wittgenstein is adverting to the possibility of non-standard models of arithmetic. Others that Wittgenstein is questioning the traditional notion of what constitutes mathematical truth. The speaker you quote is perhaps proposing the non-standard model option.

It is true to say that one can and should draw a distinction between the technical result that is G?del’s first incompleteness theorem and what we might call the philosophical significance of it. Many popular expositions of the theorem jump to the conclusion, "There are mathematical propositions that are true but unprovable". Statements like that, without qualification, are potentially misleading. We should rather speak of what is provable in specific recursively axiomatisable formal systems. And we should clarify what we mean by mathematical truth. There is room for philosophical disagreement. For example, intuitionists do not accept that it is intelligible to speak of mathematical propositions that are true but unprovable. Consequently, they have a distinctly different way of understanding the significance of the incompleteness theorems. I said something about this in my answer to this question.

Here are some references discussing the interpretation of the passage from Wittgenstein’s Remarks. There is also some useful material in the SEP article on Wittgenstein’s Philosophy of Mathematics, section 3.6.

Juliet Floyd, "Prose versus Proof: Wittgenstein on G?del, Tarski and Truth", Philosophia Mathematica, 9 (2001), pp 280-307.

Juliet Floyd and Hilary Putnam, "A note on Wittgeastein's notorious paragraph about the G?del theorem", Journal of Philosophy 97, (2000), 624-632.

Mark Steiner, "Wittgenstein as his Own Worst Enemy: The Case of G?del's Theorem", Philosophia Mathematica Vol. 9 (2001), pp. 257-279.

Timothy Bays, "On Floyd and Putnam on Wittgenstein on G?del", Journal of Philosophy, 101 (2004), pp. 197-210.

Charles Sayward, "Steiner versus Wittgenstein: Remarks on Differing Views of Mathematical Truth", Theoria 20 (3):347-352 (2005).

Answer 2

As I see it, Wittgenstein is pointing out that G?del's proof is (questionably) mapping semantics onto syntactics, something that would certainly give Wittgenstein pause. I mean, consider the following mathematical statements:

• 0 = 0
• 0 + 0 = 0
• 0 + 0 = 0 - 0
• 1 - 1 = 0 + 0
• 1 - 1 = 2 + 1 - 3

Each of these statements clearly has its own G?del number, but each of these statements — and an infinite number of others — equally clearly?evaluate to the same end-product. They all mean the same thing, despite having different structures. We can easily extend this observations to proofs. For instance, we can trivially create an infinite number of proofs that prove the same thing merely by adding unnecessary or circular steps, and it's entirely possible that there are an infinite number of unrelated and independent proofs that also prove the same thing. So we have an infinite amount of proofs with unique G?del numbers that evaluate to the same end-product.

Now, these observations don't undercut G?del's theorem by any means, but they do create some concerns. The heart of the incompleteness theory is a statement S of the form “the formula with G?del number ?? cannot be proved” constructed in such a way that ?? is the G?del number of S itself. But remember, there are ostensibly infinite statements, each with a different G?del number ??ⁿ, that are semantically equivalent — that evaluate to —?statement S. If we substitute ??ⁿ instead of ?? into S, we've changed the syntactics in such a way that the statement is no longer directly self-referential in terms of its G?del number. But we've seemingly made the same semantic statement: made a statement with the same meaning. That is something G?del did not consider or account for.