Wednesday, February 16, 2011

Reichenbach on the philosophical insignificance of G's second incompleteness theorem

Nowadays, one standard answer to various exaggerated claims about Gödel's second incompleteness theorem (the unprovability of consistency) is that even if an interesting mathematical theory could prove its own consistency, this wouldn't help us much because inconsistent theories also prove their own consistency, (so we would still have no idea whether the theory is consistent). (For instance, I think this sort of remarks, without further references, can be found in Franzen's and Smith's books, but I don't have them handy now).
In a somewhat uninspired moment of mine, I wondered why this rather straightforward observation didn't get through to the wider philosophical audience earlier and when it was formulated. The earliest mention I run into so far (although, it's not like I spent days browsing stuff systematically) is in Reichenbach's 1948 unpublished lecture notes (which, by now, have been published in 1978). The fragment comes from the first volume:  
This would mean that the proof of consistency of the language L could be given within L. A simple analysis shows that this would not improve the situation, since in this case our proof of consistency of L would be of value only if we were sure that L is consistent. In case L were not consistent, we could also deduce the statement of the consistency of L, with the qualification that then the negation of the statement were deducible too. Thus if the consistency of L were deduced within L, this fact would not prove the consistency of L.
So, one reason might be that the remark occurred in lecture notes which were unpublished for a while (by the way, if you know of Reichenbach's point being published earlier, I'd appreciate a reference), and even when they were published, it was in a two-volume set of Reichenbach's papers, which very few non-specialist would buy (or grab from a library shelf, for that matter).
But I don't think this is the whole story. The point is simple enough and certainly came to minds of many bright people who gave the issue proper consideration. The reason why this is often omitted by (hack?) philosophers is not that they are too dumb to get this point (well, at least some of them aren't). Rather it is that wishful thinking sometimes prevents people from considering objections to their view. The perspective of reaching strong philosophical conclusions by relying on some fancy mathematical apparatus, which (they think) would lend splendor to the philosophical claims themselves, is quite tempting.
Another reason is that Reichenbach's point applies to cases where one wants to claim that mathematics is somehow epistemologically flawed and that there is no mathematical certainty because mathematics cannot prove its own consistency. (... and if real mathematical knowledge is unattainable, then no knowledge is, science is useless, blah blah, blah...). It doesn't help much when one wants to suggest that since we, humans, can establish consistency claims, and mathematical "devices" cannot, our minds are not computing machines. In this argument, it seems that no deep epistemological considerations of the former sort are needed.
Incidentally, it seems that despite the fact that I've seen some philosophers using both strategies in one passage (or breath) both stances are incompatible. Either you say that mathematics is unreliable and then you cannot rely on it in an argument against computational theories of mind, or you rely on mathematics, give an argument in philosophy of mind, but cannot simultaneously deny the reliability of mathematics. (By the way, (1) even if you do the latter, you admit that mathematics can be done reliably by humans and the fact that a mathematical theory cannot prove its consistency has no bearing on whether humans can use it sensibly, (2) the sketched argument against computational theories of mind also doesn't work, but that's a whole different story). 


Nate said...

I've heard that point made in several places, including in my undergraduate course in Symbolic Logic in a Philosophy Department.

This isn't the end of the story, though. As I understand it, Hilbert was searching for a proof in arithmetic of the consistency of certain "higher" mathematical theories, so that he could justify his use of higher mathematics on the basis of arithmetic. So, one consequence of the Second Incompleteness Theorem is that you can't even prove the consistency of arithmetic within arithmetic (assuming your formal theory of arithmetic is consistent), much less the consistency of any more powerful theory. That's prima facie pretty devastating to Hilbert's Program, and an interesting Philosophical conclusion in its own right.

Rafal Urbaniak said...

Hi! Sure, the point is made even in fairly basic courses these days. And you're right, the theorem does have a philosophically important bearing on Hilbert's program (I just didn't talk about Hilbert's program). But saying that Hilbert was in trouble is one thing, and saying that mathematics is suspicious because of the theorem is another. The latter is definitely stronger, overestimates the importance of a theory proving its own consistency, and misses Reichenbach's point. In comparison, recall that for Hilbert what was devastating was not exactly that arithmetic (or set theory) cannot prove its own consistency, but rather that its consistency cannot be proven in any weaker theory.

Nate said...

Right, so perhaps the worry isn't (or, at least, shouldn't be) that ZFC (for example) being unable to prove its own consistency makes ZFC somehow suspicious. For one, ZFC is only unable to prove its own consistency if it is in fact consistent.

So, instead the worry is (or should be) simply that we're not going to be able to justify our use of more powerful mathematics on the basis of weaker mathematics. So, if you're already suspicious that ZFC might be inconsistent, I'm not going to be able to talk you out of it with a consistency proof from a weaker theory you accept (such as PA), since PA can't even prove its own consistency.

Rafal Urbaniak said...