Thursday, November 10, 2011

Ray Jackendoff and Mathematical Semantics

I went to the discussion with Jackendoff for the philosophy department today and asked him about the semantics of mathematical statements. He said that semantic mentalism does indeed commit us to a rejection of mathematical realism; "2+2=4" has no mind-independent semantic content whatsoever. He also said that this fact makes him very uncomfortable, but he thinks that just means we need to look into it further, not that we need to reject semantic mentalism. I agree with him on that. But shortly afterward I realized another implication that would probably make him even more uneasy.

Semantic mentalism doesn't only preclude mathematical realism. It precludes classical mathematics entirely.

If mathematical statements have no mind-independent semantic content whatsoever, there cannot exist in that content elements relying on mind-independent existence. Total continuous functions over the real numbers rely on lawless infinite decimal expansions (irrational numbers with no rules governing the calculation of each subsequent digit). If we are to have functions over the real numbers at all and we're not going to assume that such expansions exist mind-independently, we're forced to accept the continuity principle: for every function whose source is the rational numbers, there exists a natural number m such that for any two real numbers x and y, if x and y are identical up to the m'th digit then if (x,n) is an element of the function then (y,n) is an element of the function. Without either omniscient minds or mind-independent lawless free-choice sequences, there can be no complete total functions over the real numbers. And without those we have to get rid of lots of classical math.

Obviously intuitionism doesn't preclude realism (as our resident intuitionist realist is so fond of repeating), but phenomenology is my favorite motivation for intuitionism so far. (Just don't let McCarty catch wind of that.)

So now the question becomes: is Jackendoff committed to a conceptual semantics founded on mentalism strongly enough not only to reject mathematical realism but to adopt intuitionism or some weaker version of constructivism? Does it make less sense to him to say that sentences refer to the mind-independent world than to reject the law of the excluded middle? Or double negation elimination? Or the equivilance of a negative general and a particular negative? Or Church's thesis?

He talks like the whole point of all of this is to have a semantics that makes good intuitive sense and is in line with how people actually think, but it's immediately apparent to anyone who's tested those forbidding waters that intuititionist math is deeply unintuitive. If you're not used to thinking about this stuff (or even if you are, really), trying to get your brain around something like, "it's false that 'a thing is either true or it isn't'" is incredibly difficult. It's about as cozy in human cognition as the most mind-shattering koan. This is not good news, I think, for conceptual semantics.