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.

Monday, November 7, 2011

Squib on adverbial “at all”

As a general rule, adverbial “at all” can be used only when all of the following hold:
  • The sentence is about degree, location, number, or type.
  • The sentence is negative.
  • If the sentence is about location, number, or type, its logical form is particularly quantified.  Quantification is irrelevant to sentences about degree.

For instance...

I don’t like mushrooms at all. [aboutness: degree, charge: negative]
There are no dogs in heaven at all. [aboutness: location, charge: negative, quantification: particular]
There are no coins in my purse at all. [aboutness: number, charge: negative, quantification: particular]
*There are coins in my purse at all. [aboutness: number, charge: positive, quantification: particular]
*There isn’t every coin in my purse at all.  [aboutness: number, charge: negative, quantification: general]

There is one exception to this: when uttered in a counterfactual context, “any x at all” appends well formed formulae to yield new wff’s whenever x acts as a noun or noun phrase.  The gramaticallity of this construction depends only  on the counterfactual status and not on positivity, quantity, or aboutness.  

You might have been any animal at all.
Unicorns could live anywhere at all. (where “where” acts as a noun)
There couldn’t be any man who breathes air in space at all.

The quantity of “unicorns could live anywhere at all” is actually generalization of a particular.  In fact, it would seem that the quantity of all counterfactual sentences with adverbial “at all” is nested.  At all is a sticky problem for semanticians, because in order to understand what’s really going on we’re going to have to work out words like “could”.

What exactly is the problem with “could”?  The problem is the truth value of sentences containing it.  “Unicorns could live anywhere at all,” seems to be asserting the truth of the intended proposition just like any other statement, but if the characteristic function for the truth value of counterfactuals in the actual world exists, it is inaccessible.  It’s not that we don’t know the graph of the relevant function because cataloging all the things of such and such a class along with their truth conditions is impractical or would take an infinite amount of time--it’s that such a process can’t even begin for the class of non-actualized possibilities.  Therefore, semanticians trying to hang out in some kind of formalistic limbo will have to take a metaphysical stance on the nature of semantic practice to work out the meaning of counterfactuals and by extension the meaning of “at all”.

I see a few options for rendering “at all” meaningful.
  • Platonism (aka “the lazy way out”): the sentence has a truth value of true or false and the characteristic function exists mind-independently.  “Unicorns could live anywhere at all” means that the proposition to which it refers exists in Plato’s heaven.  It doesn’t matter that we can’t know the function’s graph; we can employ the function intentionally anyway.
  • Modal realism: the sentence has a truth value of true or false and the characteristic funciton exists mind-independently.  “Unicorns could live anywhere at all” means that for every place in the actual world there exists a possible world containing a counterpart of that place and at least one unicorn lives in it.  As above, we can use the function without being able to know its graph.
  • Conceptual semantics: much like modal realism, but fully mind-dependent.  “Unicorns could live anywhere at all” means that the pluriverse existing representationally in the mind of the speaker is such that the modal realistic interpretation is represented as true (and no relationship to a mind-independent pluriverse is relevant).
  • Intuitionism: the truth value needn’t be either “true” or “false”, in which case we get to keep sober realism and counterfactuals are unproblematic.  Obviously, this would mean a complete overhaul of semantics to account for a non-finite t domain.  I say we go for it.  Any takers?