Wednesday, May 11, 2005

A Common Bias of Uniformity

Paul Benacerraf in Mathematical Truth suggests that a semantic theory should be uniform for both mathematical discourse and everyday discourse. His idea, as I remember it, is that if we think of reference and satisfaction applying to talk of ordinary objects then we should think of them as applying to mathematical objects also. So that we should think of number terms as referential terms (at least if we think of ordinary terms as referential). I don't know much about the philosophy of math---take a look at Antimeta for some cutting edge stuff though--but I've often wondered how general such a claim should be--i.e. how much we should read ontology off surface grammar. Now I know, this isn't exactly the most original thought in the world--Quine, Carnap and all that. But my specific question is this: why should we expect a semantic theory to be uniform across different chunks of discourse. I mean I've always thought that there's one theory of truth for talk about chairs and tables and another one for talk about, say literature. For instance, when I talk about wine with some of my buddy's I tend to think that what determines the truth of what we say is very different--if equally objective--than what does when we talk about say where tables and chairs are (something we rarely do). Now I understand various ways one might get at a general worry about ontology and all that, but it doesn't seem like the idea that there has to be a uniform theory of meaning (or reference, or truth) is a very good one. In fact, I think every time one broaches a new subject the best thing to say about what the theory of truth is changes. Philosophical analysis is supposed to be the attempt at making the best sense of how we think about things--why the bias towards uniform accounts. And don't give me that science is uniform stuff. Science seems to me about as uniform as the contents of a typical issue of the Philosophical Review, think of the worlds apart between comp sci, quantum mechanics, evolutionary psychology et cetera. Well, I better let this go...

98 Comments:

Anonymous Alex said...

Hey Nat,

Just a thought: maybe Benacerraf could concede your point about non-uniformity among distinct domains. Perhaps his point is that our talk about numbers in "ordinary" discourse is really the same domain as our talk of numbers in more mathematical settings. When I say that 2+2=4 (I say this all the time), and then I say something like "I would like 2 muffins", I'm using the expression "2" in exactly the same way.

2:29 PM  
Blogger Naxos "Nat" Simeon said...

Maybe your ontology of domains of discourse is simply too conventional, too square. Maybe there's more than just mathematic domains and ordinary conversation domains, but domains subdivide and coexist. It's like the statue and the clay.

5:03 PM  
Anonymous Anonymous said...

But surely, since Hilbert's axiomatic geometry (published in 1899), mathematics itself defines what it is "about". In other words, an axiom system defines a collection of objects being precisely those objects which satisfy the axioms. A particular geometry is *about* the objects defined by its axioms, and these objects therefore constitute the semantics of that geometry.

Of course, Frege famously disagreed with Hilbert on this issue, but I think Frege misunderstood it, and was mistaken.



-- Peter

7:05 PM  
Anonymous Alex said...

Peter,

It's true that within a given axiomatic system, the axioms implicitly define what the system is about. But, in the case of mathematics, the only reason that we take a given set of axioms as constituting a system of mathematics is that we have a prior notion of what mathematics is about which we want the system to characterize. In other words, intuitively we couldn't just add whatever axioms we want and still claim that we've got an axiomatic system of mathematics. That's precisely why there's a big controversy in set theory over the Axiom of Choice. The axiom has some counter-intuitive consequences which make some think that we shouldn't include it as an axiom.

9:05 PM  
Anonymous Anonymous said...

Alex --

I responded to your post, but my response seems to have disappeared. Herewith another attempt.

Many mathematicians, much of the time, have prior notions about a branch of math which they try to formalize with an axiom system. (It is important to note in passing that lots of mathematics is not done in this manner, by the way.)

But that is a separate issue from the space of objects defined by an axiom system, which may or may not correspond to particular mathematicians' intuitions. Indeed, non-Euclidean geometry arose not because Riemann, Bolyai and Lobachevsky had prior intuitions about hyperbolic or elliptic spaces, but rather because they played with -- in a formal sense -- the axioms of Euclidean geometry. I do not believe any of these gentlemen had prior intuitions which were non-Euclidean, and all found it hard to accept the math they were developing.

This example illustrates my point: the subject matter of mathematics (ie, its meaning) comprise those spaces of objects defined by its syntax. What happens to be in mathematicians' heads is of no consequence when discussing the semantics of a formal system.

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Blogger Anthony D'Amato said...

"A particular geometry is *about* the objects defined by its axioms, and these objects therefore constitute the semantics of that geometry."

But how can we as outsiders know what the axiom system is talking about? Suppose it uses symbols that are unlike any symbols we've ever seen. Don't we need a translation manual? And who will write it?

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