Monday, June 28, 2010

FOL as the language for science

Maybe I'm missing something here...

Quine suggests that we adopt first order logic as the language for science. But, first order logic can't capture the notion of 'finitely many Fs'. It can only express the claim that there are n Fs for some particular n. Yet, we do understand the notion of finite, and use it in reasoning (e.g. if there are finitely many people at Alice's party, there is one person such that no one is taller than him) and potentially in science. Hence, we should not adopt first order logic as the language for science.

[The standard way to try to get around this, is by talking about relations to abstract objects like the numbers (There are finitely many Fs if there's a 1-1 map from the set of things that are F to the some set theoretic surrogate for the numbers). This would give you the right extension, if your scientific hypothesis could say that something had the structure of the numbers. But first order logic can only state axioms, like PA which don't completely pin down the structure of the numbers. Any first order axioms which you use to characterize the numbers will have non-standard models. This is Putnam's point in his celebrated model theoretic argument against realism. So, if you take this strategy, rather than saying that there are finitely many people at Alice's party, you can only say that the number of people is equinumerous items that satisfy a certain collection of first order axioms. And this does not rule out non-standard models.]

5 comments:

  1. That's crazy. The very question itself is confused.

    I mean what does it even mean to be the 'language for science'? Seems to me the only useful way to cache this out is that FOL captures the structure of the sort of reasoning we do in science in a robust way that makes it good for meta-level discussions about our scientific endeavors.

    Obviously any kind of logic that captured the notion of finite would not be a good language for science in this sense as it would necessarily have a non-computable system of reasoning (whether you've given a valid argument would be undecidable). Moreover in actual fact we clearly don't observe "that's something that needs to be described as actually finite in our theory not merely apparently finite" since `being able to make such observations would let us computably characterize the non-computable logic discussed above.

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    I mean in actual fact we understand, just like PA, ZFC etc.., some limited range of conclusions that we can infer from finitetude just like FOL lets us do.

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  3. To Truepath:

    Yes, certainly we will only actually *draw* certain consequences (and the ones we will draw can be finitely axiomatized in FOL), but I don't see how it follows that we can't mean that something is finite/is really an omega sequences.

    Maybe I'm misunderstanding you, but this seems analogous to saying that sense-data should be the language of science, because our observations can never include something that "needs to be described as" there actually being a cat/cannonball etc. as opposed to there just being some physical state of the world that gives rise to such and such sense data.

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  4. "What does it even mean to be the 'language for science'?"

    I readily grant that the "language for science" is as infinitely ambiguous as any other stretch of letters on a screen. But I take your asking the question to imply that this stretch is especially infinitely unclear. And there, I simply cannot agree. Wouldn't you think Quine is using it approximately like, "the language used in scientific papers"? You know, not the sort of language used by people selecting cheeses for a biology banquet, but the sort used in justificatory of argumentative scientific discourse?

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