Tuesday, January 26, 2010

Ontology of numbers and physical theory...

In a recent NDPR review (2010.01.17) of a collection of essays by Burgess, Thomas Hofweber discusses debates over nominalism in mathematics. (In particular the status of indispensability arguments.) At one point in the dialectic he writes:
"We should believe in our best [scientific] theories as they are, even if we realize that we could have done things completely differently. Approaches that show that we could have done science or mathematics differently shouldn't undermine our belief in science and mathematics as it is, or at least they shouldn't undermine it by much."
Let's leave aside the wrinkle in the last clause. I want to focus on how the authority of physical theory is being used to make claims about the ontological status of numbers (a familiar strategy following Quine-Putnam). I think there is a slide in the argument where Thomas conflates claims about what the theory is about and the way the theory is formulated.

To be clear: for the sake of argument I grant that notational variants, even excessive ones, do not undermine our belief in the truth of a scientific theory. Even conflicting theories [a la Duhem-Quine] that can account for the same current and future data need not undermine our belief in our current ruling theory. (I am very fond of Newton's fourth rule of reasoning.) [I also grant, for the sake of argument, that there is such a thing a standard ruling theory.]

But...scientific-practitioners do not take physical theories as making claims about the nature of number nor is there a reasonable way to interpret most of these theories as such. [I am open to the idea that string theory and parts of quantum mechanics may be interpreted at odds with this claim!]. As Jody Azzouni has argued, scientists don't create measuring tools to interact with numbers. And the particular math one uses is often introduced pragmatic purposes. Not to mention that the standard of "proof" in science is often very different from the standard of proof in mathematics--so here's a case where science's authority is very much questioned).

So, one cannot appeal to the *general* authority of physics to settle the question about the nature of number or mathematical objects. Rather than accepting a theory without further analysis (as a whole package), I believe one needs to do piece-meal investigation of what mathematical objects are required for (indispensable to) the empirical content of particular physical theories. (After all, the physical theory comes as a package for physics purposes, but it wasn't designed to meet all philosophical or mathematical purposes at once--note, for example, that standards of proof in physics and math can differ.)

There is more to be said about this (and I haven't reported Thomas' responses), but this has grown long enough.


  1. I gather that Azzouni's line of thinking tends toward a Wittgenstein/Carnap dismissal of platonism and nominalism as answers pseudoquestions. But it seems to me that a major role of indispensability arguments can be undermining talk of "linguistic frames" to justify making a non-arbitrary internal/external question distinction.

    In scientific explanation, numbers and physical objects mingle (so to speak), and hence they can't be so easily quarantined into different "languages". Perhaps physical theories don't straightforwardly make claims about the nature of number, but scientific explanations of physical processes do seem in some instances to appeal to mathematical facts. The Quinean point, and I think it's a good one, exposes the W/C dismissal as theft. Field's program toils to show that abstracta aren't as indispensable as they seem: nominalism naturalized. Alternatively, Burgess, following Quine, toils on with the program of situating set theory as a foundation of mathematics and semantics, responding to the appeal of reduction and systematicity: platonism naturalized.

    I think you're right to point out that standards of proof for existence claims can be very different in mathematics vs physics, but the indispensability argument seems to me to undermine the verificationist line from different standards to different senses of "exists" (relativized to linguistic frameworks).

    Still, the arguments leave open, as potentially important, questions about what "we could have done differently"; it's "indispensability" not "indispensed". Indeed, Quine's own way of doing things does things differently to a certain extent. He's a relativist structuralist with set theory as a background ontology (meaning there's lots of simple sequences in the sets any of which will do fine as an instance of the structure of N). This is by no means a consensus view among mathematicians and scientists (many are indifferent to reductionism about mathematical ontology, some are old-school platonists, some express vaguely formalist views, etc.). Further, Quine stakes out positions on questions only philosophers care about. His view, however, lets him have one domain of abstracta to handle both the role of mathematics in science and his desire to banish "creatures of darkness" for extensionalist semantics. In this regard I think Quine exhibits a desire for systematicity; the satisfaction of such cognitive aims may take the naturalist-minded philosopher beyond her scientific starting place.

    This seems lacking in Burgess's anti-nominalist writing when it is most absolute in it's insistence on philosophical deference to science... Ok, now I'm going to go home and crack open "The Oxford Handbook of Philosophy of Mathematics and Logic" to reread Maddy's and Burgess/Rosen's articles on naturalism and nominalism.

    Gone on too long already, but one more thing. Quine on going piecemeal:

    "As for the inapplicable parts of mathematics, say higher set theory, I sympathize with the empiricist in questioning their meaningfulness. We do keep their sentences as meaningful, but only because they are built of the same lexicon and grammatical constructions that are needed in application. It would be an intolerably pedantic tour de force to gerrymander our grammar in such a way as to account the inapplicable flights ungrammatical while preserving the applicable part."

  2. Dear Jeremy,
    In this context I did not (mean to) advocate talk of frameworks/frames or different languages. (You are confusing me with Azzouni or Carnap! Thanks, by the way.) I meant, rather, to suggest that physical theory often do not (tacitly) say 'all mathematical object exist'. I am objecting to a misplaced deference to science.
    In response to your concern about cutting up scientific theories--scientists do this all the time in their research and journal articles. So, why can't philosophers of mathematics learn to take close looks at how mathematical objects are used in different kinds of mathematical theory before they make general claims about mathematical existence? I am no verificationist; rather I am a naturalist reforming naturalism from within against short-cuts that reify this thing "science."

  3. Sorry about that. I sort of felt like I was spewing off a bunch of stuff just because I've been thinking a lot about the relationship between indispensibility and Quine's overall dialogue with Carnap's ideas rather than responding directly to what you were saying. Maybe the stuff towards the end of my too-long comment about systematicity is more relevant though. What do you think of the idea that in reforming naturalism from within Quine placed a value on systematicity, which lead him to find set theory appealing as a background ontology accomodating both mathematical structuralism and semantic extensionalism, the "grammar" of which ought not be "gerrymandered"? This does seem in tension with the piecemeal approach you suggest and raises a wider question about the status of cognitive aims like systematicity within naturalist philosophy.(Probably not a matter to be settled in a comment thread, I guess).

  4. Jeremy, I am no stranger to saying things in a blog that I regret later! So, no worries, I suppose.
    Yes, Quine is hostile to the piecemeal approach. In my private exchange with Hofweber I worried that his objections relied on Quine-an holism. Hofweber denied that. Instead he is worried about moves that split "truth" in a metaphysical and more deflationary parts (I suppose there are ways to read Azzouni like that). Again, that's not the intuition I wish to advocate here.
    Now it's true as you point out that against Carnap Quine has further moves to wish to challenge piece-meal approaches. As you rightly point out some of these have to do with a) systematicity (and the virtues of b) simplicity). I also suspect they have to do with c) taste (how else to read "pedantic tour de force). Finally, there seems to be a tacit argument that d) however we analyze the role of mathematical objects in science, we need to introduce a new language to do the accounting; and this allows Quine to raise the spectre of gerrymandered grammars. But as I hinted in my original post, there is a kind-of-arm-chair quality to this style of naturalism--it's far removed from the potentially messy practice of science. Scientists manage to cut upp scientific theories (in which numbers and other properties are often, as you say, intertwined) in all kinds of ways without them raising the fears that Quine alludes to. I hope this helps...

  5. All the comments were long and long. Now I just comment at your bout your posting, maybe it wouldbe nice if make your paragaraph justified...
    Thanks :-)