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Thesis The indispensability argument is the best case for numbers, and it still does not work

Emeka Anyaoha Fellow
· 5 replies
Philosophy of mathematicsAnalyticHartry FieldHilary PutnamW. V. O. Quine#fictionalism#indispensability#nominalism#philosophy-of-physics
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Where it stands

The argument usually called Quine–Putnam goes like this:

  1. We should believe in the entities our best scientific theories quantify over.
  2. Our best scientific theories quantify over mathematical objects, ineliminably.
  3. So we should believe mathematical objects exist.

It is the best argument for platonism because it doesn't rely on intuition. It turns a metaphysical question into a question about scientific practice. (Putnam, I'm told, later said he never meant it as an argument for platonism specifically, only for the objectivity of mathematics.)

I teach physics in Lagos, and I think premise 1 fails, for a reason I learned from my own subject. Physics is full of things we quantify over without believing in: frictionless planes, point masses, infinitely deep potential wells, perfectly rigid bodies, continuous fluids that are really made of molecules. Working physicists treat these as idealisations. They are indispensable to the theory as written, and nobody believes in them. Penelope Maddy made a version of this point: scientific practice doesn't treat every part of a successful theory as confirmed.

If that's right, "it appears ineliminably in our best theory" doesn't amount to "we should believe it exists". Numbers might be the deepest idealisation of all.

Hartry Field tried the other route, rewriting Newtonian gravity without numbers, and as I understand it his reconstruction works for that case but extending it to quantum mechanics is disputed. I think my route is cheaper. Tell me why I'm wrong.

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5 replies

For the thesis 2

  1. Ana Clara Ferreira

    For Fellow

    I'm with you on the conclusion but from a different direction. As an intuitionist I think mathematics is about constructions, which is a kind of activity, and I don't need numbers as objects floating anywhere for physics to use the constructions. The indispensability argument assumes that the only way for mathematics to be indispensable is to be about existing things. I don't think that assumption survives a close look at how mathematicians actually work.

    Helpful · 2
  2. Lars Eriksen

    For Fellow

    From the physics side, partly for. We regularly have mathematically different formulations of the same theory: Newtonian versus Lagrangian mechanics, or different ways of writing quantum mechanics. They quantify over quite different mathematical objects and make the same predictions. If premise 1 were right, we'd have to believe in all of them, or pick one arbitrarily. That suggests the mathematics is the language, not the subject matter.

Against the thesis 2

  1. Tomasz Szewczak

    Against Fellow

    The problem with your route is that idealisations are flagged, and mathematics isn't. A physicist knows the plane is not frictionless and can say how the real result differs. There is no corresponding "real result" that differs from the mathematical one. You cannot de-idealise away from the real numbers to something more accurate. So the analogy shows we can quantify over things we don't believe in, but not that mathematics is one of them.

    Helpful · 1
    1. Emeka Anyaoha

      Fellow

      Tomasz's reply is the one I can't shake. Idealisations come with a known way of being wrong; mathematics doesn't. I'll concede that my analogy only shows the gap between "used" and "believed in", not which side mathematics falls on. Lars's point about equivalent formulations is a better weapon than mine, I think. I'm downgrading from "it does not work" to "premise 1 is much weaker than it looks".

      Helpful · 1
  2. Derek Hollis

    Against Fellow

    Against, on practical grounds. When I verify a chip design, the arithmetic isn't idealised; it's exact, and the chip either does the arithmetic or doesn't. If you believe the physics that tells you how the transistors behave, I don't see how you avoid believing the mathematics that the physics is literally written in. "Deepest idealisation" sounds clever but I can't make it do any work.

    Helpful · 3