No Gain?

5 10 2005

I’ve been reading Mike Resnik’s book Mathematics as a Science of Patterns and have found a lot of stuff I like in it. He makes the point that if we use the indispensability argument to show that mathematical entities exist, then they shouldn’t be that different from the entities postulated by theoretical physics. I didn’t know much about the particular examples he gives from physics, but I think I would take the argument a little bit in the other direction – if mathematical entities really are indispensable to physical theory, then we might as well take them to be concrete physical objects that just happen to lack a lot of the causal and spatiotemporal properties that other physical objects have.

In addition, the indispensability argument meets the epistemological challenge, because we get epistemic access to the objects by confirming the whole theory to which they are indispensable. I’m not sure if Hartry Field discusses this point much in Science without Numbers, but Burgess and Rosen (in A Subject with no Object) seem to be puzzled by the fact that Field both gives an epistemological challenge and argues for the dispensability of mathematics. They think either one alone should be enough, if successful, and therefore the fact that he has to give both calls each into question. But if the point of the epistemological challenge is merely to show that there is no direct epistemic access to these objects, then we need to establish dispensability to show that the indispensability argument doesn’t give us indirect epistemic access. So both arguments are needed.

However, on page 109, Resnik criticizes Field, saying “whether space-time points are mathematical or physical, abstract or concrete, there will be no real gain in using them to dispense with (other) mathematical objects unless they are more epistemically accessible than the objects they replace.” This sort of objection is related to the ones that say that space-time points really are mathematical objects, and therefore Field hasn’t succeeded in nominalizing anything.

However, I think all of this misses an important point. Although Field says he’s a nominalist, it seems to me that a more important point is that he’s trying to give an internal explanation of everything in physics rather than an external one. (Resnik compares this to the contrast between synthetic geometry (where we only talk about points and lines and such) and analytic geometry (where we refer to real-number coordinates as well) and thus talks about “synthesizing” physics rather than “nominalizing” it.) Whether space-time points are concrete, physical objects or abstract, mathematical ones, and whether we have good epistemic access to them or not, they are somehow much more intrinsic to the physical system than real numbers seem to be. Real numbers are applied in measuring distances, calculating probabilities, stating temperatures, and many other things. These seem to be many different areas of the natural world, and using real numbers to explain all of them seems to involve some sort of “spooky action at a distance” as I discussed several months ago. Field’s reconstruction of Newtonian mechanics is certainly an advance on this front, whether or not it has any metaphysical, epistemological, or nominalistic gains. Thus, Resnik is wrong when he says there is no real gain in using space-time points instead of real numbers.

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6 responses

6 10 2005
Greg Frost-Arnold

Hi Kenny–

Your basic idea (viz., spacetime points “are somehow much more intrinsic to the physical system than real numbers seem to be”) strikes me as intuitively right. I am wondering about one thing though: when physicists and philosophers of physics talk about spacetime these days, they identify a spacetime with an ordered pair , where M is a manifold and g_ab a metric tensor on that manifold. But the standard definitions of a manifold appeal to the real numbers in the definiens. Do you think this creates a problem for your basic idea?

7 10 2005
Peter

Of course, in your last para, where you claim that “real numbers” are used for activities in the world such as measuring distances, you surely mean “finite approximations of real numbers”. Ain’t no one yet seen a transcendental number in the wild. A computer scientist could easily argue (and some have) that these finite entities are not best understood as approximations to infinite ones, but as things in themselves; in this view, real numbers are infinite approximations to computable numbers, not the reverse.

More importantly, on the mathematical objects indispensable for theoretical physics, how could we know otherwise, since our only knowledge of theoretical physics comes to us through its mathematics. There is no non-mathematical theoretical physics.

7 10 2005
Kenny

Greg – I don’t know if you’ve looked at Field’s stuff, but at least in Science Without Numbers, he’s just talking Newtonian theory, so he only needs R^4. But instead of axiomatizing it in a Descartes/Dedekind way by identifying points with quadruples of real numbers, he axiomatizes it in a Hilbert way just defining a few relations on the points themselves. Then he proves that (if real numbers exist then) the resulting structure is isomorphic to R^4. One would like a similar axiomatization of QM and relativity, though it’s much harder in those cases (and no one’s succeeded in 25 years of trying).

Peter – something like that is probably right. We have no terribly good justification for using real numbers instead of rationals, or at least computable reals, except that the theory of all reals is much simpler and more easily axiomatized (maybe?). And something like this does seem to be a reasonable scientific virtue. But you’re right, there is the other view that the mathematics is just an approximation to what’s out there. But as Resnik and some others point out, in order to say that this structure is an approximation, you have to say this structure exists. So we’re still postulating it, and postulating it directly in the structure of the world rather than as something that approximates that structure is a better argument for its existence (assuming that either argument works).

8 10 2005
Peter

Kenny — It may be that the theory of all reals is simpler and more easily axiomatized. On the other hand, our reason for using the reals may simpler be habit and inertia. Much of theoretical physics between 1850 and 1950 for example, posited infinite-dimensional entities as approximations for very-large-finite-dimensional entities. I think this is why and where we got hooked on using the reals. If Babbage had been successful in building his designs for programmable calculating machines, and thus computer science had arisen before statistical physics, our pure mathematical theory of real numbers would likely look very different.

9 10 2005
Kenny

That’s probably quite true, and certainly worth looking into. I’m not terribly tied to the idea of the real numbers having any physical applications – to me they still seem justified for the unification and explanation they give in mathematics.

11 10 2005
Computational Truth

The Indespensibility Argument

While I have never been really convinced about the argument for theoretical entities in physics I think the argument is even weaker for mathematical objects. At the very best I think one could use it to support the idea that ‘conditional’ mathematical…

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