### Where is the Philosophy of Physics?

#### Posted by David Corfield

As the subtitle of this blog says, we run ‘A group blog on math, physics and philosophy’. To what extent, though, do we cover all the interfaces of this triad? Well, we do some philosophy of mathematics here, and we certainly do some mathematical physics. But the question I’ve been wondering about recently is whether we should be doing more philosophy of physics.

If we followed the position that physics is the search for more and more adequate mathematical structures to describe the world, perhaps we needn’t take the philosophy of physics to be anything more than a philosophy of mathematics along with an account of how the structures which are most promising for physics are chosen. But this view of physics would be controversial.

Philosophy of physics is a large field asking questions about the interpretation of quantum mechanics (a table comparing thirteen), about the direction of time in statistical mechanics, about the relation between space, time and motion, and about much more. That such work is necessary for physics to progress is a thesis of the philosopher Michael Friedman, who has featured here before. I was reminded of his argument while reading this paper comparing Cassirer and Kuhn.

For Cassirer, says Friedman, in mathematical physics

We can…conceive all the theories in our sequence as continuously converging, as it were, on a final or limit theory, such that all previous theories in the sequence are approximate special cases of this final theory. This final theory is only a regulative ideal in the Kantian sense – it is only progressively approximated but never in fact actually realized. Nevertheless, the idea of such a continuous progression toward an ideal limit constitutes the characteristic “general serial form” of our mathematical-physical theorizing, and, at the same time, it bestows on this theorizing its characteristic form of objectivity. (p. 241)

and

…thought does not require a “substantialistic” or “ontological” identity over time of permanent “things” but merely a purely mathematical continuity over time formulated in successively articulated mathematical structures. (p. 245)

Kuhn, on the other hand,

…consistently gives the question an ontological (“substantialistic”) rather than a mathematical (“functional”) interpretation. Thus, for example, when Kuhn famously considers the relationship between relativistic and Newtonian mechanics, he rejects the notion of a fundamental continuity between the two theories on the grounds that the “physical referents” of their terms are essentially different, and he nowhere considers the contrasting idea, characteristic of Cassirer’s work, that continuity of purely mathematical structures is sufficient. Moreover, Kuhn consistently gives an ontological rather than a mathematical interpretation to the question of theoretical convergence over time: The question is always whether our theories can be said to converge to an independently existing “truth” about reality, to a theory-independent external world. (pp. 245-246)

Friedman goes on to explain how

…with Kuhn – and with the logical empiricists – that Einstein’s general theory of relativity is in an important sense incommensurable or nonintertranslatable with the Newtonian theory of universal gravitation it replaced. Whereas Newtonian theory represents the action of gravity as an external “impressed force” causing gravitationally affected bodies to deviate from straight inertial trajectories (moving with uniform or constant speed), Einstein’s theory depicts gravitation as a curving or bending of the underlying fabric of space-time itself. In this new framework, in particular, there are no inertial trajectories in the sense of the geometry of Euclid and the mechanics of Newton, and gravity is not an “impressed force” causing deviations from such trajectories. (p. 248)

So

…even after the mathematics required for Einstein’s theory was developed, it still remained fundamentally unclear what it could mean actually to apply such a geometry to our sensible experience of nature in a real physical theory. One still needed to show, in other words, that Einstein’s new theory is

empiricallyorphysicallypossible as well, and this, in turn, only became clear with Einstein’s own work on what he called the principle of equivalence in the years 1907–12. (p. 249)

What is needed is something Friedman calls *scientific philosophy* and elsewhere *meta-scientific work*:

in addition to the necessary mathematical developments (the evolution of non-Euclidean geometries, as unified and completed in Riemann’s work) and the necessary physical developments (the discovery of the constancy and invariance of the velocity of light, the numerical equality of inertial and gravitational mass underlying the principle of equivalence), we still need a set of parallel developments in contemporaneous scientific philosophy to tie together the relevant innovations in mathematics and physics and thereby effect the necessary expansion in our physical or empirical possibilities. (pp. 249-250)

In the case of general relativity, some of this meta-scientific work was done by Mach, Helmholtz and Poincaré, before Einstein.

If Friedman is right, we would expect the need for some of this meta-scientific work to allow the *n*-categorical physics revolution. I wonder what form it should take. The Café discussion which struck me as being most like contemporary philosophy of physics was that concerning Spitter, Heunen and Landsman’s work on topos theory in physics. But what of the mathematical physics we talk about mostly here?

You can see me trying to urge you knowledgeable people in this direction here. In *On Unitary Representations of the Inhomogeneous Lorentz Group*, Wigner discusses how observations made by different observers ought to be related in terms of a group acting on frames of reference, so I wondered whether a similar story existed for 2-groups.

Urs says

the Lawvere-ification of the proper center of modern theoretical physics is still to be done.

Do we need a new Mach, Helmholtz and Poincaré?

## Re: Where is the Philosophy of Physics?

It seems to me that we are getting quite a bit closer to that gooal, as of late. Not that there is nothing left to do, but signs are more hopeful than they used to be.

It’s funny that you posted this particular entry within four minutes of that conference report entry. Maybe the two are not only close in posting time.