FOUNDATIONS OF PHYSICS
Here the fundamental theories are general relativity and quantum mechanics, and the central problem is to reconcile the two. Three of the fundamental forces-electromagnetic, weak, and strong-have yielded more or less workable quantum-field descriptions, but grav- ity remains intractable. The mathematics of quantum-field theories has an annoying habit of generating impossible (infinite) values for some physical magnitudes, but the problem has been overcome by a technical trick called “renormalization” in all cases but that of grav- ity. So the problem remains: How to characterize the gravitational force at quantum distances?
Physicists engage in fascinating speculations on the source of the difficulty: for example, that it arises from the attempt to combine the essentially passive space-time of quantum theory with the dy- namic space-time of general relativity.23 But most important for our purposes is the idea that the fault might lie in our conception of space-time as a mathematical continuum. For example, Richard Feynman24 writes:
I believe that the theory that space is continuous is wrong, because we get these infinities and other difficulties . . . (ibid., p. 166).
And Chris Isham:
… it is clear that quantum gravity, with its natural Planck length, raises the possibility that the continuum nature of spacetime may not hold below this length, and that a quite different model is needed (op. cit., p. 72).
23 See Chris Isham, “Quantum Gravity,” in The New Physics, P. Davies, ed. (New York: Cambridge, 1989), pp. 70-93, esp. p. 70. I review some of the popu- lar literature on this problem in “Taking Naturalism Seriously.”
24 The Character of Physical Law (Cambridge: MIT, 1967).
286 THE JOURNAL OF PHILOSOPHY
Here the suggestion is not (as in the previous section) that the conti- nuity of space-time is a “mere idealization,” but that it does not belong in our best theory at all!
All this, as I have indicated, is quite speculative; no one yet knows what a reasonable theory of quantum gravity might be like. But the very suggestion that space-time may not be continuous is enough to add previously unimagined poignancy to our earlier conclusion of a posteriority and fallibility. What seemed a rather small concession at the end of the last section-that the grounds for the indispensability theorist’s adherence to fact could be overthrown by progress in physics-now looms as a real possibility.
Of course, the (potential) falsity of continuum mathematics in its application to space-time would be only part of the story; there are other successful scientific uses for the calculus and higher analysis. But, if science were to change so that all fundamental theories were thoroughly quantized, so that no continuum mathematics appeared there, if all the remaining applications of continuum mathematics were explicitly understood as “approximations” or “idealizations” or “models,” then even the modified indispensability theorist would retreat to some version of no-fact. The case of quantum gravity should keep us from dismissing this possibility out of hand.