I have often wondered why mathematicians have not generalized the special theory of relativity into different types of "special relativities," so to speak (not into the presently known general theory of relativity). I am sure there are other "relativities" possible in general spaces, yet hardly anything has been attempted by mathematicians. Endless papers exist on metric spaces generalizing the ordinary geometry without the dimension of time in it. Put in time and space together, and mathematicians stay out! Topologists stay with spatial spaces; they have not considered ideas which would generalize the four-dimensional time-space. This is very curious to me, epistemologically and psychologically. (I can think of one paper by van Danzig, which speculates philosophically around the notion of time topology; he says it might be a solenoidal variable. I like this, but clearly one should do much more imaginative work with time-like spaces.)
As is well known, the theory of special relativity postulates and is built entirely on the fact that light always has the same velocity regardless of the motion of the source or the observer. From this postulate alone everything follows, including the famous formula E = mc2. Mathematically speaking, the invariance of the cones of light lead to the Lorentz group of transformations. Now a mathematician could, just for mathematical fun, postulate that the frequency, for example, remains the same, or that some other class of simple physical relation is invariant. By following logically one could see what the consequences would be in such a picture of a not "real" universe.
Mathematics is now so completely different from what it was in the nineteenth century, even if ninety-nine percent of mathematicians have no feeling for physics. There are so many ideas in physics begging for mathematical inspiration — new formulations, new mathematical ideas. I do not mean the use of mathematics in physics, but the other way round: physics as a stimulant for new mathematical concepts.
Contrary to mathematics, in physics one can, in principle, keep more abreast of what is going on in research. Every physicist can know the gist of most of physics. There are very few fundamental problems now such as the problem of the nature of elementary particles or what is the nature of the physical space and time.
In present-day research in theoretical physics, even though many of the young people are very clever, ingenious, and technically superb, their fundamental ideas tend to be orthodox, and on the whole only small variations on what has been done are produced, elaboration of details, and continuation along lines that already have been started.
Perhaps this has always been so and really new ideas are exceptional.
Sometimes, half in jest, to needle contemporary young physicist friends who spend all their time examining a few very strange particles, I tell them that it is not necessarily the best way to get new inspiration about the foundations of physics and the scheme of things in space and time.
Of course, it is not a precise problem or recognized as such, but what to my mind is a first question in physics is whether there exists a true infinity of structures going down into smaller and smaller dimensions. If so it would be worthwhile for mathematicians to speculate on whether space and time change, even in their topology, in smaller and smaller regions. We had in physics an atomistic or field-structure base. If the ultimate reality consists of a field, then its points are true mathematical points and indistinguishable. There is a possibility that in reality we have a strange structure of infinitely many stages, each stage different in nature. This is a fascinating picture which becomes more physical and not merely a philosophical conundrum. Recent experiments show definitely the increasing complication of structures. In a single nucleon we may have partons, as Feynman calls them. These partons may be the hypothetical quarks or other structures. The recent theoretical attempts no longer explain the experimental models by simple quarks, but one has to involve colored quarks of different types. Perhaps one has reached a point where it might be preferable to consider the succession of structures as going on ad infinitum.
Theoretical physics is possible because there are many identical or nearly identical copies of objects and situations. If one takes the universe by definition as only one (even though it is true that galaxies resemble each other) and the world as a whole being one, the questions asked about the cosmos as a whole have a different character. The stability with respect to adding a few more elements to an already large number is no longer guaranteed. We have no way to observe or experiment with a number of universes. Therefore problems of cosmology and cosmogony have a different character from those of even the most fundamental physics.
Science would not be possible, physics would not be conceivable, if there was not this similarity or identity of vast numbers of points or subsets or groups of points in this universe. All individual protons seem to resemble each other, all electrons seem to resemble each other, the attraction between any two celestial bodies seems to be similar, depending only on distance and mass. So the role of physics appears, inter alia, to divide the existing groupings into entities of which there are very many examples that are isomorphic or almost isomorphic to each other. The hope for physics lies in the fact that one can almost repeat situations, or if not exactly repeat, the addition of one or more small changes makes relatively little difference. Whether there are twenty or twenty-two bodies does not make their behavior change radically. A belief in some fundamental stability! Somehow the hope is to describe physics in terms of simpler entities and identity of parts by some kind of union or counting. For example, physicists believed, at least until recently, that if one had many points, the behavior of their mass could be explained by two-body interactions — this means adding up the potentials between any two bodies. Otherwise, if every time through adding a few bodies one changed the behavior of the whole system, there would be no science of physics. This point is not sufficiently brought out in physics textbooks.
One can relate the notion of entropy to the notion of complexity, if one defines the distance between two algebraical structures and the total work necessary to prove a statement or a theorem as energy. Results exist stating that in given systems in order to prove such and such formula one needs so many steps. The minimum sufficient number of steps can be defined as an analogue of work or energy. This is worth thinking about. To make a sensible theory of it requires erudition, imagination and common sense. There is no axiomatic system even for the presently established body of physics.
Just as in pure mathematics, in theoretical physics we can see a dichotomy between the great new "unexpected" ideas and the great syntheses of established theories. Such syntheses are in a sense complementary or opposed to the new concepts. They summarize previous theories in a non-obvious way. Let me illustrate this distinction: the special theory of relativity is a priori a very strange and mysterious concept. It involves an almost irrational insight and an a priori implausible axiom based on the experimental fact that light velocity seems to be the same for a moving observer from a fixed emission point or vice versa. When the emission point moves away or toward the observer the velocity of light relative to the observer is the same no matter what the relative velocity is. From this alone a great theoretical edifice was built, a physical theory of space and time with so many surprising — and as we now know — technologically shattering consequences.
Quantum theory involves similarly, in a way, an a priori non-intuitive or unexpected set of concepts.
Maxwell's theory of electromagnetism would be an example of a great synthesis. It came after a great number of experimental facts were developed which were perhaps not so strange to their first discoverers. The theory that explains these observational facts in one set of mathematical equations constitutes one of the most impressive achievements of human thought. Epistemologically this theory is of a different nature, or so it seems to me, from relativity and quantum theory which were, one might say, more unexpected.