"If all mathematics disappeared," said Richard Feynman, "physics would be set back by a week" -- to which a clever mathematician retorted: "Yes, by the week of creation..."
Ralph presented a story which shows how mathematics tends to reappear in the development of physics. According to contemporary cosmological theory, an event horizon is made up of rays that move at the speed of light but stay at their place. For event horizons in space-time thus obtain ray-properties which are equivalent to those for two-dimensional curves when we ask under which condition on an inside curve the outside curve will be longer than that inside curve ("for each point on the inside curve there has to be a ray to infinity such that for any point on the ray there is no shorter line to the curve").This equality of formal constraint gives rise to the question whether there is a general area theorem for black holes, i.e., whether in post-Einsteinian cosmology apply the same mathematical principles as to Euclidean objects. The answer to this question has implications for the (seemingly?) empirical issue whether or not black holes can or must have corners.
It was Stephen Hawking who proposed the area theorem for black holes in the early '70s; the theorem was mathematically proved only in 1998. Ralph asks whether the mathematical proof has added to Hawking's physical proposal. Is it not enough, as most physicists (cf. Feynman) might argue, that the theorem allows for correct predictions and calculations? Why, then, do mathematical physicists consider it so important that the physical argument can be rendered mathematically precise?
Does physics always lead the way and are the mathematicians limited to rendering precise what is already contained in the physical arguments? How frequent and how important are cases where physical arguments are shown wrong by mathematicians (as in this case, perhaps, the empirical notion that black holes have corners was shown wrong by the mathematicians who proved Hawkins's area theorem)? To what extent, then, does "mathematical technology" influence the direction of physics? If we can't imagine Einstein's theory of relativity without Riemannian geometry, is this an instance where mathematics "directed" the development of physics? Why are the models of physics predominantly mathematical?
Philosophers might be tempted to answer Ralph's questions "platonically": If all of nature and reality has a mathematical structure, the co-incidence of physical and mathematical knowledge would not be surprising. Ralph suggested a possible explanation which does not rely on such strong ontological commitments: Maybe one has to consider the relation of physics and mathematics as a transfer of technology. As physicists import their equations from mathematics, they are thereby importing conceptualizations of their objects: The objects of physics are mathematical objects because they are conceived in mathematical terms. If this is so, the disappearance of mathematics would indeed deprive the physicists of their whole creation, of the world of physical objects.
Alfred Nordmann
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