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Relation between Mathematics and Physics

It is well known that research in mathematics stimulated research in physics and vice versa, and that this has been the case throughout the long histories of both subjects. One would expect that, consequentially, there exists a fairly large number of texts of various format on the nature of this stimulation and the results thereof, at least by the more philosophically minded among the mathematicians or the physicists and, of course, by the philosophers of either mathematics or physics. It is, therefore, even more curious to find that exactly the opposite is the case ; namely, that very few among the great names of either physics, mathematics or philosophy have devoted more than a few pages on the subject. Nowadays, the various aspects of this fundamental and not at all accidental relation are studied and for different reasons or different purposes, but still the general character of the relation is, it seems, under-appreciated and often the discussion does not move from the entrenched views of empiricism (mathematics is the language of science) or some form of platonism (physics, at least theoretical, is but a part of mathematics). However, we are no nearer to solving sometimes very old or ancient problems of the appropriatness of certain mathematical concepts for describing the material reality (eg. the ideas of continuum, differentiabilty/integrability, representation by mathematical structure etc.) or how the often uncritical import of a novel mathematical structure into physics might actually prove more harmful to decrypting the meaning of the physical theory and perhaps even be hindering its predictive success. Most importantly, can any mathematical structure be uncritically (as it more often was than not) used in modelling a physical system without somehow fundamentally distorting our view of the particular physics research and/or our approach to physcis in general? Some notable examples from history of development of fundamental physics theories are discussed in this context. However we decide to answer the last question, the problem of, in the words of the famous paper by Wigner, the (seemingly) unreasonable effectivness of mathematics as applied in physics remains the greatest challenge.

relation between mathematics and physics ; mathematical structure ; physical theory ; Wigner's problem

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