Pure mathematics has its own existence and it does not need any assumption from Physics, rather the existence of nature and laws of Nature. All the experimentally proven theories of physics till date were based on solid mathematical theories either already available or developed simultaneously.One of the most important reasons beyond the failure, at least till the present, of the new theories like String theory or Quantum Gravity is the unavailability of the mathematical language at that level. These arguments indicate that laws of mathematics don’t depend on existence of anything.
One very obvious question arises – “Does our present day pure mathematics exists in a universe which is beyond our imagination; for example in a continuous jelly like universe where nothing exist separately. What is the relevance of mathematics in the frozen universe beyond the Big Bang when nothing changed and time was undefined? Can we, being an element of present day universe, build a language, i.e. pure math, independent of our existence?
It is worth to study another question – Whether everything which exits mathematically exist physically also? Or the physical world is not completely mathematical. We have seen perfect isomorphism between mathematics and physical theories – For example between Hilbert Space and Quantum Mechanical State Vectors or between Lee Algebra and Physical Symmetries. Should the future theories of Physics will be similar isomorphism? And how nature maps the laws of nature form the many variants of mathematical theories like – why the nature have chosen three plus one manifold universe and why not seven plus two? It seems that our quest for the ultimate form of knowledge will make us engaged in getting the mapping between Mathematics and Physics.
References:
[1] Smolin, Lee, and John Harnad. “The trouble with physics: the rise of string theory, the fall of a science, and what comes next.” The Mathematical Intelligencer 30.3 (2008): 66-69.
[2] The Mathematical Universe, M Tegmark, arXiv:0704.0646[gr-qc], Founds. Phys. November 2007, 116
[3] Penrose, Roger. Cycles of time: an extraordinary new view of the universe. Random House, 2010.
[4] Carroll, Sean. From eternity to here: the quest for the ultimate theory of time. Penguin, 2010.
[5] “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960
