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## Mathematics, physics and nature

I needed to take a step back from the promising progress on explaining the hydrogen spectrum (with the Lamb shift) and digress on the question of the relation of mathematics and physics and nature.  Recently I entered into a discussion with someone regarding mathematics.  Mathematics seems to be a special sort of language effective in science by some.  But the story has to be much more complex.  Gauss considered mathematics to be the study of nature.  One view of Riemann’s ideas from 1854 that founded the study of higher dimensional geometries is that perhaps it was motivated by the physical geometry of the universe but it was still a purely abstract mathematical enterprise.  I have come to the view that in fact Riemann’s program was to study the geometry of the actual physical universe.  The suspicions of Gauss that the universe might not have Euclidean geometry was formally realized in physics by Einstein’s geometrization of gravity by early twentieth century.  I have written before about an S4 geometry of the universe, is fully tested resolves an implicit Riemann hypothesis.

I am not a fan of the Pythagorean mystery schools or mathematical mysticism personally.  At the same time, the undoubted effectiveness of mathematics in producing sophisticated models which have predictive power in the sciences suggests much more than a logical game that is being played or a linguistic game.  The fact that Einstein’s general relativity could have explanation for nontrivial physical phenomena while it is explicitly a nontrivial geometric theory along with many many other examples help to establish the idea that mathematics is effective because the actual universe has nontrivial mathematical structure.

My ideas surrounding the shape of the universe being a scaled 4-sphere arose in 2008 in no small part because of Hamilton’s obsessions with quaternions.  Much of my at the time unjustified by sober rational criteria conviction for a universe that is $S^4 = \mathbf{HP}^1$ arose from a trust in Hamilton’s obsession.  The concrete empirical evidence for this I only established in late 2012.  Of course this is still far from accepted but I hope that once the hydrogen spectrum can be modeled reasonably we can find simple ways of approaching contradiction-less unity of gravity and other phenomena.  Already we have some justification to show that an explanation of quantization by the geometry of the universe has some merit — by the simple known spectrum of the Dirac operator on a sphere and a Dirac-like equation for the photon due to Majorana.  Purely logically, since I have provided evidence via heat equation analysis that the universe must be compact, the expectation that quantum phenomena are globally determined is not surprising.

A great example of where highly nontrivial mathematics enters naturally in physics is the discovery of the Dirac operator.  The story is extremely well-known.  Dirac wanted to take the square root of the Klein-Gordon equation and derived relations of the coefficients he would need in the first order differential operator.  The relations turned out to be for the Clifford algebra which may have been considered esoterica at the time by physicists.  Then these were studied in great depth by Atiyah, Singer and others for closed manifolds for the path to the index theorem.  The Dirac operator for closed manifolds, which now I use to try to reproduce the hydrogen spectrum with Lamb shift is well-understood thanks to efforts of many physicists and mathematicians.

It is clear thus that in practice we have seen that our study of nature cannot be divorced from mathematics.  But a more serious question is whether nature itself must have mathematical structure.  I don’t know the answer to this.  In some ways the mathematical description of nature is very elementary.  Perhaps it is necessary to lay the strong foundations for an understanding of nature from which greater depth of understanding is merely eased.  Perhaps mathematics gives us to tools to eliminate the impossible from the possible but says little beyond this.

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