Abstract
It is argued that mathematics is unreasonably effective in fundamental physics, that this is genuinely mysterious, and that it is best explained by a version of Pythagorean metaphysics. It is shown how this can be reconciled with the fact that mathematics is not always effective in real world applications. The thesis is that physical structure approaches isomorphism with a highly symmetric mathematical structure at very high energy levels, such as would have existed in the early universe. As the universe cooled, its underlying symmetry was broken in a sequence of stages. At each stage, more forces and particles were differentiated, leading to the complexity of the observed world. Remnant structure makes mathematics effective in some real world applications, but not all.
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Notes
N.B. Isolated quarks cannot be observed because they are confined by the strong force (as explained in Sect. 5) but their existence can be inferred from deep-inelastic scattering experiments.
An example of S1 is the development of matrix mechanics by Heisenberg, Born and Jordan (Steiner 1998, pp. 96–97). Roughly put, they constructed quantum versions of the equations of classical mechanics by substituting matrices for the position and momentum variables. Amongst other things, this enforced the non-commutativity of position and momentum operators in the quantum equations.
For example, simple concepts of symmetry lead to group theory, of topology lead to knot theory, of aggregation lead to set theory.
The fact that it is renormalisable merely means that the coupling constants are calculable from theory rather than measured experimentally.
Despite recent travails in the development of string theory and the resort of some physicists to a multiverse explanation, I still count it as a candidate for TOE.
Locality has been questioned in the context of the Einstein–Rosen–Podolsky paradox but Griffiths (2011) has argued that it holds in the appropriate sense (i.e., a measurement carried out on one particle cannot affect the physical state of another particle elsewhere) when quantum mechanics is formulated using consistent principles.
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Acknowledgments
I would like to thank Graham Oppy for many valuable discussions in the course of preparing this paper. I also thank the anonymous referees for their help in improving its structure and clarity.
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McDonnell, J. Wigner’s puzzle and the Pythagorean heuristic. Synthese 194, 2931–2948 (2017). https://doi.org/10.1007/s11229-016-1080-6
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DOI: https://doi.org/10.1007/s11229-016-1080-6