Abstract
Following a suggestion of Wigner, we look for a deep connection between mathematics and physics. Accordingly, a mathematical foundation is proposed which leads to a field theory. In this theory, fields are constructed solely from biunique eigenfunction pieces, each piece obtained from the same general variational expression akin to least action. We show that this piecewise construction does indeed produce a finite unit of action and quantum statistics. The fundamental concept of “definability” of fields is introduced. A definable field is free of paradoxes.
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References
Wigner, Eugene P., “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Comm. on Pure and App. Math, X III, 1960.
The axiom schema of subsets is written 3u[(u = 0 V 3xx E u) n Vxx E u +-’ x Ez A x(x)], where z is any set and x(x) is a formula in which x is free and u is not free. This axiom schema creates subsets non-constructively using a formula. Proofs in ZF which are essentially indirect, such as Cantor’s Proof, require this axiom.
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Goedel, K., The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis, Annals of Math. Studies 3, Princeton, 1940. He proves that this axiom of constructibility can be adjoined consistently to ZR ZF-AR+ABR is a sub-theory of ZF (since AR—+ABR) and thus his proof applies.
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Equivalently, letting
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© 1998 Springer Science+Business Media Dordrecht
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Bendaniel, D.J. (1998). The Unreasonable Effectiveness of Mathematics in Physics. In: Hunter, G., Jeffers, S., Vigier, JP. (eds) Causality and Locality in Modern Physics. Fundamental Theories of Physics, vol 97. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0990-3_35
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DOI: https://doi.org/10.1007/978-94-017-0990-3_35
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5092-2
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