Abstract
Abstract Following the results of our recently published book [F. Lev, Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory. With Applications to Gravity and Particle Theory, Springer, 2020, ISBN 978-3-030-61101-9], we discuss different aspects of classical and finite mathematics and explain why finite mathematics based on a finite ring of characteristic p p is more general (fundamental) than classical mathematics: the former does not have foundational problems, and the latter is a special degenerate case of the former in the formal limit p → ∞ p\to \infty . In particular, quantum theory based on a finite ring of characteristic p p is more general than standard quantum theory because the latter is a special degenerate case of the former in the formal limit p → ∞ p\to \infty .
Highlights
Following the results of our recently published book [F
In [9,10], we have proposed an approach called finite quantum theory (FQT), where physical quantities belong to a finite ring or field, but wave functions are elements of a space over a finite ring or a field
As discussed in detail in [2], the concept of space–time has a physical meaning only on the classical level, i.e., when first FQT is approximated by standard quantum theory (SQT) in the formal limit p → ∞, and SQT is approximated by classical theory in the formal limit ħ → 0
Summary
The title of the famous Wigner’s paper [1] is: “The unreasonable effectiveness of mathematics in the natural sciences,” and the paper is concluded as follows: The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. (3) RT is a special degenerate case of de Sitter (dS) and anti-de Sitter (AdS) invariant theories in the formal limit R → ∞, where R is the parameter of contraction from the dS or AdS groups or Lie algebras to the Poincare group or Lie algebra, respectively In the literature, those facts are explained from physical considerations but, as shown in the famous Dyson’s paper “Missed Opportunities” [3], (1) follows from the pure mathematical fact that the Galilei group can be obtained from the Poincare one by contraction c → ∞, and (3) follows from the pure mathematical fact that the Poincare group can be obtained from the dS or AdS groups by contraction R → ∞. The main goal of this paper is to explain in the framework of Definition that: Statement: Classical mathematics is a special degenerate case of finite one in the formal limit p → ∞, where p is the characteristic of the ring in finite mathematics. The organization of this paper is clear from the titles of the sections
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