Abstract
Arithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, mathbb {R}_{+}^{4} and (−L/2,L/2)4, are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a non-Diophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms of the corresponding calculus. However, when one switches to the ‘natural’ Diophantine arithmetic and calculus, the Minkowskian character of the space is lost and what one effectively obtains is a Lorentzian manifold. I discuss in more detail the problem of electromagnetic fields produced by a pointlike charge. The solution has the standard form when expressed in terms of the non-Diophantine formalism. When the ‘natural’ formalsm is used, the same solution looks as if the fields were created by a charge located in an expanding universe, with nontrivially accelerating expansion. The effect is clearly visible also in solutions of the Friedman equation with vanishing cosmological constant. All of this suggests that phenomena attributed to dark energy may be a manifestation of a miss-match between the arithmetic employed in mathematical modeling, and the one occurring at the level of natural laws. Arithmetic is as physical as geometry.
Highlights
The idea of relativity of arithmetic follows from the observation that the four basic arithmetic operations are fundamentally non-unique, even if one assumes commutativity and associativity of ‘plus’ and ‘times’, and distributivity of ‘times’ with respect to ‘plus’
Consider real numbers R equipped with the ‘standard’ arithmetic operations of addition (+), subtraction (−), multiplication (·), and division (/)
Let us begin with the explicit form of arithmetic operations
Summary
The idea of relativity of arithmetic follows from the observation that the four basic arithmetic operations (addition, subtraction, multiplication, division) are fundamentally non-unique, even if one assumes commutativity and associativity of ‘plus’ and ‘times’, and distributivity of ‘times’ with respect to ‘plus’. The idea was extended by Benioff to real and complex numbers, and generalized in many ways, including space-time dependent fields of value functions [12, 13]. In case there is a missmatch, i.e. two different f s come into play, the conflict of arithmetics can have observational consequences Consider real numbers R equipped with the ‘standard’ arithmetic operations of addition (+), subtraction (−), multiplication (·), and division (/). Let us term these real numbers the ‘lower reals’, and denote them by lowercase symbols.
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