Non-Diophantine Arithmetics in Mathematics, Physics and Psychology

The following sections are included:

• Preface
• About the Authors
• Contents

People started using numbers and counting in prehistoric times. For thousands of years, mathematicians studied numbers and counting learning a lot in this area. Now we know two mathematical disciplines studying numbers, relations between them and operations with them: arithmetic and number theory, which is also called higher arithmetic. At the same time, numbers are used in all areas in general and in all mathematical disciplines in particular…

To build and explore non-Diophantine arithmetics, we at first describe mathematical structures called prearithmetics (Burgin, 2010), which are algebraic systems more general than Diophantine and non-Diophantine arithmetics but are in some aspects similar to arithmetics. In essence, an abstract prearithmetic is a universal algebra (algebraic system) with two binary operations and a partial order. Operations are called addition and multiplication but in a general case, there are no restrictions on these operations…

As we know, conventional (Diophantine) arithmetics of integer, rational, real and complex numbers have additional operations in comparison with the conventional Diophantine arithmetics of natural and whole numbers. That is why to build and study non-Diophantine arithmetics of real and complex numbers, we at first extend the concept of an abstract prearithmetic studied in Chapter 2 augmenting its structure by additional operations…

The credit for inventing an arithmetic-inspired non-Newtonian calculus is due to Michael Grossman and Robert Katz. They developed differential and integral calculi for functions from one non-Diophantine arithmetic of real numbers into another non-Diophantine arithmetic of real numbers. Their little book Non-Newtonian Calculus (Grossman and Katz, 1972) went, unfortunately, basically unnoticed by the mainstream mathematical community…

In this chapter, we apply the formalism developed in Chapter 4 to two old problems of fractal analysis: Fourier transform on arbitrary Cantor sets (Jorgensen and Pedersen, 1998; Jorgensen, 2006; Aerts, Czachor and Kuna, 2016), and wave equations on fractal space–times (Golmankhaneh, Golmankhaneh and Baleanu, 2015; Czachor, 2019). The analysis on Cantor sets can be easily generalized to Sierpiński-type sets (Aerts, Czachor and Kuna, 2018). The construction of the Cantor and Sierpiński sets is somewhat different from what can be found in the typical literature of the subject. On the one hand, the sets we consider do not have to possess any kind of self-similarity. On the other hand, however, we need to construct an appropriate bijection f_X : X → C as required by the non-Newtonian formalism considered in the previous chapter. The bijections always exist since the cardinality of the Cantor, Sierpiński and Koch fractals equals that of the continuum. Depending on the way we define the fractals of interest, the bijections are more or less natural. A bijection can be explicitly constructed also for the so-called F^α-calculus (Parvate and Gangal, 2005, 2009, 2011; Parvate, Satin and Gangal, 2011; Satin and Gangal, 2016), but its arithmetic aspects still await a detailed study and thus, will not be discussed here.

Fractal space–times, Rényi entropies, and the problem of dark energy are naturally related to non-Diophantine arithmetics and non-Newtonian calculi. This chapter gives an outline of results and several possible research projects in this area.

Human and animal nervous systems are information channels, creating filters between us and the physical reality. A change of a physical stimulus, x → x + Δ(x), is detectable by a subject if Δ(x) exceeds a certain just-noticeable difference Δ_jnd (Luce and Edwards, 1958). In arithmetic terms, we have the equality x + Δ = x when 0 < Δ < Δ_jnd. This equality, as we have seen, looks as an equality from a non-Diophantine arithmetic (cf. Chapter 2). Moreover, the Weber–Fechner law states that the difference between physical stimuli x + kx and x is perceived as being independent of x (Fechner, 1860; Baird and Noma, 1978). The law suggests that our nervous system implements a non-Diophantine subtraction ⊖, making (x + kx) ⊖ x independent of x. Recent psychophysical studies of acoustic stimuli (Friedrich and Heil, 2017; Friedrich, 2019) are explicitly based on a generalized arithmetic. Modern approaches to psychophysics, going in general far beyond the Fechner methods, are experimentally subtle and mathematically precise, and thus are very interesting from our non-Diophantine perspective.

Thus, we have demonstrated that, on the one hand, there is a rich and insightful theory of prearithmetics and arithmetics, which unifies and provides a common foundation for many useful mathematical structures such as rings, fields, semirings, linear algebras, ordered rings, ordered fields, topological rings, topological fields, normed rings, normed algebras, tropical and thermodynamic semirings, normed field Ω-groups, Ω-rings, lattices and Boolean algebras. Naturally, all varied utilizations of these structures are also applications of prearithmetics exhibiting their usefulness and efficacy…

To make the book easier for the reader, notations and definitions of the conventional mathematical concepts and structures used in this book are included in the appendix.

The following sections are included:

• Bibliography
• Author Index
• Subject Index