Non-Diophantine Arithmetics in Mathematics, Physics and Psychology

Czachor, Marek; Burgin, Mark

doi:10.1142/11665

“…0 of the general bijection f ðxÞ ¼ jxj q sgnðxÞ, q 2 R. Dashed curves: f ðxÞ ¼ tanh x with f ðd p Þ ¼ 0:01 (lowest), f ðd p Þ ¼ 0:15 (middle), f ðd p Þ ¼ 0:29 (upper). The bending-up occurring for the dashed lines is a consequence of bounding f(x) from above arithmetics (Burgin and Czachor 2020). The plateaus are well modeled by f ðxÞ ¼ a ln x þ b, a fact explaining why decibels correspond to a logarithmic scale.…”

Section: Arithmetic Universementioning

confidence: 92%

Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics

Czachor¹

2020

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Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ (MOND) change the dynamics, but do not alter the calculus. However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition $$\beta _1\oplus \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)+\tanh ^{-1}(\beta _2)\big )$$ β 1 ⊕ β 2 = tanh ( tanh - 1 ( β 1 ) + tanh - 1 ( β 2 ) ) , although multiplication $$\beta _1\odot \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)\cdot \tanh ^{-1}(\beta _2)\big )$$ β 1 ⊙ β 2 = tanh ( tanh - 1 ( β 1 ) · tanh - 1 ( β 2 ) ) , and division $$\beta _1\oslash \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)/\tanh ^{-1}(\beta _2)\big )$$ β 1 ⊘ β 2 = tanh ( tanh - 1 ( β 1 ) / tanh - 1 ( β 2 ) ) do not seem to appear in the literature. The map $$f_{\mathbb{X}}(\beta )=\tanh ^{-1}(\beta )$$ f X ( β ) = tanh - 1 ( β ) defines an isomorphism of the arithmetic in $${\mathbb{X}}=(-1,1)$$ X = ( - 1 , 1 ) with the standard one in $${\mathbb{R}}$$ R . The new arithmetic is projective and non-Diophantine in the sense of Burgin (Uspekhi Matematicheskich Nauk 32:209–210 (in Russian), 1977), while ultrarelativistic velocities are super-large in the sense of Kolmogorov (Technika Molodezhi 10:16–19 (11:30–33 in Russian), 1961). Velocity of light plays a role of non-Diophantine infinity. The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Treating the above example as a paradigm, we ask what can be said about the set $${\mathbb{X}}$$ X of ‘real numbers’, and the isomorphism $$f_{{\mathbb{X}}}:{\mathbb{X}}\rightarrow {\mathbb{R}}$$ f X : X → R , if we assume the standard form of Newtonian mechanics and general relativity (formulated by means of the new calculus) but demand agreement with astrophysical observations. It turns out that the observable accelerated expansion of the Universe can be reconstructed with zero cosmological constant if $$f_{\mathbb{X}}(t/t_H)\approx 0.8\sinh (t-t_1)/(0.8\, t_H)$$ f X ( t / t H ) ≈ 0.8 sinh ( t - t 1 ) / ( 0.8 t H ) . The resulting non-Newtonian model is exactly equivalent to the standard Newtonian one with $$\Omega _\Lambda =0.7$$ Ω Λ = 0.7 , $$\Omega _M=0.3$$ Ω M = 0.3 . Asymptotically flat rotation curves are obtained if ‘zero’, the neutral element $$0_{{\mathbb{X}}}$$ 0 X of addition, is nonzero from the point of view of the standard arithmetic of $${\mathbb{R}}$$ R . This implies $$f^{-1}_{{\mathbb{X}}}(0)=0_{{\mathbb{X}}}>0$$ f X - 1 ( 0 ) = 0 X > 0 . The opposition Diophantine versus non-Diophantine, or Newtonian versus non-Newtonian, is an arithmetic analogue of Euclidean versus non-Euclidean in geometry. We do not yet know if the proposed generalization ultimately removes any need of dark matter, but it will certainly change estimates of its parameters. Physics of the dark universe seems to be both geometry and arithmetic.

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“…For more on the α-arithmetic, its generalized real analysis, its fundamental topological properties related to non-Newtonian metric spaces and its calculus, including non-Newtonian differential equations and its applications, see [7,33,[44][45][46][47][48]. For gentle, thorough and modern introduction to the subject of non-Newtonian calculi, we also refer the reader to the recent book [49]. Now we proceed with our original results.…”

Section: Integralsmentioning

confidence: 99%

On a Non-Newtonian Calculus of Variations

Torres

¹

2021

Axioms

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The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.

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“…It behaves as if it assigned a different meaning to the same word. The example becomes even more intriguing if one realizes that logarithm is known to approximately relate stimulus with sensation in real-life sensory systems (hence the logarithmic scale of decibels and star magnitudes) [ 35 ].…”

Section: Non-newtonian Exponential Function and Logarithmmentioning

confidence: 99%

“…After another two decades the same idea, but in its currently most general form, was rediscovered by myself [ 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 ]. In a wider perspective, non-Newtonian calculus is conceptually related to the works of Rashevsky [ 16 ] and Burgin [ 17 , 18 , 19 , 20 ] on non-Diophantine arithmetics of natural numbers, and to Benioff’s attempts [ 21 , 22 , 23 , 24 , 25 ] of basing physics and mathematics on a common fundamental ground. Traces of non-Newtonian and non-Diophantine thinking can be found in the works of Kaniadakis on generalized statistics [ 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 ].…”

Section: Introductionmentioning

confidence: 99%

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Unifying Aspects of Generalized Calculus

Czachor

¹

2020

Entropy

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Non-Newtonian calculus naturally unifies various ideas that have occurred over the years in the field of generalized thermostatistics, or in the borderland between classical and quantum information theory. The formalism, being very general, is as simple as the calculus we know from undergraduate courses of mathematics. Its theoretical potential is huge, and yet it remains unknown or unappreciated.

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Non-Diophantine Arithmetics in Mathematics, Physics and Psychology

Cited by 22 publications

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Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics

Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics

On a Non-Newtonian Calculus of Variations

Unifying Aspects of Generalized Calculus

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