# Unifying Aspects of Generalized Calculus  
[PERSON]  
Wydzial Fizyki Technicznej i Matematyki Stosowanej, Politechnika Gdarska, 80-233 Gdarsk, Poland; [EMAIL_ADDRESS]  
28 September 20202017 October 202  
The only assumption about the domain \(\mathbb{X}\) and the codomain \(\mathbb{Y}\) is that they have the same cardinality as the continuum \(\mathbb{R}\). The latter guarantees that bijections \(f_{\mathbb{X}}\) and \(f_{\mathbb{Y}}\) exist. The bijections are automatically continuous in the topologies they induce from the open-interval topology of \(\mathbb{R}\), even if they are discontinuous in metric topologies of \(\mathbb{X}\) and \(\mathbb{Y}\) (a typical situation in fractal applications, or in cases where \(\mathbb{X}\) or \(\mathbb{Y}\) are not subsets of \(\mathbb{R}\)). In general, one does _not_ assume anything else about \(f_{\mathbb{X}}\) and \(f_{\mathbb{Y}}\). In particular, their differentiability in the usual (Newtonian) sense is not assumed. No topological assumptions are made about \(\mathbb{X}\) and \(\mathbb{Y}\). Of course, the structure of the diagram implies that \(\mathbb{X}\) and \(\mathbb{Y}\) may be regarded as Banach manifolds with global charts \(f_{\mathbb{X}}\) and \(f_{\mathbb{Y}}\), but one does not make the usual assumptions about changes of charts. Non-Newtonian calculus begins with (generalized, non-Diophantine) arithmetics in \(\mathbb{X}\) and \(\mathbb{Y}\), induced from \(\mathbb{R}\),  
\[x_{1}\oplus_{\mathbb{X}}x_{2} = f_{\mathbb{X}}^{-1}\big{(}f_{\mathbb{X}}(x_{1})+f_{\mathbb{X}}( x_{2})\big{)}, \tag{2}\] \[x_{1}\ominus_{\mathbb{X}}x_{2} = f_{\mathbb{X}}^{-1}\big{(}f_{\mathbb{X}}(x_{1})-f_{\mathbb{X}}( x_{2})\big{)},\] (3) \[x_{1}\ominus_{\mathbb{X}}x_{2} = f_{\mathbb{X}}^{-1}\big{(}f_{\mathbb{X}}(x_{1})\cdot f_{\mathbb{ X}}(x_{2})\big{)},\] (4) \[x_{1}\ominus_{\mathbb{X}}x_{2} = f_{\mathbb{X}}^{-1}\big{(}f_{\mathbb{X}}(x_{1})/f_{\mathbb{X}}( x_{2})\big{)} \tag{5}\]  
(and analogously in \(\mathbb{Y}\)). **Example 1**. : _According to one of the axioms of standard quantum mechanics, states of a quantum system belong to a separable Hilbert space. All separable Hilbert spaces are isomorphic, so state spaces of any two quantum systems are isomorphic. Does it mean that all quantum systems are equivalent? No, it only shows that mathematically isomorphic structures can play physically different roles. Similarly, the arithmetic given by (2)-(5) is isomorphic to the standard arithmetic of \(\mathbb{R}\), but it does not imply that the two arithmetics are physically equivalent._  
**Example 2**. : _The origin of [PERSON]'s special theory of relativity goes back to the observation that the velocity of a source of light does not influence the velocity of light itself, contradicting our everyday experiences with velocities in trains or football. Relativistic addition of velocities is based on a fundamental unit \(c\) and the dimensionless parameter \(\beta\), related to velocity by \(v=\beta c\). \(\beta\in\mathbb{X}=(-1,1)\) while the bijection reads \(f_{\mathbb{X}}(\beta)=\operatorname{arctanh}\beta\). The velocities are added or subtracted by means of (2) and (3),_  
\[\beta_{1}\oplus_{\mathbb{X}}\beta_{2}=\tanh(\operatorname{arctanh}\beta_{1}+ \operatorname{arctanh}\beta_{2}). \tag{6}\]  
_Interestingly, (4) and (5) are not directly employed in special relativity. The presence of the fundamental unit \(c\) is a signature of a general non-Diophantine arithmetic (which typically works with dimensionless numbers). Numbers \(\pm 1\in\mathbb{R}\) play the roles of infinities, \(\pm 1_{\mathbb{R}}=\pm\infty_{\mathbb{X}}\). The velocity of light is therefore literally infinite in the non-Diophantine sense.