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Newtonian physics is based on Newtonian calculus applied to Newtoniandynamics. New paradigms such as MOND change the dynamics, but do not alter thecalculus. Calculus is dependent on arithmetic, e.g. in special relativity weadd and subtract velocities by means of addition $\beta_1\oplus\beta_2=\tanh\big(\tanh^{-1}(\beta_1)+\tanh^{-1}(\beta_2)\big)$, althoughmultiplication $\beta_1\odot\beta_2=\tanh\big(\tanh^{-1}(\beta_1)\cdot\tanh^{-1}(\beta_2)\big)$ does notseem to appear in the literature. The map$f_\mathbb{X}(\beta)=\tanh^{-1}(\beta)$ defines an isomorphism of thearithmetic in $\mathbb{X}=(-1,1)$ with the standard one in $\mathbb{R}$. Thenew arithmetic is non-Diophantine in the sense of Burgin. Velocity of lightplays a role of non-Diophantine infinity. The new arithmetic allows us todefine the corresponding derivative and integral, and thus a new calculus whichis non-Newtonian in the sense of Grossman and Katz. Treating he above exampleas a paradigm, we ask what can be said about the set $\mathbb{X}$ and theisomorphism $f_{\mathbb{X}}:\mathbb{X}\to \mathbb{R}$, if we assume thestandard form of Newtonian mechanics and general relativity (formulated bymeans of the new calculus) but demand agreement with astrophysicalobservations. It turns out that for $f_\mathbb{X}(t/t_H)\approx 0.8\sinh(t-t_1)/(0.8\, t_H)$ the resulting non-Newtonian Friedman equation with$\Omega_\Lambda=0$ is exactly quivalent to the standard Newtonian one with$\Omega_\Lambda=0.7$, $\Omega_M=0.3$. Asymptotically flat rotation curves areobtained if `zero', the neutral element of addition, is nonzero from the pointof view of the standard arithmetic of $\mathbb{R}$. We do not yet know if theproposed generalization ultimately removes any need of dark matter, but it willcertainly change estimates of its parameters.

Non-Newtonian mathematics instead of non-Newtonian physics: Dark matter and dark energy from a mismatch of arithmetics