Comment on "Unifying Aspects of Generalized Calculus"

TL;DR

This paper critically examines Czachor's non-Newtonian calculus framework, highlighting its mathematical novelty but exposing significant conceptual and physical limitations that undermine its applicability in scientific domains.

Contribution

It provides a detailed critique of Czachor's approach, demonstrating its shortcomings and lack of physical coherence when applied to real-world scientific problems.

Findings

The framework fails with pathological bijections like the Cantor function.

It reduces to tautologies in relativity, entropy, and cosmology.

The formalism lacks predictive power and physical relevance.

Abstract

Czachor's recent proposal introduces a form of non-Newtonian calculus built by pulling back arithmetic operations through arbitrary bijections between continua. Although the idea is mathematically inventive, it runs into serious conceptual trouble when examined from a physical standpoint. Claims of universal applicability quickly unravel under scrutiny -- especially when considering pathological bijections like the Cantor function, which break the framework's core assumptions. When applied to domains such as relativity, entropy, or cosmology, the results often collapse into tautological restatements lacking real predictive power. This commentary explores these issues in depth, highlighting where and why the formalism falls short of providing a physically coherent theory.