Attempting to remove infinites from divergent series: Hardy will hardly help

TL;DR

This paper examines alternative definitions of series sum and convergence, highlighting the limitations and epistemological implications of assigning finite values to divergent series in mathematical and physical contexts.

Contribution

It analyzes the constraints and consequences of redefining series sums, especially in renormalization, and discusses the epistemological costs of such practices.

Findings

Divergent series in renormalization cannot be assigned finite sums without contradictions.

Replacements for series sum must satisfy associativity, permutation invariance, and dilution.

Accepting these practices involves significant epistemological trade-offs.

Abstract

The consequences of adopting other definitions of the concepts of sum and convergence of a series are discussed in the light of historical and epistemological contexts. We show that some divergent series appearing in the context of renormalization methods cannot be assigned finite values in a form consistent with Hardy's axioms without at the same time equating one to zero, thus destroying the mathematical building. We show that if the replacements for the concept of sum of a series are required to be associative, to be invariant under finite permutations of the terms and dilution, further restrictions emerge. We finally discuss the epistemological costs of accepting these practices in the name of instrumentalism.