Sumterms, Summands, Sumtuples, and Sums and the Meta-arithmetic of Summation

TL;DR

The paper introduces sumterms and sumtuples to formalize summation, discusses the sum splitting paradox, and surveys solutions to resolve this paradox within the meta-arithmetic of summation.

Contribution

It presents a novel formal framework for summation using syntactic and semantic entities and analyzes the sum splitting paradox with a survey of potential solutions.

Findings

Sumterms are syntactic entities for sums.

Sumtuples are semantic entities for sums.

The sum splitting paradox arises from sumterm splitting operators.

Abstract

Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) a role which can be played by a number. Sumterm splitting operators are introduced and it is argued that without further precautions the presence of these operators gives rise to instance of the so-called sum splitting paradox. A survey of solutions to the sum splitting paradox is given.