What is worthy of investigation? Philosophical attitudes and their impact on mathematical development by the example of discovering 10-adic numbers

TL;DR

This paper explores how philosophical attitudes influence the discovery and development of 10-adic numbers through a dialogue illustrating various mathematical and philosophical perspectives.

Contribution

It presents a dialogue-based approach to understanding the philosophical influences on the mathematical development of 10-adic numbers and their potential extensions.

Findings

Philosophical attitudes shape mathematical exploration and discovery.

Extensions to transfinite 10-adic numbers are considered.

Applications in computer arithmetic are discussed.

Abstract

We describe in dialogue form a possible way of discovering and investigating 10-adic numbers starting from the naive question about a `largest natural number'. Among the topics we pursue are possibilities of extensions to transfinite 10-adic numbers, 10-adic representations of rational numbers, zero divisors, square roots and 10-adic roots of higher degree of natural numbers, and applications of 10-adic number representation in computer arithmetic. The participants of the dialogue are idealized embodiments of different philosophical attitudes towards mathematics. The article aims at illustrating how these attitudes interact, in both jarring and stimulating ways, and how they impact mathematical development.