# Heuristics and Inferential Microstructures: The Path to Quaternions

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## Abstract

I investigate the construction of the mathematical concept of quaternion from a methodological and heuristic viewpoint to examine what we can learn from it for the study of the advancement of mathematical knowledge. I will look, in particular, at the

*inferential microstructures*that shape this construction, that is, the study of both the very first,*ampliative*inferential steps, and their tentative outcomes—i.e. small ‘structures’ such as provisional entities and relations. I discuss how this paradigmatic case study supports the recent approaches to problem-solving and philosophy of mathematics, and how it suggests refinements of them. In more detail, I argue that the inferential micro-structures enable us to shed more light on the informal, heuristic side of mathematical practice, and its inferential and rational procedures. I show how they enable the generation of a problem, the construction of its conditions of solvability, the search for a hypothesis to solve it, and how these processes are representation-sensitive. On this base, I argue that:- 1.
the recent development of the philosophy of mathematics was right in moving

*from*Lakatos’ initial investigation of the formal side of a mathematical proof*to*the investigation of the semi-formal (or informal), heuristic side of the mathematical practice as a way of understanding mathematical knowledge and its advancement. - 2.
The investigation of mathematical practice and discovery can be improved by a finer-grained study of the inferential micro-structures that are built during mathematical problem-solving.

## keywords

Heuristics Mathematics Representation Micro-structures Mathematical practice## Notes

### Compliance with Ethical Standards

### Conflict of interest

The author declares that he/she has no conflict of interest.

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