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Chapter 1: Introduction

Method

If I wish to keep open the possibility that there is more than one mathematics, then I have a problem in that I want to satisfy the curiosity that leads me to ask the question "What is mathematics?", but it now seems that the term "mathematics" needs some clarification. If different people use the language of mathematics in different ways, then what is the question referring to? One approach would be to take some understanding of mathematics, such as mathematics as practiced by professional mathematicians, as the mathematics; any practices that disagree with this standard are then not actually doing mathematics. This is an approach that I reject. I do acknowledge that what I would describe as the philosophy of academic mathematics is a valid inquiry in itself, but I am interested in mathematics in a much more general sense. From the point of view I am taking for this paper, saying that there is one true mathematics, and that everyone who is not following the true mathematics is just carrying out some mathematics-like practices merely hijacks the word "mathematics". It still leaves the question of what these mathematics-like practices are and how they relate to each other and to the one true mathematics, which is exactly the same question as my "What are these different mathematics and how do they relate to each other?", only the usage of the word "mathematics" has changed. I can find no grounds for claiming any special right to the word "mathematics" (or related words), so I am not prepared to prescribe what mathematics should be before I describe what people think mathematics is.

Instead of asking "What is mathematics?", I'm now left with the question "What is it that people think of as mathematics?", giving no special priority to what I think of as mathematics. This puts my approach firmly into the grounds of naturalized epistemology. There are two important points to make about this approach: the first is that the question I am attempting to answer is, on the face of it, more suited to the social sciences of psychology, sociology and anthropology than it is to analytic philosophy. We need to study people's mathematical practices to gather the data needed to back up theories of what they think mathematics is, and it is the social sciences that would be needed to gather the relevant data. However, as I said earlier, this thesis is a work of analytic philosophy, and it can be read in two ways: firstly as a work in analytic philosophy it comes to the basically sceptical conclusion that there is no way we can know if our own understanding of mathematics coincides with anyone else's, hence, we cannot justify generalising an ontology or epistemology based on introspection, and there is no way within philosophy of discerning the ontology or epistemology of the mathematics of others. The thesis can also be read as a hypothesis in the sociology/anthropology of mathematics which is testable within a scientific framework: individuals each have their own understanding of mathematics; for any two people with differing understandings of mathematics there will be questions about mathematics to which they would give different answers. If this is true then we should be able to identify the areas in which people disagree about mathematics, and the nature of these disagreements should give us clues as to the ontology and epistemology of the mathematics.

Analytic philosophy on its own is not up to the task of giving a complete account of mathematics as practised (just as it is not up to the task of giving a full account of the physical world). This thesis attempts the first step towards a fuller picture of mathematics.

© Patrick Killeen 1995