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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
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