An epistemic structuralist account of mathematical knowledge
Dissertation, University of Sydney (
2003)
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Abstract
Structuralism, the theory that mathematical entities are recurring structures or
patterns, has become an increasingly prominent theory of mathematical ontology in
the later decades of the twentieth century. The epistemically driven version of
structuralism that is advocated in this thesis takes structures to be primarily physical,
rather than Platonically abstract entities. A fundamental benefit of epistemic
structuralism is that this account, unlike other accounts, can be integrated into a
naturalistic epistemology, as well as being congruent with mathematical practice.
In justifying mathematical knowledge, two levels of abstraction are
introduced. Abstraction by simplification is how we extract mathematical structures
from our experience of the physical world. Then, abstraction by extension,
simplification or recombination are used to acquire concepts of derivative
mathematical structures.
It is argued that mathematical theories, like all other formal systems, do not
completely capture everything about those aspects of the world they describe. This is
made evident by exploring the implications of Skolem’s paradox, Gödel’s second
incompleteness theorem and other limitative results. It is argued that these results
demonstrate the relativity and theory-dependence of mathematical truths, rather than
posing a serious threat to moderate realism.
Since mathematics studies structures that originate in the physical world,
mathematical knowledge is not significantly distinct from other kinds of scientific
knowledge. A consequence of this view about mathematical knowledge is that we can
never have absolute certainty, even in mathematics. Even so, by refining and
improving mathematical concepts, our knowledge of mathematics becomes
increasingly powerful and accurate.