An Epistemic-Structuralist Account
of Mathematical Knowledge
Researcher: Lisa Lehrer Dive
- Date Complete: 2003/03
- Degree Awarded: Ph.D.
- Research Supervisors: A.D. Irvine and J. Bacon
Abstract
This thesis aims to explain the nature and justification of mathematical
knowledge using an empirical version of mathematical structuralism.
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 empirical version
of structuralism that is advocated in this thesis takes structures to be
primarily physical, rather than Platonically abstract or theoretical entities.
The primary feature of empirical structuralism is its motivation, which
is primarily epistemic.
In explicating the justification of mathematical knowledge, two notions
of abstraction are introduced. Abstraction by simplification is how
we extract
mathematical structures from our experience of the physical world, and we
use further abstraction by extension, simplification or recombination
to conceive
of derivative mathematical structures.
It is argued that mathematical theories, like all other formal systems,
do not completely capture everything about that aspect of the world
they describe.
This is made evident by exploring the implications of Skolem’s paradox
and arguing that the result demonstrates 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, mathematical knowledge
becomes increasingly
powerful and accurate in its descriptions of mathematical systems.
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