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What is mathematics about? And if it is about some sort of
mathematical reality, how can we have access to it? This is the
problem raised by Plato, which still today is the subject of lively
philosophical disputes. This book traces the history of the
problem, from its origins to its contemporary treatment. It
discusses the answers given by Aristotle, Proclus and Kant, through
Frege's and Russell's versions of logicism, Hilbert's formalism,
Godel's platonism, up to the the current debate on Benacerraf's
dilemma and the indispensability argument. Through the
considerations of themes in the philosophy of language, ontology,
and the philosophy of science, the book aims at offering an
historically-informed introduction to the philosophy of
mathematics, approached through the lenses of its most fundamental
problem.
What is mathematics about? And if it is about some sort of
mathematical reality, how can we have access to it? This is the
problem raised by Plato, which still today is the subject of lively
philosophical disputes. This book traces the history of the
problem, from its origins to its contemporary treatment. It
discusses the answers given by Aristotle, Proclus and Kant, through
Frege's and Russell's versions of logicism, Hilbert's formalism,
Godel's platonism, up to the the current debate on Benacerraf's
dilemma and the indispensability argument. Through the
considerations of themes in the philosophy of language, ontology,
and the philosophy of science, the book aims at offering an
historically-informed introduction to the philosophy of
mathematics, approached through the lenses of its most fundamental
problem.
Time and again, philosophy, in trying to untangle the issues
surrounding the an alytic-synthetic distinction, has doubted that
such a distinction can significantly be drawn at all. We think, in
face of the varied and age-old discussions on it, that such
reflections amount only to one more documentation of the tenacity
of the problems behind this distinction. We could even be justified
in promoting the thesis that this distinction refers to the complex
relationship between the universe of meanings and the universe of
objects and thus concerns each domain of human thinking where a
form of objectivity is pursued. If one accepts such a thesis, one
will find it very natural that this distinction has so frequently
occurred in the history of mathematics and in philosophical
discussions about mathematics. Since Plato, we may encounter quite
a number of interpretations of the ideas of analysis and synthesis,
which are related in one sense or other with mathematical thought.
Mathematicians of all ages have ap pealed to them in order to
distinguish different forms and styles in their argumen tation and
expositions. Philosophers have referred to them for clarification
of the specific character of mathematics in its relations to
knowledge in general. In the present volume various instances of
the analytic-synthetic distinction are discussed in relation to the
history and philosophy of mathematics, and some new perspectives
about possible interpretations and consequences are suggested."
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