4 research outputs found
Poincaré's philosophy of mathematics
The primary concern of this thesis is to investigate
the explicit philosophy of mathematics in the work of
Henri Poincare. In particular, I argue that there is
a well-founded doctrine which grounds both Poincare's
negative thesis, which is based on constructivist
sentiments, and his positive thesis, via which he retains
a classical conception of the mathematical continuum.
The doctrine which does so is one which is founded on
the Kantian theory of synthetic a priori intuition.
I begin, therefore, by outlining Kant's theory of the
synthetic a priori, especially as it applies to mathematics.
Then, in the main body of the thesis, I explain how the
various central aspects of Poincare's philosophy of
mathematics - e.g. his theory of induction; his theory
of the continuum; his views on impredicativiti his
theory of meaning - must, in general, be seen as an
adaptation of Kant's position. My conclusion is that
not only is there a well-founded philosophical core to
Poincare's philosophy, but also that such a core provides
a viable alternative in contemporary debates in
the philosophy of mathematics. That is, Poincare's
theory, which is secured by his doctrine of a priori
intuitions, and which describes a position in between
the two extremes of an "anti-realist" strict constructivism
and a "realist" axiomatic set theory, may indeed be
true
After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics
The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics