Poincaré’s philosophy of mathematics and the impossibilty of building a new arithmetic

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2019
Akçagüner, Koray
This thesis examines Poincaré’s philosophy of mathematics, and in particular, his rejection of the possibility of building a new arithmetic. The invention of non-Euclidean geometries forced Kant’s philosophy of mathematics to change, leading thinkers to doubt the idea that Euclidean postulates are synthetic a priori judgments. Logicism and formalism have risen during this period, and these schools aimed to ground mathematics on a basis other than the one that was laid down by Kant. With regards to the foundations of mathematics, Poincaré adopted Kant’s philosophy and remained an intuitionist, though naturally, he had to make significant changes in Kant’s thought. Poincaré argued that the branch of mathematics that contains synthetic a priori judgments is arithmetic, which is completely independent of experience and therefore pure. What gives arithmetic its object of knowledge and justifies the use of its fundamental principles is not experience, but a pure intuition. On the other hand, Poincaré claimed that our ideas about space and the geometric postulates are not imposed upon us, that they are not known a priori but are rather conventions, “definitions in disguise”. The role experience plays in the foundations of geometry has given us the possibility of building non-Euclidean geometries. However, since arithmetic is completely independent of experience, it is not possible for a change similar to that in geometry to take place in arithmetic, which would alter its basic concepts or principles that we consider to be true. It is argued in this thesis that it is possible to develop the intuition which lies at the basis of arithmetic and this may become the starting point of a new arithmetic. It will be shown that this is what Cantor has actually achieved when establishing transfinite ordinal arithmetic.

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Citation Formats
K. Akçagüner, “Poincaré’s philosophy of mathematics and the impossibilty of building a new arithmetic,” Thesis (M.S.) -- Graduate School of Social Sciences. Philosophy., Middle East Technical University, 2019.