Reviving the Philosophy of Geometry
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In the Anglophone world, the philosophical treatment of geometry has fallen on hard times. This chapter argues that philosophy can come to a better understanding of mathematics by providing an account of modern geometry, including its development of new forms of space, both for mathematical physics and for arithmetic. It returns to the discussions of Weyl and Cassirer on geometry whose concerns are very much relevant today. A way of encompassing a great part of modern geometry via homotopy toposes is discussed, along with the `cohesive’ variant of their internal language, known as `homotopy type theory’. With these tools in place, we can now start to see what an adequate philosophical account of current geometry might look like.
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2015 ◽
Vol 25
(5)
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pp. 1040-1070
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2014 ◽
Vol 25
(5)
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pp. 1100-1115
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