David Corfield space

Idea that Anglophone philosophy of geometry stopped prematurely, plenty of catching up to do. Can just start to sketch in some important moments in post 1930’s geometry.

Idea that with the axiomatisation of geometry by Hilbert, philosophers took the role of geometric intuition to be inessential. Then whether an axiom system concerned something geometric rather than say something analytic or algebraic hardly seems salient. With the reduction of all mathematics to set theory, what need was there to think about what was specific about geometry.

Nineteenth century

New models and gluing

Klein and new homogenous spaces as models. Riemann: sticking model patches together.

Standard logical empiricism

There is no synthetic a priori of geometry: either geometry is a priori, and then it is mathematical geometry and analytic- or geometry is synthetic, and then it is physical geometry and empirical. The evolution of geometry culminates in the disintegration of the synthetic a priori. (Reichenbach, 1951:14).

Weyl

As applying Husserlian “analysis of essence” to geometry

Cassirer

Klein Program and perception

Lautman

Arithmetic

Weil and Rosetta Stone. When did idea of an arithmetic geometry arise?

Principal bundle

Atiyah on geometry

“What is Geometry?”

Broadly speaking I want to suggest that geometry is that part of mathematics in which visual thought is dominant whereas algebra is that part in which sequential thought is dominant. This dichotomy is perhaps better conveyed by the words “insight” versus “rigour” and both play an essential role in mathematical problems.“ (293, 183 of Collected Works)

So rather like Arnold’s left/right brained. Then you enter can there be a rigour through intuitive proof debates.

What I’m after is more the intrinsic duality between algebra and geometry.

Relation of invariants to those of underlying topological space, e.g., Gauss-Bonnet more relevant in ‘The Role of Algebraic Topology in Mathematics’.

Grothendieck

Scheme, fibre functor, Galois group

Scheme as space whose points are prime ideals. Points need not be homogeneous. Functions on space to different fields at different points.

Lawvere and Cohesion

Martin-Lof

Lurie

$\infty$-toposes

Just as an ordinary scheme is defined to be “something which looks locally like Spec $A$ where $A$ is a commutative ring”, a derived scheme can be described as “something which looks locally like Spec $A$ where $A$ is a simplicial commutative ring”. (DAG V)

Cohesive homotopy type theory

As capable of capturing just about everything geometric

Check Handbook Can logicism be extended to it; history of philosophy; history of formalism.