Since a lot of the constructions in differential geometry arise from universal properties, most of them can be done by "type chasing": If the types fit, then the construction works out. Examples include:
- Pushforwards
- Pullbacks
- Directional Derivatives
It would be nice to see how much something like sledgehammer can automatically
fill in.
So, let's build Diffgeo in Coq! But do we really want to go through the suffering of constructing reals with Dedekind cuts or cauchy sequences? I for one don't enjoy analysis nearly enough to put myself through that.
Enter infinitesimal analysis_ - A clean axiomatisation of real numbers
that crucially relies on living in constructive logic to create infinitesimals
such as dx, and provides axioms to perform "physicist-style" proofs
rigorously!
Many thanks to Arnaud Spiwackfor showing me this axiomatization.
- Synthetic Differential Geometry (Blog post by Andrej Bauer)
- First steps in syntehtic computability (Math inside the effective topos).
- Axiomatize R and then move on.