This summer I've been reading one of those few books that can change your life, The Road to Reality by Roger Penrose. This is basically all of modern theoretical physics, including the mathematical fundamentals starting from first principles. I have finally gotten through all the initial mathematical chapters, which include a quick rundown of Riemannian geometry and differential forms on manifolds. Why is it that mathematical physicists make the best math teachers? I also changed my life a few years back by reading Lectures on Mathematical Physics by Robert Geroch.
I suspect that this approach to geometry could be useful for a new approach to semantics and cognitive science (see previous post on the musings of Fenstad).
Geometry is so crucial to our understanding of reality (i.e. physics), it should not be surprising if it turns out to be crucial to our understanding of language and cognition as well.
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