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© 2025. All rights reserved.
Site created with
Hakyll.
Modified theme
lanyon-hakyll
based on
Lanyon.
This somewhat disorganized collection of notes will hopefully eventually get organized into a guide for aspiring philosophers of science.
It has seemed to many people that there is something special about the theoretical representation provided by differential geometry. Some more sophisticated types would say that what’s special about differential geometry is captured by something even more general, viz. fiber bundle theory. That’s an interesting thing to reflect upon, and it connects with the philosophical questions about gauge theories — which is a huge topic. Nonetheless, here are some scattered thoughts about differential geometry.
Hermann Weyl certainly thought that there was something special about differential geometry. See the work of Tom Ryckman
S Mac Lane. Geometrical Mechanics Lectures U Chicago, Winter 1968
C Godbillon, Géométrie Différentielle et Mécanique Analytique.
From circa 1970 until today, philosophers have talked about models of the General Theory of Relativity primarily from a semantic point of view. Proposal: look at GTR syntactically, using synthetic differential geometry (SDG) for clues about how to do this. See e.g. Reyes, Koch. Compare with Mundy’s “syntactic solution” to the hole argument.
Category theory is so general that it is almost more like a language than like a theory. In other words, one can talk, or think about, almost anything in category-theoretic language. That also means that the best ways to learn category theory are by immersion, and by moving in circles where people already speak this language.
Despite these qualifications, one can still learn a lot about category theory by studying books. Here are some that I have benefitted from.
J van Oosten. Basic category theory. These notes are about as concise as can be. Plus they have a self-contained intro to categorical logic.
F Borceux. Handbook of Categorical Algebra (three volumes). Leans more toward the algebraic than the geometric or intuitive. Lots of good exercises but no solution manual.
S Mac Lane. Categories for the Working Mathematician. The classic and the ultimate. However, Mac Lane assumes that his reader has broad background knowledge in abstract mathematics. This book will be less useful to those who’ve never studied things like homology theory.
S Awodey. Category Theory. Leans toward the logical, at a more gentle pace than van Oosten or Borceux.
T Leinster. Basic Category Theory
B Pierce. Basic Category Theory for Computer Scientists
A Asperti and G Longo. Categories, Types, and Structures
Barr and Wells. Category Theory for Computing Science
See above under category theory
M Makkai and G Reyes. First Order Categorical Logic
S Mac Lane and I Moerdijk. Sheaves in Geometry and Logic
C MacLarty. Elementary Categories, Elementary Toposes
W Lawvere and S Rosebrugh. Sets for Mathematics
C Mikkelsen. Lattice Theoretic and Logical Aspects of Elementary Topoi
R Kadison and J Ringrose. Fundamentals of the Theory of Operator Algebras
VS Sunder. An Invitation to von Neumann Algebras
VS Sunder. Functional Analysis and Spectral Theory