Australian category-theorist Ross Street has an elegant, one-page summary of the first 2,500 years of western mathematics, here. This was apparently a handout given in a talk to the Macquarie University Philosophy Students Society in 1984. I found Street’s high-level view of what (some important) mathematicians have (mostly) been doing illuminating and thought-provoking, and so I reproduce it here.
A nice way to think about topoi, of course, is that due to Rob Goldblatt: a topos is the most general object that has all the properties of the category of sets.
Space, Sets and Beyond
First Cycle: General spaces advance the study of naive geometry
1. Naive geometry: Zeno, Eudoxus.
2. Axiomatic geometry (unique model intended): Euclid, Apollonius (c. 300-200 BC).
3. Algebraic techniques (coordinate geometry): Descartes 1596-1650.
4. Non-Euclidean geometry (independence of the “parallels axiom”: models without parallels axiom constructed from a model with it): Gauss, Bolyai, Lobatchewski (early 19C).
5. Locally Euclidean spaces: Riemann 1826-1866, Lie.
6. Relationships between spaces (continuity, linearity): Cauchy, Cayley, Weierstrass, Dedekind (1880-present).
Second Cycle: Toposes can be viewed as even more general spaces
1. Naive set theory: Peano, Cantor (c. 1900).
2. Axiomatic set theory (unique model intended): Hilbert, Godel, Bernays, Zermelo, Zorn, Fraenkel.
3. Abstract algebra (mathematical logic): Boole, Poincare, Hilbert, Heyting, Brouwer, Noether, Church, Turing.
4. Non-standard set theories (independence of the “axiom of choice” and “continuum hypothesis”; Boolean-valued models; non-standard analysis): Godel, Cohen, Robinson (1920-1950).
5. Local set theory (sheaves): Leray, Serre, Grothendieck, Lawvere, Tierney (1945-1970).
6. Relationships between toposes (a “topos” is a generalized set theory): 1970-present.
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