Vukutu

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.

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.