David Corfield Friedman's schema

Changes involve promotion and demotion. Observations may concern regularities which will form principles in future versions, and those which will lead to a existing principle only being seen as approximately correct.

Physics

Newton

Mathematical language: Infinite 3D Euclidean space + calculus

Coordinating principles: Newton’s Laws of motion

Empirical laws and regularities: Law of Gravitation, inertial mass = gravitational mass

Einstein

Mathematical language: 4D-pseudo Riemannian manifold + tensor calculus

Coordinating principles: Invariance of speed of light, Einstein’s equivalence principle, freefall as geodesic motion.

Empirical laws and regularities: Field equations, approximately flat time slices.

Something about QM and QFT and gauge theory

String theory

Mathematical language: Cohesive \HoTT, twisted equivariant differential cohomology

Coordinating principles: quantum gauge field theory

Empirical laws and regularities: String and M-theory

Cohomology

In 1930s

Constitutive language: algebra and topology, as set theoretic.

Theories: various defined homology and cohomology theories, associating algebraic entities to spaces

Observations: some regularities found, e.g., (simpler) spaces give the same results for any theory, homotopy invariance.

By 1952 (work done in 1940s)

Constitutive language: category theory

Theories: axiomatised (co)homology, Eilenberg-Steenrod axioms, includes some previously observed properties as axioms.

Observations: Cech ?homology? no longer a homology.

1959 The Brown representability theorem for generalized (Eilenberg-Steenrod) cohomology: Allows many new cohomologies, e.g., various cobordism theories (Thom), relating to all quarters of mathematics.

2010

Constitutive language: (∞,1)-topos theory/univalent type theory/homotopy type theory

Theories: thousands of examples of cohomology are components of Hom-space in a (∞,1)-topos; differential cohomology in cohesive (∞,1)-topos.

Observations: ? (demotions and the to-be-promoted)

Mathematics

1930s

Set theory

Every entity is a set, so can ask about membership everywhere.

Isomorphic structures behave the same

1950s and later

Category Theory

Universal properties, adjunctions, ?All concepts are Kan extensions?

Better to work with nice categories than nice objects. Need to work with operations defined up to homotopy.

2010

Homotopy Type Theory/Higher category theory

Entities are the same if equivalent at right level.

You can understand why set theory was so successful from later systems, but you wouldn’t be able to reconstruct it in all its details.

Set as well-pointed boolean topos, free co-complete on one object, occurs in a chain of 4 adjunctions.