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.

Last revised on November 30, 2019 at 10:08:54. See the history of this page for a list of all contributions to it.