User:IssaRice/Subfield of math as study of concepts preserved under transformation

See https://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/ for an articulation of the general idea.

Many of the examples are from Tao's post.

Subfield	Abstract object	Concrete representation	Transformation	Examples of concepts (invariant under transformation; doesn't depend on concrete representation)	Counterexamples of concepts (depends on concrete representation)
Probability theory	Events	Subsets of a sample space	Extension of the underlying sample space	Probability of event, Boolean operations (union, intersection, complement), equality of events	Cardinality of event
Differential geometry			coordinate change
graph theory			relabeling of the vertices
Computability theory			Computational model (Turing machine, partial recursive function, lambda calculus, register machine); also encoding	Computability
Linear algebra	Vector space	Coordinate representation with respect to a basis	I think there are multiple here. Some ideas like solution set, range, null space are preserved under row operations. Matrix similarity is defined in terms of change of coordinates (so e.g. diagonalizability is preserved under change of coordinates). Then there's orthogonal diagonalization etc.
Arithmetic	Number	Numeral	Different numeral systems	Sum, difference, product, quotient	Numerator, denominator, leading digit
Game theory	Games	extensive-form representation, normal representation?		Nash equilibrium? other solution concepts?
Decision theory	Utility	Utility function	Affine transformation (with positive slope)	Preference ordering	Utility value (the specific real number)

there is also the idea of a "funnel result" that you can prove, after which you get to think intuitively. e.g. in computability, you can prove s-m-n theorem and existence of universal function, after which you can forget about your specific encoding of partial recursive functions. also in a way, the church-turing thesis is the statement that such a funnel result exists?

in real analysis, the "funnel result" is the least upper bound property (and the other field axioms); after you get this, it doesn't matter if you started out with Cauchy sequences, dedekind cuts, etc.

See also