Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
Relate all of these to multiplication. FT = Fundamental Theorem Multiplication has systemic unit 1. Compare with 0 and blank.
Yoneda Lemma Correspondence (division) Bijection FT Noncommutative algebra. Proper subalgebra of linear transformations of a finite dimensional complex vector space accords with a nontrivial invariant subspace. (Also related to Algebra in that a proper invariant subspace exists.) FT Finite distributive lattice. Elements can be represented as sets, relations as unions or intersections. FT Galois theory. Matches intermediate fields of a field extension and the subgroups of its Galois group. Extension (of domain of addition formula for exponents) Induction FT Topos theory. The slice E/X of a topos E over its object X is likewise a topos. FT Algebraic Ktheory. How to change the ring. FT Ultraproducts. Conditions such that a first order formula is true for an ultraproduct. Noncorrespondence Cantor's theorem. A set has a power set with strictly greater cardinality. Pigeon hole principle. What happens if pigeons outnumber holes. Nonsurjectivity. Intrinsicness  Algorithm Construction FT Surface theory. Gaussian curvature is independent of the embedding. FT Central limit theorem. Sum of independent, identically distributed random variables tends toward normal distribution. Composition (primes) Substitution factors Binomial theorem. I FT Algebra. Polynomial is composed of factors. Existence of solutions to polynomial equations. FT Tessarine algebra. Polynomial of degree n has {$n^2$} roots. FT Arithmetic. Integers are composed of prime factors. FT Cyclic groups. Determines the subgroups of a cyclic group, one for each divisor of the order. FT Finitely generated abelian groups. Characterizes their structure as direct sums. FT Finitely generated modules over a principal ideal domain. Characterizes their structure as direct sums. FT Ideal theory in number fields. Unique factorization of proper ideals in terms of prime ideals. FT Symmetric polynomials. Elementary symmetric polynomials form a basis for symmetric polynomials. FT Projective geometry. Homography is the composition of a finite number of perspectivities. Classification of closed surfaces. Connected sums of tori and one or two real projective planes. components FT Vector calculus. Vector field broken down into curlfree and divergencefree components. I FT Curves. Shape of a 3dimensional curve is determined by its curvature and torsion. Extrema and interior (increasing) Examination of cases FT Linear programming. Extreme values occur at points on the boundary. FT Calculus. Integration of a function is matched with differentiation of its antiderivative. FT Lebesgue integral calculus. Absolute continuity iff is an integral of a derivative. Stokes theorem. FT Geometric calculus. Integral of a derivative over a volume equals the integral of the function over the boundary. Cauchy's integral formula. Complex analytic function, holomorphic function, on a disc is determined by its values on the boundary. Banach fixed point theorem. Contraction mapping has a unique fixed point. Universality  Unique map (uniqueness) Construction FT Homomorphisms. When a subgroup K is mapped to the identity element, then the map can be uniquely interpreted in terms of the cosets of K. Orbit stabilizer theorem. Bijection between an orbit and the set of cosets of the stabilizer subgroup. FT Projective geometry. Unique homography. FT Riemannian geometry. Unique torsionfree metric connection.
