My whirlwind tour of maybe eighty percent of all the math I know. Buckle up!
Outline
Vol. 1: Arithmetic of the Continuum.
I. The Real Number Line.
Ch. 1: Logic and Order
begins with propositional/"zeroth order" logic and applications to classical logic gates, with an emphasis on NAND. This gives us a base for specifying binary relations such as preorders and functions. The core of Ch. 1 culminates in a proof of the Schroeder Bernstein theorem, which states that double sided injections invert, via Knaster-Tarski.
Ch. 2: The Continuum, Spatially
begins with a crash course on open/closed sets, continuity, connectedness/compactness, and the major separation axioms. I frame things in terms of nets because I think nets are cool. After Top is outlined, we move to Met and construct the reals via nested Cauchy sequences. We end with six forms of completeness, all equivalent in the presence of the Archimedean Property.
II. Products of the Line.
Ch. 3: Frames of Lines
(theory of modules / linear algebra, with applications to differential equations etc)
Ch. 4: Counting in the Gaps
(from vector spaces to ordered vector spaces: measure theory and integration via the Riesz space formalism)
Ch. 5: The Fundamental Theorem
(goes from Radon-Nikodym as a definition of derivative up to Schwartz distributions and the Fourier Transform)
Ch. 6: Some Closed Quantum Systems
(applies previous chapter by focusing mainly on the quantum harmonic oscillator, ladder operators in particular, as a bridge to vol. 2)
Vol. 2: Rhapsody in C.
III. The Argand Diagram and Friends.
IV. Holomorphicity in Three Dimensions.
Vol. 3: Squaring the Circle.
V. Rotation, Orthogonality, and Spheres.
VI. Prologue in Heaven.
Previews
fair warning: much here is untested and some of it independent if not original. there may be logical errors.
| Intro to Vol. 1 |
PDF |
| Forms of Completeness |
PDF |
| Lebesgue's Number Lemma via EVT³ |
PDF |
| Intro to Vol. 2 |
PDF |