The Platonic Dream

Essays in Fundamental Mathematics

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On the Set of All Numbers

These first essays may seem redundant, because I have covered this subject at the beginning of The Map of Physics. But it well behooves us to begin a series of essays on number theory with a discussion of the subject of the theory and of the object of the theorizing.

When we talk about a set we must begin by defining the set; that is, by establishing the criterion by which we include an element in the set or exclude it from the set. Having done that for the set of all numbers, we will then develop number theory by examining the logical relationships among the members of the set and of various subsets.

GO TO: Logical Proof

GO TO: Mathematical Induction

GO TO: The Original Boolean Algebra

GO TO: The Axioms of Set Theory

GO TO: The Natural Numbers and The Integers

    GO TO: The Integers

GO TO: Elementary Infinity

GO TO: The Real Numbers

GO TO: Advanced Infinity

    GO TO: An Addendum to Advanced Infinity

    GO TO: The Semantics of Infinity

    GO TO: How Diagonalization Fails

GO TO: Hilbert's First Problem

GO TO: Infinitesimal

GO TO: The Complex Numbers

GO TO: The Fundamental Theorem of Algebra

GO TO: The Fundamental Theorem of Algebra Revisited

Algebraic Playbook

GO TO: The Algebra Playbook

GO TO: Factoring X2+1

GO TO: Everting the Doppler Shift

GO TO: Cardano's Solution of the Depressed Cubic Equation

Number Theory

GO TO: Pythagorean Triples

GO TO: The Bernoulli Numbers

GO TO: Gaussian Sums

GO TO: More on Gaussian Sums

GO TO: Gaussian Sums of Powers

GO TO: Enmeshments

GO TO: The Theorem of Finite Differences

GO TO: Katy's Theorem

GO TO: Pascal's Triangle and Prime Numbers

GO TO: Primality Testing

Infinite Series

GO TO: Euler's Theorem

GO TO: A Minor Point on Infinite Series

GO TO: Powers and Roots of The Simple Geometric Series

GO TO: Powers and Roots of The Alternating Geometric Series

GO TO: Logarithms and Their Infinite Series


GO TO: Leibniz's Theorem

GO TO: The Dirac Delta

GO TO: Dirac's Delta in Polar Coordinates

GO TO: Differentials, Exact and Inexact

GO TO: Vector Multiplication

GO TO: Vector Differentiation

GO TO: Liouville's Theorem


GO TO: Area of an Elliptic Sector

GO TO: Coordinate Transformations

GO TO: The Enclosure Element

GO TO: The Levi-Civita Tensor

GO TO: Covariant and Contravariant

GO TO: A Little Tensor Geometry

GO TO: Mathematical Properties of the Riemann Curvature Tensor

Differential Equations

GO TO: The Classical Harmonic Oscillator

GO TO: Solving The Wave Equation

GO TO: Solving Hermite's Equation

    GO TO: The Hermite Polynomials

GO TO: Solving Bessel's Equation