Help:Developers

This page contains information for developers of the SlugMath wiki. This page is under heavy construction.

=Mathematical knowledge in the SlugMath wiki=

The SlugMath wiki aims to develop a large body of mathematical knowledge. The goals of this development are described in the following sections:

Structure of knowledge
The primary units of mathematical knowledge in the SlugMath wiki are statements, definitions and structures. These are defined below:
 * Statement: A sentence or collection of sentences, which can be written in the language of set theory and proven using the Zermelo-Fraenkel axioms of set theory.
 * Definition: A sentence or collection of sentences, which can be written in the language of set theory, and which (provably) characterize a class of sets.
 * Structure: A sentence or collection of sentences, which can be written in the language of set theory, and which (provably) characterize a set.

When a statement, definition, or structure is written, a proof must be given. In this proof, logical reliance is stated using the "relies on" property. Proofs must be written in such a way to minimize logical reliance.

All mathematics within the SlugMath wiki should be written in a consistent style and notation. All proofs should be correct, written by a research mathematician, and checked by other mathematicians. Moreover, all proofs should be written in a minimalist style, and commentary should be separated from the proof.

Coverage of mathematics

 * Development of "naive" set theory from the Zermelo-Fraenkel axioms.
 * Basic first-order logic.
 * The construction of high-school mathematics from set theory.
 * Abstract algebra, including the elements of group theory, ring theory, fields, and Galois theory.
 * Basic differential and integral calculus.
 * Differential geometry, including the development of calculus on smooth manifolds with boundary and corners.
 * Classical geometry, including Euclidean and Non-Euclidean axiomatic geometry.
 * Graph theory
 * Combinatorics
 * Number theory, including quadratic reciprocity, and basic facts about binary quadratic forms.

Direct Proofs
A "direct proof" is a linearly structured argument, from a hypothesis to a conclusion, in which each sentences follows from the previous sentences via deduction and previously acquired knowledge. Direct proofs are written using the DirectProof template, as below:

Color coding and indentation are automatic.

Inductive Proofs
An "inductive proof" is a proof that a sentence $\Phi(x)$ is true for all natural numbers $x$ greater than a certain natural number $x_0$. Such a proof involves a base case and an inductive case, together with a conclusion. Inductive proofs are written using the InductiveProof template, as below:

Proofs by contradiction
A proof by contradiction is a proof of a "if-then" structured sentence via the contrapositive. Such a proof has a hypothesis (the sentence which will later be contradicted), a body, and a contradictory conclusion. Proofs by contradiction are written using the ProofbyContradiction template. A proof that $A \Rightarrow B$ is displayed below: