The SlugMath Wiki is under heavy development!
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:
- 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.
- A sentence or collection of sentences, which can be written in the language of set theory, and which (provably) characterize a class of sets.
- 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
- Number theory, including quadratic reciprocity, and basic facts about binary quadratic forms.
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:
|A direct argument|
Color coding and indentation are automatic.
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:
|$x=0$||A proof of $\Phi(0)$.|
|$x>0$.||A proof that if $\Phi(i)$ for all $i < x$, then $\Phi(x)$.|
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:
|NOT $B$ A direct argument|
|Something that contradicts $A$.|