The SlugMath Wiki is under heavy development!

# Help:Developers

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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.

## Proof Markup

### 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:

 Hypothesis A direct argument Conclusion

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:

 \$x=0\$ A proof of \$\Phi(0)\$. A proof that if \$\Phi(i)\$ for all \$i < x\$, then \$\Phi(x)\$. Conclusion

### 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\$.

### ThreeCaseProofs

##### Toolbox
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