Wikipedia:Foundations of mathematics

As of March 5, 2012, this page is an essay presenting user:Incnis Mrsi's reflexions about a perceived ill-being of articles about mathematical logic in English Wikipedia.

Introduction
In most situations it is acceptable to implicitly use equivalence or different definitions, if such equivalence became a common knowledge. Sometimes, there exist even Wikipedia articles explaining why different definitions are equivalent. But an article about any foundational concept of mathematics requires an approach different from most other mathematical topics. There, we should specify where some things are equivalent, why they are, and, sometimes, how are they equivalent.

Another problem which exists: where there are several competing approaches to foundation of mathematics. If one said something like "The space exploration is not only the American space program", then nobody would refute it, even Americans. But, sadly, today in en.Wikipedia such things as that "The material conditional can not be only classical logic's views on the material conditional" have to be explained, very patiently, then proven and defended.

Formal theories
An article on formal theory (e.g. theory (mathematical logic)) should explain both its structure as a formal system and also its intended semantics. But such wording as "this theory proves that … is a mathematical truth" should be avoided. The fact that some formal proposition is a theorem of a formal theory may also be expressed like: "if <…> were used as the foundation of mathematics, then such results as (…) would become proven". Note that a correspondence between "mathematical facts" and their representation in a formal language may pose a problem itself, see below.

Propositions, rules and metatheorems
An element of a formal theory (or several ones, see below) which has a dedicated Wikipedia article must have explicitly qualified status with respect to the theory. It may be presented as: This latter case actually represents a metatheorem, a fact about the formal system which may be proven by external means, but does not belong to the system itself. Metatheorems may not be confused with formally proven theorems.
 * An axiom or a common knowledge theorem, e.g. any tautology (logic) is a theorem of classical propositional calculus. See about different sets of axioms in equivalent theories.
 * A basic rule of inference. A rule which is postulated and does not need to be proven.
 * A rule of inference derived from basic rules and, possibly, axioms.

The same Wikipedia article unlikely perhaps may be qualified both as an axiom/theorem of some logical calculus and a rule of inference (or metatheorem). As of March 5, there are several such cases in the Category:Rules of inference.

Transtheoretical notions
There are many mathematical notions that initially were or can be used without formal definitions, but were later incorporated to formal theories. Many of such notions were incorporated in competing, essentially different theories. I will refer to such notions as to "trans⁠theoretical". Prominent examples of such are "function" (with some controversy about definitions, e.g. the predicate logic definition via relations and ∃! is unacceptable for constructive mathematics) and "Cartesian product" (with a more general meaning in category theory than in set theory).

Articles about transtheoretical notions should not focus on the definition and use in particular theory, but must give a broad picture.

Logical connectives
All logical connectives, maybe except Pierce's arrow and its dual Sheffer's stroke, are essentially transtheoretical. These articles then may not and ought not unilaterally present the topic from the classical logic perspective.

In articles about mathematical logic, a propositional formula itself should be distinguished from its derived constructs, such as truth function or, more generally, algebraic truth value. A formula written in the common propositional language (logical connectives and propositional variables) does not specify any particular logical system for its interpretation, hence such statements as "$p → q$ is logically equivalent to $¬p ∨ q$" should not be used in logical articles unless a concrete logical system, or some set of logical systems where such equivalence holds, is unambiguously determined by the context.

Entailment and rules of inference
"Entailment" and "rule of inference" are certainly transtheoretical notions. Some concrete rules of inference, such as modus ponens, are transtheoretical too. If one tries to "prove" such a rule, then it must be specified, for which formal theory or metatheory are these reasonings valid.

Theorems
Ideally, the article about a theorem should specify, which theories do prove it, or which theorems can be used to do so. In some cases it is useful to mention which theories do not prove some important result. For example, Hahn–Banach theorem cannot be proved in the most general case without the axiom of choice or equivalent postulates.

Proofs
As mentioned above, for a given "theorem" many theories which prove it can exist. Sometimes these theories are ordered by "strictness": all theorems of a weaker (but "stricter") theory belong to some other stronger (but less strict) theory. In this case Wikipedia should attempt to present those proofs for a stricter theory (if any at all) which exist in the reliable sources. This is motivated by WP:NPOV, a fundamental Wikipedia policy. A proof for a stricter theory will be valid in all stronger theories (e.g. obtained by adding extra axioms or rules of inference).

Such a proof as the Hypothetical syllogism (as of March 5) unlikely are useful, but are fundamentally flawed in its use of negation and equivalences valid in classical logic only. This problem may arise not only in mathematical logic, but in several other branches of mathematics such as algebra. Consider an identity $$2x^2 + 2(x+1)^2 = (2x+1)^2 + 1$$ and a proof deriving it from $$(a+b)^2 + (a-b)^2 = 2a^2 + 2b^2$$:
 * $$2x^2 + 2(x+1)^2 = 2((x+\frac{1}{2}-\frac{1}{2})^2 + (x+\frac{1}{2}+\frac{1}{2})^2) = 4((x+\frac{1}{2})^2 + \frac{1}{4}) = (2(x+\frac{1}{2}))^2 + 1 = (2x+1)^2 + 1$$

This proof is valid for real, complex and rational numbers, but it is invalid for any field of characteristic 2 (there is no such number as ½) and it is inapplicable to unital rings (because there is no division at all), although the identity still holds. The "universally correct" proof should be:
 * $$2x^2 + 2(x+1)^2 = 4x^2 + 4x + 2 = (4x^2 + 4x + 1) + 1 = (2x+1)^2 + 1$$

On the other hand, some facts may be explained using simplified or biased paradigms, instead of presenting a true proof. For example, the trueness of a propositional formula may be explained using truth tables, and some ideas of first-order logic – using notions of the set theory.

Proofs of axioms
Contrary to usual perception, proofs of axioms and unordered graphs of proofs are not heresy nor are they logical flaws like circulus vitiosus. If the article on the "axiom" C′ says $A, B, C ⊢ C′$ and the one about C says $A, B, C′ ⊢ C$, this means equivalence of {A, B, C} and {A, B, C′}. But it is absolutely necessary to explicitly mention from what propositions is derived a given proposition.

Tangled terminology
There are several topics where established terminology is sometimes overlapping and different from a source to source. For example, false (logic) is sometimes a propositional constant and a truth value, but in other sources the term "false" is reserved only for the truth value "0", and corresponding propositional constant (nullary connective) is referred to as the "contradiction".

Such cases may be resolved by hatnotes and indication of particular traditions and sources.