Wikipedia:Foundations of mathematics
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This page in a nutshell: Articles on foundations of mathematics require special guidelines |
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
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 #Transtheoretical notions 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:
- An axiom or a common knowledge theorem, e.g. any tautology (logic) is a theorem of classical propositional calculus. See #Proofs of axioms 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.
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.
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 "
Articles about transtheoretical notions should not focus on the definition and use in particular theory, but must give a broad picture.
Logical connectives
All
In articles about
Entailment and rules of inference
"
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
Such a proof as the Hypothetical syllogism#Proof (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 and a proof deriving it from :
This proof is valid for real, complex and rational numbers, but it is invalid for any field of
On the other hand, some facts may be explained using simplified or biased paradigms, instead of presenting a true proof. For example, the
Proofs of axioms
Contrary to usual perception, proofs of axioms and unordered graphs of proofs are not heresy nor are they logical flaws like
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.