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This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.

An axiom, also known as a presupposition, is an assumption in a logical branch or argument from which premises can be fed, implications derived, et cetera. Different sets of axioms being used are called "logical branches". The branch of classical logic, founded around 350 BCE by Aristotle, has the three axioms of:

  • The law of identity: A = A, that is, A is identical to itself.
  • The law of non-contradiction: A ∧ ~A = False, that is, A is mutually exclusive with anti-A.
  • The law of the excluded middle: A ∨ ~A, everything is either true or not true with no graduations of validity.

Branches of logic exist which add axioms or remove them. Some axioms, called presuppositions, are actually just premises in disguise, which are typically implicitly stated like an axiom is. For instance, in Alice in Wonderland:

"How am I to get in?" asked Alice again, in a louder tone.
"Are you to get in at all?" said the Footman. “That's the first question, you know."

Alice has an implicit premise that she is to get in at all, as the Footman explicitly states. This is still compatible with the laws of classical logic, but it simply adds a premise. The premises of science are explained in philosophy of science, for instance. One might be tempted to call these "axioms", and in a way, they are, but depending on the context, most axioms can be reclassified as presuppositions (although "presupposition" is typically used as a snarl word, as in presuppositionalism). A critical difference is that the axioms of classical logic or axiomatic set theory dictate forms of argument, not content: they form the basis for rules of inference, while the presuppositionalist "axiom" can only dictate the content of arguments, and not their form.

Branches of logic which remove axioms from classical logic include the various forms of fuzzy logic, which replaces the law of the excluded middle with a continuum, and paraconsistent logic, which rejects the principle of explosion (that anything validly follows from a contradiction) in various ways (for instance by rejecting the law of the excluded middle). Dialetheism, which is arguably a version of paraconsistent logic, establishes that the the law of non-contradiction is an unnecessary axiom in logic. Intuitionism or constructivism rejects the law of excluded middle as an axiom as well, though the most common versions retain explosion and do not deny that the law of excluded middle is correct (it's just not an axiom). The problem with Aristotle's affirmation of the 'law' of the excluded middle is that he was believed, almost without contradiction, for hundreds of years — leading to the rise, in the West, of the ubiquitous dichotomy, something that's useful if you need to get something done quickly, but not if you wish to have a considered understanding.

Axioms are wonderful things in logic, as they define logic outside of presuppositionalism, which has as its axioms that axioms are defined by God and what God defines as axioms are in His holy book. In some branches, axioms are seen as unquestionable, while some branches openly invite the criticism of even its foundations. It makes for a fun time when the foundations of a logical branch is criticized by a member of that branch (almost like a civil war) or when members of different logical branches get into arguments. Axioms are typically hidden, and the other guy is assumed to share the same axioms, leading to awkwardness when they don't. Examples include most debates with fundies.

Neglecting the history of science can have pernicious effects. It can mislead us into glibly asserting that principles and axioms are “self-evident” and it is with this concern that Ernst Mach writes:

Quite analogous difficulties lie in wait for us when we go to school and take up more advanced studies, when propositions which have often cost several thousand years’ labour of thought are represented to us as self-evident. Here too there is only one way to enlightenment: historical studies.
—Ernst Mach, 1911. History and Root of the Principle of the Conservation of Energy. p. 16.[1]

– Mach, later, continues in the same spirit, emphasising the importance of understanding the evolution of concepts –

We are accustomed to call concepts metaphysical, if we have forgotten how we reached them. One can never lose one’s footing, or come into collision with facts, if one always keeps in view the path by which one has come.
Idem., Op. cit., p. 17

J.L. Austin, expressed similar concerns about philosophy:

I say ‘Scholastic’ but I might as well have said ‘philosophical’; over-simplification, schematisation, and constant obsessive repetition of the same small range of jejune ‘examples’ are not only peculiar to this case but far too common to be dismissed as an occasional weakness of philosophers.”
— J.L. Austin, 1962. Sense and sensibilia[2]

– Austin then stresses the need to overcome such proclivities –

It is essential, here as elsewhere, to abandon all habits of Gleischaltung, the deeply ingrained worship of tidy-looking dichotomies.
Idem., Op. cit.

Examples of logical branches and their axioms[edit]

  • Classical logic (Aristotle, circa 350 BCE)
  • Stated above.
  • Many-valued logic (ironically also Aristotle, though he never formalized his thoughts on the matter)
  • Many-valued logics are logical systems which reject the third axiom of classical logic, the law of the excluded middle. Freed from the constraint of binary truth or falsehood for any given proposition, they include the possibility of multiple values of truth, which may be discrete or uncountably infinite. Fuzzy logics, which assign truth values on a continuum between zero (equivalent to binary falsehood) and 1 (equivalent to binary truth) are perhaps the best-known type of many-valued logic, and are closely related to Bayesian statistics, in addition to having many applications in engineering.
  • Paraconsistent logic (old, but the term dates from the early 20th century)
  • Paraconsistent logics are logical systems which discard the law of non-contradiction, usually in order to reject the principle of explosion. Different paraconsistent systems may add non-classical axioms but all share the rejection of non-contradiction.
  • Set logic (or set theory — formal study dates from the 1870s but the modern axiomatic systems are 20th-century inventions)
  • Set logics formalize the study of sets, which are very important objects in modern mathematics. The canonical set theory is Zermelo-Fraenkel set theoryWikipedia with the axiom of choice. It takes far more axioms than classical logic (and an axiom schema, which defines infinite axioms), which tend to be intuitive but whose importance is generally not particularly easy to understand without a grounding in mathematics. Set theory and its related logics may be treated as a foundation for mathematics, explaining the great effort given to, e.g., resolving Russell's paradox (from which neither Bertrand Russell's own type theory nor Zermelo-Fraenkel suffers).

These categories are neither exhaustive nor necessarily mutually exclusive. There do exist papers in the world discussing the construction of, for example, a paraconsistent set theory.

Some sets of axioms in mathematics[edit]

There are several sets of axioms in mathematics depending on which branch is being considered.

  • Euclid'sWikipedia Axioms. There we find the famous parallel axiom.
  • Hilbert'sWikipedia Axioms of Geometry. Hilbert sought to modernize Euclid and ended up with 20 axioms as well as a number of undefined objects.
  • PeanoWikipedia used five axioms to create the natural numbers.
  • Linear Vector SpacesWikipedia require 10 axioms. Since the real numbers are a linear vector space, that takes care of them.
  • Probability theoryWikipedia makes use of three axioms set forth by Andrey Kolmogorov.Wikipedia

This list is just to show that there is not one single set of axioms containing all of mathematics.

See also[edit]

  1. Mach, E., 1911. History and Root of the Principle of the Conservation of Energy. Open Court Publishing.
  2. Austin, J.L. and Warnock, G.J., 1962. Sense and sensibilia (Vol. 83). Oxford: Clarendon Press.