Axiom

https://en.wikipedia.org/wiki/Axiom

An axiom is a statement that serves as a starting point from which other statements can be logically derived.

Whether an axiom is meaningful, and, if so, what is its meaning, e.g. what it means for an axiom to be "true" is a subject of debate in philosophy and philosophy of mathematics.

Broadly, an axiom is a statement assumed to be true, to serve as a premise for further reasoning and argumentation. The word comes from the Greek ἀξίωμα (axíōma) "that which is thought worthy or fit" or "that which commends itself as evident".

However, even though some statements may be evident enough to be taken for granted (justifying this acceptance with one's intuition, experience and knowledge), not all statements fall into this category; and even the ones that do, cannot really be proven without succumbing into the infinite regression (because defining or proving a term requires defining or proving its constituent terms, and so on, ad infinitum). And as far the truths that are not (easily) found in nature and experience are concerned, the only option is to take them for granted anyway - because all theories must start somewhere, with a certain set of truths, the axioms should be understood to form the hypothesis of a conditional clause: if you assume this particular set of axioms, then this particular theorem follows. Therefore, an axiom need not be true at all; the question is not about the truth of axioms per se, but about the truth of the conditional with those axioms as the hypothesis and some theorem as the conclusion, with the conclusion logically following from the axioms.

In classic philosophy, an axiom is a statement that is so evident or well-established, that it is taken for granted, without controversy. In modern logic, an axiom is a premise or starting point for reasoning.

In mathematics, the term "axiom" is used in two related senses:

• Logical axioms are usually statements, assumed true, within some defining system of logic, denoted symbolically, e.g. A ∧ B -> A.

• Non-logical axioms are independent assertions about the elements of the domain of a specific mathematical theory, e.g. a + b = b + a in arithmetic. In this sense the term axiom is synonymous with postulate, assumption, hypothesis. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and may not even be self-evident in nature (e.g. the "parallel postulate" in Euclidean geometry).

To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences i.e. the axioms. There may be multiple ways to axiomatize a given mathematical domain.

Classic take with Euclid's postulates

Among the ancient Greek philosophers an axiom was a claim which could be seen to be self-evidently true without any need for proof.

The root meaning of the word postulate is "to demand" and so it is also used in this sense: an author demands of a reader to acknowledge some fact, that some particular thing can be achieved. For example, Euclid demands that one agrees that any two points can be joined by a straight line.

Classically, there was a slight distinction between axioms and postulates: the axioms were common across the sciences, but the postulates were particular to each science. The validity of postulates was to be established by means of real-world experience.

The classical approach is well-illustrated in Euclid's Elements, where Euclid first introduces a list of postulates (common-sense geometric facts drawn from the common experience), followed by a list of "common notions" (very basic, self-evident assertions, aka axioms).

Postulates (geometry-specific notions): 1. It is possible to draw a straight line between two points. 2. It is possible to extend a line segment continuously in both directions. 3. It is possible to describe a circle with any center and any radius. 4. It is true that all right angles are equal to one another. 5. Parallel Postulate: It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles. That is, if a line intersects two lines by making all the interior angles right angles, then the two lines are parallel (never intersect in Euclidean geometry).

Common notions (axioms, non-specific common notions):

• Things which are equal to the same thing are also equal to one another.

  transitivity of equality: (a = b ⋀ b = c) -> a = c

• If equals are added to equals, the wholes are equal.

  (a = b ⋀ c = d) -> a + c = b + d

• If equals are subtracted from equals, the remainders are equal.

  (a = b ⋀ c = d) -> a - c = b - d

• Things which coincide with one another are equal to one another.

  symmetry of equality: a = b <-> b = a

• The whole is greater than the part.

  (c = a ⋃ b) ~~> a ⊆ c

Modern development

A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning from the mathematical definitions and assertions (axioms, postulates, propositions, theorems) and that one must concede the need for primitive notions (undefined terms or concepts) in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. (and sidesteps the whole issue of truth of initial assumptions).

Axioms and infinite regression

Since a theory must start somewhere to avoid the infinite regression, it is free to make a "cut" and choose the starting point in the form of a term that is central to a particular theory, which is introduced without definition. Such starting term is supposed to be taken for granted by appealing to people's intuition as something that is obviously true. For example, the central notion in set theory is set, introduced without definition, sometimes as a mathematical primitive, other times as an axiom (e.g. "there exists the empty set").

It is usually insisted that axioms are obvious truths that are sufficiently evident to anybody with a human's intuition and perception, but this doesn't seems rigorous enough to withstand every critic. And perhaps it should be; why not just let axioms be whatever the author chooses? This is what they come to anyway.

Therefore, in a certain theory, the author comes up with a set of axioms and it is free to choose however ridiculous assumptions as they want. The whole point is this: if you accept this set of axioms as true, then these derived truths follow. I mean, for a number of theories, it already comes down to this. Axioms should be free of obligations and justification. Then, one mathematical matter would have many theories, eash one based on a different set of axiom. The problem is then to rank the competing theorems. An evident way would surely be based on a number of axioms; ceteris paribus, a theory with less axioms would surely subsume the one with more axioms. And if two competing theories have more-less the same number of axioms, perhaps the elegance would play a part, or lobbing, bribe or influencers.