An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.
Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example a + 0 = a in integer arithmetic.
[5] In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry).
[6] The word axiom comes from the Greek word ἀξίωμα (axíōma), a verbal noun from the verb ἀξιόειν (axioein), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος (áxios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper".
Among the ancient Greek philosophers and mathematicians, axioms were taken to be immediately evident propositions, foundational and common to many fields of investigation, and self-evidently true without any further argument or proof.
"[9] Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept.
[citation needed] The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference) was developed by the ancient Greeks, and has become the core principle of modern mathematics.
However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom and postulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid.
As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge.
[10] An "axiom", in classical terminology, referred to a self-evident assumption common to many branches of science.
Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates.
The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts.
However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., hyperbolic geometry).
The development of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and not as facts based on experience.
The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction.
Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions.
An early success of the formalist program was Hilbert's formalization[b] of Euclidean geometry,[12] and the related demonstration of the consistency of those axioms.
Here, the emergence of Russell's paradox and similar antinomies of naïve set theory raised the possibility that any such system could turn out to be inconsistent.
Furthermore, using techniques of forcing (Cohen) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms.
If postulates do not allow deducing experimental predictions, they do not set a scientific conceptual framework and have to be completed or made more accurate.
For instance, the introduction of Newton's laws rarely establishes as a prerequisite neither Euclidean geometry or differential calculus that they imply.
It became more apparent when Albert Einstein first introduced special relativity where the invariant quantity is no more the Euclidean length
In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification.
The 'Copenhagen school' (Niels Bohr, Werner Heisenberg, Max Born) developed an operational approach with a complete mathematical formalism that involves the description of quantum system by vectors ('states') in a separable Hilbert space, and physical quantities as linear operators that act in this Hilbert space.
For this reason, another 'hidden variables' approach was developed for some time by Albert Einstein, Erwin Schrödinger, David Bohm.
It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of the propositional calculus.
They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.
Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry.
Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms