Editing Axiom (section)

==Historical development==

===Early Greeks===
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. [[tautology (logic)|Tautologies]] excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are thus the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions ([[theorem]]s, in the case of mathematics) must be proven with the aid of these basic assumptions. 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]].<ref name=":0" />

The ancient Greeks considered [[geometry]] as just one of several [[science]]s, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's [[posterior analytics]] is a definitive exposition of the classical view.<ref>{{Cite web |date=2024-10-08 |title=Aristotle {{!}} Biography, Works, Quotes, Philosophy, Ethics, & Facts {{!}} Britannica |url=https://www.britannica.com/biography/Aristotle |access-date=2024-11-14 |website=www.britannica.com |language=en}}</ref>

An "axiom", in classical terminology, referred to a [[self-evident]] assumption common to many branches of science. A good example would be the assertion that:
<blockquote>When an equal amount is taken from equals, an equal amount results.</blockquote>

At the foundation of the various sciences lay certain additional [[Hypothesis|hypotheses]] that were accepted without proof. Such a hypothesis was termed a ''postulate''. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates.<ref>Aristotle, Metaphysics Bk IV, Chapter 3, 1005b "Physics also is a kind of Wisdom, but it is not the first kind. – And the attempts of some of those who discuss the terms on which truth should be accepted, are due to want of training in logic; for they should know these things already when they come to a special study, and not be inquiring into them while they are listening to lectures on it." W.D. Ross translation, in The Basic Works of Aristotle, ed. Richard McKeon, (Random House, New York, 1941)</ref>

The classical approach is well-illustrated{{efn|Although not complete; some of the stated results did not actually follow from the stated postulates and common notions.}} by [[Euclid's Elements|Euclid's ''Elements'']], where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of "common notions" (very basic, self-evident assertions).

:;Postulates
:# It is possible to draw a [[straight line]] from any point to any other point.
:# It is possible to extend a [[line segment]] continuously in both directions.
:# It is possible to describe a [[circle]] with any center and any radius.
:# It is true that all [[right angle]]s are equal to one another.
:# ("[[Parallel postulate]]") It is true that, if a straight line falling on two straight lines make the [[polygon|interior angles]] on the same side less than two right angles, the two straight lines, if produced indefinitely, [[Line-line intersection|intersect]] on that side on which are the [[angle]]s less than the two right angles.

:;Common notions:
:# Things which are equal to the same thing are also equal to one another.
:# If equals are added to equals, the wholes are equal.
:# If equals are subtracted from equals, the remainders are equal.
:# Things which coincide with one another are equal to one another.
:# The whole is greater than the part.

===Modern development===
A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, [[propositional logic|propositions]], theorems) and definitions. One must concede the need for [[primitive notion]]s, or 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. [[Alessandro Padoa]], [[Mario Pieri]], and [[Giuseppe Peano]] were pioneers in this movement.

Structuralist mathematics goes further, and develops theories and axioms (e.g. [[Field theory (mathematics)|field theory]], [[group (mathematics)|group theory]], [[topological space|topology]], [[linear space|vector spaces]]) without ''any'' particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., [[hyperbolic geometry]]). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility. 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.

When mathematicians employ the [[Field (mathematics)|field]] axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.

It is not correct to say that the axioms of field theory are "propositions that are regarded as true without proof." Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.

Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and [[logicism|mathematics itself can be regarded as a branch of logic]]. [[Gottlob Frege|Frege]], [[Bertrand Russell|Russell]], [[Henri Poincaré|Poincaré]], [[David Hilbert|Hilbert]], and [[Kurt Gödel|Gödel]] are some of the key figures in this development.

Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions.

In the modern understanding, a set of axioms is any [[Class (set theory)|collection]] of formally stated assertions from which other formally stated assertions follow – by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be [[consistent]]; it should be impossible to derive a contradiction from the axioms. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.

It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization{{efn|Hilbert also made explicit the assumptions that Euclid used in his proofs but did not list in his common notions and postulates.}} of [[Euclidean geometry]],<ref>For more, see [[Hilbert's axioms]].</ref> and the related demonstration of the consistency of those axioms.

In a wider context, there was an attempt to base all of mathematics on [[Georg Cantor|Cantor's]] [[set theory]]. 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.

The formalist project suffered a setback a century ago, when [[Gödel's incompleteness theorems|Gödel showed]] that it is possible, for any sufficiently large set of axioms ([[peano arithmetic|Peano's axioms]], for example) to construct a statement whose truth is independent of that set of axioms. As a [[corollary]], Gödel proved that the consistency of a theory like [[Peano arithmetic]] is an unprovable assertion within the scope of that theory.<ref>{{Citation|last=Raatikainen|first=Panu|title=Gödel's Incompleteness Theorems|date=2018|url=https://plato.stanford.edu/archives/fall2018/entries/goedel-incompleteness/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2018|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-10-19}}</ref>

It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of [[natural number]]s, an [[Infinite set|infinite]] but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern [[Zermelo–Fraenkel axioms]] for set theory. Furthermore, using techniques of [[forcing (mathematics)|forcing]] ([[Paul Cohen|Cohen]]) one can show that the [[continuum hypothesis]] (Cantor) is independent of the Zermelo–Fraenkel axioms.<ref>{{Citation|last=Koellner|first=Peter|title=The Continuum Hypothesis|date=2019|url=https://plato.stanford.edu/archives/spr2019/entries/continuum-hypothesis/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Spring 2019|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-10-19}}</ref> Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.

===Other sciences===
Experimental sciences - as opposed to mathematics and logic - also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties - either still general or much more specialized to a specific experimental context. For instance, [[Newton's laws]] in classical mechanics, [[Maxwell's equations]] in classical electromagnetism, [[Einstein's equation]] in general relativity, [[Mendel's laws]] of genetics, Darwin's [[Natural selection]] law, etc. These founding assertions are usually called ''principles'' or ''postulates'' so as to distinguish from mathematical ''axioms''.

As a matter of facts, the role of axioms in mathematics and postulates in experimental sciences is different. In mathematics one neither "proves" nor "disproves" an axiom. A set of mathematical axioms gives a set of rules that fix a conceptual realm, in which the theorems logically follow. In contrast, in experimental sciences, a set of postulates shall allow deducing results that match or do not match experimental results. 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. If the postulates allow deducing predictions of experimental results, the comparison with experiments allows falsifying ([[Falsifiability|falsified]]) the theory that the postulates install. A theory is considered valid as long as it has not been falsified.

Now, the transition between the mathematical axioms and  scientific postulates is always slightly blurred, especially in physics. This is due to the heavy use of mathematical tools to support the physical theories. 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 <math>l</math> (defined as <math>l^2 = x^2 + y^2 + z^2</math>) > but the Minkowski spacetime interval <math>s</math> (defined as <math>s^2 = c^2 t^2 - x^2 - y^2 - z^2</math>), and then [[general relativity]] where flat Minkowskian geometry is replaced with [[pseudo-Riemannian]] geometry on curved [[manifolds]].

In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The '[[Copenhagen interpretation|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. This approach is fully falsifiable and has so far produced the most accurate predictions in physics. But it has the unsatisfactory aspect of not allowing answers to questions one would naturally ask. For this reason, another '[[hidden-variable theory|hidden variables]]' approach was developed for some time by Albert Einstein, [[Erwin Schrödinger]], [[David Bohm]]. It was created so as to try to give deterministic explanation to phenomena such as [[quantum entanglement|entanglement]]. This approach assumed that the Copenhagen school description was not complete, and postulated that some yet unknown variable was to be added to the theory so as to allow answering some of the questions it does not answer (the founding elements of which were discussed as the [[EPR paradox]] in 1935). Taking this idea seriously, [[John Stewart Bell|John Bell]] derived in 1964 a prediction that would lead to different experimental results ([[Bell's inequalities]]) in the Copenhagen and the Hidden variable case. The experiment was conducted first by [[Alain Aspect]] in the early 1980s, and the result excluded the simple hidden variable approach (sophisticated hidden variables could still exist but their properties would still be more disturbing than the problems they try to solve). This does not mean that the conceptual framework of quantum physics can be considered as complete now, since some open questions still exist (the limit between the quantum and classical realms, what happens during a quantum measurement, what happens in a completely closed quantum system such as the universe itself, etc.).