Anaxiom,postulate, orassumption is astatement that is taken to betrue, to serve as apremise or starting point for further reasoning and arguments. The word comes from theAncient Greek wordἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.[1][2]
The precisedefinition varies across fields of study. Inclassic philosophy, an axiom is a statement that is soevident or well-established, that it is accepted without controversy or question.[3] In modernlogic, an axiom is a premise or starting point for reasoning.[4]
Inmathematics, anaxiom may be a "logical axiom" or a "non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A andB) impliesA), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for examplea + 0 = a in integer arithmetic.
Non-logical axioms may also be called "postulates", "assumptions" or "proper axioms".[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., theparallel postulate inEuclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain.
Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in thephilosophy of mathematics.[6]
The wordaxiom comes from theGreek wordἀξίωμα (axíōma), averbal noun from the verbἀξιόειν (axioein), meaning "to deem worthy", but also "to require", which in turn comes fromἄξιος (áxios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among theancient Greekphilosophers andmathematicians, 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.[7]
The root meaning of the wordpostulate is to "demand"; for instance,Euclid demands that one agree that some things can be done (e.g., any two points can be joined by a straight line).[8]
Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books,Proclus remarks that "Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property."[9]Boethius translated 'postulate' aspetitio and called the axiomsnotiones 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.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 (theorems, 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 termsaxiom andpostulate hold a slightly different meaning for the present day mathematician, than they did forAristotle andEuclid.[7]
The ancient Greeks consideredgeometry as just one of severalsciences, 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'sposterior analytics is a definitive exposition of the classical view.[10]
An "axiom", in classical terminology, referred to aself-evident assumption common to many branches of science. A good example would be the assertion that:
When an equal amount is taken from equals, an equal amount results.
At the foundation of the various sciences lay certain additionalhypotheses that were accepted without proof. Such a hypothesis was termed apostulate. 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.[11]
The classical approach is well-illustrated[a] byEuclid'sElements, 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 astraight line from any point to any other point.
It is possible to extend aline segment continuously in both directions.
It is possible to describe acircle with any center and any radius.
It is true that allright angles are equal to one another.
("Parallel postulate") It is true that, if a straight line falling on two straight lines make theinterior angles on the same side less than two right angles, the two straight lines, if produced indefinitely,intersect on that side on which are theangles 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.
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,propositions, theorems) and definitions. One must concede the need forprimitive notions, 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, andGiuseppe Peano were pioneers in this movement.
Structuralist mathematics goes further, and develops theories and axioms (e.g.field theory,group theory,topology,vector spaces) withoutany 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 thefield 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.
Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions.
In the modern understanding, a set of axioms is anycollection 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 beconsistent; 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[b] ofEuclidean geometry,[12] and the related demonstration of the consistency of those axioms.
In a wider context, there was an attempt to base all of mathematics onCantor'sset theory. Here, the emergence ofRussell's paradox and similar antinomies ofnaïve set theory raised the possibility that any such system could turn out to be inconsistent.
The formalist project suffered a setback a century ago, whenGödel showed that it is possible, for any sufficiently large set of axioms (Peano's axioms, for example) to construct a statement whose truth is independent of that set of axioms. As acorollary, Gödel proved that the consistency of a theory likePeano arithmetic is an unprovable assertion within the scope of that theory.[13]
It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system ofnatural numbers, aninfinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modernZermelo–Fraenkel axioms for set theory. Furthermore, using techniques offorcing (Cohen) one can show that thecontinuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms.[14] Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.
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'sNatural selection law, etc. These founding assertions are usually calledprinciples orpostulates so as to distinguish from mathematicalaxioms.
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 (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 whenAlbert Einstein first introducedspecial relativity where the invariant quantity is no more the Euclidean length (defined as) > but the Minkowski spacetime interval (defined as), and thengeneral relativity where flat Minkowskian geometry is replaced withpseudo-Riemannian geometry on curvedmanifolds.
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. 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 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 asentanglement. 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 theEPR paradox in 1935). Taking this idea seriously,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 byAlain 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.).
In the field ofmathematical logic, a clear distinction is made between two notions of axioms:logical andnon-logical (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).
These are certainformulas in aformal language that areuniversally valid, that is, formulas that aresatisfied by everyassignment of values. Usually one takes as logical axiomsat least some minimal set of tautologies that is sufficient for proving alltautologies in the language; in the case ofpredicate logic more logical axioms than that are required, in order to provelogical truths that are not tautologies in the strict sense.
Inpropositional logic, it is common to take as logical axioms all formulae of the following forms, where,, and can be any formulae of the language and where the includedprimitive connectives are only "" fornegation of the immediately following proposition and "" forimplication from antecedent to consequent propositions:
Each of these patterns is anaxiom schema, a rule for generating an infinite number of axioms. For example, if,, and arepropositional variables, then and are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata andmodus ponens, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies withmodus ponens.
Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed.[15]
These axiom schemata are also used in thepredicate calculus, but additional logical axioms are needed to include a quantifier in the calculus.[16]
Axiom of Equality. Let be afirst-order language. For each variable, the below formula is universally valid.
This means that, for anyvariable symbol, the formula can be regarded as an axiom. Additionally, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.
Another, more interesting exampleaxiom scheme, is that which provides us with what is known asUniversal Instantiation:
Axiom scheme for Universal Instantiation. Given a formula in a first-order language, a variable and aterm that issubstitutable for in, the below formula is universally valid.
Where the symbol stands for the formula with the term substituted for. (SeeSubstitution of variables.) In informal terms, this example allows us to state that, if we know that a certain property holds for every and that stands for a particular object in our structure, then we should be able to claim. Again,we are claiming that the formulais valid, that is, we must be able to give a "proof" of this fact, or more properly speaking, ametaproof. These examples aremetatheorems of our theory of mathematical logic since we are dealing with the very concept ofproof itself. Aside from this, we can also haveExistential Generalization:
Axiom scheme for Existential Generalization. Given a formula in a first-order language, a variable and a term that is substitutable for in, the below formula is universally valid.
Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, thenatural numbers and theintegers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such asgroups). Thus non-logical axioms, unlike logical axioms, are nottautologies. Another name for a non-logical axiom ispostulate.[5]
Almost every modernmathematical theory starts from a given set of non-logical axioms, and it was thought that, in principle, every theory could be axiomatized in this way and formalized down to the bare language of logical formulas.[citation needed][further explanation needed]
Non-logical axioms are often simply referred to asaxioms in mathematicaldiscourse. This does not mean that it is claimed that they are true in some absolute sense. For instance, in some groups, the group operation iscommutative, and this can be asserted with the introduction of an additional axiom, but without this axiom, we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.
This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms.
Probably the oldest, and most famous, list of axioms are the 4 + 1Euclid's postulates ofplane geometry. The axioms are referred to as "4 + 1" because for nearly two millennia thefifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. One can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist. This choice gives us two alternative forms of geometry in which the interiorangles of atriangle add up to exactly 180 degrees or less, respectively, and are known as Euclidean andhyperbolic geometries. If one also removes the second postulate ("a line can be extended indefinitely") thenelliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.
The objectives of the study are within the domain ofreal numbers. The real numbers are uniquely picked out (up toisomorphism) by the properties of aDedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires the use ofsecond-order logic. TheLöwenheim–Skolem theorems tell us that if we restrict ourselves tofirst-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied innon-standard analysis.
Adeductive system consists of a set of logical axioms, a set of non-logical axioms, and a set ofrules of inference. A desirable property of a deductive system is that it becomplete. A system is said to be complete if, for all formulas,
that is, for any statement that is alogical consequence of there actually exists adeduction of the statement from. This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation".Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system.
Note that "completeness" has a different meaning here than it does in the context ofGödel's first incompleteness theorem, which states that norecursive,consistent set of non-logical axioms of the Theory of Arithmetic iscomplete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.
There is thus, on the one hand, the notion ofcompleteness of a deductive system and on the other hand that ofcompleteness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.
Earlymathematicians regardedaxiomatic geometry as a model ofphysical space, implying, there could ultimately only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such asBoolean algebra made elaborate efforts to derive them from traditional arithmetic.Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details, andmodern algebra was born. In the modern view, axioms may be any set of formulas, as long as they are not known to be inconsistent.
^"A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a.Oxford English Dictionary Online, accessed 2012-04-28. Cf. Aristotle,Posterior Analytics I.2.72a18-b4.
^"A proposition (whether true or false)" axiom, n., definition 2.Oxford English Dictionary Online, accessed 2012-04-28.
^abMendelson, "3. First-Order Theories: Proper Axioms" of Ch. 2
^See for exampleMaddy, Penelope (June 1988). "Believing the Axioms, I".Journal of Symbolic Logic.53 (2):481–511.doi:10.2307/2274520.JSTOR2274520. for arealist view.
^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)
^Raatikainen, Panu (2018),"Gödel's Incompleteness Theorems", in Zalta, Edward N. (ed.),The Stanford Encyclopedia of Philosophy (Fall 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved19 October 2019
^Koellner, Peter (2019),"The Continuum Hypothesis", in Zalta, Edward N. (ed.),The Stanford Encyclopedia of Philosophy (Spring 2019 ed.), Metaphysics Research Lab, Stanford University, retrieved19 October 2019
