Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'When an equal amount is taken from equals, an equal amount results.Axiom of Equality. Let L {displaystyle {mathfrak {L}}} be a first-order language. For each variable x {displaystyle x} , the formula x = x {displaystyle x=x} Axiom scheme for Universal Instantiation. Given a formula ϕ {displaystyle phi } in a first-order language L {displaystyle {mathfrak {L}}} , a variable x {displaystyle x} and a term t {displaystyle t} that is substitutable for x {displaystyle x} in ϕ {displaystyle phi } , the formula ∀ x ϕ → ϕ t x {displaystyle forall x,phi o phi _{t}^{x}} Axiom scheme for Existential Generalization. Given a formula ϕ {displaystyle phi } in a first-order language L {displaystyle {mathfrak {L}}} , a variable x {displaystyle x} and a term t {displaystyle t} that is substitutable for x {displaystyle x} in ϕ {displaystyle phi } , the formula ϕ t x → ∃ x ϕ {displaystyle phi _{t}^{x} o exists x,phi } if  Σ ⊨ ϕ  then  Σ ⊢ ϕ {displaystyle { ext{if }}Sigma models phi { ext{ then }}Sigma vdash phi } An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.' The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term axiom is used in two related but distinguishable senses: 'logical axioms' and 'non-logical axioms'. Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), often shown in symbolic form, while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, 'axiom', 'postulate', and 'assumption' may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. 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). There are typically multiple 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 the philosophy of mathematics. 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 an axiom was a claim which could be seen to be true without any need for proof. The root meaning of the word postulate 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, etc. Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books, Proclus remarks that, 'Geminus held that this 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.' Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept. 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 the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems, if we are talking about 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. The ancient Greeks considered geometry as just one of several sciences, 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.