axiom

In a nutshell, the logico-deductive method is a system of inferencewhere conclusions (new knowledge) follow from premises (old knowledge)through the application of sound arguments (syllogisms, rules ofinference). Tautologies excluded, nothing can be deduced if nothingis assumed. Axioms and postulates are the basic assumptionsunderlying a given body of deductive knowledge. They are acceptedwithout demonstration. All other assertions (theorems, if we aretalking about mathematics) must be proven with the aid of the basicassumptions.

The logico-deductive method was developed by the ancient Greeks, andhas become the core principle of modern mathematics. However, theinterpretation of mathematical knowledge has changed from ancienttimes to the modern, and consequently the terms axiom andpostulate hold a slightly different meaning for the present daymathematician, then they did for Aristotle and Euclid.

The ancient Greeks considered geometry as just one of severalsciences, and held the theorems of geometry on par with scientificfacts. As such, they developed and used the logico-deductive methodas a means of avoiding error, and for structuring and communicatingknowledge. Aristotle’s http://classics.mit.edu/Aristotle/posterior.1.i.htmlPosteriorAnalyticsis a definitive exposition of the classical view.

“Axiom”, in classical terminology, referred to a self-evident assumptioncommon to many branches of science. A good example would be theassertion that

When an equal amount is taken from equals, an equal amount results.

At the foundation of the various sciences lay certain basic hypothesesthat had to be accepted without proof. Such a hypothesis was termed apostulate. The postulates of each science were different.Their validity had to be established by means of real-worldexperience. Indeed, Aristotle warns that the content of a sciencecannot be successfully communicated, if the learner is in doubt aboutthe truth of the postulates.

The classical approach is well illustrated by Euclid’s elements, wherewe see a list of axioms (very basic, self-evident assertions) andpostulates (common-sensical geometric facts drawn from ourexperience).

• Things which are equal to the same thing are also equal to oneanother.

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

• If equals be subtracted from equals, the remainders areequal.

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

• The whole is greater than the part.

• It is possible todraw a straight line from any point to any other point.

• It is possible to produce a finite straight line continuously in astraight line.

• It is possible to describe a circle with anycentre and distance.

• It is true that all right angles areequal to one another.

• It is true that, if a straight linefalling on two straight lines make the interior angles on the sameside less than two right angles, the two straight lines, if producedindefinitely, meet on that side on which are the angles less than thetwo right angles.

The classical view point is explored in more detail http://www.mathgym.com.au/history/pythagoras/pythgeom.htmhere.

A great lesson learned by mathematics in the last 150 years is that itis useful to strip the meaning away from the mathematical assertions(axioms, postulates, propositions, theorems) and definitions. Thisabstraction, one might even say formalization, makes mathematicalknowledge more general, capable of multiple different meanings, andtherefore useful in multiple contexts.

In structuralist mathematics we go even further, and develop theoriesand axioms (like field theory, group theory, topology, vector spaces)without any particular application in mind. The distinctionbetween an “axiom” and a “postulate” disappears. The postulates ofEuclid are profitably motivated by saying that they lead to a greatwealth of geometric facts. The truth of these complicated facts restson the acceptance of the basic hypotheses. However by throwing outpostulate 5, we get theories that have meaning in wider contexts,hyperbolic geometry for example. We must simply be prepared to uselabels like “line” and “parallel” with greater flexibility. Thedevelopment of hyperbolic geometry taught mathematicians thatpostulates should be regarded as purely formal statements, and not asfacts based on experience.

When mathematicians employ the axioms of a field, the intentions areeven more abstract. The propositions of field theory do not concernany one particular application; the mathematician now works incomplete abstraction. There are many examples of fields; fieldtheory gives correct knowledge in all contexts.

It is not correct to say that the axioms of field theory are“propositions that are regarded as true without proof.” Rather, theField Axioms are a set of constraints. If any given system ofaddition and multiplication tolerates these constraints, then one isin a position to instantly know a great deal of extra informationabout this system. There is a lot of bang for the formalist buck.

Modern mathematics formalizes its foundations to such anextent that mathematical theories can be regarded as mathematicalobjects, and logic itself can be regarded as a branch of mathematics.Frege, Russell, Poincaré, Hilbert, and Gödel are some ofthe key figures in this development.

In the modern understanding, a set of axioms is any collection offormally stated assertions from which other formally stated assertionsfollow by the application of certain well-defined rules. In thisview, logic becomes just another formal system. A set of axiomsshould be consistent; it should be impossible to derive acontradiction from the axiom. A set of axioms should also benon-redundant; an assertion that can be deduced from other axioms neednot be regarded as an axiom.

It was the early hope of modern logicians that various branches ofmathematics, perhaps all of mathematics, could be derived from aconsistent collection of basic axioms. An early success of theformalist program was Hilbert’s formalization of Euclidean geometry,and the related demonstration of the consistency of those axioms.

In a wider context, there was an attempt to base all of mathematics onCantor’s set theory. Here the emergence of Russell’s paradox, andsimilar antinomies of naive set theory raised the possibility that anysuch system could turn out to be inconsistent.

The formalist project suffered a decisive setback, when in 1931Gödel showed that it is possible, for any sufficiently large set ofaxioms (Peano’s axioms, for example) to construct a statement whosetruth is independent of that set of axioms. As a corollary, Gödelproved that the consistency of a theory like Peano arithmetic is anunprovable assertion within the scope of that theory.

It is reasonable to believe in the consistency of Peano arithmeticbecause it is satisfied by the system of natural numbers, an infinitebut intuitively accessible formal system. However, at this date wehave no way of demonstrating the consistency of modern set theory(Zermelo-Frankel axioms). The axiom of choice, a key hypothesisof this theory, remains a very controversial assumption.Furthermore, using techniques of forcing (Cohen) one can show that thecontinuum hypothesis (Cantor) is independent of the Zermelo-Frankelaxioms. Thus, even this very general set of axioms cannot be regardedas the definitive foundation for mathematics.