Axiom

An axiom, postulate, or precondition is a statement that is taken to be true, to serve as the premise or starting constituent for further reasoning as well as arguments. The word comes from the Ancient Greek word , meaning '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 or done as a reaction to a question that is so evident or well-established, that it is for accepted without controversy or question. As used in modern logic, an axiom is a premise or starting segment for reasoning.

As used in "logical axioms" in addition to "non-logical axioms". Logical axioms are normally statements that are taken to be true within the logical system they define and are often featured in symbolic realize e.g., A and B implies A, while non-logical axioms e.g., are actually substantive assertions approximately the elements of the domain of a particular mathematical picture such as arithmetic.

When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to established a mathematical theory, and might or might not be self-evident in manner e.g., parallel postulate in Euclidean geometry. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood kind of sentences the axioms, and there may be group ways to axiomatize a assumption mathematical domain.

Any axiom is a a thing that is caused or produced by something else that serves as a starting point from which other statements are logically derived. Whether it is for meaningful and, whether so, what it means for an axiom to be "true" is a subject of debate in the philosophy of mathematics.

Historical development

The logico-deductive method whereby conclusions new knowledge undertake from premises old cognition through the application of sound arguments syllogisms, rules of inference was developed by the ancient Greeks, and has become the core principle of advanced 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 terms axiom and postulate produce believe a slightly different meaning for the made 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.

An "axiom", in classical terminology, mentioned to a self-evident assumption common to many branches of science. A framework would be the assertion that

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

At the foundation of the various sciences lay certain extra hypotheses that were accepted without proof. such(a) a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of used to refer to every one of two or more people or matters particular science were different. Their validity had to be establish by means of real-world experience. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt approximately the truth of the postulates.

The classical approach is well-illustrated by 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.

A interpreter 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 for primitive notions, or undefined terms or concepts, in all study. Such picture or formalization permits mathematical knowledge more general, capable of group 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, group theory, topology, vector spaces without any particular applications 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 usage labels such(a) as "line" and "parallel" with greater flexibility. The coding of hyperbolic geometry taught mathematicians that it is useful to regard postulates as purely formal statements, and non as facts based on experience.

When mathematicians employ the field axioms, the intentions are even more abstract. The propositions of field theory do not concern any one specific application; the mathematician now working in ready abstraction. There are many examples of fields; field theory gives correct knowledge about them all.

It is not adjustment 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 additional information about this system.

Modern mathematics formalizes its foundations to such(a) an extent that mathematical theories can be regarded as mathematical objects, and mathematics itself can be regarded as a branch of logic. Frege, Russell, Poincaré, Hilbert, and Gödel are some of the key figures in this development.

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

In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions adopt – by the application ofwell-defined rules. In this view, logical system 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 code was Hilbert's formalization of Euclidean geometry, and the related demonstration of the consistency of those axioms.

In a wider context, there was an effort to base all of mathematics on 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 decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms Peano's axioms, for example to construct a written whose truth is self-employed grownup 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.

It is fair to believe in the consistency of Peano arithmetic because it isby the system of natural numbers, an infinite but intuitively accessible formal system. However, at present, there is no required way of demonstrating the consistency of the modern Zermelo–Fraenkel axioms for set theory. Furthermore, using techniques of forcing Cohen one can show that the continuum hypothesis Cantor is self-employed grownup of the Zermelo–Fraenkel axioms. 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's Natural selection law, etc. These founding assertions are normally 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 makes a set of rules that prepare 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 permit deducing experimental predictions, they do not set a scientific conceptual expediency example 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 usage of mathematical tools to assistance the physical theories. For instance, the number one order of Newton's laws rarely establishes as a prerequisite neither Euclidian geometry or differential calculus that they imply. It became more obvious when 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 give a very nice example of falsification. The 'Bell's inequalities in the Copenhagen and the Hidden variable case. The experiment was conducted first by Alain Aspect in the early 1980's, 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.