Early mathematicians considered axiomatic geometry as a model of physical space, and obviously there could only be such a model. The idea that alternative mathematical systems could exist was very troubling to mathematicians of the 19th century, and developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed shortly before his untimely death that these efforts were largely wasted. Ultimately, abstract parallels between algebraic systems were considered more important than details, and modern algebra was born. In the modern view, axioms can be any set of formulas until they are known to be inconsistent. Now, the transition between mathematical axioms and scientific postulates is still somewhat unclear, especially in physics. This is due to the intensive use of mathematical tools to support physical theories. For example, the introduction of Newton`s laws rarely states as a prerequisite neither Euclidean geometry nor the differential calculus they imply. It became clearer when Albert Einstein first introduced the theory of special relativity, in which the invariant quantity is no longer the Euclidean length l {displaystyle l} (defined as l 2 = x 2 + y 2 + z 2 {displaystyle l^{2}=x^{2}+y^{2}+z^{2}} ) > but the minkowski spacetime interval s {displaystyle s} (defined as s 2 = c 2 t 2 − x 2 − y 2 − z 2 {displaystyle s^{2}=c^{2}t^{ 2}- x^{2}-y^{2}-z^{2}}), then the theory of general relativity, in which flat Minkowskiian geometry is replaced by pseudo-Riemannian geometry on curved manifolds.
This would mean that there is a sequence of formulas constructed from these axioms that proves the formula that metamathematically means, « This set of axioms is coherent. » Structuralist mathematics goes further and develops theories and axioms (e.g., field theory, group theory, topology, vector spaces) without any particular application to the mind. The distinction between an « axiom » and a « postulate » disappears. Euclid`s postulates are cost-effectively motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts is based on the acceptance of basic assumptions. However, if one rejects Euclid`s fifth postulate, one can obtain theories that make sense in broader contexts (e.g., hyperbolic geometry). Therefore, you just need to be prepared to use labels such as « line » and « parallel » in a more flexible way. The development of hyperbolic geometry has taught mathematicians that it makes sense to view postulates as purely formal statements rather than facts based on experience. Almost all modern mathematical theories start from a given set of non-logical axioms, and it was thought [more explanation needed] [citation needed] that in principle any theory could be axiomatized in this way and formalized in the simple language of logical formulas. There is therefore on the one hand the concept of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The sentence of completeness and the sentence of incompleteness do not contradict each other despite their names. In fact, axioms in mathematics and postulates in experimental sciences play a different role.
In mathematics, one « proves » or « refutes » an axiom. A set of mathematical axioms gives a set of rules that define a conceptual range in which theorems logically follow. On the other hand, in the experimental sciences, a number of postulates are intended to obtain results that correspond or disagree with the experimental results. While postulates do not allow experimental predictions to be derived, they do not provide a scientific conceptual framework and must be supplemented or made more accurate. If postulates make it possible to derive predictions from experimental results, comparison with experiments makes it possible to falsify (falsify) the theory that postulates install. A theory is considered valid as long as it has not been falsified. If used in the latter sense, « axiom », « postulate » and « acceptance » can be used interchangeably. In most cases, a non-logical axiom is simply a formal logical expression used in deduction to create a mathematical theory, and may or may not be obvious in nature (e.g., parallel postulate in Euclidean geometry).
Axiomatization of a knowledge system means showing that its claims can be derived from a small set of well-understood sentences (the axioms), and that there may be several ways to axiomatize a particular mathematical field. In the field of mathematical logic, a clear distinction is made between two concepts of axioms: logical and non-logical (similar to the old distinction between « axioms » and « postulates » respectively). That is, for every statement that is a logical consequence of Σ {displaystyle Sigma }, there is actually a deduction of the statement of Σ {displaystyle Sigma }. This is sometimes expressed as « all 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 specifies the completeness of a particular common type of deductive system. Non-logical axioms are often referred to simply as axioms in mathematical discourse. This does not mean that they are claimed to be true in an absolute sense. For example, in some groups, group surgery is commutative, and this can be affirmed with the introduction of an additional axiom, but without this axiom, we can develop (the most general) group theory quite well, and we can even take their negation as an axiom for the study of noncommutative groups. Ancient surveyors distinguished axioms from postulates. While commenting on Euclid`s books, Proclos noted that « Geminus believed that this postulate [4] should not be classified as a postulate, but as an axiom, since it does not affirm, like the first three postulates, the possibility of construction, but expresses an essential property. » [8] Boethius translated « postulate » as petitio and named the Axioms common notiones, but in later manuscripts this usage was not always strictly adhered to.
The experimental sciences, unlike mathematics and logic, also have general basic claims from which deductive reasoning can be constructed to express statements that predict properties – either even more generally, or much more specialized in a particular experimental context. For example, Newton`s laws in classical mechanics, Maxwell`s equations in classical electromagnetism, Einstein`s equation in general relativity, Mandel`s laws in genetics, Darwin`s natural law of selection, etc. These founding statements are usually called principles or postulates to distinguish themselves from mathematical axioms.