Editing Axiom (section)

===Other sciences===
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 usually 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 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 ([[Falsifiability|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 when [[Albert Einstein]] first introduced [[special relativity]] where the invariant quantity is no more the Euclidean length <math>l</math> (defined as <math>l^2 = x^2 + y^2 + z^2</math>) > but the Minkowski spacetime interval <math>s</math> (defined as <math>s^2 = c^2 t^2 - x^2 - y^2 - z^2</math>), and then [[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 provide a very nice example of falsification. The '[[Copenhagen interpretation|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-variable theory|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 as [[quantum entanglement|entanglement]]. 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 the [[EPR paradox]] in 1935). Taking this idea seriously, [[John Stewart Bell|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 by [[Alain 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.).