Do axioms exist in the scientific method like in physics, chemistry, biology, …? E.g. Ockham's Razor for picking the best theory – is it an axiom? Another example is that science gathers data, analyzes it and comes to a conclusion through reasoning (usually through inductive reasoning)… but can these axioms change in the future if we find a better way? Or would they be forever set (because they are axioms)?

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    No. "Scientific method" is a loose collection of research techniques that vary from discipline to discipline and can not be formalized into "axioms", it is an art or a craft rather than theory. Only some vague general principles are common and they do change over time, sometimes they are called scientific paradigms. – Conifold Feb 4 '18 at 23:08
  • To build on Conifold's answer: for example, the consensus position among professional philosophers of biology today is that biology does not have laws in anything like the way physics is often understood to have laws. The laws (or a subset of them) would be the axioms in an formal axiomatization of biology. Since there are no laws of biology, there are no axioms of biology either. – Dan Hicks Feb 5 '18 at 2:46
  • Conifold & Dan Hicks. Superb comments, if you will accept the praise. There's more wisdom in these comments than in many answers. – Geoffrey Thomas Apr 6 '18 at 20:15
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    'Occam's Razor' is not a real thing. It is a crude 'rule of thumb' and nothing more. It can never be used to seriously rigorously defend anything. There is no actual bias toward simplicity in integration with our existing beliefs (which is what the 'razor' relies upon) in reality. – otakucode Apr 6 '18 at 22:51
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    @cbeleites - We seem to be on roughly the same page. It's not that the psycho-physical world does not exist but that is does not do so in the way that we usually imagine. Thus for instance Nagarjuna's famous proof of Buddhist metaphysics is said to prove not that nothing exists but that 'nothing really exists', where the 'really' is an important proviso. Whether he is right or wrong makes no difference to physics but makes a vast difference to philosophy. On this view solipsism would be neither strictly true or false, which would explain why it is undecidable. – PeterJ Jun 9 '18 at 11:18

cf. Hilbert's 6th problem: "Can physics be axiomized?" of his Mathematics Problems lecture delivered before the International Congress of Mathematicians at Paris in 1900:

  1. Mathematical treatment of the axioms of physics

The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.

As to the axioms of the theory of probabilities,14 it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.

Important investigations by physicists on the foundations of mechanics are at hand; I refer to the writings of Mach,15 Hertz,16 Boltzmann17 and Volkmann.18 It is therefore very desirable that the discussion of the foundations of mechanics be taken up by mathematicians also. Thus Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua. Conversely one might try to derive the laws of the motion of rigid bodies by a limiting process from a system of axioms depending upon the idea of continuously varying conditions of a material filling all space continuously, these conditions being defined by parameters. For the question as to the equivalence of different systems of axioms is always of great theoretical interest.

If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories. At the same time Lie's a principle of subdivision can perhaps be derived from profound theory of infinite transformation groups. The mathematician will have also to take account not only of those theories coming near to reality, but also, as in geometry, of all logically possible theories. He must be always alert to obtain a complete survey of all conclusions derivable from the system of axioms assumed.

Further, the mathematician has the duty to test exactly in each instance whether the new axioms are compatible with the previous ones. The physicist, as his theories develop, often finds himself forced by the results of his experiments to make new hypotheses, while he depends, with respect to the compatibility of the new hypotheses with the old axioms, solely upon these experiments or upon a certain physical intuition, a practice which in the rigorously logical building up of a theory is not admissible. The desired proof of the compatibility of all assumptions seems to me also of importance, because the effort to obtain such proof always forces us most effectually to an exact formulation of the axioms.

14. Cf. G. Bohlmann, "Ueber Versicherungsmathematik," from the collection: F. Klein and E. Riecke, Ueber angewandte Mathematik und Physik, Teubner, Leipzig, 1900.
15. E. Mach: Die Mechanik in ihrer Entwickelnng, Brockhaus, Leipzig, 4th edition, 1901.
16. H. Hertz: Die Prinzipien der Mechanik, Leipzig, 1894.
17. L. Boltzmann: Vorlesungen über die Principe der Mechanik, Leipzig, 1897.
18. P. Volkmann: Einführung in das Studium der theoretischen Physik, Teubner, Leipzig, 1900.

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    Given Hilbert's program didn't even suceed for maths, how can it be said to be valid for physics? The same Incompleteness Theorem that sank it surely applies also to physics. – CriglCragl Feb 5 '18 at 23:11
  • @CriglCragl Yes, see: "A Late Awakening to Gödel in Physics" – Geremia Jul 7 '18 at 5:18

The short answer:

  1. There is no such thing as the scientific method.
  2. Axioms exist within theories and are called postulates. However, they don't typically translate across theories.
  3. Ochman's Razor is not an axiom or postulate, but rather a guideline for pick what theory to believe when you have two or more competing theories that explain the data equally well.

It appears that you're presupposing that the axiomatic method in mathematics is the essence of mathematics and given its success in coming up with irrefutable truths one should search for ways of axiomatising other sciences; but this presupposition may not hold; here's Poincares take on this in his book Science & Hypothesis:

What is the nature of mathematical reasoning? Is it deductive as commonly thought? Careful analysis shows us it is nothing of the kind; that it participates to some extent in the nature of inductive reasoning and for that reason is fruitful.

In other words mathematics is more akin to a science and it's a false economy to turn sciences into axiomatic mathematics.

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