Axiom, an established rule or principle or a self-evident truth.

Better yet,

An Axiom is a mathematical statement that is assumed to be true

Why does math apply so well to science?

Why is 1 atom+1 atom=2 atoms?

Why does a y=mx+b work for lines on 2D surfaces in the real world?

How did math and science develop to intertwine?

Why do photons move in sin waves?

Why can we make simple equations to describe how the pressure of a gas is directly proportional to it's temperature? (P∝T)

Why does the universe follow man-made axioms of math?

Obviously I don't want you to answer any of those if you don't want to, this is the philosophy stack exchange, I just want the answer to the following question:

If you didn't get it, I'm asking why mathematical truths are also accepted as scientific truths? What makes them work so well in the real world?


The general process here is roughly:

  • We observe that the "real world" seems to have certain properties
  • We study the mathematical theory of things having those properties
  • We detect the consequences of these properties by studying the mathematical theory, and then observe them in the "real world".

It may be surprising that such a procedure works, but I interpret your question as stemming from an oversight at a more basic level — in the application of mathematics to the study of the real world, we first observed the real world and used them to inform the choice of the mathematical axioms used to describe theories of the real world, rather than the mathematical axioms coming first and then the surprising fact that the real world also agrees.

  • So it was, at least during the good times of mathematics. – Wilhelm May 18 '18 at 14:02
  • Shouldn't maths though have axioms indepedent of what we observe? – ado sar May 28 '20 at 14:22

Those mathematical axioms which are taken from experience work so well because reality works so well. Those mathematical axioms which are free and often counterfactual inventions don't work well. A prominent example is transfinite set theory which has not the least practical application. For debunked examples see: "Applications" of set theory in https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 124ff.

"How is it possible that mathematics, which is a product of human thinking independent of all experience, fits reality in such an excellent way?" [A. Einstein: "Geometrie und Erfahrung", Festvortrag, Berlin (1921), reprinted in A. Einstein: "Mein Weltbild", C. Seelig (ed.), Ullstein, Frankfurt (1966) p. 119]

Without mental images from sensory impressions and experience thinking is impossible. Without reality (which includes the apparatus required for thinking as well as the objects of thinking – we never think of an abstractum "number 3" but always of three things or the written 3 or the spoken word or any materialization which could have supplied the abstraction) mathematics could not have evolved like a universe could not have evolved without energy and mass. Therefore real mathematics agrees with reality in the excellent way it does.

Einstein answers his question in a relativizing way: "In so far the theorems of mathematics concern reality they are not certain, and in so far as they are certain they do not concern reality." [A. Einstein: "Geometrie und Erfahrung", Festvortrag, Berlin (1921), reprinted in A. Einstein: "Mein Weltbild", C. Seelig (ed.), Ullstein, Frankfurt (1966) p. 119f]

He states a contraposition (R ==> ¬C) <==> (C ==> ¬R). Both statements are equivalent. Both statements are false. To contradict them a counterexample is sufficient. A theorem of mathematics is the law of commutation of addition of natural numbers a + b = b + a. It can be proven in every case in the reality of a wallet with two pockets.

  • 1
    Nothing can be proven "in every case" in reality, it is exactly the kind of certainty that does not apply to reality. – Conifold May 17 '18 at 21:59
  • @Conifold: You are blatantly wrong. – Wilhelm May 18 '18 at 10:43

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