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Why do we rely upon scientific approach when its foundational axioms are assumed and agreed without proof?

Foundation of the scientific explorations are seem to be the mathematical axioms at its root. But if we see the definition of the axiom we find axioms are basically the propositions which requires no proof and universally accepted. If so, why do we treat science as the most reliable way to understand our reality which is based on something that itself have no proof? What are the philosophical justification of the reliability of science?

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    Wait, back up... what foundational axioms of science are you talking about? – H Walters Jun 15 '20 at 1:59
  • @HWalters mathematical axioms. Is there any way to reliably do science without mathematics? – Sazzad Hissain Khan Jun 15 '20 at 2:06
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    It seems to work. Bridges and airplanes usually stay up. We came out of caves and built all this using rationality and our human instinct to math, science and engineering. What it all means ultimately, is anyone's guess. Maybe the order we see is just an illusion, and the actual world is formless and random. It's possible. – user4894 Jun 15 '20 at 2:45
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    Most of science does not use mathematics in a substantive way, it is only used to transform data from one form into another (statistics). In this respect the "axioms" are just extensions of our linguistic conventions to counting and measurement. Where mathematics is used substantively (mathematical physics, chemistry, to a very limited extent, biology) the "axioms" are surmises from multiple observations and experiments, so they are proved (by experience), although not in a mathematical sense. And that is the "scientific approach", even without mathematics, and why it is reliable. – Conifold Jun 15 '20 at 6:38
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    We can mimick the well-known Churchill's dictum about democracy: "it has been said that [the scientific approach] is the worst form of [knowledge] except for all those other forms that have been tried from time to time.…" – Mauro ALLEGRANZA Jun 15 '20 at 7:11
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The big book of scientific knowledge is written in the language of mathematics. Four hundred years of work in this area as shown that regardless of what philosophers or physicists might think about this, it nonetheless is a fact that the use of mathematics to organize knowledge about the physical world is (to some) unreasonably effective.

As such, the reliability of science and the successes that spring from it is a circumstance which its practitioners are under absolutely no obligation to justify to the philosophy community.

For example, take the case of shooting extremely high-energy electrons at protons, as a tool for determining whether or not the protons possess any internal structure, and if so, what exactly that structure might consist of. This is the stuff of which Nobel prizes are made.

So: in what sense would this work, carried out at Stanford's SLAC facility in the late 1960's, have been in any way productively informed by having philosophers involved in the process of designing the magnetic spectroscopes that were used to make measurements on electron-proton scattering?

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