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Pure mathematics can and does stand alone as its own science.
It is the objective of pure mathematics to systematically express its content in a culture-free manner. If its truth content depends on the cultural context of its practitioners then it is assuredly not pure mathematics; in fact, if that is true then it does not qualify as mathematics at all.
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Physics, and in general science, is all about developing "models" that we hope fit to physical reality as much as possible. Fitting criteria are explanation of current observation (that is not necessarily the pure reality in its essence as the observation might well be restricted by not only the instruments but also the current paradigm and ...
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Your post seems to involve two question. A first shallower and a second deeper one.
The first question: "Why does the very mathematics work that has been axiomatically developed in the 20th century"?
The semi-obivous answer: Because it has been axiomatized exactly in order to render mathematical results which were in use in the empirical sciencies for ...
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