Can other "sciences" (it's in quotation marks, because the definition for a science is not necessarily exact) be applications of mathematics?

If other sciences, be it philosophy or economics or linguistics or philology, deal with concepts such as:

  • Existence
  • Set theoretic notions (belongs to a set, does not belong to a set, ...)
  • Generality/universality
  • Causality/relationships
  • etc.

And if these are defined "in the most pure way" in foundations of mathematics (logic etc.), then can this motivate the idea that all other sciences are applications of mathematics, even when they're not called mathematics proper?

I don't mean to ask, whether other sciences reduce to math, because that's unreasonable. Mathematics only doesn't allow one to do physics or economics or so, but my question is more about whether e.g. the notions on truth in mathematics extend to other sciences? Which I think is true?

  • What do you mean: a science ‘is an application of’ mathematics? Clearly, most sciences use mathematical concepts and techniques. Physics and chemistry do; but e.g. psychology uses a lot of statistics, too. Yet you’re presumably looking for a stronger claim than that? – Also, if a concept appears in maths and another science, that doesn’t mean that the science applied the math concept: perhaps it’s maths that applied the concept the other science offered. Arguably, this is the case with the concept of truth. (Sillier example: If you tie your shoe, that’s not ‘an application of knot theory’.) – MarkOxford Nov 17 '17 at 9:58
  • @MarkOxford Well if other sciences use mathematical theories, then they're instances of applications of mathematics? But how far are they mathematics and where does their own "science" start? – mavavilj Nov 17 '17 at 10:04
  • @MarkOxford Why can't tying ropes be about knot theory? Just because people tied shoes before knot theory does not mean that after knot theory tying ropes couldn't be about knot theory? For example, it's reasonable to believe that knot theory could figure out new knots on top of what some seamen did with their ropes? Actually knot theory could invent better knots, because they can motivate the knots by their "quantitative" physical strength for example. Not just "heuristic strength" – mavavilj Nov 17 '17 at 10:06
  • "Applications" is the wrong word: today, most "empirical" sciences are based on applied math. Maybe you are alluding to the possibility to "found" (in some sense) empirical sciences on math theories, like e.g. set theory... – Mauro ALLEGRANZA Nov 17 '17 at 10:26
  • Sure knot theory can invent new knots. I meant: just because knots are studied in maths, it doesn’t seem right to say that you are ‘applying’ maths every time you tie your shoes. The more general point was: just because there’s a conceptual overlap between two fields of study, one field isn’t automatically the applied version of the other. More generally still: without a fairly strict notion of application, the question becomes trivial. [Btw, if X is applied Y (if psychology is applied statistics, say), does it follow that Y is theoretical X (that statistics is theoretical psychology)?] – MarkOxford Nov 17 '17 at 13:17

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