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Did there exist and does there still exist a debate over which school of mathematical thought (i.e. formalism, logicism, intuitionism, etc.) had the most affinity or application for physics? In particular, I am looking at if a case can be made linking one these schools of mathematical thought can be linked to physics discoveries during the first half of the 20th century such as Einstein's discoveries of general relativity.

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One such debate opposed the historian Geoffrey Hellman and constructivists like Douglas Bridges; see e.g., the article

Hellman, Geoffrey. Quantum mechanical unbounded operators and constructive mathematics—a rejoinder to Bridges. J. Philos. Logic 26 (1997), no. 2, 121–127.

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    $\begingroup$ Could you summarize that article? It's a) behind a paywall and b) could vanish at any time, leaving your answer useless. $\endgroup$ Sep 12, 2016 at 16:50
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    $\begingroup$ @jcast do you still want a summary of the article? I can do that, since I have access. Just noticed this question. Sounds like interesting reading. $\endgroup$ Nov 12, 2016 at 21:02
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In very ancient times one could argue intuitionism was predominant. Physics was more of a tool for craftsmen, building, and navigation. In these senses it was less about discovery and more about applicability with tangible objects, and so computation with accurate calculations wasn't necessary and representing these commutative strategies in a general form (with symbols and equations) wasn't something even conceptualized. The concept of logic itself wasn't really present either in the way we understand it today. They certainly used a method of intuitionism and tradition.

Later however, as philosophy and culture surrounding academics, particularly in ancient Greece, became more about discovery and wonder and less about applicability, it certainly took upon the methods of formalism. Archimedes for example, in studying force and buoyancy, would speak of mass, velocity, volume, viscosity, etc in general terms. At this point however, math and physics still didn't use symbols to represent general values, and instead would use language to describe their ideas. Alongside this, the way physical ideas were developed could be argued as intuitionistic, such as Archimedes's "Eureka!" moment. This was however still essentially formalism, in the similar way language was used to describe the ideas. And as the use of symbols became more normalized, this use of formalism became more apparent.

In the modern day, where math is respected as a formal, axiomatic system of logic and physics is respected as a natural science of inductive reasoning, the distinction becomes clearer. It is now understood that science by definition can not use math in a logicistic sense. Pure logic, as in math, is deductive and extended from the axioms. There are no axioms for the universe however, as far as we know, and we can not study all cases at once as to be deductive, so instead physics must make use of the scientific method and be inductive and therefore not purely logical. As physics has become more abstract, it's hard to argue for any intuitionistic thinning as well, as it’s pretty hard to ever claim to have an "intution" for quantum mechanics or dark matter. Physics is accepted to use formalism. In fact the entire study of mathematical physics deals with exactly that- how to represent a physical situation with mathematical language. As physics gets more advanced this actually becomes a rather significant issue since understanding the relationships observed in quantum mechanics, for example, well enough to formalize it into mathematical notation is very difficult and not something our mathematics is necessarily prepared for. Furthermore there's actually come to be computational methods used in physics that "break" mathematical rules. For example I know of a method where the physicist essentially "crosses out" infinities and is left with the value they were in search for (??? So weird).

So to directly answer your question, while the way mathematics has been used in physics has been ever-changing, it certainly wasn't something that was "debated" upon. Being so formal in how we consider the reasoning and philosophy of logic, math, and science is in itself a fairly recent development. I doubt Archimedes ever once considered the lack of axioms for the natural world, or even how he ought to properly use mathematical language in general in his physical studies.

Looking back on your question, I noticed that while you listed logicism, intuitionism, and formalism, that you included an "etc". I'd like to apologize for not considering anything other than those three, however I do think my answer was comprehensive in its own right and that it's unnecessary to analyze any others in answering the question.

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    $\begingroup$ math is respected as a formal, axiomatic system of logic --- Sometimes when I let my mind wander a bit, I find it difficult to imagine how mathematics (or even logic) can exist, or even make sense, outside of the universe we live in. For example, see the speculations of John Stuart Mill and Augustus De Morgan in this answer. $\endgroup$ May 14 at 19:04
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    $\begingroup$ @DaveLRenfro I actually write philosophy in my free time and am currently working on a book that regards this exact idea. Essentially, if-then logic is the logic system we use within our perception, but it is impossible to justify claiming this to be necessarily true outside of our perception, in objective reality. So it may be true that math, and truly all human reasoning, makes no sense outside of the human perception we find ourselves confided in. I've expanded on the ideas a lot in my writing and am so excited to publish, I'm really happy to see you've had similar thoughts about it! $\endgroup$ May 15 at 17:04

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