Where would mathematics be without physics?
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Difference between math and physics
The reason this question is asked in the first place is because of the way math is taught in school.
For the most part, it is not mathematics itself that is taught there, but applied mathematics, i.e. the application of mathematical methods to concrete problems.
If you are not at war with mathematics anyway, you can recognize this mainly from the fact that most of the time is spent applying learned methods and calculating things explicitly, i.e. getting concrete results.
Occasionally, however, the actual math comes to the fore in school, namely when something is proven. However, even without ever having really understood a proof, i.e. without having understood the mathematics itself, one can still survive to some extent in the school subject of mathematics.
In school, one encounters applied mathematics to a noteworthy extent - apart from the subject mathematics itself - only in the subject physics, and there even to a considerable extent. Seen in this way, the subject of physics presents itself as if it were the application of mathematical methods in a special context.
What is taught as physics in school is physics. Essential characteristics of physics (experimental, empirical, natural science), at least in their main features, are consistently revealed in school.
The physics teaching content in school differs from that at university mainly in the level of difficulty (and even that applies mainly to the mathematical methods).
The following is written from a physics perspective and is unlikely to be fully shared by mathematicians.
Mathematics basically gets by without physics. The other way around, however, physics could not even begin to exist without (applied) mathematics. Mathematics is the language in which physical facts can be best described by far. It even goes so far that the "mathematical language structure" has already been (also) expanded several times by physicists. In plain language: In the past, physics has encountered scientific problems several times (especially quantum mechanics), for which mathematical methods and modes of representation first had to be "invented".
Something similar can be said of the subject of statistics: Many statistical methods owe their existence to specific problems from a wide variety of scientific fields. Mainly to be mentioned in this context are sociology, psychology and medicine (interestingly even before the natural sciences).
Mathematics is a human science, so it arises from the human mind. Logic and consistency are not only necessary, but even sufficient, i.e. everything that a person can imagine is permitted and makes sense, provided it is logical and free of contradictions. So the natural limits are very broad here. Basically, mathematics doesn't even need to be related to any kind of practice.
Physics is an empirical natural science. Empirically one can paraphrase quite well with "trial and error", and nature means that nature and not human imagination is the highest authority. Logic and consistency are also necessary in physics, but by no means sufficient: Only a fraction of what humans can imagine actually makes physical sense, although it would be logical and free of contradictions.
On the other hand, nature holds a lot of things that the human mind would never think of on its own. So nature sets both the rules of the game and the limits. It determines what is right and wrong.
From the point of view of physics (and also other disciplines), mathematics is sometimes referred to as an auxiliary science in the sense that it represents a tool kit or method construction kit, the usefulness of which is only revealed when it is used in practice.
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