Just to emphasize a few points, first, as to Step 1, mathematicians mostly did not just create some concepts. Approximately, while creating something, they often kept in mind some application (or some use of a concept already) and then developed formal apparatus for using a concept in question. Arithmetic, ancient geometry, etc. was created that way, as well as probability theory in the 17th century (using concepts from gambling). Newton was also looking for computational apparatus for dealing with continuous quantities in the analysis of motion. As far as I understand, pure mathematics is more or less the invention of 19th and 20th century (maybe except number theory).

Second, there is another way of motivation in mathematics. As Kenny pointed out, some parts of math were developed “to explain things in already-established areas of mathematics”: complex numbers, etc., even non-Euclidean geometry was being built in the context of Euclidean geometry, etc.

All this should be kept in mind to counter the speculative statement that “mathematics is developed largely on aesthetic grounds”.

In general, I think the best answer to Eugene Wigner is given by Peter: “it’s applicable because the physicists selected the mathematical concepts which are applicable”. But perhaps should be added that our world is such that we can count things, measure them, there are interesting quantitative regularities in the world and the like.

Although I would admit, that the case of group theory might need some further reflection (I am not sure about it.)

PS. I have noticed that mathematicians and mathematically-minded physicists are prone to various speculations, e.g. about “aesthetic grounds”, ideal spaces in the mind or ideal mind-independent world, etc. One physicist has told me that at school there are often a problem with pupils who are very good at mathematics but who have difficulties with physics, biology. It looks like that good formal thinking leads to a kind of inability to evaluate down-to-earth stuff.

]]>1. We are looking backwards, with the advantage of hindsight, at the history of mathematics and saying some of it models the world very well. But we are looking at the mathematics which models the world well, not all the mathematics there could be, nor even, just all the mathematics that has been articulated. Newton’s Principia, for example, proceeds through progressively refined models of planetary motion, with the later models being more accurate than the former. Is it any wonder that the later models are better fitting than the former? That’s what Newton was doing, making better-fitting models.

2. Our only way, now, to access knowledge about the world studied by physicists is through advanced mathematics, since there is no non-mathematical theoretical physics. Of course, the maths fits the physics well, because we are looking at just one object, not two.

3. The history of theoretical physics has often been as follows:

Step 1: Mathematicians create some concept, without any application in physics in mind.

Step 2: Theoretical physicists think this concept might be useful.

Step 3: The physicists (maybe with help from mathematicians) modify the concept to fit the physics domain.

Step 4: The modified concept may be taken up again by mathematicians.

Examples: probability theory (invented by mathematicians 1660s, adopted by physicists late 1800s); non-Euclidean geometry (1850s, early 1900s); category theory (1940s; 1990s).

Of course, the applicability of these examples may seem unreasonable. To which I would reply: its applicable because the physicists selected the mathematical concepts which are applicable.

The best response to Wigner’s ideas, I think, is this: Isn’t it amazing that all our trousers have two legs, so that we can wear them! Imagine if all our trousers had 1 only leg or had 3 legs!

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