"Mathematics is a part of physics."

Try telling that to a statistician, or a computer scientist, or an economist, or....

I think it's worth noting that category theory, which was initially derided as abstract nonsense even by other mathematicians, is now a central field of study in math and is also becoming a major part of theoretical physics. If we only studied things that are immediately applicable to some particular science, we would not be able to advance the sciences as quickly as we can now.

I'm not that wild about the Bourbaki style of exposition, which does emphasize rigor over intuition, but I think that there is a lot of value in having students go through that phase. Terry Tao has a good explanation of why at https://terrytao.wordpress.com/career-advice/theres-more-to-..., and I'd encourage anyone interested in the topic to read it.

> Try telling that to a statistician, or a computer scientist, or an economist, or....

It is easy to miss Arnold's point here due to imperfect expression. Which is that without reality checks and intuition about the math concepts all the elaborate abstract thought frameworks and rigorous results may be very misleading and counter-productive.

> central field of study in math and is also becoming a major part of theoretical physics

It is in vogue, perhaps, but it is hardly a major part of theoretical physics.

What Arnold is saying is that mathematics is part of the natural world, with immutable truths that can be discovered.

He is saying that the science of mathematics is part of the natural sciences. There's no reason for a statistician, or a computer scientist, or an economist to agree with that -- these fields depend on the natural truths discovered by mathematical science just as surely a biologists depends on the natural truths discovered by chemists.

Hasn't it been proven that there must be areas of math that can never be connected to other areas of math? Saying it's part of science, or the natural world, seems like saying all math is part of one great whole. That's something that's been passé since 1931 or so, I thought.

I don't know the answer to your question but want to state the a differing point of view in the same vein which is how I understood it as the contrast might be something I learn from.

There can never be practically useless mathematics. Someone will always find a practical use in the physical world for any new mathematics that is developed. The story I heard illustrating this was that in the shadow of WW1 various mathematicians who lived through that horror became (understandably) pacifists (Hardy?) and decided they wanted to make sure their work would never be useful in weaponry and so devoted themselves to the utterly useless number theory. Sadly for them in a world of crypto number theory has a major use and that is military.

This is a layman's view, but I thought Gödel's incompleteness theorem means there must be in principle mathematics that can never be connected to other parts of mathematics. And if something is part of the universe, it must be connected to something we can perceive, so I conclude that not all math can be connected to that which concerns something real.

In other words, even though math that was thought to be useless has repeatedly been applied to physics, I think the claim that it is "always" applicable is too strong and in fact proven false in principle.

That's not what GIT says. See https://plato.stanford.edu/entries/goedel-incompleteness/ for a description of what it does say.

This may or may not be helpful: Geometry is decidable per tarski's axioms of elementary geometry. As far as I know physical theories are dominantly geometric.

Unfortunately I know little in the topic, so I cannot add anything beyond noting his work.

I assure you, "abstract nonsense" is a term of affection.

I can ecpecially relate to the part where intuitively obvious things were probably invented for intuitively obvious reasons, then the derivation was thrown away and the thing was described axiomatically like the blind man describes an elephant.

The non-explanation of the determinant in my linear algebra textbook was especially infuriating: "There is this thing, right? We will later see its uses. It happens to have the following properties: ..., If you swap two columns its sign changes, ..."

The important thing is that a reasonable derivation is presented. It need not be historically accurate.

For instance, if I was teaching category theory to CS students, I would try to motivate it with topology.

Similarly, if I were teaching group theory, I would not try motivating it with solving polynomial equations.

> Similarly, if I were teaching group theory, I would not try motivating it with solving polynomial equations.

Why not? That’s a fine (and historically important) reason to study group theory, and everyone who goes through high school is fairly deeply familiar with polynomial equations.

While we are discussing Arnold, you might like this book created out of Arnold's lectures to some bright high school students in the 60s, https://www.mathcamp.org/2015/abel/abel.pdf

>Everyone who goes through high school is fairly deeply familiar with polynomial equations.

A high schooler is not nearly familiar enough with polynomials to understand the group theory component of Galois Theory. The central observation of Galois theory is that, in certain circumstances, it is possible to permute values without affecting the correctness of any polynomial (with coefficients in a certain field).

For example, suppose you have two unknown values, A and B, and knew that the following polynomial equations held:

    A + B = 0
    AB = 1
    A^2 = -1
    B^2 = -1

There are two possibilities: A=i and B=-i or A=-i and B=i.

Further, no matter what polynomials (with rational coefficients) I give you, it is impossible to distinguish these two possibilities. The group in this case is all the possible ways of permuting A and B which do not effect any polynomials (with rational coefficients). There is a lot of work involved in actually developing the theory I summarized above.

EDIT:

Perhaps more damning for using this approach to teaching group theory is that it is, in general, difficult to actually compute what the Galois group is for a given polynomial.

> There is a lot of work involved in actually developing the theory I summarized above.

Still, Galois had it licked by age 20, and was able to write it down the night before engaging in a deadly duel that took his life.

It (presumably) started with the assumption that you already know what a determinant is and does, then it described how to calculate it, commutativity properties etc... but never the geometrical interpretation, which would have helped greatly.

Imagine somebody describes a triangle to you by telling you all the mathematical properties of triangles (angle sums, pythagorean theorem, all the trig functions), but never saying it's a polygon with three edges.

This has hit the front page at least 3 times in the last year. Why?

He lost me at the beginning. Paraphrasing Wolfgang Pauli, “That is not only not right; it is not even wrong.” Certainly much of math was born from science. But the notion of pure mathematics goes back to antiquity.

I think his point is that mathematics has always been an abstraction of physical things before the effort to basically invent first principles of mathematics, axioms.

I would advise you to go and read his statement in full. His point is quite valid, even if some of the phrasing is not well-perceived.

Thanks for the response. I took your advice and was glad I did. After reading the entire piece, I agree with your statement.

Thank you for following through!