It's probably treading close to the don't be dismissive rule, but I confess it's really hard to keep reading when the piece opens with something so obviously wrong.
Historically, this is accurate. The whole concept of "pure math" is actually very recent and before that math was almost exclusively tied to real world problems that often came directly from physics.
Historically mathematics was part of accounting, both in Ancient China where it was used to calculate taxes, in Egypt and in Babylon. People used numbers to keep count and then developed advanced techniques to do sophisticated things like split non-rectangular land in equal parts.
It wasn't until much later, and only in a tiny part of the world called Europe, that physics started using mathematical models. Even at that time and in that place, there were other disciplines making advanced use of mathematics and leading to exciting discoveries, starting from Economics. Jacobi who developed utility theory around the time of Newton.
Mathematics was never a subdiscipline of physics. Important parts of mathematics were developed to support models of physics, but claiming that the parts should not exist is intellectually dishonest.
Curiously, many mathematical curricula used to include analytical mechanics, and some of the mathematics departments of universities around the world had the word "mechanics" in their names.
The first use of complex numbers was as intermediate quantities in the calculation of the roots of cubic polynomial functions. This use is analogous to using negative numbers in a ledger even though negative amounts of physical things don't make sense.
Edit: I meant physical things like apples fam. This is an important philosophical point we don't appreciate because we are so used to them. De Morgan once wrote:
"It is not our intention to follow the earlier algebraists through their different uses of negative numbers. These creations of algebra retained their existence, in the face of the obvious deficiency of rational explanation which characterized every attempt at their theory."
I am, have a PhD in it at least, and complex numbers are 100% required to do quantum mechanics, so are physically motivated. In fact, many physicists consider the complex numbers to be the preferred number system of the universe for this reason.
Historically already ancient Greeks knew pure mathematics (e.g. Euclid's Elements), and differentiated it from its applications as far as I'm concerned. That made them pretty distinct from earlier Egyptian mathematicians who indeed were only concerned with solving practical problems without much attention to logical rigor.
I am fascinated by geometric algebra but what I have seen so far seems like mathematicians trying to convince physicists and programmers to use their system, which to be honest seems much better, but is also harder to build an intuition of. I would love to read a geometric algebra for dummies/from the ground up.
Too close, in my opinion. If I said "Communication is a part of physics", it would be clear that I don't mean "all of communication is a subset of physics" and that I do mean "doing physics involves communicating your findings". I think this is the author's intention.
Edit: Actually, it does seem like the author claims mathematics is a proper subset of physics.
For Arnold, mathematics is rooted in physical intuition and experimental inquiry. Can you name some math that is completely disconnected from that intuition?
Mathematics may be rooted in that, in a historical or pedagogical sense, but areas of math can certainly be disconnected from physical intuition. Non-measurable sets (e.g. those in Banach-Tarski) and transfinite numbers cone to mind.
Non-measurable sets are precisely the kinds of things Arnold wanted marginalized in mathematical pedagogy, instead of placed front and center. They are necessary auxiliaries to the main theory, that of measures and integration but auxiliary nonetheless.
I disagree that transfinite numbers are detached from physical intuition since most of the ones you or I could write down can be easily visualized with a few ellipses here or there. But i do think Arnold would consider them marginal players. Perhaps he thought set theory was a formalist distraction from the main of mathematics!
E.g. logic? A pretty important part of mathematics that is hard to marginalize, but that can't be observed experimentally. Rather scientific observation presupposes logic ability.
It's probably treading close to the don't be dismissive rule, but I confess it's really hard to keep reading when the piece opens with something so obviously wrong.