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It’s not.
Mathematics is fundamentally about relations. Even numbers are just a type of relation (see Peano numbers).
It gives us a formal and well-studied way to find, describe, and reason about relation.
A logician / formalist would argue that mathematics is principally (entirely?) about proving derivations from axioms - theorems. A game of logic with finite strings of symbols drawn from a finite alphabet.
An intuitionist might argue that there is something more behind this, and we are describing some deeper truth with this symbolic logic.
My wife would have probably gone postal (angry-mad) if I had tried to form an improper relationship with her. It turns out that I needed a concept of woman, girlfriend and man, boyfriend and then navigate the complexities involved to invoke a wedding to turn the dis-joint sets of {woman} and {man} to form the set of {married couple}. It also turns out that a ring can invoke a wedding on its own but in many cases, it also requires way more complexity.
You might start off with much a simpler case, with an entity called a number. How you define that thing is up to you.
I might hazard that maths is about entities and relationships. If you don't have have a notion of "thingie" you can't make it "relate" to another "thingie"
It's turtles all the way down and cows are spherical.
Numbers are such a fundamental structure. I disagree with the premise that you can do mathematics without numbers. You can do some basic formal derivations, but you can’t go very far. You can’t even do purely geometric arguments without the concept of addition.
I found his book "Man and the Computer" particularly prescient.
Eh, but you can also say that about philosophy, or art, or really, anything.
What sets mathematics apart is the application of certain analytical methods to these relations, and that these methods essentially allow us to rigorously measure relationships and express them in algebraic terms. "Numbers" (finite fields, complex planes, etc) are absolutely fundamental to the practice of mathematics.
For a work claiming to do mathematics without numbers, this paper uses numbers quite a bit.
The only thing these have in common is that they are properties about other properties.
In that case, mathematics is a demonstration of what is apparent, up to but not including what is directly observable.
This separates it from historical record, which concerns itself with what apparently must have been observed. And it from literal record, since an image of a bird is a direct reproduction of its colors and form.
This separates it from art, which (over-generalizing here) demonstrates what is not apparent. Mathematics is direct; art is indirect.
While science is direct, it operates by a different method. In science, one proposes a hypothesis, compares against observation, and only then determines its worth. Mathematics, on the contrary, is self-contained. The demonstration is the entire point.
3 + 3 = 6 is nothing more than a symbolic demonstration of an apparent principle. And so is the fundamental theorem of calculus, when taken in its relevant context.
Mathematics is the study of Abstractions and Modeling using these abstractions. Entities/Attributes/Rules establishing Relationships (numerical and otherwise) all fall out of this.
The best way to understand this is through the idea of a Formal System - https://en.wikipedia.org/wiki/Formal_system All that the common man thinks of as "Mathematics" are formal systems.
A good example is this wired article How Do People Actually Catch Baseballs? - https://www.wired.com/story/how-do-people-actually-catch-bas... (archive link https://archive.is/Aarww)
"Mathematics is a field of study that discovers and organizes methods, theories, and theorems that are developed and proved for the needs of empirical sciences and mathematics itself."
In order to understand mathematics you must first understand mathematics.
For instance, here is associativity defined on addition over non-numbers a and b:
a + b = b + a
What if you add a twice?
a + a + b
To do that without numbers, you just leave it there. Given associativity, you probably want to normalize (or standardize) expressions so that equal expressions end up looking identical. For instance, moving references of the same elements together, ordering different elements in a standard way (a before b):
i.e. a + b + a => a + a + b
Here I use => to mean "equal, and preferred/simplified/normalized".
Now we can easily see that (a + b + a => a + a + b) is equal to (b + a + a => a + a + b).
You can go on, and prove anything about non-numbers without numbers, even if you normally would use numbers to simplify the relations and proofs.
Numbers are just a shortcut for dealing with repetitions, by taking into account the commonality of say a + a + a, and b + b + b. But if you do non-number math with those expressions, they still work. Less efficiently than if you can unify triples with a number 3, i.e. 3a and 3b, but by definition those expressions are respectively equal (a + a + a = 3, etc.) and so still work. The answer will be the same, just more verbose.
Why is writing a summary a bad thing?