I think the biggest mistake people make when thinking about mathematics is that it is fundamentally about numbers.

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.

Vast piles of mathematics exist without any relational objects, and not exclusively in the intuitionistic sense either. Geometers say it's about rigidity. Number theorists say it's about generative rules. To a type-theorist, it's all about injective maps (with their usual sense of creating new synonyms for everything).

The only thing these have in common is that they are properties about other properties.

I prefer a more direct formulation of what mathematics is, rather than what it is about.

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.

A former Wikipedia definition mathematics: Mathematics is the study of quantity, structure, space and change.

To form or even to define a relation you need some sort of entity to have a relation with.

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.

> I think the biggest mistake people make when thinking about mathematics is that it is fundamentally about numbers. It’s not. Mathematics is fundamentally about relations.

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.

I think of pure math as choosing a set of axioms and then proving interesting theories with them.

The most commonly used/accepted foundation for mathematics is set theory, specifically ZFC. Relations are modeled as sets [of pairs, which are in turn modeled as sets].

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.

Prime numbers are the queens/kings of mathematics though.