In the words of Kronecker: "God created the integers, all else is the work of man."

I sometimes think about the same things. As of now, my best bet is that math is one of the disciplines studying exactly these questions.

What is real? There are strong indications that what we experience as reality is an ilusion generated by what is usually refered to as the subconscious.

One could argue that knots are more real than numbers. It is hard to find two equal looking apples and talk about two apples, because it requires the abstraction that the apples are equal, while it is obvious that they are not. While, I guess, we all have had the experience of strugling with untying knots in strings.

Philosophical problems regarding the fundamental nature of reality aside, this short clip is relevant to your question:

> https://www.youtube.com/watch?v=tCUK2zRTcOc

Translated transcript:

  Physics is a "Real Science". It deals with reality. Math is a structural science. It deals with the structure of thinking. These structures do not have to exist. They can exist, but they don't have to. That's a fundamental difference. The translation of mathematical concepts to reality is highly critical, I would say. You cannot just translate it directly, because this leads to such strange questions like "what would happen if we take the law of gravitation by old Newton and let r^2 go to zero?". Well, you can't! Because Heisenberg is standing down there.

Yes these knots are real and can be experienced with a simple piece of rope.

The prime property of numbers is also very real, a number N is prime if and only if arranging N items on a rectangular, regular grid can only be done if one of the sides of the rectangle is 1. Multiplication and addition are even more simply realized.

The infinity of natural numbers is not as real, if what we mean by that is that we can directly experience it. It's a useful abstraction but there is, according to that abstraction, an infinity of "natural" numbers that mankind will not be able to ever write down, either as a number or as a formula. So infinity will always escape our immediate perception and remain fundamentally an abstraction.

Real numbers are some of the least real of the numbers we deal with in math. They turn out to be a very useful abstraction but we can only observe things that approximate them. A physical circle isn't exactly pi times its diameter up to infinity decimals, if only because there is a limit to the precision of our measurements.

To me the relationship between pi and numbers is not so unnatural but I have to look at a broader set of abstractions to make more sense of it, adding exponentials and complex numbers - in my opinion the fact that e^i.pi = 1 is a profound relationship between pi and natural numbers.

But abstractions get changed all the time. Math as an academic discipline hasn't been around for more than 10,000 years and in that course of time abstractions have changed. It's very likely that the concept of infinity wouldn't have made sense to anyone 5,000 years ago when numbers were primarily used for accounting. Even 500 years ago the concept of a number that is a square root of -1 wouldn't have made sense. Forget aliens from another planet, I'm pretty sure we wouldn't be able to comprehend 100th century math if somehow a textbook time-traveled to us.

Theres always an xkcd : https://xkcd.com/435/

To me, the least real thing in maths is, ironically, the real numbers.

As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite.

At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.

Math is about creating mental models.

Sometimes we want to model something in real life and try to use math for this - this is physics.

But even then, the model is not real, it's a model (not even a 1:1 one on top of that). It usually tries to capture some cherry picked traits of reality i.e. when will a planet be in 60 days ignoring all its "atoms"[1]. That's because we want to have some predictive power and we can't simulate whole reality. Wolfram calls these selective traits that can be calculated without calculating everything else "pockets of reducability". Do they exist? Imho no, planets don't fundamentally exist, they're mental constructs we've created for a group of particles so that our brains won't explode. If planets don't exist, so do their position etc.

The things about models is that they're usually simplifications of the thing they model, with only the parts of it that interest us.

Modeling is so natural for us that we often fail to realize that we're projecting. We're projecting content of our minds onto reality and then we start to ask questions out of confusion such as "does my mind concept exist". Your mind concept is a neutral pattern in your mind, that's it.

[1] atoms are mental concepts as well ofc

More usually, people imagine the reverse of the advanced alien civilizations: that the thing that we and they are most likely to have in common is the concept of obtaining the ratio between a circle's circumference and its diameter, whereas the things that they possibly aren't even aware of are going to be concepts like economics or poetry.

Fun historical fact: knot theory got a big boost when lord Kelvin (yeah, that one) proposed understanding atoms by thinking of them as "knotted vortices in the ether".