> is there any worse feeling that guessing your way through a problem you don't understand?

Isn't this called science?

A very small fraction of experimental science is about just doing something and seeing what happens without a strong prior. You are expected to do your homework first. As a scientist, your role is typically to read and understand all the relevant prior work in the area first, then using this theory to derive a hypothesis for the outcome of your experiment. If you are lucky, the way the experiment unfolds might be outside your understanding, given that your understanding of the prior theory was correct, you have made a new scientific discovery which will be used to create new theory.

The fragment of uncertainty in this is typically very small and the process is very far from guessing your way through, hoping to find something new (whatever you found using such a method will most likely be either already known or incorrect).

Yes, I agree with you. My comment looks dull with your explanation. Of course, I also don't advocate doing something without any hypothesis. Though, I think hands-on practice on a subject can help you learn things better. For example, I find it very useful for linear algebra (maybe this is not applicable to other subjects). I can analyze my flawed intuitions.

When I first heard about the Monty Hall problem, I didn't understand it and tried it myself. It was way easier for me to understand the flawed intuition by analyzing each line as opposed to, say, for example, the explanation of Judea Pearl (which is also good).

What I wanted to emphasize is that it is not bad guessing your way through a problem you don't understand. But yeah, of course, you should have some knowledge.

I wish I could say I have been doing science already, not yet (hopefully), but I was referring to (say) programming problems like (for example) one recently where I could see that it boiled down to asking whether there existed a linear combination of a input set of vectors with a single second vector - I was able to spot the connection, I'm not convinced I would've been able to get it without the mathematics under my belt to spot the pattern.

Regardless, I plan to be learning until the day I drop, science or not.

Is mathematics science though?

Mathematics is the exploration of the a priori. Historically some axioms have been deemed more real than others. I think Gauss rejected non-eulidean geometry for instance. But this point of view has changed. With abstract algebra, and I believe also computer science, modern mathematics is about exploring connections between structures such as they emerge from stipulated axioms and rules of inference. It is science in the sense that ideas and hypotheses can be tested experimentally. But a proof requires more than non-falsification. Then there is the complication of potentially irreducible computation problems, where essentially a kind of mining of the computational space is the only way forward. This is the new kind of science Stephen Wolfram speaks of.

I was inclined to write something similar myself as I agree in general. Mathematics is much more about the thinking than the doing, programming is more about the doing.

However, I do think that programming can be an excellent complement and aid when understanding certain concepts. Sometimes, writing a numerical simulation of a hard to grasp concept can make things fall into place. In particular in probability theory it can be quite helpful to generate a lot of random samples to validate and clarify some counterintuitive concepts.

what's the highest level math that one can theoretically get to through "just [reading] the books and [doing] the exercises"?

Field's Medal? Is that high enough?

What do you think mathematicians do?

I meant in terms of mathematical field/topic. Like the progression I did to get an engineering BS was roughly

* basic math > algebra/trig > calculus, linear algebra, differential equations

As a non-mathematician, my impression has been that mathematicians do a lot of proof-based work since (to my impression) the goal is to make new findings, not just learn concepts and formulas that they can apply to their work like an engineer. From my limited experience with proof-based work, it seems like something where you would need guidance and discussion. Whereas when I learned things like derivatives and integrals, it was very procedural and you just needed to learn a process, which is then built on by a more advanced process, etc.

Can one realistically self-learn up to low-dimensional topology or algebraic geometry just by reading books and doing the exercises?

The likelihood that you will succeed in self-learning any field of mathematics without outside help goes down as the material gets more advanced, but there's nothing fundamentally impossible about the endeavor. You can definitely get to graduate level that way if you're willing to put in the work and study the right things in the right order.

Luckily, there are plenty of resources available to someone self-studying online on various math forums.

what would you class as "graduate level" in this context? Would that be stuff like PDEs? As I said I'm not very familiar with what the 'progression' is past undergrad