You can definitely explore mathematics experimentally, we all do - computer or not, but to really learn the mathematics just read the books and do the exercises.
Being able to work things out symbolically on paper is probably one of the biggest skills you can have beyond your usual repertoire of programming skills - is there any worse feeling that guessing your way through a problem you don't understand?
Isn't this called science?
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).
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.
Regardless, I plan to be learning until the day I drop, science or not.
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 do you think mathematicians do?
* 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?
Luckily, there are plenty of resources available to someone self-studying online on various math forums.