> "The only way to approach something new is though analogy, right? I can't think of any other way."
That's certainly true to an extent, but it also strikes me as the sort of thing you're a lot more likely to hear from a philosopher than a mathematician. Sure, you could try to understand computers by drawing analogies to televisions and typewriters and everything else that they can emulate, but that approach will always miss the rather fundamental concept of what it means for a machine to be a universal computer, capable of building complex processes out of a very small set of very simple operations. The analogies only serve as our inspiration for which algorithms to implement and what meaning to ascribe to the result of a computation that the machine was always capable of performing. You can gain useful knowledge about the capabilities of computers in a bottom-up fashion starting with a simple instruction set and building things, just like mathematicians can start with a set of axioms and explore their implications ad infinitum.
That's certainly true to an extent, but it also strikes me as the sort of thing you're a lot more likely to hear from a philosopher than a mathematician. Sure, you could try to understand computers by drawing analogies to televisions and typewriters and everything else that they can emulate, but that approach will always miss the rather fundamental concept of what it means for a machine to be a universal computer, capable of building complex processes out of a very small set of very simple operations. The analogies only serve as our inspiration for which algorithms to implement and what meaning to ascribe to the result of a computation that the machine was always capable of performing. You can gain useful knowledge about the capabilities of computers in a bottom-up fashion starting with a simple instruction set and building things, just like mathematicians can start with a set of axioms and explore their implications ad infinitum.