I can logically see why a mathematical proof works by following the steps of its derivation. Although, I don't feel like I understand on a more fundamental level what is happening, why it works and how someone figured it out.
How do people gain that fundamental understanding of math (what would be called "intuition" in other areas) contrasting the pure mechanical aspect?
>>8169386
write proofs instead of just reading them.
>>8169386
Unfortunately what you desire can only be gained by being born with it.
In other words you will always be a brainlet.
>>8169386
intuition isn't intuitive. it's more based on experience.
>>8169386
it comes with time senpai. keep working
this >>8169392
write proofs that go all the way back to concepts you are familiar with
>>8169406
this. "intuitions" is decades of experience, years upon years of practice, resulting in knowledge of many general patterns as well as a plethora of special cases which lend themselves to generalization in some way or other.
this is true for most areas, not just maths.
>>8169386
Intuition is one of the non-computational properties of consciousness, one of those skills making the human mind more powerful than any Turing machine.
>>8169386
you don't have intuition until you are Cleo from Math Stackexchange
The experience of intuition unfolding is so fucking strange. It's like aliens or spirits are beaming hints to you from another dimension.
"Shit how the hell do I do this..."
>"Try writing it out on a line, anon"
Hmm. Okay, done. Now what?
>"Anon, you need to plot the center point of the line, then look at how the points might occur in relation to it."
"Okay... Oh shit there's a pattern!"
>"I think you've got it from here anon."
>>8169464
truth
>>8169386
There is logic behind it.
Look up Sequent calculus.
Also: Those proofs are missing steps that should be clear, to keep the book's length in check.
Depends on the field. Functional analytic stuff I have a better feel for, even if I don't know all the logical steps. Everything else feels like throwing darts with a blindfold on.
>>8170132
Andrew Wiles likened doing mathematics to walking around in a giant mansion with no lights on for several weeks, bumping around into the rooms and feeling the objects, then suddenly having the lights come on and seeing the whole thing.
>>8170159
Good analogy.
Keep in mind a lot of theorems were motivated by context. Searching for general patterns that explain something observed, like the fibonacci sequence.
This is usually left out of books.
Feynman had a really good explanation of relation between axioms and theorems in his messenger lectures. Check it out
