"Young man, in mathematics you don't understand things, you get used to them"
Von Neumann

“The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it.”
― Jacques Hadamard

it reminds me how zoning out allows the brain to actually work behind the scenes when one hasnt had enough sleep, in a similar manner to dreams as in the video!

Imagination is more important than Knowledge. -Albert Einstein

My whole PhD in math was

"F&^K this, I'm going fishing..."

And then solving it while threading hooks, chasing bait, and sitting on a rock watching the water lap.

Funfact: Poincaré proposed the principle of Relativity, that the laws of physics are the same in all inertial frames before Einstein. His contribution is never mentioned because his reasoning was flawed.

back in bachelor's the solution to a thermodynamics problem came to me in my dream when i gave up and slept after working for 6 hours on it

like in OPPENHEIMER, Bohr says to Oppie, do not try to understand it, Feel it.

The more rigorously trained we are, the greater bias we impose upon novel thoughts.

Back in my bachelor's, I was a really good student, and there was this result that just given to be taken as granted which usually the case for physics , so decided to derive it using the known mathematics spend weeks and 10 pages later I derived it.. Shown it to my mathematics professor and she said you should have been a math major…. Now 10 years later, am dumb as a potato 😢😢😢😢😢

I can't comment on making mathematical discoveries from experience. But when faced with having to teach myself mathematics from books, I discovered that if I took a small drink of alcohol  (I like blended Whiskies - just a finger to relax me in the evenings when I needed it) that I could make progress in what I was reading. The effect the alcohol had was the quiet down the analytic part of my brain that was trying to work with the individual pieces of the equations (going around in circles) and let me see equations and data as a coherent objects I could fit together and begin to make progress. Similar to the difference between trying to read by concentrating on the individual words rather than the sentences as a coherent thoughts which connect together to make larger thoughts.

I think there is a connection with my self-observation and dreaming. Dreams use the visualizing and imaging parts of the cortex and shut down the analytic parts. That is why, when you trying to read something is a dream, you can't. But you can still see the page with words and keep trying put it together. I believe that is source of the dreaded "Taking the Exam" dream where the questions turn into gibberish right before your eyes. My premise is that dreams work with objects (visual) rather than the pieces of the objects (analytic). It's at that level where the inspiration shows up.

By the way, I got nothing to back up any of this. It's just 50 years worth of moderately processed thinking on the subject. I'm not recommending the method to everyone (although, if you want to try, I recommend JW Black Label - but just a touch - and I would make sure you're not driving later). :)

I like how ur channel focuses on extremely unconventional topics, thanks

Points of insight after thinking REALLY hard about a problem. 1. Getting up and going to the water fountain. 2. Taking a nap. 3. (Most frustrating) Turning in the examination paper.

I work in data science / intelligence. That's always how I solve complex problems :
1) Work on it, work on it, work on it some more
2) Play around : sometimes I explore or get obssessed with more simple or trivial aspects of a problem, to get to the bottom of some pattern I suspect, almost like an autistic person.
3) Do something else and let it sediment in my brain.
4) Solution comes as intuition, in a flash.
So I agree with what is described in the video, except that think obssesive exploration and fooling around at the margins of the problem is part of the "algorithm".
It kind of feels like... side quests in an open-map game.

We have to be emotionally invested in the problem for it to generate creative results. Mathematician and physicists were some of the most passionate people

Andrew Wiles describes his work to prove Fermat's Last Theorem as taking years with him have the problem always in the back of his mind every waking moment regardless of what he was doing For him the process of discovery was like fumbling around for a light switch in the dark until you find it turn on the light and suddenly everything is clear.

John von Neumann was legendarily fast. Quitr often solutions came to him with little or no preparation. If it was particularly difficult he'd sleep on it and quite often he'd have the answer on awakening. If he didn't, he'd set the problem aside never to return to it. So I think despite his obvious gifts and contributions thiss is why he never made a discovery on the level of General Relativity or the theorems of incompleteness and consistency of arithmetic.

Finally to take a pedestrian example when I have trouble remembering a name I'll actively try not to think of it and often it will come to me

Also maybe mathematicians should practice meditation

Pasture was correct. Chance favors the prepared mind. These Mathematicians had thought about these topics for thousands of hours before taking a break and making a breakthrough. The nonexpert doesn't make a groundbreaking discovery without some preparation.

When working on my PhD thesis, I had a lot of strange ideas come into my mind... the family of problems I was working on was a new area of combinatorial design theory, and as a result, I was completely obsessed with it, and it seemed my brain was constantly churning and brainstorming in the background. I had a lot of great ideas, but it wasn't until the last year of the thesis that everything came together in a really beautiful way that classified the whole family of designs that I was looking at.

Dude I'm so glad your videos are taking off recently, you've been consistently at it for a long time, such high quality. Well done.

I learned about differential equations some years in advance and tried to solve the equation of a falling object when there is air resistence, mv´ = -mg +  cv². I had problem solving it, but suddenly one night, I realized that it's separable and could solve it. That was just a few nights before I started the Swedish "gymnasium" (comparable to high school, I think).