These videos do an incredible job of illustrating how to intuitively arrive at an answer by composing many of the parts you need to build a proof for more complex topics.
To pick an example:
Intuition is great for understanding the intermediate value theorem. If the temperature was 40 degrees at 8 am, and is 60 degrees at noon, it must have been 50 degrees somewhere in between.
Why, though? How can you guarantee that there will always have to be a time where it was 50 degrees[1]?
[1] provided that temperature is real-valued and temperature change is continuous, the debate of which I'll leave to the science people
What I believe to be true is that a solid intuitive proof will be sound, but not necessarily transcribable until formalized. Formalization of the proof is essentially the matter of making a proof communicable.
Putting it one more way: purely-intuitive proofs would be just fine if one could share one's mental state with another.
What do you think?
> Am I correct in understanding that it's not that intuition is invalid, but that it's insufficient for communicating a proof to another mathematician.
is close to correct. It's a very useful tool, but it can sometimes lead you into incorrect assumptions.
However, the rest of it incorrect. Math, and in particular, probability and statistics, are full of things that seem intuitive and easy at first glance, but are actually a bit more nuanced.
Take for example, the Monty Hall Problem. It's probably familiar to most here, but essentially you have three doors. Behind one, is a car or some other desirable object, and behind the other two are goats or something undesirable. Select a door, and you get whatever is behind that door. What's the probability of getting the car in this scenario?
Easy enough right? It's just 1/3. However, if Monty Hall opens one of the doors after you select a door, he always reveals a goat, and he offers to let you switch your door now, should you switch or keep your door?
What's the probability that you get a car if you switch? Is it 1/2 or 1/3? What's the probability that you get a car if you stay? 1/3?
Well, the somewhat surprising answer is that you have a 2/3 probability of winning the car if you switch, and a 1/3 probability of winning the car if you stay.
Often things will break down in the limits as well, so if you try to apply your finite dimensional intuition to something that corresponds to an empty object or an infinite number of objects, you'll find that you're often wrong.
No, I have had purely intuitive proofs myself that turned out to be wrong.
However what you may have picked up on is we disstain people who insist they have a brilliant idea for which someone should prove they are right. That’s like saying a car engine consists of cyclinders and piston; someone else just needs to do the work to put the pieces together.
https://terrytao.wordpress.com/career-advice/theres-more-to-...
I recommend reading the entire post - it's not too long, and links to other great materials - but this is a good relevant quote:
> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.
They do this for good reason. I had to take, as part of my philosophy concentration, a bunch of mathematical logic classes. First Order and Second Order logic, as you might imagine, are pretty simple. The rules make sense in a very intuitive way. I'd get 100 on exams just because I'm good at programming. But when I had to study metalogic, model theory, Henkin Proofs, and Godel's Theorems, that intuition quite literally flies out the window. I guess my point is that most interesting stuff is rarely intuitive.
But respectfully, ZFC came about because of logical paradoxes that couldn't be accepted as consistent.
You can create a new number system and derive all kinds of consequences but the truth is, most mathematicians care more entirely about prime number theory on the naturals that are entirely based on counting.
Most of modern math is based on the natural numbers. You can't remove all intuition. The thread that holds our love for math is also the same one that tells us we are exploring consequences that tell us a dearth about our universe.
If you think that coming up with theorems about, e.g. infinitary algebra, is intuitive, then I guess you're just a lot smarter than I am. Infinities always throw a wrench into our intuitions. Something being a "logical consequence" of something else is an incredibly simplistic way of looking at things, not to mention that I'm not exactly sure what that has to do with intuition anyway.
That 'non-intuitive' aspect of high-dimensional spheres having very small volume relative to a hypercube becomes a point that builds intuition for how those spheres behave, once you're familiar with it. And then you can start throwing out things that seem intuitively wrong, and following intuition towards new ideas.
> the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference
There is such a thing as developing your intuition.
I think your meanings are not qualitatively different.
The trick is that you need good data to feed it that makes the patterns clear. You might have not gotten a lot of that on the topic you had no intuition of. On some topics, there hasn't been much of an attempt to do that since people expect it to be hard with everyone forced to do the methods that are non-intuitive. Lots of traditions are like that. Then, there's the possibility something can't be intuitively handled at all. I don't know if those exist but they might. That many activities of people doing math seem to get an intuitive boost during their day-to-day work makes me lean towards intuitive possibilities for about everything that can be approximated (esp with heuristics).
Let's claim that writing a proof is similar to writing a program. You start from a state, your premise, and have your assumptions, input, which you process through a series of steps, maybe taking cases in between (branching) and return an output (hopefully True).
Then considering this correspondence, it will be fine to say that you, as a programmer, can guess the output/behavior/semantics of any program without running it. Similarly, a mathematician would be able to reach the conclusion that the theorem holds, without going through all the details of the proof.
Sure, in most cases you can, and as you get more and more experience and knowledge as a programmer you can say you improve your guesses. But this ends up backfiring with a good probability, when our model skipped a step, e.g. the interaction of two concurrent data structures.
Thus I don't think it is right to say "as you are able to know all axioms and rules of inference, you should be able to achieve perfect intuition". You can know the fundamentals of each layer on your stack and you would find it insane if one asked you to guess the output of any non-trivial terminating program.
Note also that the way of inference (logical consequences) can and does change (e.g. ZF vs ZFC).
() I may be misusing the word model here -- it is usually defined (at least to things I work with) as the interpretation of a theory that satisfies all theorems (inferences). () constructive proofs yes. No flame wars please :)
citation?
3Blue1Brown is better than most but far from best.
I've never been able to find a good explanation online, only in some older textbooks. I've begun to fear that there are few people who are able to read equations while the others interpret the explanations of those people, where then those same old explanations gradually mutate as they propagate without reference to their source. This Stanford lecture is the best I could find: https://www.youtube.com/watch?v=1rqJl7Rs6ps
"The Analytic Theory of Heat" Joseph Fourier who explains the realization at length.
3b1b's videos are excellent, but if they were interactive, open-source and in a similar model to Wikipedia, people could contribute and improve upon content that is already great.
There are great examples of what is possible with explorables, see Bret Victor, D3.js, distill.pub, observablehq etc. Just no encyclopedic model yet.
write one and maybe include that fact as a resume / interview item :)