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Classic anecdote of the missing proof for Shizuo Kakutani’s lemma.
Reminds me of studying my notes in school
What happened then?
Actually seems like it might be an apocryphal story, based on my googling
I’m pretty sure everyone clapped.
If you have to tell someone something, it means the something is not obvious.
Mathematicians frequently use phrases like It’s obvious or It’s easy to see, which can be profoundly discouraging for a student who does not immediately find a concept simple. In math, grappling with extremely difficult problems is part of the learning process. “A challenging experience,” Ardila told me, “can easily become an alienating one.” It’s especially important to make sure that students are not discouraged during early challenges—what’s hard to see now may become easier in time. He struck this typically demoralizing math language from his teaching.
https://www.theatlantic.com/education/archive/2021/09/bias-math-sexism-racism/620207/
This is something I’ve been thinking about recently, I’ll see something that is way too complex for me, and think “well this person is just smarter I could never do that.” After 3 months of doing simpler stuff, it now seems challenging but doable, and I even see flaws in the way they did it. Just doing something for long enough, even pretty complex things become second nature.
Most things are hard until you get them. But that’s especially true in Maths. From elementary school to university until the necessary neurons in your head connect every problem seems daunting at first. But once you see what the actual problem is, once you see what tricks can be used they become trivial to solve.
Chemistry was worse than math for me. Somehow they expected you to remember a variation of a formula from way far back, and understand that you could now use a different notation system to derive another, third formula from a new formula that you had just learned… but didn’t explain that and just threw that new third formula (with entirely different units/inputs) at you and it always was a slog to go track down how it all went together because the mental concepts just didn’t flow. I don’t even remember the name of the textbook or the professor of the class, but I still remember those stupid blue boxes in the textbook where mental mindfuck took place.
I’ve tutored calculus, and probably the biggest example I’ve seen of this is the difference quotient. The formula is exceedingly obvious once you understand it, but it takes a lot of people some time for it to “click”.
Sometimes you just need it explained like you are five or something that was explained earlier in the “concepts” section of the textbook needs to be explained explicitly during the problem demonstration sections. And on top of that repeated explanation of the same concept is another key factor in solidifying concepts but I’ve found math and physics books to be so lacking in this that it’s as if they are trying to hit the absolute minimum number of word count a textbook can have. Was very frustrating during college.
Old joke.
Professor writes a formula on the blackboard. He says “Obviously, this is…” Then he stops, looks at the formula and rushes out of the room.
The next day class resumes. the professor walks in with a big smile on his face.
“I was correct. It is obvious!”
I’m too stupid for this
Obviously…
That one is “evidently”. It wasn’t obvious until you tried.
… obviously /s
that’s how athletes talk in interviews to avoid saying anything
Ah favorite words of professors everywhere
“obviously”
“simply”
“trivially”I had a linear algebra professor who did that all the time. Never did figure out what an eigenvector is not why I would want 14 ways of finding one. Brilliant man, terrible teacher.
When you multiply a matrix and a vector, you get a new vector. An eigenvector of a matrix means the output and input vectors are pointing in the same direction.
These are important for various real-world applications, but more explanation would probably have to be context specific.
So… Like to find the optimal impact angle to send an object towards a target?
The largest eigenvector would be the most probable direction and velocity of the struck object after impact?
Usually it is something like the eigenvectors represent stable states of the system, and other states will tend to be unstable until and decay into one of those stable states.
For example, the eigenvectors of the moment of inertia tensor represent “principle axes” of rotation, and these represent the possible stable axes of rotation (usually only one or two axes is actually stable, it depends on the object).
By analyzing principle axes of inertia, you can explain why a frisbee’s rotation is very stable around one axis but unstable around all other axes. And you can predict this kind of behavior for other objects.
Another example is in quantum mechanics, eigenvectors correspond to states that result after “measurement collapse” of the wavefunction, and are useful in various quantum mechanics problems, such as predicting the behavior of atoms, molecules, or semiconductors.
The largest eigenvector would be the most probable direction and velocity of the struck object after impact?
The size of the eigenvector doesn’t really matter, because if a vector is an eigenvector, scaling it (changing its length without changing the direction) will also result in an eigenvector. It’s the direction of the eigenvector that matters.
However, the eigenvalue does matter and often has real world implications, for example, it can help you determine which of the principle axes of rotation will result in a stable rotation .
An eigenvector doesn’t change direction when it is multiplied by the matrix, but it might change its length. The amount that length changes is the eigenvalue. vM=ev where M is a matrix, v is an eigenvector of M, and e is the corresponding eigen value.
An eigenvector is just kind of the direction the matrix is pointing
Well duh, obviously.
Is this why Neo became One?
This sounds exactly like my experience with that subject in college. Makes me wonder if it’s the same guy, or if they’re just all like that. Don’t think I can remember his name anyway.
The problem is that the eigenvector is the thing that satisfies the equation he showed you. That’s what it is.
Mathematics is full of completely unsatisfying answers, and only when apply it you get any meaningful idea why those things exist. But those are not their definition.
The problem I had is that taking one assortment of numbers that had no meaning, doing a bunch of operations on them (never actually finishing the operations though, because the last steps were “obvious”) leading to a different arrangement of numbers that also meant nothing, was not a good method of teaching. The pass/fail rate of that course relative to all the others reflected that. Every other teacher/professor I had before or since would include context when introducing an entirely new concept.
Yes, people teach mathematics wrong. It should start from application, and only then get formalized.
A large part of the problem is that we put people that study pure math deciding how to teach it.
As someone both studying and teaching math: there should be two different ways to teach math - for other mathematicians and for non-mathematicians.
For mathematicians you want to use all the formal proofs and sharp definitions and so on. But we have so much fun teaching that way, we forget when we switch classes that engineers don’t like/care/are motivated to think the same way. We should pivot towards application-based, result-oriented teaching but we often just don’t. And students have to deal with it because the other class managed (pure mathematicians).
Yes this. Most math instructors teach like we’re math majors and are in it for the dirty abstract and “obvious” details that they forget most of us will never use it when working on machines or even some basic programming. Their insistence on teaching in their often inefficient way acts as a filter for so many otherwise promising engineers.
Yes, that’s exactly why many universities have classes like ‘maths for electrical engineering’
It’s not about not teaching the platonic definition.
The problem is that you don’t start at the platonic definition. Mathematicians don’t start there either, they start at a problem. The problem may even be a hole in some other platonic idea, but nothing is ever self-contained Platonism… except maybe for categories, but well, the problem it looks is how far pure Platonism can get you.
I had the opposite problem when I was learning linear algebra. The professor kept things at the most abstract and generic level, which made it hard to understand what was going on, because it felt like everything was “the thing is defined as the thing”. I don’t think it fully clicked for me until I took another class that involved some actual numerical applications of those ideas.
Not really funny, at least for me as a native German speaker. I mean the movie is great and this scene in particular but the subtitles don’t work if you understand what they say…
Might help if you mute it, but I can see how that would be jarring.
Helps a bit, thanks :)
Proof by intimidation
The proof is left as an exercise for the reader.
chatgpt werks
Trivial.
Naive.
Elegant.
“Energy”.
“Entropy”.
I mean… lasers, man… how DO they work?“I’m Too Stupid For This.”
Amen!
“What?”It’s so blatant!
the strangest yet most profound things find me while im on shrooms. or maybe that’s only when i start noticing them…
this post makes so much sense, yet the more i think about it the less sense it makes.
im gonna go touch grass and look at the electeic spiders and the strange webs they weave now
holy shit i solved it.
THE PURPOSE OF LIFE IS TO DO SOMETHING
well, i guess to do absolutely nothing and die is still something to do 🤷😅😂
