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Some Fundamental Theorems in Mathematics (2019) [pdf]

people.math.harvard.edu

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Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#41

Post by ilyagr »

hnarn wrote: post #36
ilyagr wrote: post #35

Earlier quoted context omitted.

> every equation has one solution So when you see the sun rising, cherish the moment, as it has never happened before and will never happen again.

In a philosophical manner of speaking, that's true. You never step into the same river twice.

Indeed. But the scientific approach is different -- it's all about finding patterns that do repeat and making predictions.

The original comment's "theorem" is funny, but unless you are doing a homework problem, you better have a good intuition for when to take a second look at a seemingly simple situation.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#42

Post by enchiridion »

zorked wrote: post #34
GaussBonnet wrote: post #9

Earlier quoted context omitted.

The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with M…

So the way to get kids to like Math is by forcing them to learn... proofs? Not learning how it applies to the world and solves problems, but how it can be used to impress professors who care about rigor/pedantics? Math is dirty. Math was created by businessmen who wanted to be monopolists. By gamblers who wanted to win, or at least to postpone the day when they would go broke. By scammers, and those who wanted defens…

Did you get this from somewhere?

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#43

Post by thisisnico »

idclip wrote: post #14
Konohamaru wrote: post #3

Earlier quoted context omitted.

You cannot teach people--even people at a young age--to enjoy mathematics because mathematics is based on logic and almost everyone except for a few weirdos hates logic. Mr. Spock was designed to point out how unusual people who like logic are, but as a character from a minstrel show, not as a role model....

I disagree. Mathematics is a tool to practice our hobbies .. see Feynman‘s biography. Rot mathematics is where it goes wrong. Meta mathematics, abstract, detached from real life, as alot of the academy with its publish pr perish thinking ... that produces pain. Existential voids filled with an exercise that does not help many to move forward optimally. saying weirdos hate logic is extremely dismissive to a lot of wha…

I feel like in Today's day and age it's about understanding the fundamentals of a system and using software tools that automate the heavy math to allow for creative problem solving quickly and dynamically, that wouldn't be possible on paper only or through manual computation only. I'd rather give the kid advanced tools to analyze and improve on the structure than force them to learn the already abstracted ulta-fundamental and impractical math that runs those tools...

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#44

Post by alderz »

In the subject of Artificial Intelligence it says:

> Theorem: No AI will bother after hacking its own reward function.

and then goes on:

> The picture [263] is that once the AI has figured out the philosophy of the “Dude” in the Cohen brothers movie Lebowski, also repeated mischiefs does not bother it and it “goes bowling”. Objections are brushed away with “Well, this is your, like, opinion, man”. Two examples of human super intelligent units who have succeeded to hack their own reward function are Alexander Grothendieck or Grigori Perelman.

Grothendieck abandoned the mathematical community after modernizing an entire field and eventually secluded himself in France. He is considered one of the greatest mathematicians https://en.wikipedia.org/wiki/Alexander_Grothendieck#Retirem...

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#45

Post by mechaniker »

perfunctory wrote: post #2

Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.

I'm not even convinced that schools teach math as a tool. Too few students are aware of the raw power that learning how to think mathematically and use mathematical knowledge as a problem-solving tool gives them. It provides new insight into fields as diverse as visual art and as applicable as physics. The latter of which only became the potent subject it is today after being mathematized.

Forget about about students, there are career programmers out there who only began their careers after a google search on whether they needed math to be a good programmer assured them otherwise. It's so ridiculous to me that I can't help but feel that some shadowy worldwide illuminati-esque force is intentionally sabotaging mathematical education to keep the populace unaware of how much more capable they could be when fully trained in mathematics. (Why would they do this? I don't know, but I think it has something to do with the pyramids.)

Jokes aside, I do wonder what the flaws in the mathematical education system are. Anecdotally, I've found that the truly skilled mathematics students rarely go into teaching careers, which is understandable. One of my favorite mathematics books, How to Solve It by George Polya, outlines a style of teaching that requires the teacher to be as much a confident performer as well as a skilled mathematician. It's worth saying that Polya was Hungarian, growing up at a time that mathematics teachers were expected to have serious postgraduate qualifications. Whilst it would be nice to go back to such rigorous standards, I can't imagine the resulting lack in qualified teachers such a policy would produce...

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#46

Post by baddox »

zorked wrote: post #34
GaussBonnet wrote: post #9

Earlier quoted context omitted.

The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with M…

So the way to get kids to like Math is by forcing them to learn... proofs? Not learning how it applies to the world and solves problems, but how it can be used to impress professors who care about rigor/pedantics? Math is dirty. Math was created by businessmen who wanted to be monopolists. By gamblers who wanted to win, or at least to postpone the day when they would go broke. By scammers, and those who wanted defens…

> So the way to get kids to like Math is by forcing them to learn... proofs?

I mean, there are definitely issues with the whole "forcing them to learn" part, but assuming we're given the option to change the content of the math curriculum but not to fundamentally change the education system, then yes, absolutely. You don't need a crazy level of rigor, of course, but I think learning simple proof techniques could make students experience mathematics more like solving puzzles than just completing calculations.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#47

Post by GaussBonnet »

zorked wrote: post #34
GaussBonnet wrote: post #9

Earlier quoted context omitted.

The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with M…

So the way to get kids to like Math is by forcing them to learn... proofs? Not learning how it applies to the world and solves problems, but how it can be used to impress professors who care about rigor/pedantics? Math is dirty. Math was created by businessmen who wanted to be monopolists. By gamblers who wanted to win, or at least to postpone the day when they would go broke. By scammers, and those who wanted defens…

Proofs are only an example of what grade school teachers don't know. They are obviously missing all of the modern applications as well. Whether you want to call those applications "math" or something more specific ("engineering", "cryptography", etc) is a matter of taste I guess. The point still stands that kids are learning hardly anything of theory OR practice in their grade schools, and this is due to the teachers not knowing those things either.

We can talk about why the teachers don't know these things, but I think the real issue is American culture still sees math and science as something for pitiful, nerd-virgin dorks. You can try to sell math as this sexy, dangerous thing you use to win wars, but ultimately you will have to reckon with the image that most Americans have of mathematicians and scientists, which is illustrated in shows like the Big Bang Theory: they're ridiculous laughing stocks who use science as a kind of cope for not being socially successful.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#48

Post by ellis-bell »

itcrowd wrote: post #39
dan-robertson wrote: post #31

I recall learning the fundamental theorem of applied maths. The approximate phrasing is “in applied maths, if it looks right then it is” and a more precise phrasing is that “in applied maths, all reasonable series converge, all functions are continuous, differentiable and smooth almost everywhere, all limits (and integrals) exist, all sums or integrals commute, all Taylor series are good approximations, all of the si…

Do you remember where you read it? It's a nice and funny formulation but a quick attempt at googling didn't bring this up for me.

i wonder if it's a folk theorem kind of thing? i heard something similar from one of my physics professors back in undergrad

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#49

Post by mechaniker »

zorked wrote: post #34
GaussBonnet wrote: post #9

Earlier quoted context omitted.

The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with M…

So the way to get kids to like Math is by forcing them to learn... proofs? Not learning how it applies to the world and solves problems, but how it can be used to impress professors who care about rigor/pedantics? Math is dirty. Math was created by businessmen who wanted to be monopolists. By gamblers who wanted to win, or at least to postpone the day when they would go broke. By scammers, and those who wanted defens…

I mean, that's mostly a nonsense sentiment, but I really enjoyed the allegory so I will not hold it against you. The widest ranging theorem in mathematics was created by a bunch of greek cultists who believed that the natural numbers were at the heart of existence. The most important mathematical textbook, one that was taught almost exclusively for over a millenia in the west, was steeped in the traditions of said cult. That tradition is still there and has, for the most part, always been there (although, as an extremely applied mathematician, I find it's practioners to be alien).

Plus, tell me: how do you learn proofs? They aren't statements in a textbook that you just learn and recite like poetry. You have to learn to see them, to understand how all those moving parts fit together. When done right, teaching proofs is an exercise in illumination. I find it is more enjoyable all around when performed as a creative, exploratory exercise: give the students the axioms, set them lose on a conjecture.

Of course, for that you need a) Teachers who can serve as guides in the mathematical worlds and b) Students who aren't culturally predisposed to dislike mathematics.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#50

Post by pmiller2 »

ilyagr wrote: post #35
dan-robertson wrote: post #31

I recall learning the fundamental theorem of applied maths. The approximate phrasing is “in applied maths, if it looks right then it is” and a more precise phrasing is that “in applied maths, all reasonable series converge, all functions are continuous, differentiable and smooth almost everywhere, all limits (and integrals) exist, all sums or integrals commute, all Taylor series are good approximations, all of the si…

> every equation has one solution So when you see the sun rising, cherish the moment, as it has never happened before and will never happen again.

The funny thing is, besides this one you picked out, pretty much all the rest of these either are continuity assumptions or follow from continuity. I think that's because physicists tend to dismiss discontinuities and singularities as "unphysical," but it does make good intuitive sense.

I suspect that "all equations have one solution" is a lot easier to satisfy when all functions are continuous, or otherwise nice, as well.

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