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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]

#51

Post by dan-robertson »

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.

It’s basically just a joke. I got it from a professor of mine. But like all good jokes, there’s some element of truth in it: most problems in applied maths relate to something physical, and in (non-quantum) physics, something happens and only one thing happens. Therefore your problems are either bad models of reality or they have exactly one solution. It often turns out that even if your methods aren’t technically correct or properly justified, you can still reach that solution.

It is reminiscent of what Euler called “the universality of analysis” (note that over the history of mathematics, analysis has been used to refer to just about everything that isn’t geometry. In this case it means roughly the sort of mathematics a physicist or engineer might know well, so non-abstract algebra with integration and differentiation). He could do crazy things like solve combinatorics problems using infinite expansions of series in ways that didn’t really make sense (I think this example is well known but I can’t think what it is) but get to the right answer. A non-Eulerian example would be Dirac delta functions which aren’t functions at all and can be tricky to formalise but also just magically work within the framework of abstract symbol manipulation that is used in applied maths without significant modification (except when e.g. some fundamental property of your model of the universe depends on the value that the Heaviside step function takes at 0)

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

#52

Post by DOYOUWERETOLD »

JackFr wrote: post #26
cinntaile wrote: post #4

Earlier quoted context omitted.

(Here I am focusing on math outside of the math taught in pure mathematics programs.) If anything the focus on math being a tool should be even bigger than it is today if you ask me. Most if not all of the math you get taught in school is math that was invented to solve a problem, it was not created as art. Introducing this historical context would probably help students in understanding why learning math is useful i…

Replace arithmetic with “math appreciation”? No thanks. I’m still disappointed by the way music and art are taught at the elementary and middle school levels. Singing, playing an instrument, drawing are all things which can effectively be taught and are so much more useful than useless watered down “art” and “music” I had to take in grade and middle school.

Q: Do you were told 'Music' using Sound, painting Pictures ?

A: (I do:) 'FOR [Insert:Abstract] AND [Insert:Verb]'... so math ? (-;

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

#53

Post by asgard1024 »

alderz wrote: post #44

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 huma…

I think it just happened: https://news.ycombinator.com/item?id=24061653

GPT-3 just wrote an essay arguing that intelligence is doing nothing.

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

#54

Post by efavdb »

pmiller2 wrote: post #50
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.

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…

That's mostly true -- however, physicists devote a lot of time to studying phase transitions and characterizing the kinds of discontinuities that are possible etc in that context.

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

#55

Post by ur-whale »

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…

Sounds like many of my physics teachers.

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

#56

Post by pmiller2 »

efavdb wrote: post #54
pmiller2 wrote: post #50

Earlier quoted context omitted.

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…

That's mostly true -- however, physicists devote a lot of time to studying phase transitions and characterizing the kinds of discontinuities that are possible etc in that context.

Ah, yes, I forgot about phase transitions. Those are actually interesting discontinuities. It seems to me the uninteresting ones are the ones that have roughly the same behavior all around them (e.g. poles of complex functions are not that interesting).

Would you say that's an accurate characterization?

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

#57

Post by ur-whale »

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.

Very, very strongly disagree with you.

First, some of the most beautiful math stemmed from an attempt at solving an actual, practical problem, and the path the "discoverers" of such math followed to get there is often profoundly different from the way it's taught.

Your view, which I've been subjected to by most of my math teachers my entire curriculum is the exact reason why most people hate maths because it feels like a bunch of very complicated and hard to learn stuff that seems to serve no purpose whatsoever.

When you've been learning a bunch of very theoretical stuff for decades and finally find an actual application / use for it, my experience is that:

     - you do *then* see the beauty of it
     - you curse your math teachers and their ancestors to the seventh generation for not having shown you the practical application of the (typically over-generalized) theory he/she was force-feeding you back then.

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

#58

Post by codethief »

laichzeit0 wrote: post #7
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 convinced that if Book of Proof [1] was taught in high school, most people would (a) understand math, and (b) not hate it. http://www.people.vcu.edu/~rhammack/BookOfProof/index.html

This is a good book but I'm not sure it's low-level enough for those people in high school who don't like or haven't learned enough math – he mentions matrices, sin/cos, real numbers, … right in the first chapter!

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

#59

Post by Koshkin »

mechaniker wrote: post #45
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…

> I'm not even convinced that schools teach math as a tool

V.I.Arnold has written an excellent rant about this: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

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

#60

Post by Koshkin »

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…

That's not what history of mathematics tells us.
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