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Fermat's Little Theorem (2013) [video]

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Re: Fermat's Little Theorem (2013) [video]

#11

Post by baggers »

CurtMonash wrote: post #8

I love Fermat's Little Theorem, both for its own sake and because it's an intermediate step in proving the Two Squares Theorem. My equivalent to "counting sheep" used to be to review the proof of the TST. The simplest proof I know of Fermat's Little Theorem is induction, assuming that we already know the Binomial Theorem. Suppose a^p == a. (== is my symbol for "is congruent to mod p" in this post.) Expand (a+1)^p, an…

This question is as someone who is mathematically curious but not yet adept.

In programming we slowly gain a big grab-bag of patterns and approaches to certain problems and build an intuition of what to apply where. It doesn't nearly cover your full experience but helps break problems down.

Do you find there is an analog to this with theorems? If so what's the essentials from your 'grab bag' and, beyond just reading more, what practices help build your feeling of where to use them?

Re: Fermat's Little Theorem (2013) [video]

#13

Post by ColinWright »

clear-as-mud wrote: post #12

This is why I despise mathematicians. Everything was going smoothly, until they obscured the whole concept behind that awful notation. Why? Notation is the destroyer of understanding. Letter, pipe, letter, superscript. Yeah, real nice. Clear as mud.

If you choose not to invest the time learning the notation, you will forever be limited to writing everything out in longhand. Try reading maths from the 1600s or 1700s and you will come to appreciate the value of notation.

And don't start down the road of thinking everything can be made unambiguous. It is context dependent, and always will be. Look at computer code, look at something deep in the bowels, and tell me what the "+" sign means. Is it integer? Float? List? Or has it been over-ridden by some library you've loaded somewhere else? Even finding where it's been defined can be a challenge. You have to trust that the author has played fair, and that your intuitive understanding that "+" is doing something consistent with the concept of "+" in numbers is close enough.

You could write out all the math notation in longhand, but it won't aid your understanding. If you actually follow the video then you will follow the notation. But there's a reason why we say "read like math" - it takes time and effort to gain understanding. It's not a novel, it's not a comic.

    The only way to learn mathematics
        is to do mathematics.
            -- Paul Halmos

    "There is no Royal Road to geometry,"
        -- Euclid, in reply to King Ptolemy's
           request for an easier way of learning
           mathematics.

Re: Fermat's Little Theorem (2013) [video]

#14

Post by ColinWright »

baggers wrote: post #11
CurtMonash wrote: post #8

I love Fermat's Little Theorem, both for its own sake and because it's an intermediate step in proving the Two Squares Theorem. My equivalent to "counting sheep" used to be to review the proof of the TST. The simplest proof I know of Fermat's Little Theorem is induction, assuming that we already know the Binomial Theorem. Suppose a^p == a. (== is my symbol for "is congruent to mod p" in this post.) Expand (a+1)^p, an…

This question is as someone who is mathematically curious but not yet adept. In programming we slowly gain a big grab-bag of patterns and approaches to certain problems and build an intuition of what to apply where. It doesn't nearly cover your full experience but helps break problems down. Do you find there is an analog to this with theorems? If so what's the essentials from your 'grab bag' and, beyond just reading…

Yes, there is, but I've never thought of it in those terms, and I can't enumerate them easily. There are techniques to apply, approaches to try, and connections to make that can help. But I don't know any way of building intuition other than by actual doing.

Tim Gowers writes well about how to build mathematical knowledge, techniques, and a library of tools.

Re: Fermat's Little Theorem (2013) [video]

#15

Post by ColinWright »

JadeNB wrote: post #7
dkarapetyan wrote: post #2

Reminds me of Burnside's Lemma ( https://en.wikipedia.org/wiki/Burnside%27s_lemma ).

Indeed, the idea of the proof of Burnside's lemma is essentially the same as what is often called the necklace-counting proof of Fermat's little theorem: https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_th... .

Yes, and the submitted video is that necklace counting proof, hence the video reminding them of it.

Re: Fermat's Little Theorem (2013) [video]

#16

Post by S4M »

Am I the only one who doesn't get it? My background is in maths and I am (was) able to prove formally Fermat's little theorem, but I got lost around minute 3:08, and am not clear why the fact that 5 is prime means that any combination must take 5 rotations to return to itself. I can see that it is true but the reason is not clear to me.

Re: Fermat's Little Theorem (2013) [video]

#17

Post by jonsen »

ColinWright wrote: post #13
clear-as-mud wrote: post #12

This is why I despise mathematicians. Everything was going smoothly, until they obscured the whole concept behind that awful notation. Why? Notation is the destroyer of understanding. Letter, pipe, letter, superscript. Yeah, real nice. Clear as mud.

If you choose not to invest the time learning the notation, you will forever be limited to writing everything out in longhand. Try reading maths from the 1600s or 1700s and you will come to appreciate the value of notation. And don't start down the road of thinking everything can be made unambiguous. It is context dependent, and always will be. Look at computer code, look at something deep in the bowels, and tell me…

Well said and all, but notation is a destroyer of understanding, if you are not well versed in reading math. There is a very abrupt jump in the video from the pleasant down to earth demonstration using picture language when the "cold" math notation is just thrown at you in the end.

Re: Fermat's Little Theorem (2013) [video]

#18

Post by ColinWright »

jonsen wrote: post #17
ColinWright wrote: post #13

Earlier quoted context omitted.

If you choose not to invest the time learning the notation, you will forever be limited to writing everything out in longhand. Try reading maths from the 1600s or 1700s and you will come to appreciate the value of notation. And don't start down the road of thinking everything can be made unambiguous. It is context dependent, and always will be. Look at computer code, look at something deep in the bowels, and tell me…

Well said and all, but notation is a destroyer of understanding, if you are not well versed in reading math. There is a very abrupt jump in the video from the pleasant down to earth demonstration using picture language when the "cold" math notation is just thrown at you in the end.

If you are not well versed in reading math then you have two options. One is to forever remain fuzzy and imprecise in your understanding, the other is to learn the notation you need, see how it matches your internal understanding, and then use it as an enabler.

If the notation destroys your understanding, then you didn't understand it in the first place, you only had a vague sense of satisfaction with a pseudo-understanding. This is akin to thinking you understand an algorithm, but not being able to code it.

I have every sympathy with people who find the notation a barrier, because so did I. Over the years, indeed, the decades, I have come to appreciate notation as a way of concisely expressing deep and complex ideas. Try not using notation - your brain quickly turns to mush as you try to follow hundreds of words that could otherwise be expressed in a mere dozen symbols.

Archimedes in "Measurement of the circle" wrote:

    The area of any circle is equal to a
    right-angled triangle in which one of
    the sides about the right angle is equal
    to the radius, and the other to the
    circumference, of the circle.
Today we would write:

    Area = (1/2)⋅r⋅(2πr)= π⋅r^2
People have a phobia about notation, but the solution isn't to avoid notation, it's to show that it is powerful, and that overcoming the phobia is one way of gaining access to that power.

Edited to change the example from Newton's Principia to Archimedes

Re: Fermat's Little Theorem (2013) [video]

#19

Post by jonsen »

ColinWright wrote: post #18
jonsen wrote: post #17

Earlier quoted context omitted.

Well said and all, but notation is a destroyer of understanding, if you are not well versed in reading math. There is a very abrupt jump in the video from the pleasant down to earth demonstration using picture language when the "cold" math notation is just thrown at you in the end.

If you are not well versed in reading math then you have two options. One is to forever remain fuzzy and imprecise in your understanding, the other is to learn the notation you need, see how it matches your internal understanding, and then use it as an enabler. If the notation destroys your understanding, then you didn't understand it in the first place, you only had a vague sense of satisfaction with a pseudo-unders…

My point was, that the way notation is just thrown at you in the video can only add to the phobia for the uninitiated. The video is clearly aiming at basic pedagogic explanation, and it is all destroyed in the end. The math versed who easily read that math without explanation do not need the bead juggling. The pedagogy fails in the end.

Re: Fermat's Little Theorem (2013) [video]

#20

Post by ColinWright »

jonsen wrote: post #19
ColinWright wrote: post #18

Earlier quoted context omitted.

If you are not well versed in reading math then you have two options. One is to forever remain fuzzy and imprecise in your understanding, the other is to learn the notation you need, see how it matches your internal understanding, and then use it as an enabler. If the notation destroys your understanding, then you didn't understand it in the first place, you only had a vague sense of satisfaction with a pseudo-unders…

My point was, that the way notation is just thrown at you in the video can only add to the phobia for the uninitiated. The video is clearly aiming at basic pedagogic explanation, and it is all destroyed in the end. The math versed who easily read that math without explanation do not need the bead juggling. The pedagogy fails in the end.

The explanation of the theorem has got as far as it can go by that point. There's a choice to be made. Avoiding notation means that those who could now connect the explanation with things they have already seen is lost. Avoiding notation also means that it's just all been a show, and there's no real take away except that, well, this guy played with beads an counted things.

If someone is truly notation-phobic then there is nothing to be done.

Having the notation means that those in the middle ground have a chance to see real maths in action, and may be inspired to do more. The genuinely notation-phobic will never advance in math. For those who know the theorem this is a cute visualisation. People will only learn when they are ready, and all pedagogy needs to be targeted at those who are ready for it.

I suspect we agree more than people reading this might think, but we might disagreed over the perceived purpose of the video. I see it as getting people engaged with the thought processes, and then showing that it's really math, and look, here are the formulas that come out of it. It's an opportunity to learn about the formulas. For the genuinely notation-phobic, there is no real hope, except for them to see that this is a cool thing their phobia is preventing them from understanding better, so maybe getting over the phobia would be worthwhile.

I could do a complete case analysis, but this isn't the place, and I don't have the time. Would you suggest not having the notation at all? What would then be the point of the video? Who would gain?

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