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⬐ clear-as-mud

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.

⬐ ColinWright

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.

⬐ jonsen

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.

⬐ ColinWright

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

⬐ jonsen

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.

⬐ ColinWright

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?

⬐ jonsen

Yes, we agree a long way. I'm sure the video works well in its intended greater context, as you describe here. My points are about taking the video at face value out of context, which is probably what lead to clear-as-muds comment.

⬐ britcruise

Hey there, I created this video. It was just one part of a longer lesson I was trying on "Random Algorithms" because it involved Fermat's primality test...so in this case we really needed that formula to try it out which is why I added it at the end.

https://www.khanacademy.org/computing/computer-science/crypt...

⬐ CurtMonash

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, and note that every binomial coefficient except the first and last has p in the numerator but not the denominator. Hence (a+1)^p == a^p+1 == a+1.

Cleaning up the proof from there is trivial.

⬐ baggers

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?

⬐ ColinWright

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.

⬐ baggers

Thanks, that's good to know.

⬐ CurtMonash

I haven't been a mathematician for decades, and in particular I don't recall the details in the examples I'm about to give. :) But I'd say yes up to a point. In particular, one matches problems to generalized patterns, which in math often has the feel of transforming a problem into another form.

1. If you're going to take a test for school, you can recognize patterns pretty easily. For example, Harvard has a 3-day Qualifying Exam as a PhD requirement, given twice a year. Once, after looking at the exam pages for the first two days, I said "Hmm, there hasn't been a Schwarz Reflection Principle question yet."

The next evening, I got a nice hug from Lisa Mantini. :)

2. If you're doing actual research math, it's of course much harder. For example, the theorem in my thesis was scooped by a few months by Neyman and Mertens. What's more, they did it as a special case of a fixed-point theorem, while I just proved it by brute force.

And now some simpler anecdotes from my undergraduate days.

3. In my E&M course, we were supposed to work out fields for a cylindrical wire with an off-center cylindrical hole. This was a LOT easier in curvilinear coordinates than Euclidean. When I showed that to the professor in his office, he literally stood up and applauded.

4. In an abstract algebra course, there was a typo in the book for one of the exercises as to whether a certain polynomial was reducible. I solved the hard form by transforming the problem into one about a polynomial with matrix variables.

I took that story with me when I went to graduate school. Ron Livne's eyes lit up, and he transformed it again into something about -- well, into something about transformations on the polynomial's complex roots.

⬐ dkarapetyan

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

⬐ JadeNB

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... .

⬐ ColinWright

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

⬐ JadeNB

Heh, sorry about that. I was on a low bandwidth connection, and couldn't watch the video.

⬐ 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.

⬐ gjm11

Suppose the smallest (positive) number of rotations it takes to return is d. Then the numbers of rotations that have that effect are 0, d, 2d, 3d, etc. But clearly 5 rotations take you back where you started, so 5 is one of those numbers; that is, 5 is a multiple of d. Since 5 is prime this means either d=1 or d=5.

My second sentence above is a bit of a handwave. If you want to get formal about it, here's one way: suppose d isn't a multiple of 5; then note that there's an integer a such that ad = 1 (mod 5). Then ad rotations have the same effect as 1 rotation; but ad rotations do nothing, so d=1.

Now the handwaving is concentrated in the "then note that ..."; if that isn't sufficiently obvious, consider ad for a=0,1,2,3,4 and note that no two can be equal mod 5 because if ad=bd mod 5 then (a-b)d is a multiple of 5, but neither factor is a multiple of 5 and 5 is prime. So: five numbers, all of them different mod 5, so we must have one each of each congruence class mod 5; in particular, one of them is 1 mod 5.

⬐ S4M

Yes, that's a good explanation (I suspect your first paragraph is at the core of Fermat's little theorem's demonstration), it's just a shame it was left out of the video - but maybe because this part would require its own video.

⬐ mrcactu5

Fermat Little Theorem is one of my favorite results. Here is a short proof by Lionel Levine it is only 2 pages http://www.math.cornell.edu/~levine/fermat.pdf

He also get formulas like: pq divides a^pq - a^p - a^q + a

⬐ JadeNB

> He also get formulas like: pq divides a^pq - a^p - a^q + a

Incidentally, assuming you mean p and q to be distinct primes, this latter formula is almost just another instance of Fermat's little theorem: we have that q divides (a^p - a)^q - (a^p - a) and p divides (a^q - a)^p - (a^q - a) (that's the little theorem), and then that q divides (a^(pq) - a^q) - (a^p - a)^q and p divides (a^(pq) - a^p) - (a^q - a)^p (that's essentially the binomial theorem); so both p and q divide a^(pq) - a^p - a^q + a.

⬐ zurtri

There is an awesome book entitled "Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem"

(you can google it for a link to a supplier)

Well worth the read. Tells the whole story of the famous "Fermat's last Theorum" problem and is extremely well written.

⬐ CurtMonash

My best friend from graduate school (which was the Harvard Math department at a time Wiles was an assistant professor) used to stay with me when he visited his employer in NYC after moving to Israel. He would take an overnight flight and arrive, quite blearly, early in the morning. The day of a visit, the NYT had the story. And so I greeted him at the door to my apartment, not with my usual warm hello, but rather the words "Wiles proved Fermat's Last Theorem".

⬐ jacderida

Simon Singh, the guy who wrote that book, also made a very cool documentary on it. It used to be on YouTube, but it was removed by the BBC (they have a very annoying habit of removing educational content from YouTube). It's on the BBC iPlayer.

There is also an excellent long interview with Ken Ribet, who proved a result that inspired Wiles to take on the proof: https://www.youtube.com/watch?v=nUN4NDVIfVI

⬐ curryhoward

You may know this, but for those who might not: Fermat's Last Theorem and Fermat's Little Theorem are two totally different theorems (both about number theory).

Fermat's Little Theorem: For any integer a and prime p, a^p is congruent to a mod p.

Fermat's Last Theorem: For any integer n > 2, there are no integers a, b, and c such that a^n + b^n = c^n.

The Little Theorem has been known for hundreds of years and is easy to prove. The Last Theorem, on the other hand, was only finally proven in 1994 by Andrew Wiles after countless failed attempts by mathematicians before him.

⬐ mixedmath

There is an elementary number theory textbook by the famed number theorist and Ramanujan scholar Dr. George Andrews called Number Thoery, http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/...

It contains proofs of very many results from elementary number theory, like Fermat's Little Theorem here. The reason this book is different from any other is because the proofs are all combinatorial, like counting beaded necklaces or otherwise. It is also a Dover book, meaning it costs very little. It is also one of the books I used as reference to create the elementary number theory course that I sometimes teach.