7 ms·

What does it feel like to invent math? (2015) [video]

106 points by espeed 10y ago

kbd 10y ago
As seems typical, this video goes on for a while and is straightforward until it completely loses you all of a sudden. I have no idea what would even motivate the separation of numbers into "rooms" like the video shows, let alone how that explains why an infinite positive sum = -1.
- espeed 10y ago
  The infinite sum makes sense when using the p-adic number system [1] -- and measuring distance using the p-adic metric -- which forms an ultrametric space [2]. An interesting related concept is the Bruhat–Tits building [3]. For an intro on p-adic numbers, read these two short articles "A first introduction to p-adic numbers" [4] and "A Tutorial on p-adic Arithmetic" [5] or see the short video "Introduction to p-adic Numbers" [6]: [1] https://en.wikipedia.org/wiki/P-adic_number https://en.wikipedia.org/wiki/P-adic_number [2] https://en.wikipedia.org/wiki/Ultrametric_space https://en.wikipedia.org/wiki/Ultrametric_space [3] Bruhat–Tits building https://en.wikipedia.org/wiki/Building_(mathematics) https://en.wikipedia.org/wiki/Building_(mathematics) [4] A first introduction to p-adic numbers http://www.madore.org/~david/math/padics.pdf http://www.madore.org/~david/math/padics.pdf [5] A Tutorial on p-adic Arithmetic https://koclab.cs.ucsb.edu/docs/koc/r09.pdf https://koclab.cs.ucsb.edu/docs/koc/r09.pdf [6] Introduction to p-adic Numbers https://www.youtube.com/watch?v=vdjYiU6skgE https://www.youtube.com/watch?v=vdjYiU6skgE
  - cderwin 10y ago
    I think the problem is that the introduction of the p-adic metric was poorly motivated. It's not clear why we would want 1+p+p^2+p^3+... to converge to -1, and introducing the p-adic metric to do so doesn't show why the p-adic numbers are useful in general (all it's really saying is that the partial sums are 2^(n+1)-1). I'm sure you can derive all sorts of weird metrics so that various weird identities are true; that alone fails to make them interesting. Based on this video alone it's not clear that either the identity or the p-adic norm are interesting in any non-trivial sense of the word. The result is that the introduction of the p-adic metric is hard to follow and the resulting identity seems arbitrary, even if you manage to follow the bit about the metric. (And these combined with a lack of rigor where it's needed seem to be recurring problems in 3Blue1Brown videos.)
    - t-ob 10y ago
      > I'm sure you can derive all sorts of weird metrics so that various weird identities are true On the rational numbers, at least, the p-adic metrics are more or less your whole lot, according to Ostrowski's Theorem [1]. There is a kind of cognitive hurdle everyone who studies these numbers has to clear, in that things that should be "large" turn out to be very small indeed, when viewed under a p-adic lens. I think it's more instructive to build up the ring of p-adic integers first [2, chapter 2], and construct the p-adic numbers from there. I can assure you they are very useful, though! A general theme in number theory is to take a "global" problem, defined over the integers, and to translate it into infinitely many "local" ones (over the p-adics, for each prime p). These are sometimes easier to solve and, if you're lucky, offer insight into the global solution you're looking for. [1]: https://en.wikipedia.org/wiki/Ostrowski's_theorem https://en.wikipedia.org/wiki/Ostrowski's_theorem [2]: http://www.springer.com/gb/book/9780387900407 http://www.springer.com/gb/book/9780387900407
    - posterboy 10y ago
      >doesn't show why the p-adic numbers are useful in general well, you certainly are neither more help either ;)
    - stablemap 10y ago
      My short advertisement is that it has proven very useful to study Diophantine equations by reducing mod n. The Chinese remainder theorem [1] tells you to focus on reducing mod p^r, where p is a prime. If r = 1 then you are working in a field but in the ring Z/4, for example, I know 2 ≠ 0 and yet 2·2 = 0. To make the p-adics Z_p I stitch all of these Z/p^r together: an element is a choice of a_r in each Z/p^r and these have to be compatible: a_2 reduces mod p to a_1 and so on. The resulting Z_p has no "zero divisors", and if I allow myself to invert p I get a field Q_p. This is a huge improvement, a foundation on which to build analysis and geometry as we did over R. [1] https://en.wikipedia.org/wiki/Chinese_remainder_theorem https://en.wikipedia.org/wiki/Chinese_remainder_theorem
- taneq 10y ago
  "Step 2: Draw the rest of the owl."
Thoreandan 10y ago
I especially like the 3Blue1Brown videos on Hanoi/binary-counting and explaining (visually) Euler's Identity.
risefromashes 10y ago
I didn't understand the proof for 0.5 + 0.25 + ... = 1 itself. By his visual number line analogy, wouldn't 2/3 + (2/3)^2 + ... = 1 too? Visually, that too seems to "approach" 1.
- evanb 10y ago
  It approaches 2. 2/3+(2/3)^2 = 2/3 + 4/9 = 6/9+4/9=10/9 > 1, and all the terms are positive, so as you add more terms it will get farther from 1. If you do 10 terms you get to 1.96532somethingsomething. If you do 100 terms you get 1.[seventeen 9s]508somethingsomething.
  - BillBohan 10y ago
    leereves is correct. The sequence is 2/3 + 2/9 + 2/27 + 2/81 + 2/243 + ... and it does approach 1 In your second term you should take 2/3 of the remaining 1/3, not (2/3)^2
- Houshalter 10y ago
  I like the algebraic approach: x = 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ... 2x = 1 + 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ... 2x = 1 + x x = 1
- leereeves 10y ago
  The number line analogy does work for 2/3, in the same way it worked for 9/10. It represents the repeating trinary number 0.22222... = 1. But the analogy only corresponds to p + p^2 + p^3 ... = 1 when p = 1/2 because that's the only time the remainder = p. For 2/3, the number line analogy instead gives 2/3 + 1/3 * 2/3 + (1/3)^2 * 2/3 ...
- ssivark 10y ago
  Coming from someone who is "used to" this idea for a long time, I'm not sure how helpful my explanation would be to a learner, but I decided to take a shot at it, just in case it helps someone. That said, in my experience, a significant part of "understanding" involves getting used to things and expanding the scope of what you consider "natural". -- Suppose you wish to walk from 0 to 1. You'll have to walk half the distance. Then half of the remaining half. Then half of the remaining quarter. Then half of the remaining eighth. If you keep doing this "asymptotically", you will get very very close to 1 and (almost) reach there. That's the basic idea. This is also known as Zeno's paradox: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_of_motion https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_o... Instead of halving step sizes on each iteration, you could reduced by a different fraction. In general, this is called a geometric series: https://en.wikipedia.org/wiki/Geometric_series https://en.wikipedia.org/wiki/Geometric_series When the ratio is (2/3), after two steps you overshoot 1. (You can see that (2/3) = 0.66, (2/3)^2 = 0.44 )
afarrell 10y ago
Does anyone know of a book, channel, or subreddit which is focused on critiquing the explanations in videos and blogs like this and talking about the art of explaining technical concepts?
- posterboy 10y ago
  I don't think that would be straight forward, because of competing ideas.
  - afarrell 10y ago
    Hence why I suspect it would make more sense as a subreddit or other forum, where you have people debating what does or does not make a good explanation. I guess it would be kinda like literary analysis, but with a much clearer practical point.
    - posterboy 10y ago
      for children or adults, general or math specific, school/university/doctorate context or just plain how to make things understood? Because the latter is basically just what mathematics should be, see http://etymonline.com/index.php?term=mathematic http://etymonline.com/index.php?term=mathematic
- allemagne 10y ago
  This isn't directly what you're asking for, but the book "How To Solve It" by George Polya is a great tour of how to be a teacher of mathematical concepts and how to "simulate" that teacher in your mind if you're encountering a problem no one else has. https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069116407X/ref=dp_ob_title_bk https://www.amazon.com/How-Solve-Mathematical-Princeton-Scie...
allengeorge 10y ago
Note that he does have a webpage [0] and you can support his continued work via Patreon [1]. [0]: https://www.3blue1brown.com/ https://www.3blue1brown.com/ [1]: https://www.patreon.com/3blue1brown https://www.patreon.com/3blue1brown
khana 10y ago
Been there. It's beautiful. Do it now.

106 points by espeed 10y ago

kbd 10y ago

As seems typical, this video goes on for a while and is straightforward until it completely loses you all of a sudden. I have no idea what would even motivate the separation of numbers into "rooms" like the video shows, let alone how that explains why an infinite positive sum = -1.

espeed 10y ago
The infinite sum makes sense when using the p-adic number system [1] -- and measuring distance using the p-adic metric -- which forms an ultrametric space [2]. An interesting related concept is the Bruhat–Tits building [3]. For an intro on p-adic numbers, read these two short articles "A first introduction to p-adic numbers" [4] and "A Tutorial on p-adic Arithmetic" [5] or see the short video "Introduction to p-adic Numbers" [6]: [1] https://en.wikipedia.org/wiki/P-adic_number https://en.wikipedia.org/wiki/P-adic_number [2] https://en.wikipedia.org/wiki/Ultrametric_space https://en.wikipedia.org/wiki/Ultrametric_space [3] Bruhat–Tits building https://en.wikipedia.org/wiki/Building_(mathematics) https://en.wikipedia.org/wiki/Building_(mathematics) [4] A first introduction to p-adic numbers http://www.madore.org/~david/math/padics.pdf http://www.madore.org/~david/math/padics.pdf [5] A Tutorial on p-adic Arithmetic https://koclab.cs.ucsb.edu/docs/koc/r09.pdf https://koclab.cs.ucsb.edu/docs/koc/r09.pdf [6] Introduction to p-adic Numbers https://www.youtube.com/watch?v=vdjYiU6skgE https://www.youtube.com/watch?v=vdjYiU6skgE
- cderwin 10y ago
  I think the problem is that the introduction of the p-adic metric was poorly motivated. It's not clear why we would want 1+p+p^2+p^3+... to converge to -1, and introducing the p-adic metric to do so doesn't show why the p-adic numbers are useful in general (all it's really saying is that the partial sums are 2^(n+1)-1). I'm sure you can derive all sorts of weird metrics so that various weird identities are true; that alone fails to make them interesting. Based on this video alone it's not clear that either the identity or the p-adic norm are interesting in any non-trivial sense of the word. The result is that the introduction of the p-adic metric is hard to follow and the resulting identity seems arbitrary, even if you manage to follow the bit about the metric. (And these combined with a lack of rigor where it's needed seem to be recurring problems in 3Blue1Brown videos.)
  - t-ob 10y ago
    > I'm sure you can derive all sorts of weird metrics so that various weird identities are true On the rational numbers, at least, the p-adic metrics are more or less your whole lot, according to Ostrowski's Theorem [1]. There is a kind of cognitive hurdle everyone who studies these numbers has to clear, in that things that should be "large" turn out to be very small indeed, when viewed under a p-adic lens. I think it's more instructive to build up the ring of p-adic integers first [2, chapter 2], and construct the p-adic numbers from there. I can assure you they are very useful, though! A general theme in number theory is to take a "global" problem, defined over the integers, and to translate it into infinitely many "local" ones (over the p-adics, for each prime p). These are sometimes easier to solve and, if you're lucky, offer insight into the global solution you're looking for. [1]: https://en.wikipedia.org/wiki/Ostrowski's_theorem https://en.wikipedia.org/wiki/Ostrowski's_theorem [2]: http://www.springer.com/gb/book/9780387900407 http://www.springer.com/gb/book/9780387900407
  - posterboy 10y ago
    >doesn't show why the p-adic numbers are useful in general well, you certainly are neither more help either ;)
  - stablemap 10y ago
    My short advertisement is that it has proven very useful to study Diophantine equations by reducing mod n. The Chinese remainder theorem [1] tells you to focus on reducing mod p^r, where p is a prime. If r = 1 then you are working in a field but in the ring Z/4, for example, I know 2 ≠ 0 and yet 2·2 = 0. To make the p-adics Z_p I stitch all of these Z/p^r together: an element is a choice of a_r in each Z/p^r and these have to be compatible: a_2 reduces mod p to a_1 and so on. The resulting Z_p has no "zero divisors", and if I allow myself to invert p I get a field Q_p. This is a huge improvement, a foundation on which to build analysis and geometry as we did over R. [1] https://en.wikipedia.org/wiki/Chinese_remainder_theorem https://en.wikipedia.org/wiki/Chinese_remainder_theorem
taneq 10y ago
"Step 2: Draw the rest of the owl."

Thoreandan 10y ago

I especially like the 3Blue1Brown videos on Hanoi/binary-counting and explaining (visually) Euler's Identity.

risefromashes 10y ago

I didn't understand the proof for 0.5 + 0.25 + ... = 1 itself. By his visual number line analogy, wouldn't 2/3 + (2/3)^2 + ... = 1 too? Visually, that too seems to "approach" 1.

evanb 10y ago
It approaches 2. 2/3+(2/3)^2 = 2/3 + 4/9 = 6/9+4/9=10/9 > 1, and all the terms are positive, so as you add more terms it will get farther from 1. If you do 10 terms you get to 1.96532somethingsomething. If you do 100 terms you get 1.[seventeen 9s]508somethingsomething.
- BillBohan 10y ago
  leereves is correct. The sequence is 2/3 + 2/9 + 2/27 + 2/81 + 2/243 + ... and it does approach 1 In your second term you should take 2/3 of the remaining 1/3, not (2/3)^2
Houshalter 10y ago
I like the algebraic approach: x = 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ... 2x = 1 + 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ... 2x = 1 + x x = 1
leereeves 10y ago
The number line analogy does work for 2/3, in the same way it worked for 9/10. It represents the repeating trinary number 0.22222... = 1. But the analogy only corresponds to p + p^2 + p^3 ... = 1 when p = 1/2 because that's the only time the remainder = p. For 2/3, the number line analogy instead gives 2/3 + 1/3 * 2/3 + (1/3)^2 * 2/3 ...
ssivark 10y ago
Coming from someone who is "used to" this idea for a long time, I'm not sure how helpful my explanation would be to a learner, but I decided to take a shot at it, just in case it helps someone. That said, in my experience, a significant part of "understanding" involves getting used to things and expanding the scope of what you consider "natural". -- Suppose you wish to walk from 0 to 1. You'll have to walk half the distance. Then half of the remaining half. Then half of the remaining quarter. Then half of the remaining eighth. If you keep doing this "asymptotically", you will get very very close to 1 and (almost) reach there. That's the basic idea. This is also known as Zeno's paradox: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_of_motion https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_o... Instead of halving step sizes on each iteration, you could reduced by a different fraction. In general, this is called a geometric series: https://en.wikipedia.org/wiki/Geometric_series https://en.wikipedia.org/wiki/Geometric_series When the ratio is (2/3), after two steps you overshoot 1. (You can see that (2/3) = 0.66, (2/3)^2 = 0.44 )

afarrell 10y ago

Does anyone know of a book, channel, or subreddit which is focused on critiquing the explanations in videos and blogs like this and talking about the art of explaining technical concepts?

posterboy 10y ago
I don't think that would be straight forward, because of competing ideas.
- afarrell 10y ago
  Hence why I suspect it would make more sense as a subreddit or other forum, where you have people debating what does or does not make a good explanation. I guess it would be kinda like literary analysis, but with a much clearer practical point.
  - posterboy 10y ago
    for children or adults, general or math specific, school/university/doctorate context or just plain how to make things understood? Because the latter is basically just what mathematics should be, see http://etymonline.com/index.php?term=mathematic http://etymonline.com/index.php?term=mathematic
allemagne 10y ago
This isn't directly what you're asking for, but the book "How To Solve It" by George Polya is a great tour of how to be a teacher of mathematical concepts and how to "simulate" that teacher in your mind if you're encountering a problem no one else has. https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069116407X/ref=dp_ob_title_bk https://www.amazon.com/How-Solve-Mathematical-Princeton-Scie...

allengeorge 10y ago

Note that he does have a webpage [0] and you can support his continued work via Patreon [1]. [0]: https://www.3blue1brown.com/ https://www.3blue1brown.com/ [1]: https://www.patreon.com/3blue1brown https://www.patreon.com/3blue1brown

khana 10y ago

Been there. It's beautiful. Do it now.