
What does it feel like to invent math? (2015) [video] - espeed
https://www.youtube.com/watch?v=XFDM1ip5HdU
======
kbd
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
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
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
> 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)

------
allengeorge
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)

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risefromashes
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
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
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

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afarrell
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
I don't think that would be straight forward, because of competing ideas.

~~~
afarrell
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
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)

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

