
A mysterious connection between number theory, algebra and string theory? - Billesper
https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/
======
blrgeek
And one of the connections is Ramanujan's letters. Both the FIRST letter that
he wrote to Hardy, and the LAST letter he wrote on his death-bed talking about
'mock-theta functions' that are now called mock-modular functions.

> Ken Ono, of Emory University in Atlanta, Ga. “Without either letter, we
> couldn’t write this story.”

Weird, and fascinating!

------
rrss1122
String theorists should stop referring to themselves as physicists and just
accept they're pure mathematicians already.

String theory makes for an interesting branch of mathematics, but an awful
excuse for a physical theory.

~~~
j2kun
The problem is physicists tend to use "intuition" to prove things. I don't
know any string theorists, but if they were trained to think like physicists
then it's probably not any different.

~~~
privong
> The problem is physicists tend to use "intuition" to prove things.

That is not correct (or I am completely misunderstanding what you intended to
say). There's a huge component of physics that is experimental, and those
experiments are used to develop and test models and theories. Initution comes
into play when developing theories or thinking about new directions to take
experiments, but intution is not and cannot be a substitue for experimental
evidence.

~~~
chriswarbo
There's a distinction between "mathematical proof" and "empirical proof".
Physicists are certainly good at empirical proof, but I think the 'appeal to
intuition' complaint was aimed at the mathematical side (ie. the Maths in
Physics isn't very rigorous).

As a former Physicist and current Computer Scientist, I would agree with that
complaint (although our use of empirical evidence is FAR below that in
Physics).

For example, in Programming Language Theory it's amazing how many _different_
notions of "equal" there are (isomorphism, definitional, judgemental,
propositional, extensional, etc.). In Physics, there's just "=" :)

~~~
raverbashing
That "intuition" is what brought us Special and General Relativity

"the Maths in Physics isn't very rigorous"

Well, of course it isn't, because of the physical limitations. You can't
expect to get Newton's second law and make it work with any kind of
mathematical object. (if that's what you mean)

> In Physics, there's just "="

Scalar equality? Vector equality? Magnitude equality?

Also leptons are 'equal' following the Pauli exclusion principle?

~~~
chriswarbo
> That "intuition" is what brought us Special and General Relativity

Yes. Intuition is a good thing. It just shouldn't be relied on for proofs. To
compare, Mathematicians followed their intuitions to bring us all kinds of
results (eg. ). It's a good thing. It just shouldn't be used to prone to
making mistakes when but shouldn't be relied on for (Mathematical) proof.

> You can't expect to get Newton's second law and make it work with any kind
> of mathematical object.

That's not rigour, that's generality. For an example of rigour, compare the
treatment of (infinitesimal) calculus in Physics and in pure Mathematics (eg.
see
[http://en.wikipedia.org/wiki/Calculus#Foundations](http://en.wikipedia.org/wiki/Calculus#Foundations)
).

>> In Physics, there's just "="

> Scalar equality? Vector equality? Magnitude equality?

I would say the first two are "instances" of the same principle (in the same
way that, for example, intensional equality can be "instantiated" for
integers, arrays, matrices, etc.). Magnitude equality composes the absolute
value operation with "the" equality operation, so I would call it a shorthand
rather than a distinct equality principle.

> Also leptons are 'equal' following the Pauli exclusion principle?

What an excellent example of a non-rigorous statement ;) (ie. what is your
model of leptons?)

For reference, some of the examples I mentioned are described at
[http://ncatlab.org/nlab/show/equality](http://ncatlab.org/nlab/show/equality)

~~~
DougMerritt
>> Also leptons are 'equal' following the Pauli exclusion principle?

> What an excellent example of a non-rigorous statement

That is completely uncalled for. The difference between leptons (as a kind of
fermion, which obey the P.E.P.) and bosons (which do not) is no more and no
less than the different definition of identity/equality in the two cases.
Rigorously.

[http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistic...](http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics)

[http://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics](http://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics)

------
cschmidt
Interesting this is a reprint from Quanta magazine, from March 12.

[https://www.quantamagazine.org/20150312-mathematicians-
chase...](https://www.quantamagazine.org/20150312-mathematicians-chase-
moonshines-shadow/)

The original version has some photographs missing from the Scientific American
verison.

~~~
sctb
Thanks, we updated the link from
[http://www.scientificamerican.com/article/mathematicians-
cha...](http://www.scientificamerican.com/article/mathematicians-chase-
moonshine-s-shadow/).

------
ganeumann
Both physics and the unreasonable success of math at explaining the world make
much more sense if you just assume we are living in a math equation.

~~~
JonnieCache
What would that even mean though? People throw around these "universe is a
computation" ideas without ever making any predictions.

Do you think you're living in the matrix? Or do you just think that quantum
mechanics can be fully simulated on a classical turing machine?

Here is the almighty Scott Aaronson to set us right on the subject:

[http://www.closertotruth.com/series/the-cosmos-
computer#vide...](http://www.closertotruth.com/series/the-cosmos-
computer#video-1677)

[http://www.closertotruth.com/series/what-does-quantum-
theory...](http://www.closertotruth.com/series/what-does-quantum-theory-
mean#video-1680)

EDIT: lets throw in this full length lecture at IBM for good measure, even
though its not 100% related
[https://www.youtube.com/watch?v=Z7lv4-Bah5c](https://www.youtube.com/watch?v=Z7lv4-Bah5c)

~~~
wyager
>What would that even mean though?

That all physical systems are governed by equations rather than, say, gnomes
pulling levers.

>Or do you just think that quantum mechanics can be fully simulated on a
classical turing machine?

This seems quite reasonable. There's no good reason this couldn't be the case,
and quantization of fundamental units like Energy-time/Momentum-distance is
certainly something that a programmer might reasonably do for a simulation.

~~~
kordless
Don't forget worrying about hackers.

~~~
VLM
Have some fun with mixing automata theory and the universe as a machine, where
there's whole classes of machine that can or cannot be simulated by other
machines or classes of machines and algorithms to convert from one machine to
another or run one class of machine on others, and limitations and all kinds
of big O notation results.

Not entirely different from the idea of being able to convert from magnetic to
electric field and vice versa via a collection of moving bits and bobs as
discovered a century or two ago.

It could be useful, in some peculiar way. Or maybe not.

~~~
kordless
> limitations and all kinds of big O notation results

Or blockchains.

> as discovered a century or two ago

Give it up to Michael Faraday!

------
AlexCyrlex
The first equation on page 8 of the arXiv paper
[http://arxiv.org/abs/1503.01472](http://arxiv.org/abs/1503.01472) seems
strange to me. It looks like saying "function F(t) given by F(t) +
Sum(something)." Shouldn't it use ':=' symbol instead of '+', like "F(t) :=
Sum(something)"? Or maybe this is the case when math notation is so complex
that common symbols have completely different meaning?

~~~
impendia
You are quite correct. (Professional mathematician myself, and indeed a former
grad student of Ken Ono, so I have at least a passing familiarity with the
subject matter.)

------
dimillian
So I think this is more a philosophical question but ...

As humans, we created mathematics, this is something we created from
"nothing"(?) in order to explain, rationalise things we observe. Now, how can
we assume that the universe is logical, and that it can be explained by
mathematical equations that WE create?

This is bugging my mind every time I think about it.

~~~
heed
We didn't create mathematics, we discovered it.

~~~
dimillian
Is this backed? I mean, this is still and will forever be a debate, do we
discovered or invented mathematics?

~~~
VLM
Geography analogy, some people can discover a continent for some bunch of
people, but the only thing you can invent or create is a map.

[http://en.wikipedia.org/wiki/Map%E2%80%93territory_relation](http://en.wikipedia.org/wiki/Map%E2%80%93territory_relation)

~~~
taeric
But would you instead argue that you discovered how to draw the map?

------
xvilka
I think this can be connected to the recent work (more like a side product of
the main paper) on the ABC Conjecture by S. Mochizuki.

[1] [http://mathoverflow.net/questions/106560/philosophy-
behind-m...](http://mathoverflow.net/questions/106560/philosophy-behind-
mochizukis-work-on-the-abc-conjecture)

[2]
[http://michaelnielsen.org/polymath1/index.php?title=ABC_conj...](http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture)

------
alricb
The M24 and Monster groups mentioned in the article are examples of the
"sporadic" groups, which are 26 exceptional finite simple groups. John Conway
(mentioned in the article, inventor of his game of life) discovered four of
these groups, Co0, Co1, Co2 and Co3.

It's notable that the Mathieu Groups (like M24) were discovered in the 19th
century, long before any other sporadic group was known.

[http://en.wikipedia.org/wiki/Sporadic_group](http://en.wikipedia.org/wiki/Sporadic_group)

------
crb002
There is nothing mysterious. Endofunctions on sets are transformations. Think
permutations but you are allowed to have repeats. If you iterate them f(x),
f(f(x)), f(f(f(x))) ... the function forgets elements until you have a stable
partition which cycles.
[http://chadbrewbaker.github.io/combinatorics/transformations...](http://chadbrewbaker.github.io/combinatorics/transformations/permutations/2014/08/07/endo.html)

~~~
sweezyjeezy
That in no way explains why these K3 surfaces have anything to do with mock
theta functions...

~~~
crb002
It explains why iterated functions turn into group like structures and the
Mosnster is present.

------
littletimmy
This is so exciting. Pity one has to devote one's life to study such things to
even start to make sense of these discoveries.

~~~
myth_buster
Well if it excites you then perhaps you can take it as a hobby or side project
and get some satisfaction and sense of achievement as you progress on this
path.

~~~
littletimmy
Yeah that's what I do, actually. I sneak in a little reading of math from time
to time on lunch breaks or after work.

------
31reasons
>>It took several more years before mathematicians succeeded in even
constructing the monster group, but they had a good excuse: The monster has
more than 10^53 elements, which is more than the number of atoms in a thousand
Earths.

Did they create a set of numbers the size of 10^53 ? That seems impossible
since you need more capacity then all atoms of 1000 Earths!

~~~
AngrySkillzz
That's not quite what they mean. By "constructed" they mean "demonstrated that
it exists and has the specified properties." They didn't write down every
element explicitly.

~~~
wilsynet
That's right. For example, you can construct the integers Z without writing
all of them down. Which would take a long time ...

------
lotsofmangos
The naming is fantastic, I mean, _Umbral Moonshine Conjecture_.

By the time we actually find a theory connecting all this stuff, it will
probably have aggregated a name so ridiculous that nobody will believe it.

------
oafitupa
Math and physics look like a lot of fun. Does anyone know if the real fun only
starts after like a decade of studying?

~~~
saraid216
It depends on what you find fun.

I get a lot of enjoyment out of simply solving problems that I find in the
wild. (I spent some time proving that my particular walking pattern was, in
fact, more efficient than an alternative because the actual distance traveled
was shorter.) If that's sufficient for you, then all it requires is a little
studying and a little imagination.

If you want to contemplate the cutting edge, then yeah, a decade of studying
is necessary mostly because it's hard to actually comprehend all the
implications on the cutting edge until you've done so.

On the upside, there are plenty of MOOCs on physics and math these days; if
you find a set you like, you can absorb plenty that way.

~~~
itsjareds
I guess we find the same problems in the wild enjoyable. I spent a few nights
procrastinating on homework to brainstorm how I could try to find the most
efficient walking path between points on campus. I figured that I would need
to be able to represent the campus on some sort of plane where each point has
a value referring to its elevation, and then finding the geodesic. I figured I
could refer to some physiology literature and find if anyone has tabulated
average energy expenditures for walking at different grades (downhill, flat,
or uphill). Now, my math skills are really sub-par (haven't gotten past single
variable calculus), so this question had me asking various physicist friends
how to solve the problem. I just found myself learning along the way.

I'm curious how you determined efficiency for each of the candidate walking
patterns. Did you compare only the traveled distance, or did you also take
into account the energy spent per unit distance? I think that could make a
difference if, for example, you had these walking strategies:

Walking strategy S => A "normal" human gait, except you travel in an squiggle
path (i.e. not a straight line) to your destination.

Walking strategy T => Do continuous jumping jacks while walking, but continue
in a straight path.

S may travel a longer distance, but will exert less energy overall and
therefore be more efficient than T. Now, that's a pathological scenario, but I
wonder if your walking pattern could have the same issue where you are
actually exerting more energy despite walking a shorter distance. It'd be
interesting to do more research on the biophysics of how your body moves.

