
How computers could change pure mathematics - j2kun
http://www.slate.com/articles/health_and_science/science/2015/03/computers_proving_mathematical_theorems_how_artificial_intelligence_could.html?wpsrc=sh_all_dt_fb_top
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brandonb
And on the flip side, computers can suggest beautiful theorems to
mathematicians by making it easy to run virtual experiments.

There's now a burgeoning field of pure math called "Experimental Mathematics,"
where they do just that. A good intro on how computer experiments can be used
in algebraic geometry is here:
[http://math.uga.edu/~noah/files/spheres.pdf](http://math.uga.edu/~noah/files/spheres.pdf)

(Disclosure: Noah was my college roommate and we published a paper with Henry
Cohn on experimental mathematics for high-dimensional sphere packing, so I'm a
biased participant.)

~~~
spacehome
I did my PhD in complex dynamical systems, and in that field, computer
experimentation is invaluable. The favorite way to generate hypotheses is to
explore with computer tools. Once you notice a pattern, then you trot out the
theoretical tools to try to solve it.

~~~
cb18
Could you give some examples of the computer tools/software that are commonly
used?

Could you provide or maybe point me somewhere where I could get an expansion
of the ideas in your last sentence? Is it basically creating a simulation
along certain non-linear parameters then running the simulation and watching
what results? Like, "hmm... that looks like a phase transition, let me see if
I can workout what is going on"?

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xyzzy123
We're already in a world where mathematicians publish proofs that only a few
other people in the world can (and have the time to) understand or verify. I'm
thinking of e.g. Shinichi Mochizuki.

I think a lot of the art will be getting the computer to emit proofs in a form
amenable to human understanding.

~~~
fsk
99.9%+ of all papers published by Math professors are incomprehensible to all
but a few experts.

You know all those famous problems that were solved recently? How do you know
they were actually solved? Maybe the Math professors are just pretending to
check each others' proofs, playing a gentleman's game, and they don't want to
admit that these achievements were wrong?

How do you know? Do you really want to spend 5-10 years learning enough to
check one of these proofs?

If it's a computer-generated proof, and people checked the source code but
can't read the proof (because it's terabytes of details), is that a valid
proof? It isn't really leading to any greater understanding.

~~~
acadien
>99.9%+ of all papers published by Math professors are incomprehensible to all
but a few experts.

That's just flat out not true. In fact I would hazard a guess that the reverse
is true (0.1% of math papers are incomprehensible to most). Not to say that
math is simple but that just because something is highly specialized does not
mean it is incomprehensible. Of course my only evidence is that the vast
majority of specialized physics papers that I read only require some
background in quantum mechanics, statistical mechanics and some of the problem
domain. I read some papers in math journals too and feel like they're about on
par in general.

There are certainly some mathematicians that publish in super highly
specialized fields that are remarkably difficult to understand but these are a
teensy tiny minority.

~~~
skierscott
Understanding papers depends on what your field is.

For example, I'm interested in optimization/signal processing/machine learning
and finishing up my education. In the related journals I can barely understand
the notation and miss the higher level concepts.

On the other hand I'm taking a class that required me to read a journal paper
written by the professor. It was strange: I could understand it. I followed it
and most of the finer details too.

I would say that OPs statement _is_ true: 99.99% of papers published by math
profs are incomprehensible to almost everyone. Not only do they use foreign
notation there's also a lot the don't say because it's assumed the reader
knows enough to prove it -- hence "left as an exercise for the reader..."

~~~
acadien
Just because someone doesn't understand how a jet engine works doesn't mean
its incomprehensible to them. It just means they haven't spent the time and
effort to understand it. There are very very few topics out there in this
world that you cannot understand if you've had the proper training.

Granted a lot of topics may take a couple of years to understand, or even a
decade. But guess what, it took the author that long to understand it too.

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KhalilK
It is worth noting that according to the Curry-Howard isomorphism[0], programs
are equivalent to proofs and vice versa.

0.[https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...](https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence)

~~~
wolfgke
You have to pay close attention what system of logic the proofs are written
in. So, for example, it is an open problem, how and whether proofs written in
homotopy type theory can be interpreted computationally (source:
[http://en.wikipedia.org/wiki/Homotopy_type_theory#Computer_p...](http://en.wikipedia.org/wiki/Homotopy_type_theory#Computer_programming)).

~~~
kmicklas
I thought this was more or less solved with the recent developments on the
cubical sets models? There is even a toy interpreter with univalence and
higher inductives:

[https://github.com/simhu/cubical](https://github.com/simhu/cubical)

~~~
jonsterling
They've done a lot of good work, but it's not quite done yet. The original
conjecture has not been answered in the affirmative, and there are also
further questions...

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graycat
The scenario in the OP is not very promising:

E.g., we can study the natural numbers and the integers, rationals,
algebraics, and reals, but why the reals? Well, they are the only complete,
Archimedean ordered field.

Why do we care? We want the order because we want to consider bigger and
smaller. We want a field because we want to be able to do basic arithmetic.

But why do we want completeness? Because it says that pi, e, square root of 2,
derivatives, and the Riemann integral exist with few or no more additional
assumptions.

Why? With completeness we get compactness, continuity, and uniform continuity
and, thus, know that the Riemann integral of a real valued continuous function
on a closed interval of finite length exists and we also get the fundamental
theorem of calculus.

Why do we care about calculus? Because it is the main way to discuss, analyze,
and understand continuous change, e.g., Newton's second law and planetary
motion, fluid flow, heat flow, electro-magnetism, etc.

And why do we care about such analysis? Because it has been one of the pillars
of physical science, engineering, the industrial revolution, and the growth of
technology and economic productivity since Newton -- that is, we care about
_applications_.

So, to understand mathematics and why we do it and why some math is important,
need also to understand applications to the betterment of human life on this
planet, and I'm not holding my breath while waiting for the automation of
that!

Apparently now there is a big theme, climb on the bandwagon, ride the wave
along with everyone else trying to get people up on their hind legs to grab
their eyeballs for ad revenue, of with _artificial intelligence, the machines
are about to take over_.

We've had many waves before -- they come and they go. And we've had that wave
of the computers about to take over before, and apparently we will more times
before computers can get even remotely close to any such thing.

Ah, basic rule: "Always look for the hidden agenda.". Here, it feels to me
like some people are grabbing at me, by the heart, the gut, maybe below the
belt, definitely below the shoulders and not between the ears.

~~~
bjwbell
Sometimes I wonder if the usefulness of calculus is overstated compared to
other areas of math. Linear algebra & discrete math pop up a lot more than
integrals in CS.

I loved my real & complex analysis courses but my courses on linear algebra,
abstract algebra, & discrete math have been much more useful.

~~~
graycat
I can understand and essentially agree with everything you said except your
first sentence, and I'm not sure the rest of what you said does much to
support your first sentence!

One point is, CS isn't the only area for _applications_!

E.g., in the software for my startup, sure, I have some matrix theory, right
there in the code, but it turns out the matrix theory is what is left for the
actual code after some earlier derivations very much in _calculus_!

When I was a prof in a B-school and teaching linear programming, right, awash
in linear algebra, I mentioned to my students, all of whom had had the
required courses in calculus, that I regarded it as a "pillar of Western
Civilization". I still do.

Without Newton's second law, Maxwell's equations, etc., I strongly suspect
that Western Civilization would be a very different and much less good place.

E.g., my father in law eventually slowed down his farming and got a job in
town. He was head of the REMC -- Rural Electric Membership Cooperative. So, it
was the local electric utility. They handled only the last few miles and
bought their electric power from the _grid_ , really from one private power
company.

Some of his customers were factories, and at one point he asked me why his
engineers put large capacitors outside some of the factories. Well, I'd been a
ugrad math major but, except for one course I wanted instead of another that
would have been required, also a physics major, and had done well in ordinary
differential equations, so had see the differential equations of basic passive
AC circuit theory, that is, with resistors, capacitors, and inductors.

So, sure, the factories had a lot of big electric motors with a lot of
inductance. So, the utility pushed current to the motors but half a cycle
later the motor pulled more current. So, net, the utility was moving a lot
more electrons than necessary to deliver the power it was getting paid for
and, thus, was getting more power losses in its lines. So, put a capacitor
just outside the plant, and then the plant looks like a pure resistor to the
electric company and all the extra electron moving is just between the motors
and the capacitor just outside the plant. Ah, _applied calculus_!

There are many more such examples; the examples say that calculus is really
important but don't really settle your question about "overstated"; for that
question, I don't know what to say!

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mixedmath
I recall an experiment that Gowers was doing when he had a program that would
write proofs to basic theorems and questions that an undergraduate math major
might encounter in a first analysis or linear algebra course. His experiment
was to see if other could distinguish between proofs written by students and
proofs written by the program.

You can see some of that in [1] and [2]. Although I was not fooled (nor do I
think that people who both understand analysis and have a bit of foresight
into the structural approaches such a program would take would be fooled), it
was interesting to see.

[1]:[https://gowers.wordpress.com/2013/04/02/a-second-
experiment-...](https://gowers.wordpress.com/2013/04/02/a-second-experiment-
concerning-mathematical-writing/)
[2]:[https://gowers.wordpress.com/2013/04/14/answers-results-
of-p...](https://gowers.wordpress.com/2013/04/14/answers-results-of-polls-and-
a-brief-description-of-the-program/)

------
enupten
Also,
[http://www.math.rutgers.edu/~zeilberg/Opinion137.html](http://www.math.rutgers.edu/~zeilberg/Opinion137.html)

(Doron Zeilberger's opinion can only be gleaned from a large sample of his
writings.)

