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
(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.)
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.
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"?
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.
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.
>> "Do you really want to spend 5-10 years learning enough to check one of these proofs?"
The other replies here all seem to be "Pssh, I can already read math, and I'm a layman."
As a counterpoint, I often try to read my friends' crypto or applied math papers and literally cannot imagine understanding any of it, even given infinite time. Some stuff is just so opaque to me that I'm pretty sure it's permanently unattainable for someone with my intelligence. At the very least I'm much further than 10 years from following the state of the art in big-boy mathematics.
I would gladly use computerized comprehension-improving tools if I could get them, but then again I'm just a dumb engineer.
>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.
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..."
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.
I think you are seriously overestimating the capabilities of 'most'. In my experience, it is impossible to convince 'most' of truly trivial (to the mathematically schooled) things such as
- there are as many odd integers as integers.
- 0.99999999... equals 1.
But it also depends on what you call 'math'. Applied math often is 'easy'. Modern abstract math, I am convinced 90+% will never be able to read, and 99% of the rest would need serious study. I know I can't make heads or tails from many papers, and I do have a M. Sc. in math. That's partly because I am rusty, but there also are lots and lots of terms in areas that I never heard about in areas such as algebra that I barely studied. For an example, look at http://mathworld.wolfram.com/search/?query=cohomology&x=0&y=..., pick any page, and check whether 'most' would be able to comprehend it.
I'd second this, if only because I have nothing further than an undergrad CS degree and I can read computational complexity papers just fine. You don't need much extra education to understand a paper, you might to understand some of the implications of it
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...).
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:
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...
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.
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.
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!
> 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.
Well, and the complex numbers, they're the only algebraically closed field.
> Well, and the complex numbers, they're the only algebraically closed field.
That is very, very far from being true! There are algebraically closed fields of every characteristic and of arbitrarily large cardinality. What did you mean?
> > Well, and the complex numbers, they're the only algebraically closed field.
> That is very, very far from being true! There are algebraically closed fields of every characteristic and of arbitrarily large cardinality. What did you mean?
Although BrainInAJar seems simply to have misspoken (https://news.ycombinator.com/item?id=9254999), you have just explained one of the things that the remark could have meant: namely, that ℂ is the unique characteristic-0, algebraically closed field with the cardinality of the continuum.
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.
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
(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.)