How computers could change pure mathematics
slate.com
slate.com
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.)
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"?
I think a lot of the art will be getting the computer to emit proofs in a form amenable to human understanding.
"A fully automatic problem solver with human-style output"
M. Ganesalingam, W. T. Gowers
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.
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.
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..."
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.
- 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.
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.
0.https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
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.
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?
> 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 loved my real & complex analysis courses but my courses on linear algebra, abstract algebra, & discrete math have been much more useful.
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!
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-... [2]:https://gowers.wordpress.com/2013/04/14/answers-results-of-p...
(Doron Zeilberger's opinion can only be gleaned from a large sample of his writings.)