> A meeting on how to extend the GPY method was immediately organized at the American Institute of Mathematics in San Jose, California. As a bright-eyed and bushy-tailed grad student, I felt extraordinarily lucky to be there among the world’s top experts. By the end of the week, the experts agreed that it was basically impossible to improve the GPY method to get bounded prime gaps. Fortunately, Yitang Zhang did not attend this meeting. Almost a decade later, after years of incredibly hard work in relative isolation, he found a way around the impasse and proved the experts wrong. I guess the moral of my story is that when people organize meetings on how not to solve the Riemann hypothesis (as they do from time to time), don’t go!
Distantly related question. It was claimed that Alpha Zero taught itself to play incredibly high level chess from the very basic point of playing with itself millions of times after having been fed the basic building block rules of chess and nothing else.
Now I'm wondering if something like this can be made to work for discovering new mathematical relationships or solving existing hypothesis by training a neural network with what we know of maths so far and then letting it "play" with the equations?
There is a sense in which that is a great idea, and a sense in which it is a not so great idea. If AlphaProof could prove theorems as superhumanly as AlphaGo wins at Go, that would really be something, they might be able to claim all of the millennium prizes. However, the longterm usefulness to mathematics may be somewhat limited. Mathematicians are interested in proofs not so much because of the statements, but because of what those proofs add to our knowledge of the way math fits together. It isn't just about "true" or "false," it's about why. If AlphaProof generated a long and unreadable proof with no discernible structure, it would not be such a large contribution to mathematics as if a human figured out "why" and wrote it in a way other humans could understand.
Of course, it is possible that clever users could tease useful information out of the black box truth oracle. Perhaps by modifying the statement of the conjecture and checking each modification, they could discover some parts of the structure of the problem, which would help inspire a useful proof. Furthermore, it would be helpful to never again waste time trying to prove a false conjecture.
I don't think so, but I think the trick will be how to score a new mathematical relationship as "interesting". Basically, a fitness metric. With Chess, you have the obvious win/loss metric, and also number of turns and possibly time. With math, how do you quantify the level of interestingness between 2+2=4 and p=np?
This isn't a snarky question, by the way. I'm genuinely interested in how this might be done.
Adding an extra point, one thing that makes this difficult is that a lot of this kind of research is not about applying existing ideas but coming up with something that has never been done before, and neutral networks are a lot worse with creativity than with pattern matching. It's also hard to get enough training examples, especially since you'd either want these to be in strict logical language or train a system to recognize whether a hundred page proof in natural language is rigorous or not. So it's like a very long maze where noody knows if you're getting closer to the exit or not, and in most rooms you have to come up with something original or discover a new dimension. And there's the aspect whereby a lot of significant research is: create a new abstraction, realize and convince others that this abstraction is useful and gives you new intuition that lets you solve new problems.
The trick is coming up with a reward function that drives exploration when "played" against itself. Chess has an easy one, win the game. General mathematics doesn't have anything so easily specified.
Your comment reminds me of Gödel's proof of his incompleteness theorem. Basically, he said that if you treat a mathematical statement as a sequence of tokens, and then assign a number to each token, then you can transform the statement into a big number. Then, going one step further, if a mathematical proof is a sequence of statements then it can be too transformed into a really long number. Then you can come up with rules to manipulate these numbers to create new valid statements and proofs.
I don't know if there was much work done on this idea beyond Gödel's proof, but I think it's a really neat idea that gives a way to manipulate mathematical proofs as objects and create new ones from a very high level view.
you might be able to do something like "predict the next prime number" or "predict the next zero of the Riemann zeta function".
you could try something like this for statements in a formal axiomatic system, but know that you're running up against things like the halting problem / entscheidungsproblem / godel incompleteness. so it may be possible to train a neural net to decide the veracity of a statement and do so more quickly than a human might, but you would inevitably be running up against things that are truly undecidable in nature. which is not like go or chess where although they are difficult, they are decidable.
But it fails on two counts. One is that it doesn't say what the title claims. We know now what the Rieman Hypothesis is, but how did the author learned to love, and more importantly to "fear" the RH? What's there to fear? The name of the famous movie (with the bomb) doesn't even have "fear" in it, so the author had something in mind with "fear", but after watching the video, I have no clue what.
Second is the promise in the beginning of the video to give us a hint why the RH is important. What we learn is that somehow each zeta zero adds another harmonic to some type of series approximation of the (cousin of the) prime counting function. Which is great; if we add enough harmonics we get to approximate this function as precisely as we want. But why is it important that these zeros are on the vertical line Im z = 0.5 ? I have absolutely no clue.
Don't get me wrong, I find that watching this video was a very good investment of 16 minutes of my life. But there is no need to overpromise, especially since the video delivers a lot as is.
> But why is it important that these zeros are on the vertical line Im z = 0.5
Great question.
The formula works whether or not the zeros have real part 0.5, but its implications are different.
If you're counting primes up to x you will get the main term (roughly x/log x) and each zero p will give an oscillating error term ~ x^p/p. If the real part is always 0.5 then these error terms are all on sqrt(x) scale, which is what you would expect from random variation. If on the other hand you have a single zero right of the line, say at real part 3/4 (and mirror image one at real part 1/4) then now you get a secondary error term of size x^(3/4) that dominates all of the other error terms, giving some extra structure to the primes: they're no longer random, you can predict where they are more or less likely.
> but how did the author learned to love, and more importantly to "fear" the RH? What's there to fear?
It's made abundantly clear: they learned to love it through the course they were fortunate enough to take as an undergraduate, and they learned to fear it later, as a researcher, when they realised that what there was to fear was wasting their career attacking something that was probably just too big and too hard to be sensible for them to attack.
If one could predict with pinpoint accuracy all the prime numbers approaching infinity with certainty and with trivial resources (without having to test each one for prime-ness), would it have any impact on modern day cryptography?
If so much technology is built around a hypothesis, it is essentially building a large house of cards if the hypothesis turns out to be incorrect.
Imagine if the world set up a monetary system and what would happen after 100 years if suddenly everyone could print money when a fatal flaw in a system was discovered.
The vertical scaling of the counting function with log(p) went a little quick, does someone know an easy to digest resource to this?
also what would be the logic for now including the p^n of primes, p ??
I felt that way about most of the video. I never took advanced math theory classes, but I have enough background that those classes would probably have been the next step for me if I had continued on.
Seemed like every time I felt like he was about to explain something in a way that would click for me, then all the sudden he left out the last two sentences and moved on.
Off topic, but does anyone know where one gets started on making motion graphics like in that video? This combination of cool maths and animated visuals is so interesting to me, but I wouldn't even know where to begin. What kind of software is used to generate this kind of visualisation? Are there free or affordable online courses? Is there a job market for someone with this ability and also with software development chops?
3Blue1Brown on YouTube has a number of great videos into math that help give a visual understanding without needing to sit down and do the pure number crunching. I'm not sure he has one on the Riemann Hypothesis in particular, but he does cover a number of items that would be somewhat related and help build a level of comfort with being exposed to the math, even if you can't do the pure number crunching.
Mathologer would be my second recommendation. Once again not sure he has touched the Riemann Hypothesis, but he does a number of great videos where you can start expanding being comfortable with often untouched fields of math (like modular arithematic) while introducing a how seemingly unrelated fields end up having connections.
Videos can go between 20 minutes to an hour and I often enjoy watching one every few days.
Depends on your level of {experience, intended engagement, patience for pedantics, intended outcomes}.
I've found that 3Blue1Brown (youtube) gives a great starting point for many topics. Provided you actively participate in them, the videos will build up some intuition so you know what questions to ask (and why they're worth asking).
The author usually links to followup resources so you know where to go next. I've found his resources a bit hit-or-miss, but they give me enough to know if I want to continue exploring. If I do, I usually check out a good textbook (I refer to https://news.ycombinator.com/item?id=17617825 often).
The accompanying video popped up on my YouTube recommendation list last night and it is really good. Like, you married a numberphile video with 3blue1brown, good.
Still nice of the video in the original post to give a bit more of a historical context, instead of focusing entirely on the mathematics. It could have done without the building analogy around the beginning though, that didn't come up later and was just distracting.
That was a fantastic video. I was left wondering a bit about the difference between the Riemann Zeta function and the original Zeta function, but I found this 3Blue1Brown video which explains exactly that: https://youtu.be/sD0NjbwqlYw
Matt Haig's "The Humans" talks about humanity finding the answer to the Riemann Hypothesis being feared universally - it's simultaneously very important to the plot, and a convenient MacGuffin.
"Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable" By Lars Ahlfors was very enjoyable to me. However, I read it after I already knew Complex Analysis.
i second tristan needham's visual complex analysis.
some other good ones:
* the one we used in my undergrad course was fisher's complex variables which is great if you're learning for the purposes of applications. it's a cheap dover book.
* rudin's real and complex analysis (if theory is your thing. note that rudin's books, while great, do require a good background in math).
* as the article mentions, eli stein has a series of books on the four main branches of analysis. i believe the second book is on complex analysis.
[+] [-] rfurmani|5 years ago|reply
> A meeting on how to extend the GPY method was immediately organized at the American Institute of Mathematics in San Jose, California. As a bright-eyed and bushy-tailed grad student, I felt extraordinarily lucky to be there among the world’s top experts. By the end of the week, the experts agreed that it was basically impossible to improve the GPY method to get bounded prime gaps. Fortunately, Yitang Zhang did not attend this meeting. Almost a decade later, after years of incredibly hard work in relative isolation, he found a way around the impasse and proved the experts wrong. I guess the moral of my story is that when people organize meetings on how not to solve the Riemann hypothesis (as they do from time to time), don’t go!
[+] [-] rfurmani|5 years ago|reply
The lecture videos are online and are a good way to get both historical context and connection to how people are thinking about it currently.
This seminar talk on how to fail to prove RH was also great https://www.math.rutgers.edu/news-events/list-all-events/ica...
[+] [-] Santosh83|5 years ago|reply
Now I'm wondering if something like this can be made to work for discovering new mathematical relationships or solving existing hypothesis by training a neural network with what we know of maths so far and then letting it "play" with the equations?
Am I being too naive here?
[+] [-] whatshisface|5 years ago|reply
Of course, it is possible that clever users could tease useful information out of the black box truth oracle. Perhaps by modifying the statement of the conjecture and checking each modification, they could discover some parts of the structure of the problem, which would help inspire a useful proof. Furthermore, it would be helpful to never again waste time trying to prove a false conjecture.
[+] [-] fooqux|5 years ago|reply
I don't think so, but I think the trick will be how to score a new mathematical relationship as "interesting". Basically, a fitness metric. With Chess, you have the obvious win/loss metric, and also number of turns and possibly time. With math, how do you quantify the level of interestingness between 2+2=4 and p=np?
This isn't a snarky question, by the way. I'm genuinely interested in how this might be done.
[+] [-] rfurmani|5 years ago|reply
[+] [-] Bjartr|5 years ago|reply
[+] [-] mensetmanusman|5 years ago|reply
‘Proofs’ in human parlance are those which we believe to be interesting or aesthetic or useful.
[+] [-] ajnin|5 years ago|reply
I don't know if there was much work done on this idea beyond Gödel's proof, but I think it's a really neat idea that gives a way to manipulate mathematical proofs as objects and create new ones from a very high level view.
[+] [-] ellis-bell|5 years ago|reply
you could try something like this for statements in a formal axiomatic system, but know that you're running up against things like the halting problem / entscheidungsproblem / godel incompleteness. so it may be possible to train a neural net to decide the veracity of a statement and do so more quickly than a human might, but you would inevitably be running up against things that are truly undecidable in nature. which is not like go or chess where although they are difficult, they are decidable.
[+] [-] credit_guy|5 years ago|reply
But it fails on two counts. One is that it doesn't say what the title claims. We know now what the Rieman Hypothesis is, but how did the author learned to love, and more importantly to "fear" the RH? What's there to fear? The name of the famous movie (with the bomb) doesn't even have "fear" in it, so the author had something in mind with "fear", but after watching the video, I have no clue what.
Second is the promise in the beginning of the video to give us a hint why the RH is important. What we learn is that somehow each zeta zero adds another harmonic to some type of series approximation of the (cousin of the) prime counting function. Which is great; if we add enough harmonics we get to approximate this function as precisely as we want. But why is it important that these zeros are on the vertical line Im z = 0.5 ? I have absolutely no clue.
Don't get me wrong, I find that watching this video was a very good investment of 16 minutes of my life. But there is no need to overpromise, especially since the video delivers a lot as is.
[+] [-] rfurmani|5 years ago|reply
Great question. The formula works whether or not the zeros have real part 0.5, but its implications are different. If you're counting primes up to x you will get the main term (roughly x/log x) and each zero p will give an oscillating error term ~ x^p/p. If the real part is always 0.5 then these error terms are all on sqrt(x) scale, which is what you would expect from random variation. If on the other hand you have a single zero right of the line, say at real part 3/4 (and mirror image one at real part 1/4) then now you get a secondary error term of size x^(3/4) that dominates all of the other error terms, giving some extra structure to the primes: they're no longer random, you can predict where they are more or less likely.
[+] [-] gimboland|5 years ago|reply
It's made abundantly clear: they learned to love it through the course they were fortunate enough to take as an undergraduate, and they learned to fear it later, as a researcher, when they realised that what there was to fear was wasting their career attacking something that was probably just too big and too hard to be sensible for them to attack.
[+] [-] wodenokoto|5 years ago|reply
The article talks about the personal story of how the author first met the Riemann Hypothesis.
[+] [-] Covzire|5 years ago|reply
[+] [-] mensetmanusman|5 years ago|reply
If so much technology is built around a hypothesis, it is essentially building a large house of cards if the hypothesis turns out to be incorrect.
Imagine if the world set up a monetary system and what would happen after 100 years if suddenly everyone could print money when a fatal flaw in a system was discovered.
[+] [-] red_trumpet|5 years ago|reply
[+] [-] nabla9|5 years ago|reply
Discussions based on the title are useless. Good articles frequently have titles that are bad or not even relevant. Title of an article:
1. May not be chosen by the author. Its often editor decision.
2. Article can have multiple changing titles. For clickbait reasons.
Discussions based on title are generally worthless. In the HN you can ask someone to change the title to be more relevant.
[+] [-] jbj|5 years ago|reply
[+] [-] war1025|5 years ago|reply
Seemed like every time I felt like he was about to explain something in a way that would click for me, then all the sudden he left out the last two sentences and moved on.
Still a neat video though.
[+] [-] mdoms|5 years ago|reply
[+] [-] xevrem|5 years ago|reply
[+] [-] odyssey7|5 years ago|reply
[+] [-] SkyBelow|5 years ago|reply
Mathologer would be my second recommendation. Once again not sure he has touched the Riemann Hypothesis, but he does a number of great videos where you can start expanding being comfortable with often untouched fields of math (like modular arithematic) while introducing a how seemingly unrelated fields end up having connections.
Videos can go between 20 minutes to an hour and I often enjoy watching one every few days.
[+] [-] ryanianian|5 years ago|reply
I've found that 3Blue1Brown (youtube) gives a great starting point for many topics. Provided you actively participate in them, the videos will build up some intuition so you know what questions to ask (and why they're worth asking).
The author usually links to followup resources so you know where to go next. I've found his resources a bit hit-or-miss, but they give me enough to know if I want to continue exploring. If I do, I usually check out a good textbook (I refer to https://news.ycombinator.com/item?id=17617825 often).
[+] [-] phamilton|5 years ago|reply
[+] [-] rfurmani|5 years ago|reply
[+] [-] wodenokoto|5 years ago|reply
[+] [-] st1x7|5 years ago|reply
Still nice of the video in the original post to give a bit more of a historical context, instead of focusing entirely on the mathematics. It could have done without the building analogy around the beginning though, that didn't come up later and was just distracting.
[+] [-] legel|5 years ago|reply
[+] [-] slavik81|5 years ago|reply
[+] [-] mpettitt|5 years ago|reply
[+] [-] spicymaki|5 years ago|reply
[+] [-] frankfrankfrank|5 years ago|reply
"Denen die Got dienen müssen
Alle dinge zum besten dienen."
Translated:
"All those that serve God
All things best serve them."
[+] [-] jhncls|5 years ago|reply
Which it suggests translating as "For those who love God, all things must work together for the best".
https://en.wikipedia.org/wiki/Bernhard_Riemann
[+] [-] st1x7|5 years ago|reply
[+] [-] lldbg|5 years ago|reply
[+] [-] ellis-bell|5 years ago|reply
some other good ones:
* the one we used in my undergrad course was fisher's complex variables which is great if you're learning for the purposes of applications. it's a cheap dover book.
* rudin's real and complex analysis (if theory is your thing. note that rudin's books, while great, do require a good background in math).
* as the article mentions, eli stein has a series of books on the four main branches of analysis. i believe the second book is on complex analysis.
[+] [-] rfurmani|5 years ago|reply
[+] [-] Moeg|5 years ago|reply