You have to be very careful with these "plausibility arguments" that physicists like to make. It may well happen, for example, that there are counterexamples to the Riemann hypothesis, but that they are so rare that they have probability zero (similar to, for example, the zero probability that a random real number in some interval is a rational number). So, a plausibility argument saying how unlikely counterexamples are would prove nothing. Or, it could happen, like the famous disproved Riemannian conjecture on the crossing of pi(x) and Li(x) (see: Skewe's number) that the counterexample exists but it is ridiculously large. The usual kind of ethos in physics is to be a bit too loose with approximations or to disregard very extreme cases.
I don't understand the particulars of the spectrum of this operator that they have constructed, but I have heard others describe this approach as a simple reformulation of one hard problem in terms of another equally hard problem.
What specifically is your first paragraph attacking? I just read their paper and have no idea what you're going after. They present a concrete strategy for a rigorous proof, but with gaps left. They also check that the suggested strategy has the properties we expect given what they want it to do. They do not make a plausibility argument like you suggest, except in so far as that they argue that their proof strategy might plausibly work. The distinction, I think, is very significant; the end result here would be a rigorous proof.
Oh, I'm probably wrong and just arguing from my prejudices. I obviously haven't tried reading their paper. What is their argument for why the strategy would work?
Uh, I'm out of my field so I'm not sure I can summarize it better than the article did, but I can try. They prove that the Riemann hypothesis is equivalent to a property of a specific operator, however they can't prove that property (that's it's self-adjoint under some inner product) since they don't know what the appropriate domain of the operator is. But the operator does do several things: first, it's classical version is what was expected. Secondly, they suggest an approach towards getting an inner product that would the operator self-adjoint. Lastly, the authors conclude "The fact that iH [the operator in question] is PT symmetric, with a broken PT symmetry, offers a fresh and optimistic outlook" which I don't understand well enough to add to.
Taking this down to an even more basic level that doesn't require much knowledge of this stuff - the basic strategy of this program is to find a quantum system that they can prove (rigorously) has energy levels that map directly to the Riemann zeta zeros (well, the non-trivial ones) under a simple shift/rotation in the complex plane.
Since energy levels of these systems are always real (and in a straight line on the complex plane), this would establish that the zeros are all in a straight line as well, namely the one where Re(z) = 1/2.
The tough bit is finding that system. These researchers seem to think they've made headway; I'm withholding judgment for now, since by my reading the whole motivation for reaching for a physical system (real eigenvalues) is exactly what they still need to establish about their proposed system. They might have just succeeded in recasting the Riemann hypothesis in a different but no more tractable form, we'll have to see.
There is no dichotomy here. Mathematicians also use heuristics to figure out what is probably true. This is a very important part of mathematical activity - generating good, not yet worked out ideas based on rough insights. What the physicists are doing here is also how the Fermat's Last Theorem was approached - a conjectural analogy to a different field(elliptic curves).
The only difference is that they are more careful about separating heuristics from proofs. If you read physics textbooks, they are less interested in being formally precise.
Mathematical activity is not all about formal reasoning, though that is the output.
There is a theory that many famous unproved conjectures are true but unprovable with the traditional methods you hold dear. Of course Godel showed there are true unprovable theorems. But those are weird paradoxical self referencing things. That can sort of be proved if you use a different system of axioms. But it's possible there are unprovable true theorems that aren't like that.
Freeman Dyson created an example of one. He conjectured that no power of 5, it's base 10 representation reversed, is a power of 2. This is almost certainly true. There is no particular correlation with powers of numbers and their base 10 representation. They are basically random except for the last digit. There is no pattern to it. And because the base 10 representations get longer and longer, it gets increasingly unlikely there exists a counterexample. You can calculate the probability and it's very low.
The thing is that statement is very likely to be unprovable. If it's true, it's not true for any particular reason. Nothing at all says a counterexample can't exist. It just happens to be increasingly unlikely to happen. A mathematical coincidence.
You can prove some famous conjectures using probabilistic arguments. For instance Goldbach's conjecture that every even number is the sum of 2 primes. Because primes are distributed basically randomly, as numbers get bigger, it gets more and more likely there are two primes that add to get that number. There are an awful lot of prime numbers after all. But again, there's not any reason to suspect there's a mathematical reason a counterexample can't exist. It's just unlikely.
This is really hard to formalize. Obviously prime numbers are deterministic, not random. Obviously base 10 representations aren't really random. It'd be more precise to say that they aren't correlated with the particular properties we are interested in. But even that is pretty hard to pin down precisely.
Mathematicians strongly disrespect anything that isn't a rigorous proof. But if you do that you might miss out on a lot of interesting things. And there is absolutely no guarantee that the universe will always give us that!
> I don't understand the particulars of the spectrum of this operator that they have constructed, but I have heard others describe this approach as a simple reformulation of one hard problem in terms of another equally hard problem.
I don't know if it is something inherent in the RH, or just because it has received so much attention, but it seems to have a propensity for having equivalent formations in areas that do not have any obvious connection to it.
A couple examples:
• RH is equivalent to the assertion that for integers n >= 3, |log lcm(1,2,...,n) - n| < sqrt(n) log^2(n).
• For integers n >= 1, let s(n) = sum of divisors of n. E.g., s(6) = 1 + 2 + 3 + 6 = 12. Let H(n) = 1 + 1/2 + 1/3 + ... + 1/n. Then RH is equivalent to the assertion that s(n) <= H(n) + exp(H(n)) log(H(n)), with equality only at n = 1.
I think you're mistaken in thinking that the arguments Bender et al are making are physical. Operator theory, Hilbert spaces, Lebegue measures, etc. are pure mathematical tools used to construct and derive insight from Quantum Mechanics. The tools themselves are grounded in the usual sense of mathematical rigor. The inspiration to attack this conjecture, the Riemann hypothesis, framed as a quantum system might come from physical intuition, but everything else is rigor.
I wasn't thinking that they're physical, just that physicists are used to playing fast and loose with approximations because it's a technique that usually works in physics. Or to not really care about what a Dirac distribution really is. Or that defining a tensor (field) as something that transforms like a tensor is a satisfying definition.
I am skeptical of the methods that they're used to. But it doesn't look like they're doing that, so I'm probably wrong.
I'm not a mathematician, nor a physicist, but I don't understand this statement: similar to, for example, the zero probability that a random real number in some interval is a rational number. Is it just a consequence of rational numbers being countable and real numbers not being countable? Seems like an odd way to define probability in this instance.
It is a consequence of countability, but I like to think about it this way: a real number is, roughly, an infinite sequence of "random" digits. What is the probability that your infinite sequence of random digits will randomly settle into an infinite repeating pattern? That's obviously, zero, right? No way that a random process will produce an infinitely repeating pattern.
But just because it's "probabilistically impossible" to pick a rational number doesn't mean that they don't exist.
It is a consequence of the rationals being countable, in that all countable sets necessarily have this property.
The intuition (well, one way of looking at it) is that if you have a line of length 40, and a subset of that line which has total length 2, then a randomly-chosen point on the line has a 5% chance (2 in 40) of being part of that subset. It's not important that the subset be connected; the intervals (3,4) and (28,29) have length 1 each, length 2 in total. We want to think in terms of intervals because they're very easy to measure: the interval (low,high) is (high-low) amount of stuff.
It turns out that there are so few rational numbers that you can define a series of intervals with the following properties:
- every rational number is contained within one or more of the intervals
- the (infinite) sum of the lengths of each interval is arbitrarily small
So for any value, you can capture the entire set of rationals in a set of intervals whose total length is less than that value. This means the "amount of stuff" in the set of rational numbers cannot be bounded away from 0; removing the rationals from a line of probability-stuff leaves no less probability-stuff than was there before.
It comes from how we define infinity/the number line. Any number divided by infinity is 0, that's easy. But between any two points there is an infinite number of rational numbers and between any two rational numbers you can derive an infinite number of irrational numbers. I know this isn't kosher, but it is pretty much infinity/(infinity ^ infinity). That leads us back to the probability of picking a rational number as 1/infinity, because for every rational number there are infinity irrational numbers that we also could have chosen.
David Foster Wallace had a really good book called Everything and More where he explored the history of infinity. Might be of interest to you.
That's a really bad argument, since between any two irrational numbers, there's an infinite number of rational numbers.
Ed: For an explicit construction of such numbers --
Suppose we have a<b, two irrational numbers.
Let b-a = d, the difference between them.
Then there exists N, an integer, such that 10^(-N) < d.
Let b1 be b truncated at the (N+1)th digit past the decimal point.
Then b1 is rational (finitely many digits) and b-b1 < d (and b1 < b), so b1 is in (a,b).
You can then add back one digit of b at a time, to get b2, b3, etc which form an infinite sequence of rational numbers converging to b from below. (We get infinite unique numbers from the fact b is irrational. Not all the b_n need be distinct.)
I wish there was a third alternative to upvote/downvote -- "sounds like a good chap acting in good faith, but knows less than nothing of this subject".
I don't understand why Quanta magazine keeps getting promoted on HN. Guys, it's sensational writing. Not scientific. Please understand this. Quanta is like Wired. Poor academic quality and sensational writing. Neither are based in scientific rigor.
I'm majoring in Pure Maths and this is annoying to see yet another poor scientific article on Math.
It's most certainly not scientific writing. But I think that is not the point of these publications -- they are meant for a wide audience.
So I think I should interpret your criticism as saying that quanta does a poor job in maintaining rigor while trying to explain advanced topics in mathematics and theoretical physics to a wide audience. My impression, on the other hand, is that their reporting on these topics beats similar publications hands down. I am therefore happy to recommend it to interested laypeople, for example to my (few...) friends outside of academia and to people here on HN.
By the way, I hope you realize that these two viewpoints are not completely orthogonal.
> By the way, I hope you realize that these two viewpoints are not completely orthogonal.
You win the Fields Medal. :)
I appreciate your feedback and you seemed to have interpreted my text words as intended. I disagree with your conclusion but you do understand my position. I'm grateful for that. So much can be misunderstood through the web. I point others to your comment to hopefully elaborate.
I do think Quanta does a poor job at maintaining the rigor and no they aren't as bad as other publications. But we should hold them to a higher standard (heck all publishers). They are closer to what I think is needed in the industry but it's too much hype for me and not enough rigor.
> I'm majoring in Pure Maths and this is annoying to see yet another poor scientific article on Math.
I dislike personal qualifications-based arguments, so I'm loathe to contribute to them. However, for people who are persuaded by mentions of qualifications: I have a PhD in mathematics and I think Quanta magazine is more-or-less the best popular-level writing available on the subject.
As such, perhaps the article would benefit from some history. As mentioned on the Wikipedia page https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_con... observations regarding the distribution of zeros on the critical line led directly to early results in random matrix theory. This 2009 interview with Freeman Dyson discusses this connection, as well as this approach to the Riemann Hypothesis (Dyson refers to it as "a fourth joke of nature"): http://www.ams.org/notices/200902/rtx090200212p.pdf
I recommend reading that interview in its entirety, or at least the section on "jokes of nature", for Dyson's thoughts on a lot of subjects.
> I dislike personal qualifications-based arguments, so I'm loathe to contribute to them.
Umm. There is a difference being being treated for an infection by a MD and a PhD in Philosophy. Context is key. I'm not trying to 'puff up' and make it an ego competition.
Instead, I'm trying to provide some context with my feedback. Nor do I intend to speak for 'all' of anything, let alone mathematicians. (Specifically, I'm not just a layman providing feedback on this math/physics article)
I appreciate you taking the time to provide feedback. I'm most interested in the pdf. When I get a brief moment, I'll check it out. Ciao!
Was that article on symplectic geometry in Quanta?
My work is with algorithms that simulate conservative systems and that sounded pretty good to my ears. Of course they were with the human angle by the end, but still they did a damn good job in explaining what a "cotangent bundle" is.
My bad. Hopefully this [1] comment and this [2] one helps clear up my position. They are accurate, yes. But poor in the sense that it's all conjecture, not evidence based.
From what I've read on quanta, they write remarkably clear articles for their target audience. And I've even read the referenced paper by Bender et al. I think the author has done a good job in describing what the heck the paper was about.
First, even though Math and Physics are similar, they aren't. To me, it's like saying the book '1984' prevented dictatorial rule. Literature can have a societal impact but we shouldn't be championing a new book, that hasn't been written by authors very few have heard of. (Mr. Berry and Mr. Keating may have laid the ground work but Mr. Bender, Brody and Muller have only say 'if we write this book, we'll change society'. Maybe...but maybe not.)
Second, nothing in math is proved by physics. Math is separate. The rigor of math is a high bar to prove Riemann's hypo but physics will not and cannot pure any mathematical idea. It can demonstrate something but it can't PROVE anything.
Third, namely due to my second reason. I've grown a distaste for Quanta's mag. It's sensational. Not rigorous. And heck, some how this is published in a journal "If the analysis presented here can be made rigorous to show that ^H is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true" ( I added for emphasis).
If I were to say 'hey I have this thesis, if I can prove it....I proved it'. Umm, why is this news? Why is this something that many gravitate towards?
Let's stop 'hype'ing the news. Let's report it. This kind of publishing is garbage and doesn't foster additional scientific discoveries. They should focus on 'if Riemann's hypo is true' they have discovered 'x, y, z'. Many things have been derived by assuming Riemann's true. These new ideas are pushing our scientific understanding. Let's focus on that, not the hype of bs.
I think you're mistaken in thinking that the arguments presented by Bender et al are "physical", in the sense that you mean. Operator theory, Hilbert spaces, Lebegue measures, etc. are all purely mathematical constructs that are incidentally used to model Quantum Mechanics and derive physical insight. The constructs themselves are grounded in the usual sense of mathematical rigor. The inspiration to attack this conjecture, the Riemann hypothesis, framed as a quantum system might come from physical intuition, but everything else is rigorous.
I think Terry Tao gave an example of this sort of "inspired" thinking when he described the proof of the prime number theorem as listening to the "music" of the primes: "We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin Transform. The next and most difficult step is to prove that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem."
> The constructs themselves are grounded in the usual sense of mathematical rigor.
Yes but it doesn't work backwards. Math is routinely used to bring insights in the physical world. Nothing about the Riemann hypothesis is rigorous, it's an unproven proof. Most mathematicians assume it's true but the proof evades them. Physics doesn't lead insights into math. It would be nice but math is just a way to talk about abstraction. The moment you try to 'physical' ideas, you start thinking the earth is the center of the universe. Rigor, in this case, is tremendously hard. The idea that physics "could" lead to insights...is weak and sensational writing. Which brings me back to my original problem/comment I said.
I'm a bit late, but I wanted to comment on a number of things. The Riemann hypothesis is a conjecture, so I'm not sure what you mean when you say "nothing about the Riemann hypothesis is rigorous." Do you mean 'not rigorous' = 'unproven'? I think your comment that "Physics doesn't lead to insights in maths" is just plain wrong. There a number of topics in mathematics that were hugely influenced by the study of physics. In fact, much of 17th and 18th-century mathematics became inseparable from physics. That said, I'm a bit confused when you try to distinguish the two subjects because I feel that large parts of them are entwined. So, what is 'physics' exactly according to you? For me, the Wikipedia definition suffices:
Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force.
To study physics, we need maths. So does your definition of 'physics' include maths?.. or is 'physics' just observations? Take for example the subject of Quantum Mechanics. There are models that we use to make predictions about what we observe. And these models are mathematical. See what I'm trying to get at? The language of physics is inherently mathematical in nature (pun intended).
I'm not sure why I'm not being misunderstood. Math is boundless. Physics is bounded by natural elements. Math can help physics but very rarely would a bounded logic structure help the boundless.
I agree with a lot of what everyone has been saying to me.
But two big objections to your reply:
1. Riemann hypothesis isn't and hasn't been proved. Many mathematicians choose to 'assume it's true' then they come up with other ideas. Also, not rigorous in math = unproven. You can't make those leaps in math. Socially maybe but not in math. If someone were to prove RH is false, then many works will become void because it's based on RH.
2. Physics has limitations. Maths don't. That my point. I find it tough to see a situations where observations (and T. Physics) with it's natural science bound, will aid in any understanding with Maths, let alone RH. I'm sure there are some limited cases but this sensational article acts like 'it's about to happen'. Sigh.
If you're going to use 'bounded', define in what sense you mean that.
Yes, Riemann HYPOTHESIS is a conjecture.
I don't think you're getting what I'm getting at. Let me put it another way. Which parts of the paper have anything to do with physics? They're just talking about operators on linear vector spaces and their eigenvalues.
i'm sorry if i was unclear.. i am aware of who bernhard reimann is and was using his name as shorthand for the hypothesis originating in his 1859 paper(o): .."and it is very probable that all roots are real."; of which this entire article is about
what's my motivation for my intention? I am unsure exactly.. personally? I derive significant joy from working on mathematics; socially? to solve the problem would solidify assumed validity across a varying subset of mathematics and its consequents
The article is very interesting and serve one purpose very well - trigger my curisoity of what is that. Both heard of. Not understand much. But somehow it is linked. And strangely talked in a way not totally out of reach.
If someone like me with only some basic maths/stats/QM background and interest to know more, any pointer to understand this.
This reminds me of story I heard about a pure math PhD student who wrote a deep proof for their dissertation. Unfortunately, they relied on certain math shortcuts that only work in a physics context. Their advisor didn't catch it, and I recall it ended badly for both the student and the advisor.
That sounds bizarre. Pure Math is more theoretical and has less limitations than that of Theoretical Physics. Maybe this person you knew just went down the wrong path and should've focused on Theoretical Physics and not math.
If I understand the parent comment correctly, this is the advanced version of "proofs" of propositions in economics that just assume that dy/dx is defined in such a way that dy/dz = (dy/dx)(dx/dz). A mathematician would say "Let y: U\to V where U,V are open subsets of R^k and y\in C^1 (or C^\infty). Then..." and define clearly what's a function, what's a variable/number/vector and so forth.
Most humorous comments are downvoted into oblivion, but occasionally something humorous will strike a chord with the mob and it'll get upvotes. But if there's a pattern regarding what gets a favourable response, versus what doesn't, I've never been able to identify it.
Humour is fine in principle, but good comments should attempt to further the discourse - e.g. expanding explanations, adding related information, explaining opposing viewpoints, making constructive criticism, fact-checking, or asking questions.
For example, it's very unclear to me what your original comment has to do with trying to prove the Riemann hypothesis. Assuming there is a link, adding an explanation might have improved its reception.
Is this another attempt at contributing to the discussion by being "humorous"? Trying to be funny while attempting to explain the joke, and failing at both, should be sending a clear signal.
We get it. Math has driven people nuts in the past. And others in that time period successfully joked solving those hard problems must be as easy as talking to the divine.
Normally I wouldn't join a conversation this far gone, but I'm curious - are you defining words that you consider to be basically the same as "shut the fuck up" to be violence?
Consider that now that the original joke isn't visible anymore, all these other comments of yours about why you got downvoted are boring to read indeed.
These physicists observations won't achieve anything other than to corroborate the existing hypothesis. Proof requires a lot more than just observation.
There are no physicists' observations here, as I get it, it is all theoretical work, which is essentially mathematics. It is not rigorous mathematics, otherwise we would actually have a proof, but just a proposed reasoning that in time might be made rigorous and become a real proof. Future will tell us.
It's not physics, it's just a reformulation of the problem into linear algebra with a new intuition about why it should be true, namely that if it were not true, then we'd be able to observe imaginary energy levels.
If the intuition you're referring to is the idea of finding a self-adjoint operator with eigenvalues exactly corresponding to the nontrivial zeroes, then yes. The existence of such an operator would guarantee that the zeroes are all on the critical line.
If you're referring to the heuristic arguments for why their proposed not-quite-self-adjoint operator should have only real eigenvalues, then no.
yeah, physics proofs (and the scientific method in general) rely on empirical data whereas mathematical proofs rely on deductive reasoning from existing proven statements. It's literally the opposite approach.
The article makes no mention of any proposed experiments. It sounds like the physicists are just using their understanding of the mathematics of quantum physics along with their intuition about such physical systems to guide the search for a rigorous proof. (cf. the article's discussion of PT-symmetric matrices)
I don't understand the particulars of the spectrum of this operator that they have constructed, but I have heard others describe this approach as a simple reformulation of one hard problem in terms of another equally hard problem.