I'm majoring in Pure Maths and this is annoying to see yet another poor scientific article on Math.
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.
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 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.
Still, I will attempt to contribute on issues of substance. I thought that everything in this article was fine. It's an old idea, as mentioned in this comment: https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_con...
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.
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!
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.
1 - https://news.ycombinator.com/item?id=14046334 2 - https://news.ycombinator.com/item?id=14046131
The the heuristic linking the Riemann hypothesis to a quantum system is actually taken pretty seriously as an avenue of attack on the problem.
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 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."
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.
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 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.
Not science.
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.