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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, Le

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."

heimatau 10y ago
> 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.
- adyavanapalli 10y ago
  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).
  - heimatau 10y ago
    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. Not science.
    - adyavanapalli 10y ago
      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.

heimatau 10y ago

> 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.

adyavanapalli 10y ago
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).
- heimatau 10y ago
  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. Not science.
  - adyavanapalli 10y ago
    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.