This discussion centers on the perceived inadequacies of physics books in presenting mathematical concepts, particularly in relation to quantum electrodynamics (QED) and the historical context of mathematical rigor. Participants argue that while physicists often simplify mathematics for practical predictions, this approach can lead to a lack of rigor that some find philosophically unsatisfactory. The conversation highlights the distinction between the intuitive methods used by physicists, exemplified by Euler's work, and the strict logical frameworks demanded by mathematicians. Ultimately, the group acknowledges the intimate relationship between physics and mathematics, suggesting that rigorous formulations may not always be necessary for scientific progress.
PREREQUISITESPhysicists, mathematicians, educators, and students interested in the interplay between mathematics and physics, particularly those examining the philosophical aspects of mathematical rigor in scientific theories.
disregardthat said:How on Earth would one ever arrive at the intricacies of the current mathematics used in physics without continually backing the process up rigorously? And that's not mentioning how you would even get started without the growing mathematical generalizations which inspire and allows for new ideas to form.
Arsenic&Lace said:Intriguing, my own opinion of the field wasn't based upon experience, I had simply heard from a peer working in quantum computing at IBM that the theorists/experimentalists there generally felt that it was purely academic and impractical.
You seem to be unable to understand my reasoning, so I will repeat it one more time:Arsenic&Lace said:I'm not saying that you don't need to have arguments to support your propositions, that is a caricature of what I am saying which is challenging to agree with if you carefully read my statements. What I am stating is that the levels of rigor are irrelevant and unnecessary.
disregardthat said:What you're suggesting is about as ridiculous as saying that no evidence is ever necessary in physics, because our current theories seem to work pretty well without it.
Arsenic&Lace said:Intriguing, my own opinion of the field wasn't based upon experience, I had simply heard from a peer working in quantum computing at IBM that the theorists/experimentalists there generally felt that it was purely academic and impractical. Has anyone attempted to recast it in a more physical light, rather than in terms of formal, obtuse topology? Or is this inefficient/impossible? It was quite a while ago but Feynman's contributions to our understanding of supercooled helium were due to taking a very mathematically convoluted theory from the condensed matter group and trying to make it as simple as possible, in so doing obtaining everything they had and more. But that might not be the case here.
I am not saying that this demonstrates that rigor is always useless, but I think this debate would end extremely quickly if somebody could find a specific example of where, had it not been for formal mathematical rigor, progress in science or engineering would grind to a halt or follow false paths. Grand claims have been made that theories in physics would be a mess without rigor, but no actual evidence has been presented that this is the case. Indeed, I can even provide evidence to the contrary, given that QFT is still not that mathematically rigorous of a theory (to my knowledge).
So here's my concrete challenge: Show me, how we can analyze the speed of convergence of finite elements methods without using rigorous mathematics.
For the first paragraph:ZombieFeynman said:In regards to the first paragraph above:
Topological insulators are still a long way from practical significance. That does nothing to take away from the interest in them the "fundamental research" point of view. There are topologically DISTINCT states of matter, recently predicted, experimentally confirmed and only now (in the last decade) being explored. The only TRULY SOLVED area is essentially the free electron case. The interplay between strong e-e interactions and spin-orbit coupling is a largely unexplored and extremely exciting (if difficult) area of research (and happens to be my current primary area of interest). Although applications would be wonderful, the primary excitement for me is the exploration not only of a new state of matter but a fundamentally different TYPE of state of matter. That you fail to appreciate this point is troubling.
To your second:
Having mathematical tools that we know are logically self consistent is extremely useful. Not having to ensure that these tools are logically consistent on our own is extremely convenient. I am completely unsure of how you could fail to realize this. I am curious at to where you are at in your physics journey? Are you actively involved in research? What kind? I am somewhat baffled by your responses here.
No, that's wrong. If the algorithm converges fast in one situation, it might be totally inaccurate in another situation with the same number of iterations. No company has the time to test the algorithm for every concrete situation before they use it. That would be pointless. You want to know in advance, which method is better suited for your concrete problem and which method isn't. You don't want to run the algorithm 10 times until you think that the result is close enough to the exact solution. (Of course, you can't know that either without a proof.)Arsenic&Lace said:I mean, I can implement the algorithm in my language of choice (Python or chicken Scheme if I can get away with it, Fortran or C/C++ if I preferred to suffer/needed the performance) and benchmark how much time it takes to converge. If this exceeds my optimization constraints (i.e. if Pointy Haired Boss wants me to have it run in <20 minutes or something) I need to consider implementing a different algorithm or attempting to optimize my existing implementation.
In other words, I could care less if it takes *insert expression here* steps/terms/increments to converge, I only care about the time it takes, a question which can be determined with brute force.
Well, it answers the question in the sense that it tells me that you have no idea what you are talking about, if that is what you wanted to know.Does this answer your question or do I still not understand it? In short, the answer is that I only care about real time, not number of steps/increments/terms.
rubi said:No, that's wrong. If the algorithm converges fast in one situation, it might be totally inaccurate in another situation with the same number of iterations. No company has the time to test the algorithm for every concrete situation before they use it. That would be pointless. You want to know in advance, which method is better suited for your concrete problem and which method isn't. You don't want to run the algorithm 10 times until you think that the result is close enough to the exact solution. (Of course, you can't know that either without a proof.)
We don't even need a PDE example here (I'm really too lazy to think of one, but i could probably pick any PDE i wanted with a some free parameter that I'd vary). The problem occurs already for ODE's. Simulate a harmonic oscillator with a low frequency and one with a high frequency with the same ##\Delta t## using Euler's method. The higher the frequency, the faster the solution will diverge.Arsenic&Lace said:Provide a concrete example, otherwise I have no idea if you are merely speculating or not.
rubi said:We don't even need a PDE example here (I'm really too lazy to think of one, but i could probably pick any PDE i wanted with a some free parameter that I'd vary). The problem occurs already for ODE's. Simulate a harmonic oscillator with a low frequency and one with a high frequency with the same ##\Delta t## using Euler's method. The higher the frequency, the faster the solution will diverge.
micromass said:Right. Or other things to consider: Who says the algorithm will converge at all? Who says the algorithm will converge to the right solution? For example, Newton-Rhapson or fixed point algorithms will not always converge and if they do, they might not give the right solution. Theory is needed to see which is the case.
Or if you want to solve systems of linear equations, who says the very solution you get can be trusted? There are many subtle caveats in these cases where you have ill-conditioned systems. How would you know what ill-conditioned even is without theory?
ZombieFeynman said:One needs only to read through Stone and Goldbart's Mathematics for Physics to see many examples of the things we need to prove to make our operators well behaved.
micromass said:Awesome, I'll be sure to check out this book since it looks quite good.
ZombieFeynman said:The only TRULY SOLVED area is essentially the free electron case. The interplay between strong e-e interactions and spin-orbit coupling is a largely unexplored and extremely exciting (if difficult) area of research (and happens to be my current primary area of interest).
atyy said:What's the status of symmetry protected topological order? I'd heard it proposed as the concept for the interacting case.
ZombieFeynman said:As far as I'm aware, Xiao-Gang Wen has put out some very nice papers on SPT order in bosonic systems. I must admit, my own focus is quite narrowly in transition metal oxide systems.
atyy said:I googled "topological transition metal oxide" and got:
http://arxiv.org/abs/1212.4162
http://arxiv.org/abs/1109.1297
Is it stuff like that you're working on?
ZombieFeynman said:At the risk of giving away too many bits of personal information than I'd prefer to, those papers are in very close proximity to my interests.
To learn that Hilbert spaces are complete(a completely irrelevant fact)? To prove Parseval's theorem (invented in 1799... put on rigorous foundations more than a century later!)? To learn how to force the delta function to be consistent with the function space framework using the convoluted framework of distributions(in spite of the fact that we can use it perfectly fine for its intended purpose without ever worrying about this)?One needs only to read through Stone and Goldbart's Mathematics for Physics to see many examples of the things we need to prove to make our operators well behaved.
Arsenic&Lace said:I see none of you actually work in computational fields.
it is not possible, merely by studying the structure of the algorithms utilizing powerful mathematics, to make useful predictions about performance
To learn that Hilbert spaces are complete(a completely irrelevant fact)?
Here's a challenge: Find some actual evidence that a). the completeness of Hilbert spaces posed a serious question to physicists at some point, b). doubts about Parseval's theorem posed a serious question to physicists/engineers at some point and c). that the "inconsistencies" of the delta function resulted in spurious results or prevented physicists from actually advancing physics.
Nope, not trolling; you really can't make useful predictions about performance in the real world using pure mathematics.micromass said:Are you actually serious or just trolling at this point?
I guess you don't know what a Hilbert space is. It's complete by definition. And its completeness is used in QM all the time, although it is usually just swept under the carpet.
Ah, the classic http://en.wikipedia.org/wiki/Straw_man
Software like ANSYS just implements algorithms that have been discussed by mathematicians. Of course, they rely on rigorous results proved by mathematicians. They even employ mathematicians. You have to be blind to not see this. Additionally, of course they need to benchmark their software. Software development consists of more than just implementing algorithms. The greatest performance gain is due to the use of efficient algorithms, however. If you use an algorithm of complexity ##O(n^2)## instead of ##O(\log(n))##, then you can optimize as much as you want, it will always be inferior.Arsenic&Lace said:Earlier in the thread, rubi made statements about how industries somehow rely on brave mathematicians to analyze their algorithms ahead of time to give them theoretical information about convergence so that they can pick the most accurate and timely tools, miraculously ahead of actually using them, without needing to rely on benchmarks.
You couldn't even come up with the obvious harmonic oscillator counterexample on your own. I still think you have absolutely no clue what you are talking about....and in the case rubi declared I "didn't know what I was talking about"...
I never said that they rely purely on mathematics. However, they rely heavily on it....but to assume that they rely on pure mathematical theory to avoid this problem is simply untrue; they rely on experiential knowledge.
You seem too fixated on how your group does things. In my group (computational) we do make use of rigorous results, such as the guarantee that metadynamics converges asymptotically, or the simple fact that certain algorithms scale like O(n^a). We don't just blindly use any numerical solver, we do pick the ones that are known to work better.Arsenic&Lace said:Nope, not trolling; you really can't make useful predictions about performance in the real world using pure mathematics.
AnTiFreeze3 said:Arsenic, abandon all of your miniscule points of argument. We're debating a broader topic than what you're meandering about. You have actual physicists arguing with, and denying what you're saying. You have actual mathematicians arguing with, and denying what you're saying.
ZombieFeynman said:I am also in disagreement with A&L, however, I don't think an argument from authority is a good way to proceed.
Arsenic&Lace said:Nope, not trolling; you really can't make useful predictions about performance in the real world using pure mathematics.
The argument I was making regarding Hilbert spaces is that completeness is indeed one of their properties, but that it is a useless property to learn about as a physicist and utterly irrelevant to physical theory.
Nope, not a straw man either, or at least not an intentional one. In general, ZF is stating that rigorous details found in Stone and Goldbart represent useful mathematical definitions or proven theorems for which the level of rigor presented in S&G is necessitated; I contend that this is not the case. For instance, the grotesquely convoluted discussion of the Dirac delta function in the second chapter serves no useful purpose for... anything, really.
Char. Limit said:Not all arguments from authority are fallacious arguments. When the authority is a relevant authority, it's a fairly good argument overall. And in this case, the authority in question is about as relevant as you can get.
ZombieFeynman said:I am also in disagreement with A&L, however, I don't think an argument from authority is a good way to proceed.