Fabulous paper, I had no idea that Ramanujan's divergent series sum 1 + 2 + 3 ... = -1/12 is used in the derivation of the equation of the Cashmir effect! (p. 2, eq. 3)
I don't defend physicists mathematical laxity, we are very lazy, but I feel like we can't get dinged for something that came out of Ramanujan, he's solidly in the math canon. Arguing that we're using it out of context, maybe, but I've never heard his work should be considered unsound.
Something not very intuitive is there's no ab initio (i.e. "blank slate", a priori, purely theoretical, etc.) notion of "legitimacy" in mathematics, or even physics. There was a lot of debate on whether complex numbers were "legitimate". The truth is, they work well in several physical models; that's where their legitimacy comes from. We assume that any crazy axiomatic system can model reality, well, almost any actually. There are some properties excepted of any axiomatic system, like self-consistency, and of course the ability to generate some kind of complete (or locally complete at some domain) model of reality. What we then do is try to come up with models and infer the correct or most useful ones from experiments.
This means if a physical model yields a sum like 1+2+3+..., we of course expect the result to be unphysical and the model invalid. However, if experimental data is consistent with a result like -1/12, that would be a clue to explore a different arithmetical system for our model, one which generalizes summations (to Ramanujam summation or other kinds of sums). If this new model reproduces experimental data, you're on a very good track :) [1]
In mathematics the situation is even more interesting: the legitimacy is driven by taste of researchers, which will blend physical and philosophical relevance, and even aesthetics, curiosity and interest in part of the mathematician. (This is why we understand mathematics to be almost art, although a very particular art of course)
[1] Note: I believe the theory of learning (Machine Learning) developed recently is the best current formal understanding of this procedure, and how not to get it wrong. The overview is that whatever your model is, it shouldn't have too many parameters (certainly not more than data you're trying to fit), and that a reliable test for a theory is (a) either making a novel prediction on unseen data that turns out correct (validation); (b) Use the theory to model a part of the data, and see how it fares on the unseen part (for when you can't conduct new experiments; this is cross-validation).
its only unsound if you are using standard summation. That is not the only kind, nor is it the only useful kind. The same can be said about zero or negative numbers, or the real number line. Those are also "fanciful fictions"
Divergent series have no finite sum, by definition. It's self-contradictory to proceed as though they have a sum anyway, even if it gives "meaningful" results. I think that's different from zero, negative numbers, real numbers, etc., which aren't self-contradictory.
I mean, do you really think the sum of the positive integers is -1/12? It's very easy to prove that it isn't. (The sum of any two positive integers is guaranteed to be greater than either one of them. Therefore, no sum of positive integers can be negative.)
> Divergent series have no finite sum, by definition. It's self-contradictory to proceed as though they have a sum anyway, even if it gives "meaningful" results.
It's self-contradictory in the definition that you're using for sum. It'd be like saying:
1 apple + 1 orange != 2 fruits because addition only works across the same units.
one thing to keep in mind, is that infinity as a physical entity may not even exist. we have constructed infinity from ZFC, and deduced consequences of those axioms.
Many many examples in Math go against natural intuition but they are still useful
for example, you say that the real number isn't "self-contradictory". Look at the banach-tarski paradox, that certainly contradicts intuition. A large part of mathematics is foregoing natural intuitive, and just follow the logical deductions of axioms. That is how we got hyperbolic geometry, which is essential to special relativity.
It is not unsound, and it does not assume that a divergent series has a finite value.
It is true, that on real numbers addition is an RxR->R function, and you can only add 2 numbers, and you can't add 3 or even infinitely many numbers. But then you can define an another function that can take a whole series as an argument, and call it addition too.
You may forbid mathematicians naming another function 'addition', and talk about infinite sums, but then who cares. You can even die on this hill. Again, who cares.
you're essentially passing the ball to the mathematicians:
seems like you're saying "we physicist got this from the math people, take it up with them"
but what irks me, is the attitude that disregards the proper understanding of things which you are making use of (in this sentence: 'you' is a informally defined: 'physicists from community')
Sure, but I've always viewed Ramanujan's summation of divergent series to be a curio, I'd never have expected to see it turn up in physics. Pondering this a little, I think what disturbs me about this is the irreducible discontinuity it introduces. If you were to tell me that because of reasons, F = ma is wrong, it should have been F = m^{1.00000000001}a all along, then the resulting model of physics doesn't change that much; but if the Casimir equation should use ζ(-1+ε) rather than ζ(-1), then the sum is finite but as large as you like and the smaller that ε is, the larger this term is.
This paper criticizes renormalization as illegitimate but completely fails to mention the interpretation of renormalization that was given by Wilson and others (see "renormalization group" for details).
There’s also earlier strict mathematical study of renormalisation procedure by Bogolubov and Parasyuk. QFT Green functions are distributions and their product is ill-defined, thus their naive use results in apparent infinities.
This interpretation of renormalization supposes that there is a more fundamental theory at a higher energy scale from which the quantum field theories arise. These then have an effective field theory which is evolved to the energies that are used in current experiments. During this evolving the non-renormalizable interactions go down but not the renormalizable ones which suggests an explanation why we only seem to need the renormalizable ones. However, if one starts out with the idea that something more fundamental exists at higher energy scales one has already proven that QED is 'wrong' in the sense that it is not a fundamental theory.
> However, if one starts out with the idea that something more fundamental exists at higher energy scales one has already proven that QED is 'wrong' in the sense that it is not a fundamental theory.
Isn't that a given for all physics theories? Nobody thinks that physics is 100% complete and we will never have a new and broader theory one day.
It is kind of hard to take someone seriously who is so obviously dripping with bitterness, ending their blog post with "Maybe some adventurous person outside the degenerate welfare-queen anglosphere will figure it out"
it's hard to stay positive when people used 'credential-based' reasoning to disregard your questioning/criticism; this is very standard in academia.
and it's not like you can say "let's see who does better in the real products marketplace" with many of these research things (over 20 years before viable commercialization, if ever).
> it's hard to stay positive when people used 'credential-based' reasoning to disregard your questioning/criticism; this is very standard in academia.
Life is hard. Doesn't change the fact that a weird rant full of bitter ad hominems is almost certainly going to be ignored as being a crazy person wanting some sort of revenge.
This is pretty kooky and bitter, probably not worth giving it much thought. Also he seems, and admits, that he is even less knowledgeable on the field than the people whose contrarianism he apparently takes a t face value.
Vixra is a huge red flag for those not aware. It means the research is so actively questionable that it's not allowed on the non-peer reviewed arxiv server. That's an extremely low bar to clear. There are a lot of crackpots on arxiv, you need something special to get kicked off.
It's also worrisome to see a paper that's concerned with mathematical correctness using both commas and periods as decimal points interchangeably. See Tables 1 and 2, for example.
Many countries use them the opposite of what you're probably used to, and it's hard to keep this in check when writing in a foreign language (in the author's case English).
This isn't worrisome at all.
Definitely not the same people. I think it's by someone who was bitter that 'mainstream' science didn't respect their crackpottery, or something like that.
There can be quality papers on Vixra, sure. But I'm not about to sift through all of the bad ones to find them. That's not an appeal to authority fallacy, that's a useful heuristic.
We're discussing a single paper posted on Hacker News by some user called Certhas, whom I'm not familiar with. So that doesn't improve my desire to view the paper. Am I meant to assume this person has the necessary background knowledge to vet the paper? And if not, why should I waste my time looking at it?
The internet is absolutely full of nonsense, and nobody is obligated to look through it in the name of open-mindedness. Most of us have things to do.
Mind you, I'm not saying the paper is bad. I'm saying it needs to pass certain filters before I spend time giving it a try.
It doesn't matter. If the content of the paper were worth even looking at, it should exist in channels that do discriminate more.
Yes, the venue or the speaker matters, not just the message.
Calling that an appeal to authority, or failing to keep an open mind, is just something the quack cries to explain their lack of impact. That cry is just another thing it is entirely reasonable and defensible to utterly ignore.
But, whether or not your point of view is legitimate—in my case, I agree with your point of view, and won't give serious consideration to anything only available on vixra, and even just finding out an otherwise reputable-seeming paper was on vixra would give me pause—it is literally credentialism.
I'm fairly sympathetic to both of you here. Even without the vast trove of bullshit that is the Internet, the published scientific literature is already far more than anyone could possibly hope to read with care. Using a credentialism filter is one possible step in the process of narrowing down the set of papers available to read to a subset small enough that you can read them all. It's a popular method that has been practiced in some form for as long as there have been organized intellectual pursuits. And while it's evidently good for maintaining some approximation of a local optimum, I suspect it's also why the adage that scientific progress comes one funeral at a time rings so true.
We should be clear though it's the narrowing process that has the value. Credentialism is just one possible means of narrowing that scores highly on ease of use, but perhaps not so highly for learning radically new things.
There's no getting around it. Are you going to invest in the guy trying to sell his engine that runs on water? He's not in any reputable channels either. Maybe it's suppressed by oil companies.
I think it's good enough to rely on the fact that the world has a lot of people in it, and there are more than enough people, not even including the idiots and misguided, but more than enough provably rational people who just have decided it's their hobby or even their vocation, to investigate even wacko claims, purely for academic reasons without necessarily caring about the claims themselves. They will bring anything worth our attention to us. Maybe we are all one of those people on occasion or on specific topics. To me that is quite sufficient way to allow for the possibility that someone did actually discover a way to burn water essentially like gasoline.
No it's not. It's a useful heuristic to adjust your priors before engaging with the work.
It's also a useful bit of information when making the decision what to spend time on. Obviously this is a decision that you have to make before reading the paper.
If you're a bit familiar with typical physics crackpottery there are many other red flags that might tell you this is not worth your time.
I wonder: are there good reasons no one has tried to formalize QFT and the standard model up to an axiomatic mathematical model? I believe this would largely override any doubts and possibly point out to important inconsistencies in the theory. Can the theory give unique predictions (or unique distributions) for physical systems?
Because it is hard and an active area of mathematical research. Which relates to the second question, yes, QED is the most accurate physical theory ever created and its unique predictions have been validated over the past 50 years. The issue is that physicists are a little hand wavy with mathematical formalism, so mathematicians are left trying to formalize , and that is worthwhile because of how effective it has been as a physical theory
Mathematicians absolutely have been trying to formalise QFT for decades, with excellent success in many areas! Have a quick look at the nLab page for quantum field theory: https://ncatlab.org/nlab/show/quantum+field+theory . It is a complicated and hairy problem, and one that's frankly not even well-defined, because QFT can be approached from different perspectives. Often, this is done by using analogies with simpler quantum mechanical theories or classical field theories, but these end up not holding up to mathematical scrutiny when you try to implement them in a naive way.
But this is really quite normal for mathematics. Nobody argues that quantum mechanics is poorly-defined, even though the quantum mechanics that physics undergraduates learn is certainly poorly-defined, and most physicists will never learn about rigged Hilbert spaces or know enough functional analysis to define it properly. This never results in any actual disagreements. All trained physicists will come to the same predictions if they start from the same assumptions, even if they don't know the strict, mathematical formalities.
Definitional issues almost always arises because of infinities, and true infinities are rare in physics -- it's almost always just an approximation for "very big" or "very many". QFT has a lot of infinities. Quantum mechanics promotes classical theories with finite numbers of degrees of freedom into infinite-dimensional wave functions. QFT attempts to do the same, but with classical theories having uncountably infinite degrees of freedom. It's no surprise that there are definitional issues.
And even if you have an axiomatic mathematical model, you'd be surprised how little added value that is to physics as a whole. It's certainly an interesting and important area of research, but the fact that it's ongoing doesn't somehow invalidate the predictions made using formal methods, because those methods are reproducible. Renormalisation isn't magic, it's just a set of rules that you apply, and in fact, those rules have a quite natural physical interpretation (in terms of bare vs effective theories).
The abstract sounds crack-potty tbh. "Suspicious coincidences" is a weird phrase. I didn't actually read the paper, but what could they possibly be arguing where "coincidences" matter. Is the accusation that they made up a fake theory that just happens to be right all the time?
A comment on some of the presentation of the history (‘staunton has already noted the strangeness of bellyaching in 2021 about renormalization without mentioning the renormalization group)
In section 4.3 the author incorrectly describes Fermi’s comments to Dyson as a criticism of Dyson’s work on QED. At the time of this meeting, Dyson had already given up on his QED program and was working on the strong interaction. Fermi’s comments that Dyson had neither a “clear physical picture” or a “precise and self-consistent mathematical formalism” were regarding some (pre-quark) pion theory Dyson had been working at at Cornell starting in 1951, not QED.
Also the author describes Oppenheimer’s initial resistance to Dyson’s work, but fails to mention that Oppenheimer was eventually convinced that Dyson’s work was valuable and gave Dyson a lifetime appointment at the IAS.
The paper is written in scathing tones, by page 11 I was laughing at each new report of QED's 'perfect' agreement with whatever was the latest experimental result for the electron's anomalous magnetic moment although as it continued as far as page 14, it became sobering. However, the recent discrepancies are way down in the decimal points of the result now; when these calculations require 1000s of complicated Feynman diagrams and pages and pages of calculation, errors seem inevitable. My hope is that an alternative approach can be found that avoids these complicated calculations and in fact, researchers already seem to be working on this [1].
Now they are. People (maybe 10 worldwide who care, I would guess) now discus disagreements of results (in some very low order decimal) due to software bugs in numeric integration code, most of which is not public...
The situation seems hardly different except which decimal place they talk about. That's how this kind of science currently progresses.
the fact that the calculations always seemed to be chasing the experimental result seemed the most damning thing to me
i remember from physics undergrad a million years ago that great value was put on predicting new things -- you can always find a way to fit existing numbers!
an epicycles vs ellipses kind of thing
(but of course w/o actually being expert in the topic, it's hard to really know what to think)
I have read only half of the paper and I feel like the author is holding a grudge or something.
But, in general, I wonder why old papers (that are basically a foundation to other theories) are not critically re-reviewed to see if they still hold to scrutiny today? Hmm.
60 comments
[ 0.19 ms ] story [ 126 ms ] threadIn other words, if 1 + 2 + 3 + ... had a finite value, then that value would be -1/12. But it doesn't have a finite value, because it diverges.
It's like saying that if pigs had wings, then they could fly. That's not something I'm happy to base physics on.
The mathematically sounds footing for this is unfortunately routinely not taught in undergrad physics.
This means if a physical model yields a sum like 1+2+3+..., we of course expect the result to be unphysical and the model invalid. However, if experimental data is consistent with a result like -1/12, that would be a clue to explore a different arithmetical system for our model, one which generalizes summations (to Ramanujam summation or other kinds of sums). If this new model reproduces experimental data, you're on a very good track :) [1]
In mathematics the situation is even more interesting: the legitimacy is driven by taste of researchers, which will blend physical and philosophical relevance, and even aesthetics, curiosity and interest in part of the mathematician. (This is why we understand mathematics to be almost art, although a very particular art of course)
[1] Note: I believe the theory of learning (Machine Learning) developed recently is the best current formal understanding of this procedure, and how not to get it wrong. The overview is that whatever your model is, it shouldn't have too many parameters (certainly not more than data you're trying to fit), and that a reliable test for a theory is (a) either making a novel prediction on unseen data that turns out correct (validation); (b) Use the theory to model a part of the data, and see how it fares on the unseen part (for when you can't conduct new experiments; this is cross-validation).
Some references in this area: Computational Learning Theory https://en.wikipedia.org/wiki/Computational_learning_theory
VC Theory https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_th...
Statistical learning theory https://en.wikipedia.org/wiki/Statistical_learning_theory
I mean, do you really think the sum of the positive integers is -1/12? It's very easy to prove that it isn't. (The sum of any two positive integers is guaranteed to be greater than either one of them. Therefore, no sum of positive integers can be negative.)
It's self-contradictory in the definition that you're using for sum. It'd be like saying:
1 apple + 1 orange != 2 fruits because addition only works across the same units.
1. The sum of any two integers is guaranteed to be bounded.
2. Therefore, all sums of integers are bounded.
Many many examples in Math go against natural intuition but they are still useful
for example, you say that the real number isn't "self-contradictory". Look at the banach-tarski paradox, that certainly contradicts intuition. A large part of mathematics is foregoing natural intuitive, and just follow the logical deductions of axioms. That is how we got hyperbolic geometry, which is essential to special relativity.
It is true, that on real numbers addition is an RxR->R function, and you can only add 2 numbers, and you can't add 3 or even infinitely many numbers. But then you can define an another function that can take a whole series as an argument, and call it addition too.
You may forbid mathematicians naming another function 'addition', and talk about infinite sums, but then who cares. You can even die on this hill. Again, who cares.
seems like you're saying "we physicist got this from the math people, take it up with them"
but what irks me, is the attitude that disregards the proper understanding of things which you are making use of (in this sentence: 'you' is a informally defined: 'physicists from community')
Surely just a typo, but, just to make sure, it's Casimir (https://en.wikipedia.org/wiki/Hendrik_Casimir)—nothing to do with Kashmir (https://en.wikipedia.org/wiki/Kashmir).
https://news.ycombinator.com/item?id=31431033
https://news.ycombinator.com/item?id=34873133
Isn't that a given for all physics theories? Nobody thinks that physics is 100% complete and we will never have a new and broader theory one day.
https://old.reddit.com/r/BrilliantLightPower/comments/ffnnxw...
Hydrinos took me down another rabbit hole.
and it's not like you can say "let's see who does better in the real products marketplace" with many of these research things (over 20 years before viable commercialization, if ever).
Life is hard. Doesn't change the fact that a weird rant full of bitter ad hominems is almost certainly going to be ignored as being a crazy person wanting some sort of revenge.
The internet is absolutely full of nonsense, and nobody is obligated to look through it in the name of open-mindedness. Most of us have things to do.
Mind you, I'm not saying the paper is bad. I'm saying it needs to pass certain filters before I spend time giving it a try.
Yes, the venue or the speaker matters, not just the message.
Calling that an appeal to authority, or failing to keep an open mind, is just something the quack cries to explain their lack of impact. That cry is just another thing it is entirely reasonable and defensible to utterly ignore.
As jjgreen points out (https://news.ycombinator.com/item?id=34937266), it does.
But, whether or not your point of view is legitimate—in my case, I agree with your point of view, and won't give serious consideration to anything only available on vixra, and even just finding out an otherwise reputable-seeming paper was on vixra would give me pause—it is literally credentialism.
We should be clear though it's the narrowing process that has the value. Credentialism is just one possible means of narrowing that scores highly on ease of use, but perhaps not so highly for learning radically new things.
I think it's good enough to rely on the fact that the world has a lot of people in it, and there are more than enough people, not even including the idiots and misguided, but more than enough provably rational people who just have decided it's their hobby or even their vocation, to investigate even wacko claims, purely for academic reasons without necessarily caring about the claims themselves. They will bring anything worth our attention to us. Maybe we are all one of those people on occasion or on specific topics. To me that is quite sufficient way to allow for the possibility that someone did actually discover a way to burn water essentially like gasoline.
It's also a useful bit of information when making the decision what to spend time on. Obviously this is a decision that you have to make before reading the paper.
If you're a bit familiar with typical physics crackpottery there are many other red flags that might tell you this is not worth your time.
https://news.ycombinator.com/item?id=34873133
But this is really quite normal for mathematics. Nobody argues that quantum mechanics is poorly-defined, even though the quantum mechanics that physics undergraduates learn is certainly poorly-defined, and most physicists will never learn about rigged Hilbert spaces or know enough functional analysis to define it properly. This never results in any actual disagreements. All trained physicists will come to the same predictions if they start from the same assumptions, even if they don't know the strict, mathematical formalities.
Definitional issues almost always arises because of infinities, and true infinities are rare in physics -- it's almost always just an approximation for "very big" or "very many". QFT has a lot of infinities. Quantum mechanics promotes classical theories with finite numbers of degrees of freedom into infinite-dimensional wave functions. QFT attempts to do the same, but with classical theories having uncountably infinite degrees of freedom. It's no surprise that there are definitional issues.
And even if you have an axiomatic mathematical model, you'd be surprised how little added value that is to physics as a whole. It's certainly an interesting and important area of research, but the fact that it's ongoing doesn't somehow invalidate the predictions made using formal methods, because those methods are reproducible. Renormalisation isn't magic, it's just a set of rules that you apply, and in fact, those rules have a quite natural physical interpretation (in terms of bare vs effective theories).
In section 4.3 the author incorrectly describes Fermi’s comments to Dyson as a criticism of Dyson’s work on QED. At the time of this meeting, Dyson had already given up on his QED program and was working on the strong interaction. Fermi’s comments that Dyson had neither a “clear physical picture” or a “precise and self-consistent mathematical formalism” were regarding some (pre-quark) pion theory Dyson had been working at at Cornell starting in 1951, not QED.
Dyson describes this meeting in https://www.webofstories.com/play/freeman.dyson/94 but the required context is in the prior segment https://www.webofstories.com/play/freeman.dyson/93
Also the author describes Oppenheimer’s initial resistance to Dyson’s work, but fails to mention that Oppenheimer was eventually convinced that Dyson’s work was valuable and gave Dyson a lifetime appointment at the IAS.
https://arxiv.org/abs/2010.10345
how can it be in any way 'scientific' to base results on unpublished material??
> The analytic results we accept as true are partially unpublished, and the people who did the original calculation are long dead.
this ain't science, it's (American) imperialism applied through the cultural-institution of academia.
[1] Nima Arkani-Hamed - The End of Spacetime (Sep 2022) https://www.youtube.com/watch?v=GL77oOnrPzY
The situation seems hardly different except which decimal place they talk about. That's how this kind of science currently progresses.
i remember from physics undergrad a million years ago that great value was put on predicting new things -- you can always find a way to fit existing numbers!
an epicycles vs ellipses kind of thing
(but of course w/o actually being expert in the topic, it's hard to really know what to think)
But, in general, I wonder why old papers (that are basically a foundation to other theories) are not critically re-reviewed to see if they still hold to scrutiny today? Hmm.