4 ms·

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 t

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.

jjgreen 4y ago
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.
munchler 4y ago
It's unsound because it assumes that a divergent series has a finite value. It's a kind of fanciful fiction, no matter who came up with it. In 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.
- bmacho 4y ago
  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.
- Certhas 4y ago
  There is solid reasoning behind it though. Divergent perturbation series can be summed in a meaningful way depending on context. Eg https://en.wikipedia.org/wiki/Borel_summation https://en.wikipedia.org/wiki/Borel_summation The mathematically sounds footing for this is unfortunately routinely not taught in undergrad physics.
  - munchler 4y ago
    [flagged]
    - gnramires 4y ago
      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). Some references in this area: Computational Learning Theory https://en.wikipedia.org/wiki/Computational_learning_theory https://en.wikipedia.org/wiki/Computational_learning_theory VC Theory https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_theory https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_th... Statistical learning theory https://en.wikipedia.org/wiki/Statistical_learning_theory https://en.wikipedia.org/wiki/Statistical_learning_theory
- gmadsen 4y ago
  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"
  - munchler 4y ago
    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.)
    - karpierz 4y ago
      Also, if finite sums of integers have a property, that does not mean that infinite sums of integers have the same property. Ex: 1. The sum of any two integers is guaranteed to be bounded. 2. Therefore, all sums of integers are bounded.
    - gmadsen 4y ago
      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.
    - karpierz 4y ago
      > 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.
donnowhy 4y ago
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')

jjgreen 4y ago

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.

munchler 4y ago

It's unsound because it assumes that a divergent series has a finite value. It's a kind of fanciful fiction, no matter who came up with it. In 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.

bmacho 4y ago
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.
Certhas 4y ago
There is solid reasoning behind it though. Divergent perturbation series can be summed in a meaningful way depending on context. Eg https://en.wikipedia.org/wiki/Borel_summation https://en.wikipedia.org/wiki/Borel_summation The mathematically sounds footing for this is unfortunately routinely not taught in undergrad physics.
- munchler 4y ago
  [flagged]
  - gnramires 4y ago
    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). Some references in this area: Computational Learning Theory https://en.wikipedia.org/wiki/Computational_learning_theory https://en.wikipedia.org/wiki/Computational_learning_theory VC Theory https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_theory https://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_th... Statistical learning theory https://en.wikipedia.org/wiki/Statistical_learning_theory https://en.wikipedia.org/wiki/Statistical_learning_theory
gmadsen 4y ago
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"
- munchler 4y ago
  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.)
  - karpierz 4y ago
    Also, if finite sums of integers have a property, that does not mean that infinite sums of integers have the same property. Ex: 1. The sum of any two integers is guaranteed to be bounded. 2. Therefore, all sums of integers are bounded.
  - gmadsen 4y ago
    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.
  - karpierz 4y ago
    > 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.

donnowhy 4y ago

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')