9 ms·

Echoes of Electromagnetism Found in Number Theory

178 points by EA-3167 3y ago

swatcoder 3y ago
Terrible headline for an interesting article. “Echoes” suggest a causal relationship, which would be a fascinating to see occur emanating from physical properties and reflecting in mathematical abstractions. What this actually sounds like is “Reflecting on work by physicists helps mathematicians find analogous relationships in their own field” — which is cool and makes for a good story, but is very much not the same thing.
- deleted 3y ago
  [deleted]
- resters 3y ago
  While this is a valid nitpick, I think people sometimes overdo it. Mathematical objects can be both abstractions and systems in and of themselves. They are abstract "enough" to be useful as models, but anyone who infers causality has a fundamental misunderstanding of what mathematics is. I think this comes from our priority in the educational system of teaching mathematical techniques applicable to specific subjects rather than focusing on mathematics as the art of using and manipulating mathematical objects. On the other hand, I had an interesting chat with sommeone who had earned a pH.D in mathematics and physics (and one other discipline which I forget) and he regretted wasting his time learning about mathematical systems that were not applicable to physics research. I didn't retort, but if I had I would have said simply that one never knows how useful links might be found and that there have been a lot of surprises.
  - godelski 3y ago
    Well I think part of the problem is that our education system results in many people thinking mathematics is anything other than a language. Most often people think that math is something we are uncovering about the universe. The OP's critique makes perfect sense from the perspective that physicists (and other sciences) utilize mathematical language. The "causal" relationship is as interesting as "English linguists discover new insights about colors by learning Russian." Similarly I would say physics is a language, though much more grounded in the observable world (since this is explicitly what physicists are trying to do: model our world/universe/reality). > and he regretted wasting his time learning about mathematical systems that were not applicable to physics research. Which shows the above claim about language. Though I'm a bit surprised since these ideas often connect due to the abstractions. 3 PhDs is a bit insane. 2 is even odd. Why go back to school when you already know how to research?
    - jebarker 3y ago
      > Why go back to school when you already know how to research? There's a couple of reasons I can think of. One is that if you want to do fundamental research in a particular field then you probably need other people to work with and that's mostly going to be in academia and PhD is the pathway. The other is that if you take a break from academia after a PhD it's really hard to come back. Three PhDs does seem extreme though! I'm sort of in the latter position. I did a PhD in math, have spent 15 years in govt and industry but a part of me wishes I could go back to pure math research. I think the only path to do that would probably be to do a PhD in a related field such as theoretical physics.
      - godelski 3y ago
        Interesting. I totally get the desire to have collaborators but at the same time I'm particularly burnt out on academia. And most of my PhD has been teaching myself things so tbh I'm not sure what I need the institution for. Not that there aren't useful things that they can provide, but there's also problems that come with it too.
    - fn-mote 3y ago
      > many people thinking mathematics is anything other than a language. I disagree with the claim that mathematics is only language, if by language you mean terms and definitions. The EASY part of mathematics is the language. There are deep truths, too, and they are not just language. For a (kind of) popular example, the question of "Where do the zeros of the Riemann zeta function lie?" is way way more than just vocabulary. There is a 20 minute video by 3Blue1Brown on the subject. [1] [1]: https://www.youtube.com/watch?v=sD0NjbwqlYw https://www.youtube.com/watch?v=sD0NjbwqlYw
      - godelski 3y ago
        > The EASY part of mathematics is the language. This is where I highly disagree. Doing the math is doing the language. I'm not sure which part is "easy". Even setting up problems is often incredibly difficult. This is where many physics students often struggle, despite having strong math skills there's a challenge to describe the problem to be solved. Even strong mathematicians often struggle to translate between mathematics and their native spoken language. Though 3B1B is quite good at doing this, though there is quite a lot of simplification to goes on due to the target audience. Like I said to the sibling comment, do not mistake how powerful and impressive language is. It created our modern world and there's a reason it is more important to us than even our phones. But all languages, including math, are never complete. They cannot describe everything, unfortunately. But I thought my name would have indicated that I was well aware of this notion. > "Where do the zeros of the Riemann zeta function lie?" is way way more than just vocabulary And a language isn't just vocabulary. That's like saying a programming language (which remember has a direct correspondence. They're equivalent) is just tokens. Idk if you want to look into CH Correspondence, Lambda Calculus, Abstract Algebra, Category Theory, or if I should just quote Poincare about mathematics being the study of relationships.
  - mjan22640 3y ago
    Mathematics is overdone nitpicks.
- Ar-Curunir 3y ago
  Have you ever heard of poetic license? Mathematics doesn’t have to be boring and pedantic all the time.
gumby 3y ago
Echoes of the predictive power of Noether’s theorem. Nice!
mattdeboard 3y ago
What a well-written, approachable article. if I understand right, the summary is basically, there is a certain way, an abstraction from geometry, you can use to mathematically describe electromagnetic fields. You can also use this certain geometric abstraction to describe how different types of mathematical abstractions are entangled, like electricity and magnetism are entangled
- immmmmm 3y ago
  Did a PhD on dualities in physics, i both find these findings deeply elegant and this article exceptionally well written!
shuiling 3y ago
[dead]
jebarker 3y ago
Coincidentally I just put down Edward Frenkel's "Love & Math" before opening HN to see this. In particular I was reading the chapter about the early 2000s meetings at IAS using DARPA funding to explore the links between the Langlands Program and physics. Highly recommend the book for those interested in an accessible overview of this area of math.
XzAeRosho 3y ago
Numberphile published an hour long video [1] about the Langland program, which dives a bit into this "grand unified theory of mathematics". I'm not a mathematician, but I find these topics fascinating. [1] https://youtu.be/4dyytPboqvE?si=UyEl-N0sIkQw5HEm https://youtu.be/4dyytPboqvE?si=UyEl-N0sIkQw5HEm
- maze-le 3y ago
  I can also recommend Edward Frenkel's (the guest of this video) book "Love and Math" on the subject. He writes as well as he talks, the book is very approachable, not just for a mathematically literate audience, but also for those who "don't like math". Its about the Langlands Program, but also about his own personal history of becoming a mathematician in the Soviet Union and his experiences with antisemitism during that time. Fascinating Stuff, even though it only barely touches the surface of his research.
  - nairboon 3y ago
    Frenkel was also recently a guest on Lex Fridman's podcast, very interesting.
- Koshkin 3y ago
  One of the well-known category theorists has said [0], > I’ve never succeeded in understanding the slightest thing about it. [0] https://golem.ph.utexas.edu/category/2010/08/what_is_the_langlands_programm.html https://golem.ph.utexas.edu/category/2010/08/what_is_the_lan... (Category Theory, of all things, being the best tool for formalizing dualities and all sorts of analogies between seemingly different objects of study.)
fooker 3y ago
I wonder if it's the opposite. Like electromagnetism behaves a certain way because of fundamental properties of numbers.
- diogenes4 3y ago
  Sorry, opposite of what? You seemed to have inserted causation where none is implied.
  - fooker 3y ago
    The word echo implies causation.
    - bowsamic 3y ago
      So you are just commenting on the title and not the contents of the article?
    - layer8 3y ago
      You mean (logical) implication, not causation.
- smokel 3y ago
  There might well be a connection between the two. However, there are much more obvious connections between electromagnetism, or all of physics for that matter, and numbers [1]. Even though I prefer to interpret things differently, the paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" [2] might be a good avenue to think about these things. I wouldn't be surprised if numbers and physics go hand in hand, in the sense that we can only observe these, and not the more irregular patterns that make up the total chaos that we might exist in. [1] https://en.m.wikipedia.org/wiki/Maxwell%27s_equations https://en.m.wikipedia.org/wiki/Maxwell%27s_equations [2] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...
  - 3y ago
nologic01 3y ago
It is getting very hard to popularize modern research, especially when it is of the more abstract type (as in mathematics). The author does his best but I am not sure it is even a solvable problem. The challenge is what level of familiarity with mathematics to assume of the reader. To transmit a sense of the actual accomplishment of the work, the actual concepts need to link to internalized concepts available to the readers (a sort of mathematical mapping that follows the pattern: A is to B what x is to y). So its important to have a coherent set of x,y to work with (the assumed level of audience knowledge) and that set has to be expressive enough to convey the essence of the new development and its importance / limitations. It feels still possible to achieve this using book-level amount of effort from author and audience (where entire early chapters lay the groundwork for the main course), but an article length might be prohibitevely short. Something that could help, given the online format, is hyperlinks to further explanations (but they must be short and at a consistent level).
gdprrrr 3y ago
> Hamiltonian Ğ-spaces (pronounced G-hat spaces). G-hat is Ĝ. It's not even a typo because it's consistent.
- smokel 3y ago
  For those too lazy to look it up: ˘ is called a "breve", and it is rounded, not even pointy as the typical "hat" is in physics and mathematics literature. Obviously, a rounded hat like that will stay on top of a human head only for a brief amount of time. Not sure if that is the actual reason (people in this thread are fairly pedantic about causality), but it may help to remember that a breve is used to signify short vowels.
  - mejutoco 3y ago
    Breve means short (in duration, as in brevity) too, if it helps to remember it.
danbruc 3y ago
The title is clickbait - they found a connection between two mathematical objects by using a third mathematical object to translate between the two. The third mathematical object also happens to be useful in physics.
- Koshkin 3y ago
  Yeah, I have always wondered, though, how much physics is in the modern theoretical physics.
  - danbruc 3y ago
    That probably depends heavily on the field, there is certainly a huge difference whether you are trying to better understand established physics or whether you are trying to come up with a new theory in an area where we are lacking experimental data.
    - Koshkin 3y ago
      Well, what I mean is that (to give you perhaps an extreme example) classical mechanics, while not "lacking experimental data," has for the last 200 years been effectively considered part of (applied) mathematics (known as "analytical mechanics"). Thermodynamics, electromagnetism, quantum field theory, general relativity - they all seem to be purely mathematical now.
      - danbruc 3y ago
        Then I probably misunderstood you, I thought you were talking about how closely the math is tied to the actual world, say all the reformulations of classical mechanics as opposed to some supersymmetric theory. In the end physics is applied mathematics, once you have a theory that describes your observations, you can just study the mathematical structure and look for interesting things, only occasionally going back to the lab to verify your predictions, improving confidence that you picked the correct model.
- lukeplato 3y ago

178 points by EA-3167 3y ago

Terrible headline for an interesting article. “Echoes” suggest a causal relationship, which would be a fascinating to see occur emanating from physical properties and reflecting in mathematical abstractions. What this actually sounds like is “Reflecting on work by physicists helps mathematicians find analogous relationships in their own field” — which is cool and makes for a good story, but is very much not the same thing.

deleted 3y ago
[deleted]
resters 3y ago
While this is a valid nitpick, I think people sometimes overdo it. Mathematical objects can be both abstractions and systems in and of themselves. They are abstract "enough" to be useful as models, but anyone who infers causality has a fundamental misunderstanding of what mathematics is. I think this comes from our priority in the educational system of teaching mathematical techniques applicable to specific subjects rather than focusing on mathematics as the art of using and manipulating mathematical objects. On the other hand, I had an interesting chat with sommeone who had earned a pH.D in mathematics and physics (and one other discipline which I forget) and he regretted wasting his time learning about mathematical systems that were not applicable to physics research. I didn't retort, but if I had I would have said simply that one never knows how useful links might be found and that there have been a lot of surprises.
- godelski 3y ago
  Well I think part of the problem is that our education system results in many people thinking mathematics is anything other than a language. Most often people think that math is something we are uncovering about the universe. The OP's critique makes perfect sense from the perspective that physicists (and other sciences) utilize mathematical language. The "causal" relationship is as interesting as "English linguists discover new insights about colors by learning Russian." Similarly I would say physics is a language, though much more grounded in the observable world (since this is explicitly what physicists are trying to do: model our world/universe/reality). > and he regretted wasting his time learning about mathematical systems that were not applicable to physics research. Which shows the above claim about language. Though I'm a bit surprised since these ideas often connect due to the abstractions. 3 PhDs is a bit insane. 2 is even odd. Why go back to school when you already know how to research?
  - jebarker 3y ago
    > Why go back to school when you already know how to research? There's a couple of reasons I can think of. One is that if you want to do fundamental research in a particular field then you probably need other people to work with and that's mostly going to be in academia and PhD is the pathway. The other is that if you take a break from academia after a PhD it's really hard to come back. Three PhDs does seem extreme though! I'm sort of in the latter position. I did a PhD in math, have spent 15 years in govt and industry but a part of me wishes I could go back to pure math research. I think the only path to do that would probably be to do a PhD in a related field such as theoretical physics.
    - godelski 3y ago
      Interesting. I totally get the desire to have collaborators but at the same time I'm particularly burnt out on academia. And most of my PhD has been teaching myself things so tbh I'm not sure what I need the institution for. Not that there aren't useful things that they can provide, but there's also problems that come with it too.
  - fn-mote 3y ago
    > many people thinking mathematics is anything other than a language. I disagree with the claim that mathematics is only language, if by language you mean terms and definitions. The EASY part of mathematics is the language. There are deep truths, too, and they are not just language. For a (kind of) popular example, the question of "Where do the zeros of the Riemann zeta function lie?" is way way more than just vocabulary. There is a 20 minute video by 3Blue1Brown on the subject. [1] [1]: https://www.youtube.com/watch?v=sD0NjbwqlYw https://www.youtube.com/watch?v=sD0NjbwqlYw
    - godelski 3y ago
      > The EASY part of mathematics is the language. This is where I highly disagree. Doing the math is doing the language. I'm not sure which part is "easy". Even setting up problems is often incredibly difficult. This is where many physics students often struggle, despite having strong math skills there's a challenge to describe the problem to be solved. Even strong mathematicians often struggle to translate between mathematics and their native spoken language. Though 3B1B is quite good at doing this, though there is quite a lot of simplification to goes on due to the target audience. Like I said to the sibling comment, do not mistake how powerful and impressive language is. It created our modern world and there's a reason it is more important to us than even our phones. But all languages, including math, are never complete. They cannot describe everything, unfortunately. But I thought my name would have indicated that I was well aware of this notion. > "Where do the zeros of the Riemann zeta function lie?" is way way more than just vocabulary And a language isn't just vocabulary. That's like saying a programming language (which remember has a direct correspondence. They're equivalent) is just tokens. Idk if you want to look into CH Correspondence, Lambda Calculus, Abstract Algebra, Category Theory, or if I should just quote Poincare about mathematics being the study of relationships.
- mjan22640 3y ago
  Mathematics is overdone nitpicks.
Ar-Curunir 3y ago
Have you ever heard of poetic license? Mathematics doesn’t have to be boring and pedantic all the time.

gumby 3y ago

Echoes of the predictive power of Noether’s theorem. Nice!

mattdeboard 3y ago

What a well-written, approachable article. if I understand right, the summary is basically, there is a certain way, an abstraction from geometry, you can use to mathematically describe electromagnetic fields. You can also use this certain geometric abstraction to describe how different types of mathematical abstractions are entangled, like electricity and magnetism are entangled

immmmmm 3y ago
Did a PhD on dualities in physics, i both find these findings deeply elegant and this article exceptionally well written!

shuiling 3y ago

[dead]

jebarker 3y ago

Coincidentally I just put down Edward Frenkel's "Love & Math" before opening HN to see this. In particular I was reading the chapter about the early 2000s meetings at IAS using DARPA funding to explore the links between the Langlands Program and physics. Highly recommend the book for those interested in an accessible overview of this area of math.

XzAeRosho 3y ago

Numberphile published an hour long video [1] about the Langland program, which dives a bit into this "grand unified theory of mathematics". I'm not a mathematician, but I find these topics fascinating. [1] https://youtu.be/4dyytPboqvE?si=UyEl-N0sIkQw5HEm https://youtu.be/4dyytPboqvE?si=UyEl-N0sIkQw5HEm

maze-le 3y ago
I can also recommend Edward Frenkel's (the guest of this video) book "Love and Math" on the subject. He writes as well as he talks, the book is very approachable, not just for a mathematically literate audience, but also for those who "don't like math". Its about the Langlands Program, but also about his own personal history of becoming a mathematician in the Soviet Union and his experiences with antisemitism during that time. Fascinating Stuff, even though it only barely touches the surface of his research.
- nairboon 3y ago
  Frenkel was also recently a guest on Lex Fridman's podcast, very interesting.
Koshkin 3y ago
One of the well-known category theorists has said [0], > I’ve never succeeded in understanding the slightest thing about it. [0] https://golem.ph.utexas.edu/category/2010/08/what_is_the_langlands_programm.html https://golem.ph.utexas.edu/category/2010/08/what_is_the_lan... (Category Theory, of all things, being the best tool for formalizing dualities and all sorts of analogies between seemingly different objects of study.)

fooker 3y ago

I wonder if it's the opposite. Like electromagnetism behaves a certain way because of fundamental properties of numbers.

diogenes4 3y ago
Sorry, opposite of what? You seemed to have inserted causation where none is implied.
- fooker 3y ago
  The word echo implies causation.
  - bowsamic 3y ago
    So you are just commenting on the title and not the contents of the article?
  - layer8 3y ago
    You mean (logical) implication, not causation.
smokel 3y ago
There might well be a connection between the two. However, there are much more obvious connections between electromagnetism, or all of physics for that matter, and numbers [1]. Even though I prefer to interpret things differently, the paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" [2] might be a good avenue to think about these things. I wouldn't be surprised if numbers and physics go hand in hand, in the sense that we can only observe these, and not the more irregular patterns that make up the total chaos that we might exist in. [1] https://en.m.wikipedia.org/wiki/Maxwell%27s_equations https://en.m.wikipedia.org/wiki/Maxwell%27s_equations [2] https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.m.wikipedia.org/wiki/The_Unreasonable_Effectivene...
- 3y ago

nologic01 3y ago

It is getting very hard to popularize modern research, especially when it is of the more abstract type (as in mathematics). The author does his best but I am not sure it is even a solvable problem. The challenge is what level of familiarity with mathematics to assume of the reader. To transmit a sense of the actual accomplishment of the work, the actual concepts need to link to internalized concepts available to the readers (a sort of mathematical mapping that follows the pattern: A is to B what x is to y). So its important to have a coherent set of x,y to work with (the assumed level of audience knowledge) and that set has to be expressive enough to convey the essence of the new development and its importance / limitations. It feels still possible to achieve this using book-level amount of effort from author and audience (where entire early chapters lay the groundwork for the main course), but an article length might be prohibitevely short. Something that could help, given the online format, is hyperlinks to further explanations (but they must be short and at a consistent level).

gdprrrr 3y ago

> Hamiltonian Ğ-spaces (pronounced G-hat spaces). G-hat is Ĝ. It's not even a typo because it's consistent.

smokel 3y ago
For those too lazy to look it up: ˘ is called a "breve", and it is rounded, not even pointy as the typical "hat" is in physics and mathematics literature. Obviously, a rounded hat like that will stay on top of a human head only for a brief amount of time. Not sure if that is the actual reason (people in this thread are fairly pedantic about causality), but it may help to remember that a breve is used to signify short vowels.
- mejutoco 3y ago
  Breve means short (in duration, as in brevity) too, if it helps to remember it.

danbruc 3y ago

The title is clickbait - they found a connection between two mathematical objects by using a third mathematical object to translate between the two. The third mathematical object also happens to be useful in physics.

Koshkin 3y ago
Yeah, I have always wondered, though, how much physics is in the modern theoretical physics.
- danbruc 3y ago
  That probably depends heavily on the field, there is certainly a huge difference whether you are trying to better understand established physics or whether you are trying to come up with a new theory in an area where we are lacking experimental data.
  - Koshkin 3y ago
    Well, what I mean is that (to give you perhaps an extreme example) classical mechanics, while not "lacking experimental data," has for the last 200 years been effectively considered part of (applied) mathematics (known as "analytical mechanics"). Thermodynamics, electromagnetism, quantum field theory, general relativity - they all seem to be purely mathematical now.
    - danbruc 3y ago
      Then I probably misunderstood you, I thought you were talking about how closely the math is tied to the actual world, say all the reformulations of classical mechanics as opposed to some supersymmetric theory. In the end physics is applied mathematics, once you have a theory that describes your observations, you can just study the mathematical structure and look for interesting things, only occasionally going back to the lab to verify your predictions, improving confidence that you picked the correct model.
lukeplato 3y ago