18 ms·

Why is Maxwell's theory so hard to understand? (2007) [pdf]

292 points by sytelus 7y ago

sleavey 7y ago
So not so much hard to understand (at least for trained physicists at the time), rather too much modesty from Maxwell himself to shout about the importance of his discovery.
bronlund 7y ago
Not one word about how they butchered Maxwell's 20 equations down to 4 - and what may have been lost in the process.
lordnacho 7y ago
With a lot of theories I find the breakthrough in discovery is separate from the breakthrough in explanation. How quickly was it that the familiar four vector equations were published? When you come across them now in an undergrad class, it seems so simple because a lot of work went into distilling the insight into the bits that matter. When the theory was still being fleshed out, there were probably a load of intermediate steps acting like a scaffolding. Likewise with relativity, you can pick up a book and read about it because a lot of people have looked at it over the years and figured out which explanations actually work.
- wwweston 7y ago
  I think it was Oliver Heaviside who made Maxwell’s equations accessible, maybe even in something like the vector calc forms frequently used today.
- jart 7y ago
  Explanations that work for whom? I can't speak for electromagnetism or physics. What I can say is that for years I felt like I didn't have the least bit of clarity on color modeling until I referred to Maxwell's primary materials, which are cited by just about no one sadly.
  - pas 7y ago
    Could you explain this a bit? (Excuse me if I'm asking too much, I don't really know what you mean by color modeling, and how it connects to Maxwell - or maybe you are talking about wavelengths?)
    - jart 7y ago
      One of Modest Maxwell's hobbies was color photography. I wouldn't want to rob you of the joys of discovering the rest.
    - abecedarius 7y ago
      https://en.wikipedia.org/wiki/Color_triangle#Maxwell's_disc https://en.wikipedia.org/wiki/Color_triangle#Maxwell's_disc
zethraeus 7y ago
This is a lovely essay. Wonderful layers-of-reality analogy to quantum mechanics.
- elbear 7y ago
  Yes. That's the reason why I saved it. I don't know much about quantum mechanism, but I could feel a vague intuition developing after reading this essay.
sytelus 7y ago
I actually have happened to delved quite a bit to "really" understand the Maxwell's equations. I've bought original treatise, books with its commentary and plain old "for idiots" sort of books. His original treatise is super dense and unapproachable. Right now we can wear Maxwell's equations on t-shirt but their original form were forbidding. Even with modern form you "really" need to get concepts of differential geometry if you want to just play beyond abstract. There are tons of hand wavy explanations of div and curl out there but almost all can be broken with crafting clever question like “ok, so what you think curl of that would look like?”. I don't think even today I can claim I really get these concepts. In any case, some of my biggest takeaways were these: 1. There is no such thing as "proof" of Maxwell's equations. Just like Einstein's field equations, Newton's laws and many other things in Physics, Maxwell equations are also simply laid out as lets assume these. Vast majority of “greatness” in Physics is simply assuming something without needing to fully understand it and then cross your fingers to see if some good predictions comes out of it. 2. The major achievement of Maxwell's equations is that you can predict velocity of light using other physical constants that have seemingly nothing to do with light. A consequence that we only later realized was that this was literally a constant and not relative to who is measuring it! This is easily one of the most non-obvious achievement in Physics.
- Certhas 7y ago
  You should try to understand differential forms, if you have those then Maxwells equations in vaccuum really aren't strange any more. But Differential forms aren't that strange really. They are the mathematical objects that allow you to integrate along surfaces and curves. Of course their theory hadn't been developed when Maxwell wrote them. And Maxwell was very much concerned with EM in matter, which mixes the properties of the EM fields and materials, and that can thus be expected to get a bit messy. But I don't see what you are striving for when you say "really" understand them. I think this is a psychological category, rather than a hard criterion. Can you apply the formalism to calculate consequences? That's the main issue. Maybe you can have a better or worse intuition about the consequences, but that is often mainly due to practice. You can't expect to correctly intuit all possible consequences of a system as rich as EM. > Young man, in mathematics you don't understand things. You just get used to them. -- John von Neumann
ThePhysicist 7y ago
Modern notation has contributed greatly to making Maxwell's equations more understandable. I dare you to have a look at Maxwell's original paper (https://royalsocietypublishing.org/doi/10.1098/rstl.1865.0008 https://royalsocietypublishing.org/doi/10.1098/rstl.1865.000...) and try to understand it, I'd say even for a seasoned physicist it's not trivial. Compare that to the modern differential form of the equations (e.g. https://en.wikipedia.org/wiki/Maxwell%27s_equations https://en.wikipedia.org/wiki/Maxwell%27s_equations), which (IMHO) are really easy to understand even for second-year Physics students (given that you understand the underlying vector operators). Personally, I found the "flow analogy" always the most intuitive, and there are some books that teach electrodynamics in that way. Typically one would start with electrostatic problems and work one's way to the more complicated stuff like magnetic fields. I guess to derive a "complete" understanding of classical electrodynamics you need to understand the concept of relativity as well. Magnetism is a consequence of relativity and the finite speed of light, so if you accept that it becomes easier to understand (IMHO). Of course you then have to "understand" relativity, which just moves the problem to a different area. But then again, it's always like that in Physics :)
- jjdredd 7y ago
  > Magnetism is a consequence of relativity and the finite speed of light That doesn't make any sense.
  - rovolo 7y ago
    Here's a video explainer: https://www.youtube.com/watch?v=1TKSfAkWWN0 https://www.youtube.com/watch?v=1TKSfAkWWN0 Magnetism affects moving charges, but not stationary charges. A magnetic field in your reference frame is really just an electric field in the moving charge's reference frame.
raxxorrax 7y ago
> Modesty is not always a virtue Seriously... where and when in the last 50 years in modern society was modesty ever a virtue? I certainly wouldn't expect anything like that at Princeton in the US. Otherwise an interesting essay nevertheless, but I certainly don't agree with all the points. > Mendel's modesty setback the progress of biology by fifty years. Hilarious conjecture.
mannykannot 7y ago
This is a wonderful essay, and I wish I had come across it as a freshman. For example: "Just as in the Maxwell theory, the abstract quality of the first-order quantities is revealed in the units in which they are expressed. For example, the Schrödinger wave-function is expressed in a unit which is the square root of an inverse cubic meter. This fact alone makes it clear that the wave-function is an abstraction, for ever hidden from our view. No-one will ever measure directly the square root of a cubic meter." That, of course, is an observation about quantum mechanics, and it is only on rereading the essay today that I noticed its structure: Dyson begins with an anecdote about Maxwell giving an address that spent most of its time discussing some features of Helmholz's "splendid hydro-dynamical theorems" (with a nod to Kelvin, who I guess was present), before briefly drawing analogies to his electrodynamical theory. Dyson brings this up within an essay in which he uses Maxwell's theory to make a point about coming to understand quantum mechanics, though he spends rather more time on his ultimate topic than Maxwell did in his talk.
deleted 7y ago
[deleted]
m0zg 7y ago
Theory of electromagnetic field (of which Maxwell's is a part) was one of my favorite topics in the university. The other being digital signal processing. I don't have very good memory, so I have to rely on things having a lot of consistency and internal structure so they could be efficiently "compressed" and "recovered". Both of these fields have that - the math is consistent, beautiful, and somewhat more intuitive than in other fields. Theory of electromagnetic field does assume that your integral and differential calculus is pretty good though.
UncleSlacky 7y ago
I've seen the "another theory which I prefer" quote mentioned elsewhere (possibly with reference to this paper) as being a form of modesty. I would say that it looks to me like typical British understatement, which is often misunderstood (particularly by Americans) and may have been simply an attempt at humour on his part. https://en.wikipedia.org/wiki/English_understatement https://en.wikipedia.org/wiki/English_understatement
- BeetleB 7y ago
  The author of this piece, though, is British.
jiggawatts 7y ago
Maxwell's theory is hard to understand because it's based on an almost-but-not-quite-appropriate algebra. It's like stringly-typed programming. It works, but it's messy and hard to understand. Maxwell's set of coupled differential vector electromagnetic equations simplify to a hilariously short single equation in Geometric Algebra. A random Google turned up this paper comparing classic EM and the GA formulation. It's not even the simplest possible representation, because that uses natural units and a 4D GA to basically condense the entire set of EM theory into about 4 characters worth of equation that is fully relativistic for free: https://www.researchgate.net/publication/47524066_A_simplified_approach_to_electromagnetism_using_geometric_algebra https://www.researchgate.net/publication/47524066_A_simplifi... It's almost a joke. To me it's reminiscent of looking at beginner programmers. You see them do crazy things like calculate a date "next month" by taking apart the pretty-printed date string, parsing to find the month, realising that sometimes the day part is one digit and sometimes it's two, then having to worry about m-d-y or d-m-y formats, building little tables of "days per month" and leap years... and so on. They can write pages and pages of error free code and it's still Wrong because the correct way is to just call "thedatevar.AddMonth(1);" and be done with it. PS: 3D game engine vector algebra libraries have all of this in common with the physics maths. Things like the cross product being bizarre, having to pick a basis, not being able to interpolate rotations, gymbal lock, rounding error, having different maths for 2D and 3D, a bunch of special cases to worry about, and so forth... Watch Enki Mute's Siggraph 2019 presentation on Geometric Algebra. It's mind-blowing how many stupid little quirks of vector algebra just evaporate if you're prepared to step outside of your comfort zone: https://youtu.be/tX4H_ctggYo https://youtu.be/tX4H_ctggYo
- neutronicus 7y ago
  Do you know if they've done work on formulating the Vlasov equation (coupled evolution of EM fields and phase-space density of populations of charged particles) in this way?
  - petschge 7y ago
    I know that Glasser and Qin at Princeton are working on geometrical view of plasma physik. They aim to get a view of particle-in-cell methods (basically a way to discretize the Vlasov equation using sparse sampling) using a geometric view point expressed using discrete exterior calculus. But I am not sure how much of that is ready or published.
ajeet_dhaliwal 7y ago
Very interesting article, I think there's a lot of parallels there to business and startups. I'm a technical person who has always struggled to promote, and it seems scientists/researchers face the same issues. The ideal is to be a showman and technically brilliant.
antman 7y ago
Maxwell's equations are an artifact of the Mathematical Formulation and they can be written in a perhaps easer way. If one describes them in Geometrc Algebra terms it is just one equation [0] and one can understand it in a natural way. During the nineteenth century and the discussions on different formal ways to describe vector fields Hamilton's quartenion system prevailed. Maxwell had reservations and presented his treatise of electricity and electromagnetism as "the introduction of the ideas, as distinguished from the operations and methods of Quaternions" [1], thus the formulation we know. [0] Mathematical descriptions of the electromagnetic field https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t... [1] The vector algebra war a historical perspective https://arxiv.org/pdf/1509.00501.pdf https://arxiv.org/pdf/1509.00501.pdf
- llamaz 7y ago
  Currently with Maxwell's equations you can point to each one and couple it with an experiment. If you simplify the equations any further, it will be a result of mathematical elegance rather than fundamental undergraduate-level physics.
  - jiggawatts 7y ago
    > and couple it with an experiment. Only a handful of special cases that have obvious "paradoxes" that cannot be explained with Maxwell's equations. It's not even a fully general set of equations classically, it can only handle a certain constrained motions at low velocities, short distances, and generally without accelerations. I mean sure, a map is simple to understand and is fine for navigating a city, but let's not pretend the Earth is flat and then only introduce its spherical geometry in 2nd year studies. That's not the right pedagogical approach.
    - pdonis 7y ago
      > Only a handful of special cases that have obvious "paradoxes" that cannot be explained with Maxwell's equations. What are you talking about? > It's not even a fully general set of equations classically, it can only handle a certain constrained motions at low velocities, short distances, and generally without accelerations. Huh? Maxwell's Equations are relativistically invariant and cover all classical electrodynamic phenomena. The only thing they don't cover is quantum mechanics (although in quantum field theory Maxwell's Equations are still the field equations of the quantum electromagnetic field, so even there they play a role).
vincent-toups 7y ago
Try SU(3) Yang Mills!
- jjdredd 7y ago
  Why not SUSY then? Once you understand the formalism you can derive any field theory to your liking.
state_less 7y ago
> Mathematics is the language that nature speaks. Nature seems to speak in forces and had been doing so long before we arrived and started honing our math skills. Nature made an imprint on our math. Newton came up with new math to make more accurate physical predictions. There is a good reason to be an adherent to the mathematical descriptions. We might want the photon to go through only one slit, To help make nature more classical and understandable, but that’s not the nature we know and observe. So we keep to the math and let it speak to us, lest our work-a-day understandings of the world lead us astray.
scottlocklin 7y ago
A lot of the early days failure of Maxwell was his notation was insane. As others notice below; Heaviside notation[1] made it a lot easier to follow and calculate things. Notation is underrated as increasing human power: I'm not sure General Relativity would have been possible without Christoffel notation. FWIIW fun stuff to read; Whittaker's theories of Aether and Electromagnetism gives the detailed history of all this. Cheap from Dover books. Mind blowing in how theoretical physics used to work. [1] https://books.google.com/books?id=nRJbAAAAYAAJ&pg=PA109#v=onepage&q&f=false https://books.google.com/books?id=nRJbAAAAYAAJ&pg=PA109#v=on...
- UncleSlacky 7y ago
  The 1st edition of his "History of the Theories of Aether and Electricity" is available for free here: https://archive.org/details/historyoftheorie00whitrich/page/n6/mode/2up https://archive.org/details/historyoftheorie00whitrich/page/...
- SilasX 7y ago
  Yes! Even the Pythagorean Theorem is a mouthful if you express it in Latin prose rather than algebra. And I assume universal gravitation is the same way.
- Balgair 7y ago
  Einstein Notation is similar in this regard. It makes the 'mechanics' of tensor mathematics much easier to deal with. Kinda like how arabic numerals are a lot easier to manipulate than roman numerals. I'm not sure sure, but I believe that Helen Dukas, Einstein's secretary, is the one who came up with Einstein notation originally. She was just trying to get through his notes faster when writing things up and that method of notation later became accepted. I can't find that citation though, so calling it Dukas Notation is for the moment erroneous.
  - harry8 7y ago
    To hell with it. So much great work from lowly support, frequently women, has been robbed of its rightful place. Call it Dukas notation until proved otherwise. Crick and Watson still have their ill-gotten novel prizes and most are unaware they are little more than thieves. (Wildly racist and sexist, sure, hard to deny that nowadays so we don't talk about it as loudly as we should). tl;dr "Dukas notation" just sounds good.
montalbano 7y ago
Previous interesting HN discussion on the same paper: https://news.ycombinator.com/item?id=18837677 https://news.ycombinator.com/item?id=18837677
0xff00ffee 7y ago
This paper captured what I encountered as a TA in the 90's: students could solve the integral forms, but had trouble with the differential due to fields. I think this is because some students have a very difficult time going from the mental image of fields as 2D arrows, to div, curl and grad in practice. There's a big step function there in manipulating the equations. I wasn't a very good TA because I didn't have much luck explaining to the struggling students (still feel bad about that, sorry folks, I was getting my master's degree and had no choice to be a TA). And students who didn't understand this in second-year physics had even more trouble when their EE classes went into field and waves classes (conductors, antennae, etc.). TL; DR: I think this is more a problem with teaching advanced differential calculus concepts than Maxwell.
jwmerrill 7y ago
This is a lovely essay, but I think it goes a little too far in claiming that the electromagnetic field is intangible or immeasurable, in slightly forced analogy with quantum mechanics. > We now take it for granted that electric and magnetic fields are abstractions not reducible to mechanical models. To see that this is true, we need only look at the units in which the electric and magnetic fields are supposed to be measured. The conventional unit of electric field-strength is the square-root of a joule per cubic meter... This does not mean that an electric field-strength can be measured with the square-root of a calorimeter. It means that an electric field-strength is an abstract quantity, incommensurable with any quantities that we can measure directly. A more conventional way to think of the dimensions of the electric field is [force]/[charge] (e.g. units of Newtons/Coulomb), and you can observe the electric field by observing the force it exerts on a charged particle through f=qE for example by observing the trajectory of an electron in a cloud chamber. Dyson says that the square of the field is a measurable energy density, but it’s arguably harder to measure an energy density than it is to measure a force. Dyson’s point is much more true for quantum mechanics, where the only measurable things seem to be quadratic combinations of the wave function, and I do like the analogy he points out with electromagnetism, but I think he oversimplifies a bit to make his point.
- mncharity 7y ago
  > not reducible to mechanical models [...] is much more true for quantum mechanics Yes, but... a caveat. How big is an atom? "Unimaginably small" is an oft repeated phrase. What is an atom? "Definitions [...] models [...] skill at switching between models". Electron behavior? Quantum... "unintuitive... the equation is understanding". So how well is "small" taught? Horribly, even by the lackluster baseline of current science education research. Asking first-tier medical school graduate students how big cells are, is not happy thing. But hey, maybe cells are "unimaginable" too. So how well are atoms taught? One challenge in teaching high-school stoichiometry, is students not thinking of atoms as real, as physical objects. But hey, maybe that's a failure to "switch models". So how well is electron behavior taught? Well, when students use the many realistic interactives emphasizing molecular electron density... oh wait. Well, when students view the many molecular dynamics videos showing electron density... oh wait. They do exist... now find them without using google scholar and sending people email. :/ But hey, if students ever do see them some year, maybe no understanding will result, given how unintuitive it all is. Punchline? Teaching things badly seems associated with failure attribution errors. As with education research that's "we taught atoms really badly... surprisingly that didn't work... so we draw the obvious conclusion... students of this age aren't developmentally able to understand atoms". And physics side... there often seems a blurred vision of objectives and their properties. There are a great many plausible learning objectives between "atoms are real" and "i∂_{t}ψ=Hψ". And the usefulness of "mechanical" models varies greatly among them. So "the equation is understanding" gets repeated, in contexts where it's inappropriate, and where it distracts from a broad long-term societal failure to improve wretched science education content.
steve76 7y ago
Maxwell and Faraday saw the world as made up of little pneumatic tubes? A pipe segment breaks and shoots out, a network of pipes collapse in? Einstein came along and said you don't need material. Your watch ticking is medium enough. The consensus today is thermal transfer does not hit everything at once. Try to trace it, you can only get so accurate before you smudge the picture or send feedback to the microphone. What motivates me is: - Atoms as just really fine grains in material - Reversing spectroscopy to create atoms - Reversing telescopes to create planets - Everything's just the Sun Odd, and will be laughed at, but at least it's something to work against when learning.
arkj 7y ago
Does the professor’s claim that the “origin of species” is more important than Maxwell’s equations got any real takers? Or is it just his personal opinion?
Vysero 7y ago
" We, with the advantage of hindsight, can see clearly that Maxwell's paper was the most important event of the nineteenth century in the history of the physical sciences. If we include the biological sciences as well as the physical sciences, Maxwell's paper was second only to Darwin's ``Origin of Species''." By what metric? How has Darwin's ``Origin of Species`` been even half as prolific as Maxwell's work? Maxwell's work revolutionized our understanding of the physical world and had lead to the creation of thousands upon thousands of technological advances. It has literally forged the modern world.
- AbrahamParangi 7y ago
  I think Darwin gets special note for the fact that while there are many incredible works of theory in math and physics, there's really only one in biology: On the Origin of Species by Charles Darwin.
  - Vysero 7y ago
    Certainly not saying that it's not noteworthy, but if I had to make a top 5 list he wouldn't be on it. Apparently, my metric and the authors metric are different.
- CamperBob2 7y ago
  One answer to your question: https://en.wikipedia.org/wiki/Nothing_in_Biology_Makes_Sense_Except_in_the_Light_of_Evolution https://en.wikipedia.org/wiki/Nothing_in_Biology_Makes_Sense... Darwin:Maxwell :: biology:physics is a very apt comparison.
LatteLazy 7y ago
Once you understand VC operators (grad, div and curl) and the associated intergals, Maxwells Equations are actually really intuitive and quit beautiful. I was lucky that the maths course I was on covered those a few weeks before we started electrodynamics. Plus I had Mary Boas Mathematical Methods in the Physical Sciences which teaches this material (and Greens theorem etc) very well.
dschuetz 7y ago
Ah, yes. It's not the ignorant physicists' fault that Maxwell's theory hasn't got the attention it deserved. It's Maxwell's modesty's fault that set back the science of physics two decades. What a ridiculous notion.
tagrun 7y ago
It appears no one here has actually read the essay. He's not talking about mathematical complexity of Maxwell's equations (and no, you can use differential forms, Clifford algebras, quaternions, vector calculus, but no matter how you write it, it's still the same thing, and at the end of the day, you will end up solving exactly the same set of coupled partial differential equations, to the letter --there's no magical mathematical notation that makes this go away). The difficulty he's referring to is in the physics (and not mathematics) associated with the idea that fields are real, fundamental physical entities, and cannot be reduced to mechanical models with "gears and wheels" permeating the space (which was a popular idea back then).
- mjfisher 7y ago
  You prompted me to read the actual essay. It was considerably more interesting than the comments here would have suggested; which is not a surprise, given it was penned by Freeman Dyson. His insight that Maxwell's contemporaries lacked even the language to fully describe what a transformative idea he had is acutely interesting. Once again language both shapes and traps thinking. He makes the same point with quantum mechanics as well - that we're constrained by not having the proper language to describe the most fundamental behaviour. I think it's fair to say that point still stands today.
peter_d_sherman 7y ago
Excerpt: "Maxwell explained how the ancient theory that matter is composed of atoms ran into a logical paradox. On the one hand, atoms were supposed to be hard, impenetrable and indestructible. On the other hand, the evidence of spectroscopy and chemistry showed that atoms have internal structure and are influenced by outside forces. This paradox had for many years blocked progress in the understanding of the nature of matter. Now finally the vortex theory of molecules resolved the paradox. Vortices in the aether are soft and have internal structure, and nevertheless, according to Helmholtz, they are individual and indestructible. The only remaining task was to deduce the facts of spectroscopy and chemistry from the laws of interaction of the vortices predicted by the hydrodynamics of a perfect fluid."
bearsbearsand 7y ago
what happened to Maxwells original quaternion based equations. the changes made against his will by lorrentz and others to simplify after his death were an affront to science and have set back humanity a hundred years.
known 7y ago
"Physics is to math what sex is to masturbation" --Feynman
gumby 7y ago
This is fundamentally a sociological (or philosophy of science) essay rather than about why Maxwell's equations are still so hard to understand for the student. In that it's quite interesting but I found the jump to a parallelism between the model of Maxwell's and quantum mechanical models a bit of a stretch.
robochat 7y ago
We've all gotten used to the idea that Electric and Magnetic fields are real but then we learn about the Aharonov-Bohm effect and suddenly, maybe it's the electric potential and magnetic vector potential are what is 'real'. But then, there's all the issue with gauges. I think that the best way of teaching electromagnetism might still be in the future.
creed98 7y ago
What equation links Newton's universe to maxwells? Here you will find your answer to 2nd layer product/square by another ;) Again this is the same guy who created maxwells demon. There are no accidents

292 points by sytelus 7y ago

So not so much hard to understand (at least for trained physicists at the time), rather too much modesty from Maxwell himself to shout about the importance of his discovery.

bronlund 7y ago

Not one word about how they butchered Maxwell's 20 equations down to 4 - and what may have been lost in the process.

lordnacho 7y ago

With a lot of theories I find the breakthrough in discovery is separate from the breakthrough in explanation. How quickly was it that the familiar four vector equations were published? When you come across them now in an undergrad class, it seems so simple because a lot of work went into distilling the insight into the bits that matter. When the theory was still being fleshed out, there were probably a load of intermediate steps acting like a scaffolding. Likewise with relativity, you can pick up a book and read about it because a lot of people have looked at it over the years and figured out which explanations actually work.

wwweston 7y ago
I think it was Oliver Heaviside who made Maxwell’s equations accessible, maybe even in something like the vector calc forms frequently used today.
jart 7y ago
Explanations that work for whom? I can't speak for electromagnetism or physics. What I can say is that for years I felt like I didn't have the least bit of clarity on color modeling until I referred to Maxwell's primary materials, which are cited by just about no one sadly.
- pas 7y ago
  Could you explain this a bit? (Excuse me if I'm asking too much, I don't really know what you mean by color modeling, and how it connects to Maxwell - or maybe you are talking about wavelengths?)
  - jart 7y ago
    One of Modest Maxwell's hobbies was color photography. I wouldn't want to rob you of the joys of discovering the rest.
  - abecedarius 7y ago
    https://en.wikipedia.org/wiki/Color_triangle#Maxwell's_disc https://en.wikipedia.org/wiki/Color_triangle#Maxwell's_disc

zethraeus 7y ago

This is a lovely essay. Wonderful layers-of-reality analogy to quantum mechanics.

elbear 7y ago
Yes. That's the reason why I saved it. I don't know much about quantum mechanism, but I could feel a vague intuition developing after reading this essay.

sytelus 7y ago

I actually have happened to delved quite a bit to "really" understand the Maxwell's equations. I've bought original treatise, books with its commentary and plain old "for idiots" sort of books. His original treatise is super dense and unapproachable. Right now we can wear Maxwell's equations on t-shirt but their original form were forbidding. Even with modern form you "really" need to get concepts of differential geometry if you want to just play beyond abstract. There are tons of hand wavy explanations of div and curl out there but almost all can be broken with crafting clever question like “ok, so what you think curl of that would look like?”. I don't think even today I can claim I really get these concepts. In any case, some of my biggest takeaways were these: 1. There is no such thing as "proof" of Maxwell's equations. Just like Einstein's field equations, Newton's laws and many other things in Physics, Maxwell equations are also simply laid out as lets assume these. Vast majority of “greatness” in Physics is simply assuming something without needing to fully understand it and then cross your fingers to see if some good predictions comes out of it. 2. The major achievement of Maxwell's equations is that you can predict velocity of light using other physical constants that have seemingly nothing to do with light. A consequence that we only later realized was that this was literally a constant and not relative to who is measuring it! This is easily one of the most non-obvious achievement in Physics.

Certhas 7y ago
You should try to understand differential forms, if you have those then Maxwells equations in vaccuum really aren't strange any more. But Differential forms aren't that strange really. They are the mathematical objects that allow you to integrate along surfaces and curves. Of course their theory hadn't been developed when Maxwell wrote them. And Maxwell was very much concerned with EM in matter, which mixes the properties of the EM fields and materials, and that can thus be expected to get a bit messy. But I don't see what you are striving for when you say "really" understand them. I think this is a psychological category, rather than a hard criterion. Can you apply the formalism to calculate consequences? That's the main issue. Maybe you can have a better or worse intuition about the consequences, but that is often mainly due to practice. You can't expect to correctly intuit all possible consequences of a system as rich as EM. > Young man, in mathematics you don't understand things. You just get used to them. -- John von Neumann

ThePhysicist 7y ago

Modern notation has contributed greatly to making Maxwell's equations more understandable. I dare you to have a look at Maxwell's original paper (https://royalsocietypublishing.org/doi/10.1098/rstl.1865.0008 https://royalsocietypublishing.org/doi/10.1098/rstl.1865.000...) and try to understand it, I'd say even for a seasoned physicist it's not trivial. Compare that to the modern differential form of the equations (e.g. https://en.wikipedia.org/wiki/Maxwell%27s_equations https://en.wikipedia.org/wiki/Maxwell%27s_equations), which (IMHO) are really easy to understand even for second-year Physics students (given that you understand the underlying vector operators). Personally, I found the "flow analogy" always the most intuitive, and there are some books that teach electrodynamics in that way. Typically one would start with electrostatic problems and work one's way to the more complicated stuff like magnetic fields. I guess to derive a "complete" understanding of classical electrodynamics you need to understand the concept of relativity as well. Magnetism is a consequence of relativity and the finite speed of light, so if you accept that it becomes easier to understand (IMHO). Of course you then have to "understand" relativity, which just moves the problem to a different area. But then again, it's always like that in Physics :)

jjdredd 7y ago
> Magnetism is a consequence of relativity and the finite speed of light That doesn't make any sense.
- rovolo 7y ago
  Here's a video explainer: https://www.youtube.com/watch?v=1TKSfAkWWN0 https://www.youtube.com/watch?v=1TKSfAkWWN0 Magnetism affects moving charges, but not stationary charges. A magnetic field in your reference frame is really just an electric field in the moving charge's reference frame.

raxxorrax 7y ago

> Modesty is not always a virtue Seriously... where and when in the last 50 years in modern society was modesty ever a virtue? I certainly wouldn't expect anything like that at Princeton in the US. Otherwise an interesting essay nevertheless, but I certainly don't agree with all the points. > Mendel's modesty setback the progress of biology by fifty years. Hilarious conjecture.

mannykannot 7y ago

This is a wonderful essay, and I wish I had come across it as a freshman. For example: "Just as in the Maxwell theory, the abstract quality of the first-order quantities is revealed in the units in which they are expressed. For example, the Schrödinger wave-function is expressed in a unit which is the square root of an inverse cubic meter. This fact alone makes it clear that the wave-function is an abstraction, for ever hidden from our view. No-one will ever measure directly the square root of a cubic meter." That, of course, is an observation about quantum mechanics, and it is only on rereading the essay today that I noticed its structure: Dyson begins with an anecdote about Maxwell giving an address that spent most of its time discussing some features of Helmholz's "splendid hydro-dynamical theorems" (with a nod to Kelvin, who I guess was present), before briefly drawing analogies to his electrodynamical theory. Dyson brings this up within an essay in which he uses Maxwell's theory to make a point about coming to understand quantum mechanics, though he spends rather more time on his ultimate topic than Maxwell did in his talk.

deleted 7y ago

[deleted]

m0zg 7y ago

Theory of electromagnetic field (of which Maxwell's is a part) was one of my favorite topics in the university. The other being digital signal processing. I don't have very good memory, so I have to rely on things having a lot of consistency and internal structure so they could be efficiently "compressed" and "recovered". Both of these fields have that - the math is consistent, beautiful, and somewhat more intuitive than in other fields. Theory of electromagnetic field does assume that your integral and differential calculus is pretty good though.

UncleSlacky 7y ago

I've seen the "another theory which I prefer" quote mentioned elsewhere (possibly with reference to this paper) as being a form of modesty. I would say that it looks to me like typical British understatement, which is often misunderstood (particularly by Americans) and may have been simply an attempt at humour on his part. https://en.wikipedia.org/wiki/English_understatement https://en.wikipedia.org/wiki/English_understatement

BeetleB 7y ago
The author of this piece, though, is British.

jiggawatts 7y ago

Maxwell's theory is hard to understand because it's based on an almost-but-not-quite-appropriate algebra. It's like stringly-typed programming. It works, but it's messy and hard to understand. Maxwell's set of coupled differential vector electromagnetic equations simplify to a hilariously short single equation in Geometric Algebra. A random Google turned up this paper comparing classic EM and the GA formulation. It's not even the simplest possible representation, because that uses natural units and a 4D GA to basically condense the entire set of EM theory into about 4 characters worth of equation that is fully relativistic for free: https://www.researchgate.net/publication/47524066_A_simplified_approach_to_electromagnetism_using_geometric_algebra https://www.researchgate.net/publication/47524066_A_simplifi... It's almost a joke. To me it's reminiscent of looking at beginner programmers. You see them do crazy things like calculate a date "next month" by taking apart the pretty-printed date string, parsing to find the month, realising that sometimes the day part is one digit and sometimes it's two, then having to worry about m-d-y or d-m-y formats, building little tables of "days per month" and leap years... and so on. They can write pages and pages of error free code and it's still Wrong because the correct way is to just call "thedatevar.AddMonth(1);" and be done with it. PS: 3D game engine vector algebra libraries have all of this in common with the physics maths. Things like the cross product being bizarre, having to pick a basis, not being able to interpolate rotations, gymbal lock, rounding error, having different maths for 2D and 3D, a bunch of special cases to worry about, and so forth... Watch Enki Mute's Siggraph 2019 presentation on Geometric Algebra. It's mind-blowing how many stupid little quirks of vector algebra just evaporate if you're prepared to step outside of your comfort zone: https://youtu.be/tX4H_ctggYo https://youtu.be/tX4H_ctggYo

neutronicus 7y ago
Do you know if they've done work on formulating the Vlasov equation (coupled evolution of EM fields and phase-space density of populations of charged particles) in this way?
- petschge 7y ago
  I know that Glasser and Qin at Princeton are working on geometrical view of plasma physik. They aim to get a view of particle-in-cell methods (basically a way to discretize the Vlasov equation using sparse sampling) using a geometric view point expressed using discrete exterior calculus. But I am not sure how much of that is ready or published.

ajeet_dhaliwal 7y ago

Very interesting article, I think there's a lot of parallels there to business and startups. I'm a technical person who has always struggled to promote, and it seems scientists/researchers face the same issues. The ideal is to be a showman and technically brilliant.

antman 7y ago

Maxwell's equations are an artifact of the Mathematical Formulation and they can be written in a perhaps easer way. If one describes them in Geometrc Algebra terms it is just one equation [0] and one can understand it in a natural way. During the nineteenth century and the discussions on different formal ways to describe vector fields Hamilton's quartenion system prevailed. Maxwell had reservations and presented his treatise of electricity and electromagnetism as "the introduction of the ideas, as distinguished from the operations and methods of Quaternions" [1], thus the formulation we know. [0] Mathematical descriptions of the electromagnetic field https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t... [1] The vector algebra war a historical perspective https://arxiv.org/pdf/1509.00501.pdf https://arxiv.org/pdf/1509.00501.pdf

llamaz 7y ago
Currently with Maxwell's equations you can point to each one and couple it with an experiment. If you simplify the equations any further, it will be a result of mathematical elegance rather than fundamental undergraduate-level physics.
- jiggawatts 7y ago
  > and couple it with an experiment. Only a handful of special cases that have obvious "paradoxes" that cannot be explained with Maxwell's equations. It's not even a fully general set of equations classically, it can only handle a certain constrained motions at low velocities, short distances, and generally without accelerations. I mean sure, a map is simple to understand and is fine for navigating a city, but let's not pretend the Earth is flat and then only introduce its spherical geometry in 2nd year studies. That's not the right pedagogical approach.
  - pdonis 7y ago
    > Only a handful of special cases that have obvious "paradoxes" that cannot be explained with Maxwell's equations. What are you talking about? > It's not even a fully general set of equations classically, it can only handle a certain constrained motions at low velocities, short distances, and generally without accelerations. Huh? Maxwell's Equations are relativistically invariant and cover all classical electrodynamic phenomena. The only thing they don't cover is quantum mechanics (although in quantum field theory Maxwell's Equations are still the field equations of the quantum electromagnetic field, so even there they play a role).

vincent-toups 7y ago

Try SU(3) Yang Mills!

jjdredd 7y ago
Why not SUSY then? Once you understand the formalism you can derive any field theory to your liking.

state_less 7y ago

> Mathematics is the language that nature speaks. Nature seems to speak in forces and had been doing so long before we arrived and started honing our math skills. Nature made an imprint on our math. Newton came up with new math to make more accurate physical predictions. There is a good reason to be an adherent to the mathematical descriptions. We might want the photon to go through only one slit, To help make nature more classical and understandable, but that’s not the nature we know and observe. So we keep to the math and let it speak to us, lest our work-a-day understandings of the world lead us astray.

scottlocklin 7y ago

A lot of the early days failure of Maxwell was his notation was insane. As others notice below; Heaviside notation[1] made it a lot easier to follow and calculate things. Notation is underrated as increasing human power: I'm not sure General Relativity would have been possible without Christoffel notation. FWIIW fun stuff to read; Whittaker's theories of Aether and Electromagnetism gives the detailed history of all this. Cheap from Dover books. Mind blowing in how theoretical physics used to work. [1] https://books.google.com/books?id=nRJbAAAAYAAJ&pg=PA109#v=onepage&q&f=false https://books.google.com/books?id=nRJbAAAAYAAJ&pg=PA109#v=on...

UncleSlacky 7y ago
The 1st edition of his "History of the Theories of Aether and Electricity" is available for free here: https://archive.org/details/historyoftheorie00whitrich/page/n6/mode/2up https://archive.org/details/historyoftheorie00whitrich/page/...
SilasX 7y ago
Yes! Even the Pythagorean Theorem is a mouthful if you express it in Latin prose rather than algebra. And I assume universal gravitation is the same way.
Balgair 7y ago
Einstein Notation is similar in this regard. It makes the 'mechanics' of tensor mathematics much easier to deal with. Kinda like how arabic numerals are a lot easier to manipulate than roman numerals. I'm not sure sure, but I believe that Helen Dukas, Einstein's secretary, is the one who came up with Einstein notation originally. She was just trying to get through his notes faster when writing things up and that method of notation later became accepted. I can't find that citation though, so calling it Dukas Notation is for the moment erroneous.
- harry8 7y ago
  To hell with it. So much great work from lowly support, frequently women, has been robbed of its rightful place. Call it Dukas notation until proved otherwise. Crick and Watson still have their ill-gotten novel prizes and most are unaware they are little more than thieves. (Wildly racist and sexist, sure, hard to deny that nowadays so we don't talk about it as loudly as we should). tl;dr "Dukas notation" just sounds good.

montalbano 7y ago

Previous interesting HN discussion on the same paper: https://news.ycombinator.com/item?id=18837677 https://news.ycombinator.com/item?id=18837677

0xff00ffee 7y ago

This paper captured what I encountered as a TA in the 90's: students could solve the integral forms, but had trouble with the differential due to fields. I think this is because some students have a very difficult time going from the mental image of fields as 2D arrows, to div, curl and grad in practice. There's a big step function there in manipulating the equations. I wasn't a very good TA because I didn't have much luck explaining to the struggling students (still feel bad about that, sorry folks, I was getting my master's degree and had no choice to be a TA). And students who didn't understand this in second-year physics had even more trouble when their EE classes went into field and waves classes (conductors, antennae, etc.). TL; DR: I think this is more a problem with teaching advanced differential calculus concepts than Maxwell.

jwmerrill 7y ago

This is a lovely essay, but I think it goes a little too far in claiming that the electromagnetic field is intangible or immeasurable, in slightly forced analogy with quantum mechanics. > We now take it for granted that electric and magnetic fields are abstractions not reducible to mechanical models. To see that this is true, we need only look at the units in which the electric and magnetic fields are supposed to be measured. The conventional unit of electric field-strength is the square-root of a joule per cubic meter... This does not mean that an electric field-strength can be measured with the square-root of a calorimeter. It means that an electric field-strength is an abstract quantity, incommensurable with any quantities that we can measure directly. A more conventional way to think of the dimensions of the electric field is [force]/[charge] (e.g. units of Newtons/Coulomb), and you can observe the electric field by observing the force it exerts on a charged particle through f=qE for example by observing the trajectory of an electron in a cloud chamber. Dyson says that the square of the field is a measurable energy density, but it’s arguably harder to measure an energy density than it is to measure a force. Dyson’s point is much more true for quantum mechanics, where the only measurable things seem to be quadratic combinations of the wave function, and I do like the analogy he points out with electromagnetism, but I think he oversimplifies a bit to make his point.

mncharity 7y ago
> not reducible to mechanical models [...] is much more true for quantum mechanics Yes, but... a caveat. How big is an atom? "Unimaginably small" is an oft repeated phrase. What is an atom? "Definitions [...] models [...] skill at switching between models". Electron behavior? Quantum... "unintuitive... the equation is understanding". So how well is "small" taught? Horribly, even by the lackluster baseline of current science education research. Asking first-tier medical school graduate students how big cells are, is not happy thing. But hey, maybe cells are "unimaginable" too. So how well are atoms taught? One challenge in teaching high-school stoichiometry, is students not thinking of atoms as real, as physical objects. But hey, maybe that's a failure to "switch models". So how well is electron behavior taught? Well, when students use the many realistic interactives emphasizing molecular electron density... oh wait. Well, when students view the many molecular dynamics videos showing electron density... oh wait. They do exist... now find them without using google scholar and sending people email. :/ But hey, if students ever do see them some year, maybe no understanding will result, given how unintuitive it all is. Punchline? Teaching things badly seems associated with failure attribution errors. As with education research that's "we taught atoms really badly... surprisingly that didn't work... so we draw the obvious conclusion... students of this age aren't developmentally able to understand atoms". And physics side... there often seems a blurred vision of objectives and their properties. There are a great many plausible learning objectives between "atoms are real" and "i∂_{t}ψ=Hψ". And the usefulness of "mechanical" models varies greatly among them. So "the equation is understanding" gets repeated, in contexts where it's inappropriate, and where it distracts from a broad long-term societal failure to improve wretched science education content.

steve76 7y ago

Maxwell and Faraday saw the world as made up of little pneumatic tubes? A pipe segment breaks and shoots out, a network of pipes collapse in? Einstein came along and said you don't need material. Your watch ticking is medium enough. The consensus today is thermal transfer does not hit everything at once. Try to trace it, you can only get so accurate before you smudge the picture or send feedback to the microphone. What motivates me is: - Atoms as just really fine grains in material - Reversing spectroscopy to create atoms - Reversing telescopes to create planets - Everything's just the Sun Odd, and will be laughed at, but at least it's something to work against when learning.

arkj 7y ago

Does the professor’s claim that the “origin of species” is more important than Maxwell’s equations got any real takers? Or is it just his personal opinion?

Vysero 7y ago

" We, with the advantage of hindsight, can see clearly that Maxwell's paper was the most important event of the nineteenth century in the history of the physical sciences. If we include the biological sciences as well as the physical sciences, Maxwell's paper was second only to Darwin's ``Origin of Species''." By what metric? How has Darwin's ``Origin of Species`` been even half as prolific as Maxwell's work? Maxwell's work revolutionized our understanding of the physical world and had lead to the creation of thousands upon thousands of technological advances. It has literally forged the modern world.

AbrahamParangi 7y ago
I think Darwin gets special note for the fact that while there are many incredible works of theory in math and physics, there's really only one in biology: On the Origin of Species by Charles Darwin.
- Vysero 7y ago
  Certainly not saying that it's not noteworthy, but if I had to make a top 5 list he wouldn't be on it. Apparently, my metric and the authors metric are different.
CamperBob2 7y ago
One answer to your question: https://en.wikipedia.org/wiki/Nothing_in_Biology_Makes_Sense_Except_in_the_Light_of_Evolution https://en.wikipedia.org/wiki/Nothing_in_Biology_Makes_Sense... Darwin:Maxwell :: biology:physics is a very apt comparison.

LatteLazy 7y ago

Once you understand VC operators (grad, div and curl) and the associated intergals, Maxwells Equations are actually really intuitive and quit beautiful. I was lucky that the maths course I was on covered those a few weeks before we started electrodynamics. Plus I had Mary Boas Mathematical Methods in the Physical Sciences which teaches this material (and Greens theorem etc) very well.

dschuetz 7y ago

Ah, yes. It's not the ignorant physicists' fault that Maxwell's theory hasn't got the attention it deserved. It's Maxwell's modesty's fault that set back the science of physics two decades. What a ridiculous notion.

tagrun 7y ago

It appears no one here has actually read the essay. He's not talking about mathematical complexity of Maxwell's equations (and no, you can use differential forms, Clifford algebras, quaternions, vector calculus, but no matter how you write it, it's still the same thing, and at the end of the day, you will end up solving exactly the same set of coupled partial differential equations, to the letter --there's no magical mathematical notation that makes this go away). The difficulty he's referring to is in the physics (and not mathematics) associated with the idea that fields are real, fundamental physical entities, and cannot be reduced to mechanical models with "gears and wheels" permeating the space (which was a popular idea back then).

mjfisher 7y ago
You prompted me to read the actual essay. It was considerably more interesting than the comments here would have suggested; which is not a surprise, given it was penned by Freeman Dyson. His insight that Maxwell's contemporaries lacked even the language to fully describe what a transformative idea he had is acutely interesting. Once again language both shapes and traps thinking. He makes the same point with quantum mechanics as well - that we're constrained by not having the proper language to describe the most fundamental behaviour. I think it's fair to say that point still stands today.

peter_d_sherman 7y ago

Excerpt: "Maxwell explained how the ancient theory that matter is composed of atoms ran into a logical paradox. On the one hand, atoms were supposed to be hard, impenetrable and indestructible. On the other hand, the evidence of spectroscopy and chemistry showed that atoms have internal structure and are influenced by outside forces. This paradox had for many years blocked progress in the understanding of the nature of matter. Now finally the vortex theory of molecules resolved the paradox. Vortices in the aether are soft and have internal structure, and nevertheless, according to Helmholtz, they are individual and indestructible. The only remaining task was to deduce the facts of spectroscopy and chemistry from the laws of interaction of the vortices predicted by the hydrodynamics of a perfect fluid."

bearsbearsand 7y ago

what happened to Maxwells original quaternion based equations. the changes made against his will by lorrentz and others to simplify after his death were an affront to science and have set back humanity a hundred years.

known 7y ago

"Physics is to math what sex is to masturbation" --Feynman

gumby 7y ago

This is fundamentally a sociological (or philosophy of science) essay rather than about why Maxwell's equations are still so hard to understand for the student. In that it's quite interesting but I found the jump to a parallelism between the model of Maxwell's and quantum mechanical models a bit of a stretch.

robochat 7y ago

We've all gotten used to the idea that Electric and Magnetic fields are real but then we learn about the Aharonov-Bohm effect and suddenly, maybe it's the electric potential and magnetic vector potential are what is 'real'. But then, there's all the issue with gauges. I think that the best way of teaching electromagnetism might still be in the future.

creed98 7y ago

What equation links Newton's universe to maxwells? Here you will find your answer to 2nd layer product/square by another ;) Again this is the same guy who created maxwells demon. There are no accidents