
Why is Maxwell's theory so hard to understand? (2007) [pdf] - fanf2
http://www.damtp.cam.ac.uk/user/tong/em/dyson.pdf
======
ncmncm
It is unfortunate that Dyson neglects the role of Oliver Heaviside, again.
This is in a long tradition of English neglect, ultimately traceable to
Heaviside's status as a commoner. Heaviside invented the mathematical tools we
still use to understand and teach Maxwell, and most of the important
consequences of the theory, but Pupin, Hertz, Marconi, and deForest used his
methods and <del>took</del> got the credit.

Today Heaviside's method is taught as Laplace transforms, with Heaviside's
name scrubbed off. We only hear of him as an alternative name for the step
function, the integral of the Dirac impulse function, and of the "Heaviside
layer", the ionosphere that makes transcontinental radio actually possible,
but we would have waited decades longer without him.

An excellent reference for the importance of Heaviside in the ultimate success
of application if Maxwell's theory is Paul J. Nahin, "Oliver Heaviside: The
Life, Work, and Times of an Electrical Genius of the Victorian Age",
[https://www.amazon.com/Oliver-Heaviside-Electrical-Genius-
Vi...](https://www.amazon.com/Oliver-Heaviside-Electrical-Genius-
Victorian/dp/0801869099/ref=mp_s_a_1_1?ie=UTF8&qid=1546801611&sr=8-1&pi=AC_SX236_SY340_QL65&keywords=nahin+heaviside)

~~~
iorrus
Exactly, one of the main reasons why maxwell is so hard to understand is that
everything is expressed using quarternions unlike Heaviside who expressed the
equations using the vector notation we see them expressed in today. In reality
‘Maxwell’s’ equations are in fact Heaviside’s.

~~~
jacobolus
Arguably the reason that generations of STEM students have been horribly
confused about 3-dimensional vectors and rotations (including
electric/magnetic fields), etc. is that they were reframed in the confused and
non-generalizable Gibbs/Heaviside language, instead of in Grassmann/Clifford’s
formalism in which vectors and bivectors can be properly described as separate
types of objects.

It can be so much nicer.
[http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf](http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)

~~~
MaxBarraclough
Reminds me, I've been meaning to read this [0] blog post for a while now. (See
also the HN discussion [1].) My clueless intuition tells me its points may be
analogous to your linked document. (It mentions Heaviside, at least.)

[0] [https://www.gamedev.net/articles/programming/math-and-
physic...](https://www.gamedev.net/articles/programming/math-and-physics/do-
we-really-need-quaternions-r1199/)

[1]
[https://news.ycombinator.com/item?id=18365433](https://news.ycombinator.com/item?id=18365433)

~~~
jacobolus
Perhaps start with [https://www.shapeoperator.com/2016/12/12/sunset-
geometry/](https://www.shapeoperator.com/2016/12/12/sunset-geometry/) for a
concrete example.

~~~
MaxBarraclough
Looks good, thanks.

------
wmnwmn
E&M is far from "simple". It contains special relativity, for starters. Also
it is incompatible with thermodynamics: solving this problem is why Planck
invented quantum mechanics. Also point charges have infinite energy: this
problem leads to renormalization theory. Also it introduces gauge invariance,
an essential but complex part of all modern theories. And lastly, the
mathematics of E&M is a big step up from Newtonian theory.

~~~
nonbel
>"E&M is far from "simple". It contains special relativity, for starters."

A theory (which is just a set of assumed first principles and rules of logic)
can be simple but allow you to deduce vast complexity from it. In fact, it is
ideal for a theory to be as simple as possible.

The "game of life" is not really a theory, but it demonstrates that simple
rules can lead to surprising complexity:
[https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life](https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life)

EDIT:

I can't begin to imagine why this would trigger a downvote. Are there people
out there who prefer complex theories that make things more difficult to
understand than necessary? Perhaps because it makes them feel smart or
something, I don't know.

But that is basically saying you dislike science, because the point of science
is to synthesize information into a small set of simple "laws" that allow us
to deduce accurate and precise predictions.

~~~
aj7
I reflexively downvote the game of life. I have never heard of a single useful
application or analogy of “finite automata.” Yes, there are speculations about
extremely small Planck lengths, by no new physics has panned out. Finally I am
sad that one of the greatest computer scientists and polymaths, Ed Fredkin,
got sucked into this. We all have weaknesses.

~~~
ckocagil
Surely you mean cellular automata, not finite automata.

~~~
aj7
Yes, you are right. My ignorance shows, but I am still hanging on to my
opinion of cellular automata.

------
gumby
I am struck by the clarity and simplicity of Dyson's writing.

As to the substance: yes, hiding your work under a bushel does nobody any
favor, but it's understandable when it feels like the alternative is the
strident trumpeting of trivial "advancements".

But also cautionary is the constraints due to the wrong metaphors. We often
see this today in linear predictions as to the impact of some new idea. It's
easy to mock Ballmer when he laughed at the iphone back in 2007 (back when
this essay was written). But his comment made sense _for his customer base_.
He (and nor could Apple) couldn't really see that the game was changing.

Dyson points out that Maxwell himself had the same problem.

~~~
hopler
Why would you say Apple, whose main business was an extensive traxk record in
creating, promoting, and selling next generation technology, that their major
investment in a new field was blind speculation?

In contrast to Microsoft whose business was providing stable solutions and
extracting monopoly rents.

~~~
gumby
Not sure I understand your point. Mine was that in science or business we are
often held back from understanding by the limitations of our models, even when
we ourselves are the inventors.

I thought I used an example well known to HN to make the point. Apple’s stated
hope was to eventually have 1% of the market (and they didn’t originally plan
for an App Store). Like Maxwell, _they_ didn’t know they had a blockbuster.

Microsoft listened very closely to their customers, who were and are mainly
corporate I/T and something like the iphone wasn’t something they said they
wanted. In fact it honestly had no affordances that IT wanted.

Maxwell understood the math but his own mechanical metaphor distracted him,
and other physicists too, so that they couldn’t really see the larger
implications.

~~~
icebraining
_an example well known to HN_

I think the issue is that example doesn't seem to be true.. Jobs said at the
time of launch that it was a revolutionary and magical product which was five
years ahead of everyone else and would change everything. And the hope wasn't
to have 1% eventually, but by the end of 2008.

------
k9s9
Interesting. And makes me wonder what else we are missing and
misunderstanding, as we waste so much time trying to express the complexity of
the world in spoken language rather than math. Not just physics but in
sociology, psychology, economics etc.

Faraday purely from experiments 30-40 years before Maxwell, intuitively
understood electromagnetism. Nobody serious believed it because he didn't have
the training to express it mathematically. Dyson is saying it took Physicists
another 20-30 years post Maxwell to get it. So basically ~70 years wasted.

~~~
tim333
Spoken language has a lot going for it. I note the featured article and it's
discussion are all words not maths.

By the way looking at Maxwell's paper it seems awful complicated
[https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1865...](https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1865.0008)

compared to the modern form of the equations
[https://ethw.org/w/images/d/d6/Maxwell_image_02.jpg](https://ethw.org/w/images/d/d6/Maxwell_image_02.jpg)

~~~
ncmncm
The modern form of the equations was worked out by commoners, so had to be
filtered through the work of others before it could be taught.

~~~
scabarott
You're really caught up on this commoner/non-commoner dynamic (from your other
comments). Was it really that important? - many of the well known scientists
of that time were 'commoners'.

~~~
ncmncm
I'm not, but they were. But it was not discussed openly in print at the time,
and so must be inferred.

~~~
tim333
I just read his Wikipedia and he was sponsored by his uncle Sir Charles
Wheatstone of Wheatstone bridge fame and ended up a Fellow of the Royal
Society so was not so badly situated.

------
killjoywashere
It's not until now, over 20 years later, that I realize what a magician my E&M
prof was. After months of careful development of basic E&M, Maxwell's
equations simply fell out. As I recall, it was capacitance that was the
impetus. It was as though we had struggled in a dark tunnel for months and
suddenly came around a turn and there was light. Not a little, but sheets of
daylight and all the equations just fell together, one from the next. All the
equations revealed themselves from the middle of one lecture to the middle of
the next lecture.

And then, in E&M 2, with a different professor, I spent the entire semester
coming to grips with the fact that the implications of these formulas would
challenge my grasp on reality for the rest of my life.

~~~
amatthew
In what ways were they that important later in life?

~~~
killjoywashere
Because my undergrad is in physics and I'm still in science, E&M opportunities
are evident to me, but I know I would struggle to really go after them myself.
So I get help, but I sure wish I could do it on my own.

------
amelius
Some people hold the view that a better way of teaching electromagnetism is
through geometric algebra. See:
[https://arxiv.org/abs/1010.4947](https://arxiv.org/abs/1010.4947)

~~~
Koshkin
Also, the approach based on differential forms is very appealing; for a list
of various formulations of electromagnetism, see
[https://en.m.wikipedia.org/wiki/Mathematical_descriptions_of...](https://en.m.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field)

~~~
19f191ty
Aren't the two very closely related, if not equivalent? Differential forms and
Geometric Algebra I mean.

~~~
jules
Yes, they operate on the same objects. The main operations on differential
forms are the exterior derivative d and the wedge product f /\ g. These are
independent of the metric. If you add a metric you get one extra operation
called the Hodge star ⋆, which is a prefix operator, so it operates as ⋆f.

Alternatively, if you have a metric on your space you can define the geometric
algebra derivative D and the geometric product. These are closely related to
the differential forms operations, e.g. Df = df + ⋆d⋆f. Using it, Maxwell's
equations become one equation, namely DF = J where F is the electromagnetic
field tensor and J is the four current.

So in my opinion it's indeed best to not put this as differential forms vs
geometric algebra, but rather as a single theory in which some operations
don't depend on the metric (d, /\\) and some do (⋆, D, geometric product).

Unfortunately, geometric algebra has been overhyped by its proponents, and the
papers they write are less than rigorous, so some people are under the
impression that it's crackpottery, so it's safer to call it Clifford algebra
:)

~~~
earthicus
Is it possible to work with geometric algebra effectively without a metric
using some more complicated construction?

~~~
jacobolus
Sure. What model to use depends on you are trying to model.

[http://geocalc.clas.asu.edu/pdf-
preAdobe8/PGwithCA.pdf](http://geocalc.clas.asu.edu/pdf-
preAdobe8/PGwithCA.pdf)

[http://geocalc.clas.asu.edu/pdf-
preAdobe8/DLAandG.pdf](http://geocalc.clas.asu.edu/pdf-preAdobe8/DLAandG.pdf)

etc.

------
bsder
People didn't really get Maxwell because Maxwell's equations don't need a
medium aka "aether".

Waves propagating without a medium is a shocking message in this time frame.

Maxwell published his equations in 1861/1862 in first form.

In 1887, the Michelson–Morley experiment gave the first experimental disproof
of the "aether".

It also doesn't help that you have things like "displacement current" and have
to find strange integration surfaces to make Maxwell's equations work out for
things like capacitors or motors. (The Heaviside-Hertz pedagogy that everybody
is _STILL_ taught is particularly problematic to use for motors.)

~~~
NotAnEconomist
> In 1887, the Michelson–Morley experiment gave the first experimental
> disproof of the "aether".

In what way is LIGO not a bigger, and successful, version of the Michelson-
Morley experiment?

I'm honestly asking, because from my limited understanding, it seems to be.

From Wiki:

> This result is generally considered to be the first strong evidence against
> the then-prevalent aether theory, and initiated a line of research that
> eventually led to special relativity, which rules out a stationary aether.

It seems that LIGO merely confirmed properties of an existent aether, not that
we no longer believe in one:

> Aether theories (also known as ether theories) in physics propose the
> existence of a medium, the aether (also spelled ether, from the Greek word
> (αἰθήρ), meaning "upper air" or "pure, fresh air"[1]), a space-filling
> substance or field, thought to be necessary as a transmission medium for the
> propagation of electromagnetic or gravitational forces.

Naively, spacetime and quantum fields are both forms of aether theories.

~~~
bsder
If LIGO confirmed "aether", then the measurements in the perpendicular arms
would vary depending upon time (orientation of Earth in rotation, orientation
of Earth around Sun, etc.).

As far as I know, they very much do _NOT_ vary--to an absolutely _amazing_
precision.

As far as I can tell, if the "aether" existed and we could detect it, we
basically would have no hope of detecting gravitational waves.

> Naively, spacetime and quantum fields are both forms of aether theories.

Not ... really.

The existence of "aether" implies a preferential frame of reference. And every
experiment we have done to attempt to establish such has failed.

~~~
NotAnEconomist
But the arms are predicted to vary in those cases by the same theory that LIGO
is confirming: our rotation causes frame dragging, and there's some weird Sun-
Earth orbital relativistic effects as well, right?

I know the Sun-Jupiter orbit produces a large portion of the estimated 5,000
watts of gravitational emissions given off by our solar system. My mental
model of this is that it perturbs the aether ("spacetime") through which it
travels, radiating waves which carry energy out of the system. Is there a
different one?

It's just the power output in those waves is incredibly, incredibly low --
LIGO can only hear much larger events, such as large astral bodies colliding.

So I'm not sure I understand -- there seems to be at least a gravitational
aether.

~~~
axilmar
By aether, do you mean an underlying medium that allows forces to be
propagated? I guess that using that definition, there is an 'aether'.

The difference with the rejected concept of aether is that this "aether" is
deformable by gravity, whereas the rejected one is not.

~~~
raattgift
> underlying medium that allows forces to be propagated

General Relativity is a local theory concerned with the mechanisms that
generate the metric, the geodesics implied by the metric, and the coupling of
objects to those geodesics.

The relevant forces are those which accelerate objects into non-geodesic
motion (or boost them from one geodesic to another). Those are local[1] as
well: electromagnetism and the nuclear interactions. There's nothing like a
luminiferous aether or underlying medium in the Standard Model, even if you
look funnily at the gauge bosons -- they obey Lorentz covariance.

I made a couple of sibling comments to yours, one of which deals directly with
your comment's parent's idea of a gravitational aether.

\- --

[1] RM Wald, _Quantum Field Theory on Curved Spacetime and Black Hole
Thermodynamics_ (University of Chicago Press, 1994). Cf. the top of page 6 of
Hollands & Wald 2014,
[https://arxiv.org/abs/1401.2026](https://arxiv.org/abs/1401.2026)

~~~
NotAnEconomist
Isn't every "field" in QFT basically an aether?

~~~
raattgift
Please tell me what you mean by aether, as rigorously as you can, and then I
can give you an answer to the question which seems to be bedevilling you.

Meanwhile I can guess at what you're asking:

The Standard Model ( _a_ QFT) has interacting quantum fields obeying purely
local dynamics. The behaviour "here-and-now" depends on the field-values
"here". It does not depend on field-values "now" but far from "here".
Moreover, the Standard Model is Lorentz-invariant, meaning its laws hold in
any inertial frame of reference, thus the scare quotes in the previous
sentence.

The luminiferous aether that Michelson & Morley were looking to measure was
motivated by finding a single special (and universal, or at least covering the
whole solar system) inertial frame of reference in which Maxwell's equations
hold exactly. No such frame exists; there is a huge democracy of inertial
frames in which relativistic electrodynamics is exact in the classical limit.
(
[https://en.wikipedia.org/wiki/Preferred_frame](https://en.wikipedia.org/wiki/Preferred_frame)
)

------
paulpauper
Maxwell did not use vector calculus, which actually makes it pisser to
understand because it bypasses having to convert the vectors to coordinates. I
find short hand notion more confusing than writing it all out even if the
latter takes more room. I did not understand general relativity at all until I
saw an example where everything was written out and then it made much more
sense

~~~
earthicus
This is something of a myth. Maxwell actually uses _both_ to ease the
mathematical pain for his readers - he writes the equations out coordinate by
coordinate, and he also compresses them using Hamilton's (quaternion) model of
vectors. In fact Maxwell was the one who introduced the gradient, curl, and
divergence (actually he used convergence) operators, and he addressed
precisely your concern in his book by giving both formulations!

~~~
tntn
Quaternions are not a "model of vectors."

Maxwell used per coordinate equations and quaternion equations, but never
wrote out what are now known as "Maxwell's equations," which were first
formulated by heaviside using Gibbs-Heaviside vector calculus.

Gibbs-Heaviside vector calculus formulations proved far more valuable to
engineers and scientists than the equivalent quaternion formulations - despite
the criticisms from Hamilton and Tait.

~~~
srean
> Quaternions are not a "model of vectors.

I dont think I agree. Yes they are not the Gibbsian vector but they are a
pretty damn good model of what we want to model as vectors. Its already a
little closer to geometric algebra / calculus than Gibbsian vectors. Its a
pity that Grassman and Clifford's work gained attention much later.

Once javascript is out of the bag one cant do much about it. Thats how I feel
about Gibbsian vector calculus and geometric algebra. Not that javascript or
Gibbsian vectors are bad, quite the contrary, they are astonishing in their
own right and scope, but one still feels they could have been so much better.

~~~
tntn
I think geometric algebra is a very cool and elegant approach as well, but it
has not yet proved useful or valuable to working engineers.

If we want to compare to programming languages, perhaps Haskell vs. C. One is
beautiful and elegant, the other is down to earth and conceptually simple for
"normal" people. One is primarily used by academics and researchers, the other
is used to build the world. Both are admirable.

~~~
srean
> has not yet proved useful or valuable

They would have, had they been the "first mover". Hence my Javascript analogy

Had a better language been released at that time it would have been just as
useful and valuable. In fact much more so. But as is usual, a
"worse_is_better" first mover accrues so much head start in the mind share and
in the tooling / literature that it becomes impractical to switch, especially
when both can express the same set of things (just that one does it more
concisely than the other). Another shallower example: imperial vs metric
units.

I dont think its like Haskel::C at all. Haskell is way more mathy and has a
steeper learning curve than C. One who can master curl divergence and vector
stoke's theorem can master geometric algebra with less effort

------
mikorym
I have a copy of Maxwell's two volumes on electromagnetism and what makes it
difficult to read for me is that his "equations" are actually a long list of
disparate equations that describe a unified theory. It's not a simple "answer"
such as E=mc^2, it is a series of such descriptor answers.

Mathematically, it is easier for me to focus on one aspect and study that
aspect's equations (at a time).

------
UncleSlacky
"The phrase “Another theory of electricity which I prefer” seems deliberately
intended to obscure the fact that this was his own theory."

To me, this just sounds like typical British understatement, which I'm sure
the audience at the time would have understood.

------
naringas
>We still have passionate arguments between believers in various
interpretations of quantum mechanics, the Copenhagen interpretation, the many-
worlds interpretation, the decoherence interpretation, the hidden-variables
interpretation, and many others.

lukcily there has been slight progress since 2007
[https://www.quantamagazine.org/frauchiger-renner-paradox-
cla...](https://www.quantamagazine.org/frauchiger-renner-paradox-clarifies-
where-our-views-of-reality-go-wrong-20181203/)

~~~
almaya
It seems that this arguably new paradox has some issues:
[https://www.scottaaronson.com/blog/?p=3975](https://www.scottaaronson.com/blog/?p=3975)

------
weedwarrior
The theory is very easy to understand. Its the differential notation and
manipulation of said notation that is challenging for people who dont math
often

------
aj7
Say what? Start with the integral representations of the equations a la
Halliday and Resnick. You can do tons of useful problems. Once you understand
vector calculus, you will understand the equations as written in differential
form. Then, pick up a copy of Purcell to see how magnetism comes out of the
Lorentz transformation of electrostatics. If you are mainly interested in
applied problems, go through Corson and Lorraine. Finally: 1. Realize that the
treatment of ferromagnetism is much too limited in most E&M texts. 2. Optics
is also worthless in those books. It is a field unto itself, with theoretical,
applied, and quantum parts all handled by different books. 3. You don’t use
Feynman’s Lectures to learn anything the first time. You use them to see if
you understand physics as well as Feynman (you don’t.). All the problems have
to have been done BEFORE this step.

~~~
twtw
You seem to be responding to the title, not the essay.

Dyson describes the theory as "simple and intelligible" once you accept the
concept of fields, which was very much foreign in Maxwell's time.

~~~
aj7
You’re right. I’m fairly embarrassed.

