# Math development and under-appreciation of Maxwell's Equations

Freeman Dyson expresses the opinion in his 1972 essay titled "Missed Opportunities" that Maxwell's equations could have played a much bigger role, one that is comparable to classical mechanics, in advancing mathematics (and then in turn, advancing physics).

He says mathematicians in the late 19th century failed to grasp the opportunity, and that

"If they had taken Maxwell's equations to heart as Euler took Newton's, they would have discovered, among other things, Einstein's theory of special relativity, the theory of topological groups and their linear representations, and probably large pieces of the theory of hyperbolic differential equations and functional analysis.

A great part of twentieth century physics and mathematics could have been created in the nineteenth century, simply by exploring to the end the mathematical concepts to which Maxwell's equations naturally lead." $$\quad$$(emphasis mine)

Dyson spends quite some paragraphs addressing this line of thought in "Missed Opportunities" (which I totally enjoyed). More discussions can be seen in another article of his titled "Why is Maxwell's Theory So Hard To Understand" where he acknowledges that historically, even among the physicists, "It was regarded as an obscure speculation without much experimental evidence to support it... (quoting Pupin) how few were the physicists who had caught the meaning of the theory, even twenty years after..."

# Here's My question:

Where do researchers stand on this? Have any authors remarked somewhere that Dyson's proposal seems more of a wishful thinking (if not naive) than a plausible suggestion of an alternative history?

(Researchers/authors include: historians, mathematicians, physicists, as well as journalists, pop science writers, social commentators, etc)

Dyson's "Missed Opportunities" is a non-technical essay that is often cited by mathematicians and physicists (it seems). I'm not sure how to dig through the list to find useful information for my inquiry.

This question requires a certain level of expertise in both history and math/phys. Currently within hsm StackExchange I can find only one post that is remotely related.

I'm not sure if will be a good idea to phrase this question sufficiently differently to cross post on physics StackExchange or MathOverflow, asking about the plausibility of Dyson's idea more in terms of the pure logic of math and physics.

Edit:

To put some comments in context, they involve the original title "Did math lag by 40 years due to under appreciation of Maxwell's Equations?", which was problematic thus replaced.

• This is a question of opinion. Mathematicians were occupied with other things, and late 19 century was actually a very fertile period. One cannot say that mathematics "lagged". But mathematicians really have not read Maxwell (and still very few read him). While Einstein could extract special relativity from his study of Maxwell in his student years. Historians unlike Dyson usually do not consider the questions of the type "What would happen if...?" – Alexandre Eremenko Dec 5 '17 at 13:54
• I am afraid the question as is does not fit this SE. "Did math lag by 40 years" and "plausible suggestion of an alternative history" are explicitly opinion-based and hence off-topic, "where do historians (or mathematicians and physicists) stand on this" might be ok but assumes that people widely ponder such questions which I doubt. It might just be Dyson. His idea that one could extract special relativity from Maxwell's equations is at variance with history, as Lorentz-Poincare's alternative interpretation of them shows, expecting group representations in 19th century is equally anachronistic. – Conifold Dec 5 '17 at 19:41
• I understand that you and Dr.Eremeko are telling me that this is most likely not an issue that would have been picked up by someone who takes history inquiries with some degree of seriousness. This musing of Dyson's interests me so I gotta ask, and I won't know this to be "not really a history question" had I not asked. – Lee David Chung Lin Dec 5 '17 at 21:24

Maxwell's equations as they are expressed today, with modern vector notation and not Maxwell's quaternion notation, were first written by Oliver Heaviside in a few papers in the mid- to late 1880s, but it wasn't until 1893 that the first volume of his Electromagnetic Theory appeared (and vol. 2 in 1899 and vol. 3 in 1912), so these delays in better notation could have stalled the mathematical appreciation of Maxwell's equations.

See:

• Whoa I didn't realize the quaternion-vs-vector development (also a missed opportunity in Dyson's essay) was so intertwined with Maxwell's equation! Thanks for the reference. – Lee David Chung Lin Dec 7 '17 at 6:12
• In Maxwell's 1861 paper "On the physical lines of force" no use is made of "quaternion notation" and what later came to be called Maxwell's equations are written component by component (i.e. the divergence is written as a sum of partial derivatives). Maxwell didn't use vector notation at that point, but otherwise his presentation of his equations is quite easy to read for a modern reader. – Dan Fox Sep 26 '18 at 7:26

I think Maxwell's theory was well received by the physicists of the 19th century, see this list of references most of which date from the 1890s.

The time lag of 20 years after the publication of Maxwell's equations in book form is certainly not longer than the time lag between Newton and Euler.

It is true, however, that there is only one mathematician on this list, Poincare, whose treatise of Maxwell's theory leaves something to be desired, because, oddly for a mathematician, "mathematics and abstruse reasoning are avoided".

• The reference to the encyclopedia is great. Thanks. I guess in this instance Poincare was in more of a physicist mentality, as he wanted the readers (and himself) to be convinced that Maxwell's theory is reasonable. Very interesting. – Lee David Chung Lin Dec 5 '17 at 18:16