While Einstein was extremely good at math, he didn't invent the math that he used. For example tensor calculus was invented by Tullio Levi-Civita. Einstein learned it from Levi-Civita's textbook, and then Einstein and Levi-Civita had a lengthy correspondence in which a number of Einstein's misconceptions were cleared up.

But including Feynman in that list is an egregious joke. Richard Feynman was one of the top mathematical talents to grace physics. He was a lightning fast calculator. I know of two math competitions that he was in (one from NYU, the other the Putnam). He won both by wide margins. He was the only student ever to get a perfect score on both the math and physics sections of the Princeton graduate entrance exams. He invented a whole series of mathematical techniques, Feynman diagrams being only the most famous.

Einstein was merely extremely good at math. Feynman was phenomenal. Neither is a good example of physicists not needing to be good at math.

Yes, I didn't even get to Feynman example. Feynman was a certifiable genius and asking whether he was good or bad at math is "not even wrong".

Einstein was an exceptionally strong mathematician. He mastered integral and differential calculus by age 15, and gained ample knowledge in more advanced topics (topology, geometry, etc.) in the following years. Note that this was much more impressive back then than it is today. That being said, he did also command impressive intuition and observational skills.

By the way, the exact same can be said for Feynman, with the exception that Feynman was a stronger mathematician yet.

why do you say this was more impressive back then than it is today? I thought all the areas you mentioned were well-understood. if anything I'd think this kind of math is more daunting today because it just takes 1 click to get to a wall of text on wikipedia in impenetrable notation. I'd think back then any kid who picked up a text could just read it without potentially diving into 100 years of abstract math.

A normal school curriculum back then did not go so far as to teach calculus. When Einstein taught himself vector calculus, he used the textbook written by the guy who invented it because it was the only textbook. Then he personally wrote to the guy to clear up his misconceptions.

Today, you can learn this material in open university courses, and if you have a question, consult a plethora of freely available resources on the internet.

Accessibility of information matters.

But that's not what the article says at all! Here's the quote:

even the greatest theoretical physicists of the twentieth century including Einstein, Fermi, Feynman and Bohr were really known for their physical intuition than for formidable mathematical prowess. Einstein's strength was to imagine thought experiments, Fermi's was to do rough back-of-the-envelope calculations. So while mathematics is definitely key to making advances in fields like particle physics, even in those fields what really matters is the ability to imagine physical phenomena and make sense of them. The history of physics presents very few examples – Paul Dirac’s work in quantum mechanics and Hermann Weyl's work in group theory come to mind – where mathematical beauty and ability alone served to bring about important scientific progress.

There is nothing there that attempts to imply in any way at all that Einstein was bad at math. Infact, it matches your characterization of him ("Einstein was indeed a very good mathematician" ~= "mathematics is definitely key to making advances in fields like particle physics") quite well.

I think what it is trying to say is that it is quite rare for a break through in Physics to require new maths. While we can all think of exceptions (and indeed the article mentions them) I think it's a reasonable point to make.

I actually think this is a pretty good article, and attempts to make subtle points. Dismissing it like this by mischaracterizing what it is saying does it a dis-service IMHO.

> So while mathematics is definitely key to making advances in fields like particle physics, even in those fields what really matters is the ability to imagine physical phenomena and make sense of them.

Isn't that really what that paragraph is getting at? Modern physics is not intuitive so it doesn't matter how hard you try to imagine space-time or wave functions there is nothing to grab onto without the mathematics. In fact being imaginative in those fields means coming up with new mathematical theories and theorems. That is why I'm dismissive. The article is not making a subtle point and it is doing a disservice by presenting "imagining physical phenomena" as anything other than intuition guided by mathematics. Especially so when it comes to relativity and quantum mechanics.

So the answer is affirmatively yes. You do need very deep mathematical understanding to make strides like Einstein and Feynman and can not just "imagine" your way to the answer without having a proper grasp of the mathematical foundations.

I also don't know what new mathematics means. The interplay between modern physics and mathematics is very strong and Ed Witten and people like him constantly make new connection within mathematical fields that mathematicians did not think of making. Is that considered new mathematics? I think yes.

My interpretation might be wrong but then article isn't making much of a point.

Nah, advanced physics still requires a strong physical intuition. I'll wager that coming up with novel and interesting solutions to physics like amplituhedron [1] requires a large dose of intuition. It also requires formidable mathematical chops, but not nearly as much as similar methods in string theory and similar require. Basically there's a base level of "good mathematics" which helps with physics, after which it's handy but not necessary. Sort of like IQ vs income, where greater IQ tends to mean greater income up to IQ's of about 120-130. Afterwords it's a wash.

What the OP means by saying that Einstein (or Feynman) didn't excel at mathematics mean they didn't excel vis-à-vis noted mathematicians or other physicists at raw mathematical formalism. One example of this would be the different approaches to study quantum electrodynamics by Schwinger and Feynman [2]. There's a funny version Feynman recounts in one of his memoirs, but essentially he developed (Feynman) diagrams to calculate particle probabilities while Schwinger preferred (formal) algebras.

PS I'd recommend reading "Sure you're joking, Mr Feynman!" if you want anecdotal stories about physics and insight in Feynman's famed "intuition".

[1]: https://www.quantamagazine.org/20130917-a-jewel-at-the-heart... [2]: https://en.wikipedia.org/wiki/Julian_Schwinger#Schwinger_and...

> Nah, advanced physics still requires a strong physical intuition.

True, but without mathematics, physical intuition can become a dead end, because it doesn't lead to theories expressed clearly enough to forge consensus among practitioners. Consider Nicola Tesla, who had plenty of physical intuition, but whose record in science is sparse because he didn't express his ideas mathematically to a degree sufficient to built lasting theories.

As to the present, string theory (granted its limited testability) is built entirely on novel mathematics, and physical intuition has a limited role.

Also, it can be argued that a physical intuition guided by mathematical concepts is more productive than the raw, unvarnished kind that Tesla showed.

I interpret that paragraph as being able to intuitively understand the relationship between the maths and the physics.

By 'new' math I mean development of new mathematical tool and techniques to deal with new physics theory. Generally physicists are applying existing techniques to new problems, which is why that understanding of the relationship between maths and your theory is important.

"I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot" - Einstein to Levi-Civita

He could grok the math, which was difficult and very new at the time, because he had the best possible teachers.

Einstein "physical intuition" is just an euphemism for lack of references and citations in papers of Einstein.

https://en.wikipedia.org/wiki/Relativity_priority_dispute

Special relativity is pretty much math-free though (in Einstein's formulation, not in Minkowski's).

I think Einstein himself wrote that to move to general relativity he had to go to great lengths in studing fields of mathematics he didn't know - not least because the were just being discovered at the time.

This lends credence to the idea that is genius was foremost guided by physical (I would say also philosophical) intuition, though of course you can' t discount the role of math in GE.

> Special relativity is pretty much math-free though (in Einstein's formulation, not in Minkowski's).

True, but only because Einstein hadn't yet acquired the essential mathematical background. Instructive in this connection is a key equation from his first relativity paper (https://www.fourmilab.ch/etexts/einstein/specrel/www/):

β = 1/sqrt(1-v^2/c^2))

(sorry -- the original paper doesn't number its equations)

It should seem obvious that this equation describes a relation between orthogonal dimensions, something that Einstein didn't realize, but that Minkowski certainly did. Minkowski expanded and clarified Einstein's mathematics in order to express time as a separate dimension, then made his famous remark about spacetime: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality".

Einstein then remarked, "Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore." But in subsequent years Einstein learned much more mathematics, and the general theory is much more dependent on mathematical ideas than the special theory was, indeed the general theory can't be meaningfully expressed without advanced mathematics.

I think you are a little hasty in assuming that the author had the popular misconception that Einstein was bad at math. Certainly, Einstein was much better at math than the average man, but among his fellow physicists, not to mention the handful of physicists as celebrated as he is, his mathematical skills pales in comparison.

In other words, his contributions to science arose from his exceptional physical insights, and not his mathematical brilliance.

> In other words, his contributions to science arose from his exceptional physical insights, and not his mathematical brilliance.

True about special relativity, certainly false about general relativity. Between 1905 and 1915, Einstein learned the mathematics required to formulate and write the general theory.

Einstein was hardly the best mathematician at the time, simply because there were people who spent as much time on theoretical math as he did on physics.

The parent poster claimed that one of Einstein's strengths was his mathematical prowess. Whether or not Einstein was among the best mathematicians of his day is irrelevant.