Over at Cosmic Variance, Julianne (not JoAnne, as I originally typed) has a very nice post about the cult of genius in physics, and its relationship to research on the problems caused by excessive praise. Doug Natelson also has some comments. There's some fascinating stuff in the articles about praise, with some likely relevance to once and future education arguments, but I need to think about it more before I comment.

My reason for posting, though, is found in the comments. Specifically, comment #10, by "mclaren", which reads in part:

Sure, I've got a degree in physics, but basically I'm too stupid to do anything useful. Linear homoegeon partial differential equations I can handle, conformal mapping and so forth, but that's kiddy math. It's trivial. That's not real math. That undergrad stuff. The vast majority of the human race, including me, can't do _real_ math. Case in point: I took a course on algebraic topology in college...

If you want to find a pernicious meme that's a blight on the face of physics, it's not the over-idolization of "genius," exemplified by Einstein and Feynman, it's this crap. Too many people approach physics as if there's some sort of Great Chain of Being, with the most abstract theoretical particle physics at the very top and low-energy experimentalists down at the bottom, just above biologists and rude beasts incapable of speech.

This drives me right up the wall.

There's no inherent moral worth to working on more "fundamental" and mathematical physics. A lack of familiarity with algebraic topology is not a defect in character, or a sign of gross stupidity. Low-energy physics is **different** than high-energy theory, but not inferior to it.

Lots of people have responded to mclaren's comment, mostly with "Don't feel bad, lots of other people are too dumb to be high-energy theorists, too," and while I'm sure that's some comfort, it misses the real point. It's still buying into the myth that high-energy particle theory is the highest possible calling. This is as poisonous for the profession as the idea that, as Doug puts is, "if you get a physics PhD but don't end up a full professor at Harvard, you're a plodder."

In fact, I have a much higher regard for phenomenologists than "pure" theorists, and an even higher regard for experimentalists. They don't get to set inconvenient constants equal to one, or approximate away all the hard problems, but instead have to deal with the world as it really is. In my opinion, there's a lot more ingenuity involved in figuring out clever ways to measure the dependence of gravity at very short length scales, or electric dipole moments at the 10^{-30} level than there is in figuring out yet another of the 10^{500} possible ways of wrapping hypothetical extra dimensions around hypothetical warped manifolds in whatever the toy model of the moment is.

And yet there's this persistent myth that the highest form of achievement in physics is whatever is farthest removed from reality. Students learn to idolize people who noodle around with abstract math, while the people who do the genuinely difficult work of connecting abstract math with experimentally measurable reality are dismissed as mere tinkerers who couldn't cut it in the world of high-energy theory.

The worst part of this is that it's not even particularly consistent with the "cult of genius." The thing about Feynman that was really remarkable was not his facility with abstract math, but rather his physical intuition. The people who are justly hailed as among the very best physicists of the modern era-- Feynman and Einstein, as well as people like Bohr and Fermi-- had a gift for making abstract physics concrete, not for doing algebraic topology.

Einstein's "miraculous year" of 1905 wasn't founded on great mathematical leaps-- Lorentz and FitzGerald had come up with time dilation and length contraction earlier. Einstein's great accomplishment was more in making relativity obvious and acceptable, in coming up with a clear and concrete physical picture that showed **why** the bizarre consequences of relativity are correct, and moreover **have** to be correct. Feynman's theory of QED doesn't have the mathematical elegance or rigour of Schwinger's theory, but Feynman is better known because he provided an intuitive way of thinking about the theory that makes it useful. And it's worth noting that the pinnacle of Feynman mythology is an experiment, not a theoretical development-- his O-ring demonstration at the Challenger hearings.

It's absolutely crazy for mclaren to be beating himself up for not knowing algebraic topology, and yet, that's what the cult of theory does-- it rates the most abstract and incomprehensible material above the concrete and practical, in complete defiance of reality. It's absolutely maddening.

(Even if you restrict the discussion to the theory side, I don't buy it. People like John Bell, or Peter Shor, or Woijciech Zurek are (or were, in the case of Bell) scary smart, and don't need algebraic topology to do fundamental theoretical physics. And I'd match their accomplishments against anything Ed Witten ever dreamed up.)

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People like the Great Romance of the Gentleman Physicist, I think. The reality of people working hard and picking up things as they go along, getting really into a problem then solving it and afterwards realising that it was, in fact, pretty obvious (like most things are when you know how to solve them), that's not very glamourous. If someone like an Einstein or a Feynman has panache and can explain the Big Ideas with a certain style, that's not hurting anyone.

Not that they weren't exceptionally clever people, mind you. Feynman, in particular, though, is worshipped in large part for things other than his (exceptional) body of research work.

Also, along with Shor, Zurek, Bell, I'd put Reinhard Werner. He is also extraordinarily clever (and fits in with that crowd, particularly Bell, quite nicely) but I am not sure to what extent he is known outside of the QIT/Foundations field.

Jeez. All this ragging on algebraic topology. I don't suppose it's worthwhile to point out that its use in the study of defects is just as important at low energies as at high energies? Math is used in physics because it makes things

easieralthough it might not seem so from the initial investment. Most people aren't particularly interested in seeing abstraction thrown around if it doesn't actually let you learn something new.And the O-ring thing? Not Feynman's idea.

Even as a theorist, I'm often hacked off at this 'cult of theory' because of the implication that some people 'have it' (i.e. are super geniuses who find everything easy) and some people don't. This annoys me for two reasons:

1. It indirectly denigrates those very smart people like Einstein and Feynman and Newton who worked very, very hard on their studies. It is undeniable that they were smart, but they also spent years thinking hard about their problems (Einstein spent a decade on GR). It's as silly as people who emphasize that Michael Jordan is a great natural athlete (true), but who seem to gloss over the fact that he wouldn't have gotten far in the NBA if he hadn't worked his rear off every day at practice.

2. People often seem to use this as an excuse for not challenging themselves intellectually. "Some people have a knack for math - I don't" is a great excuse for not trying at all. Again, that isn't to say that people don't have different aptitude levels, but in my experience most people don't bother to put in the effort to learn a difficult topic at all. In other words, there are natural boundaries, but most people seem to give up long before they hit it.

"The people who are justly hailed as among the very best physicists of the modern era-- Feynman and Einstein, as well as people like Bohr and Fermi-- had a gift for making abstract physics concrete, not for doing algebraic topology."

I fully agree with this view. The most amazing thing about Einstein's work is that it is accessible for the most part to anyone with some basic physical concepts and algebraic skills. When I teach relativity to undergraduates, I encourage them to go read Einstein's original papers on relativity, because they already have everything they need to understand them - even his first paper on GR is comprehensible!

They don't get to set inconvenient constants equal to one, or approximate away all the hard problems, but instead have to deal with the world as it really is.Heh. I would think that what one wants from a theorist is a decent ability to strip away as much complexity as possible (but no more than that), while successfully modeling real phenomena.

I have pretty much equal regard for great theorists and great experimentalists. My previous field of ultrafast dynamics in liquids would have been vastly poorer for the lack of either.

(It may be worth noting that my background is Chemistry, not Physics. Where experimental physicists are generally expected to be (and are!) pretty decent theoreticians as well, historically there has been a much stronger distinction between theorists and experimentalists in physical chemistry.)

I agree with Chad Orzel. I intentionally add no Feynman anecdotes here.

There is extreme prejudice that I've seen against professors of sciences at liberal arts colleges and community colleges, who are treated with a certain knee-jerk contempt in comparison with professors at major research universities. So far as I know, only one Nobel prize in a science ever went to a community college teacher. Sorry, don't have a citation here.

I also see the prejudice that Chad expresses so well here about a Great Chain of Being in Math and Science (great analogy, by the way).

Partly as a side-effect of my having been a teacher (details omitted), I mostly focus on doing Math and Physics which is:

(1) Nontrivial;

(2) rather Elementary (thus accessible to nonscience undergraduates);

(3) Original.

Yet I do not feel inferior to people who do Math and Physics which is:

(1) Nontrivial;

(2) rather Advanced;

(3) Original.

I feel that, per the Scientific Method, if research penetrates the potential barriers of Peer Review, then it is real research. Einstein Envy, and the closely related Anti-Einstein fervor, are, in my opinion, pathologies.

There is confusion when equating certain talents with "smartness". Case in point, an engineer I've worked with on some experiments was super-clever; I can tell him I wanted to "do this" or "build that" and he could immediately see all the problems and solutions, and think up some really ingenious methods. My esteem of him was well above most theoreticians for the reason that most Physicists simply couldn't do what he does.

Likewise, I saw the movie "Word Play" this weekend. You might think that champion crossword solvers would be very smart, erudite, and well-read. You have to be to have such a large vocabulary, right? Not really true, they just are good at pattern recognitions, and the ability to tell that certain letter combinations are probably real words, without even knowing the meaning.

Not 10^500 acceptable string theory vacua, 10^1000 - bumped up a tad for the landscape. Theory can predict anything,

http://www.nature.com/news/2007/070219/full/070219-4.html

"The researchers essentially put such a contraction into their model 'by hand', although Frampton says..."

No theory survives reproducible falsifying observation. We are fools if we do not look at something theoretically absurd that is consistent with prior observation. Theory changes; one observed reality exists.

What we do not know does not hurt us nearly so much as what we know to be true that isn't.

Preach it, brother. Though, to be fair, the good theorists I've known are well aware that experimentalists aren't "failed" theorists. I can't stand the idea of some sort of Great Chain of Intellect where pure mathematicians are at the top, followed by high energy theorists, then the Great Unwashed physicists, then (horrors) chemists and biologists. Parameterizing something as rich as "what it means to be a good scientist" by a single number (IQ? H-parameter?) is demonstrably unwise. There are raw aptitudes, acquired skills, and (physical/chemical/biological) intuition, as well as extrinsic factors like access to resources.

Great post - right on!

Maybe a little off-topic, but here's a great article on failure of reductionism (that drives "Theory of Everything" and "theory is the king" approach) and a number of real problems that exist in condensed matter, materials physics, biology, chemistry and other medium-scale (atomic-macro), low-energy sciences.

Sorry, here's the link to the paper:

The Theory of Everything,

R. B. Laughlin and David Pines

PNAS Vol. 97, Issue 1, 28-31, January 4, 2000

http://www.pnas.org/cgi/content/full/97/1/28

Hear hear. As a mathematician, I guess I should be flattered at algebraic topology being on top... But really, this whole idea of a hierarchy is pernicious and absurd. Math, physics, chemistry, biology - in all of these, we are going up against nature, and they can all be damn hard. But each takes a very different way of thinking. And within each, there is theory and experiment, there are different subfields, all of which take a different way of thinking. And even in each subfield, there are people who make progress by approaching problems in very different ways...

What I've learned, both in teaching and doing research, is that there are as many kinds of "smarts" as there are personalities. This isn't a wishy-washy "you're all smart in your own way" view - watching some very good mathematicians do very bad physics or biology confirms this, I think. Intelligence is not like running, where there really is a fastest, a second fastest, and so on. In fact, far from being linearly ordered, I would argue that it is close to the opposite extreme.

Further, there's really not that much inborn about it, but that's another argument.

I say all this as a not so Scarily Smart Person, who has talked to and worked with a number of SSPs.

One concession I would make: it does seem that it is easier to get by doing crappy work in some fields (*cough* applied math *cough*cough*) than others, due in part to differing cultures and funding levels. But doing good or great work in any science seems to be of equal difficulty, requiring equal (but different) talents.

And not to slight the non-sciences, I just don't know anything about them.

"lack of familiarity with algebraic topology is not a defect in character, or a sign of gross stupidity. Low-energy physics is

differentthan high-energy theory, but not inferior to it...myth that high-energy particle theory is the highest possible calling. This is as poisonous for the profession as the idea that, as Doug puts is, 'if you get a physics PhD but don't end up a full professor at Harvard, you're a plodder.'""In fact, I have a much higher regard for phenomenologists than "pure" theorists, and an even higher regard for experimentalists...lot more ingenuity involved...than there is in figuring out yet another of the 10^500 possible ways of wrapping hypothetical extra dimensions around hypothetical warped manifolds in whatever the toy model of the moment is."

Are these consistent?

How are these not consistent?

Higher value assigned to "practical" science that deals with the real world, as opposed to "abstract" science, which is borderline metaphysics, AFAIAC. In this worldview it is condensed, molecular, atomic and optical physics (in addition to materials/biology/chemistry) and experimental side of it that gets the highest marks in terms of "truthiness", followed closely by theory in the same fields, followed by nuclear, astronomy, high energy, phenomenology/model builders, and finally abstract multi-dimensional mathematicians dealing with overhyped and untestable claims.

The Great Chain of Being idea is made explicit in the following old joke:

Molecular biologists look down on other biologists.

Chemists look down on molecular biologists.

Physicists look down on chemists.

God looks down on physicists.

...and mathematicians look down on God.

Well, it's certainly unusual to follow the claim that it's "poisonous" to think of physicists as being on a hierarchy based on specialization with an assertion of precisely that form.

As it currently is, the post reads like it's wrong to say particle theorists are more ingenious than AMO experimentalists, but perfectly kosher to say the reverse.

Well, I think it's silly to get into pissing contests over IQ and smartness and all that.

Even though I suspect that many theorists would have the slight edge in IQ tests or physics exams or Physics GRE or whatever - but this is only because theory is closely matched to skills required in tests. If you tested people on changing oil or fixing a capuccino machine, I also suspect experimentalists would take an edge.

The point I would make is to look at how much various fields contribute to figuring out the world around us - most of which consists of real things like rocks, air, flowers, etc.

Reductionism is an extremely narrow-minded way of looking at these problems. Yes, Schroedinger equation explains everything. Or does it?

Suppose all interactions are unified today - is this really the end of science (or even physics)? Is this going to explain anything "real" about the world we live in? Do we actually understand anything by reducing everything to a few simple laws?

I find that idolizing theory, especially the "Grand Unified Theory" comes primarily from inability to grow beyond the "Einstein" quest and find your own questions to answer, which requires a lot of creativity.

I also agree with other responses (here and in original thread) that being smart means nothing unless you also can use your brains to do something meaningful. A lot of condensed matter theorists I know can deal with vague generalizations, but often are unable (or unwilling) to pin down a specific problem or issue or perform a calculation.

Personally, I'd have loved to be a concert pianist. But I know that the reason why I'm not is not that I'm too dumb.

Everyone is good at something.

In fact, I have a much higher regard for phenomenologists than "pure" theorists, and an even higher regard for experimentalists.Chad's statement was not specific to AMO physics. The examples were, but that's merely a statement of what's at the forefront of Chad's mind I'd guess.

Various replies, collected into one comment:

Adam:

Also, along with Shor, Zurek, Bell, I'd put Reinhard Werner. He is also extraordinarily clever (and fits in with that crowd, particularly Bell, quite nicely) but I am not sure to what extent he is known outside of the QIT/Foundations field.I don't know his work that well, but I didn't intend that to be an exhaustive list. Those are just the smartest people I could think of off the top of my head this morning.

Aaron:

Jeez. All this ragging on algebraic topology.It's the specific class that mclaren mentioned, and I'm using it as shorthand for higher math in general. And the main point isn't the utility of abstract math, but rather the attitude that holds that unless you're working on the most fundamental and mathematical type of particle physics, you're not a

realphysicist.D:

Well, it's certainly unusual to follow the claim that it's "poisonous" to think of physicists as being on a hierarchy based on specialization with an assertion of precisely that form.The second of those was intended as a statement of personal opinion. I personally am vastly more impressed by, say, Eric Adelberger of the Eot-Wash group than, say, Ed Witten, but I wouldn't claim that as a universal truth, or claim that physics students who can't come up with anything as clever as the modified torsion pendulum they use in Washington should feel like they're second-rate physicists.

As it currently is, the post reads like it's wrong to say particle theorists are more ingenious than AMO experimentalists, but perfectly kosher to say the reverse.Again, I wasn't trying to provide an exhaustive list of physicists whose ingenuity I admire-- it's just what came to mind when I was typing.

Chad: I just put Werner in because he is chief on my list of underappreciated physicists and because he's in the same game as the other guys that you listed. I agree that there are many, many more.

I've always said the smartest guys in our physics building are the machinists.

biologists and rude beastsGrunt. Snuffle. Grunt.

*farts*

*looks around*

Wha'?

:-)

It won't do any good, most likely, to mention that it was never my intent to idolize pure theory. In fact, the trend toward abstraction divorced from observables has arguably wrecked many fields, including music theory, economics (the collapse of Long Term Capital Management is the most spectacular example, along with Rational Choice Theory), pomo litcrit "theory," and string "theory."

That said, it remains a fact that if you want to do serious physics at the level of a research scientist, you had better be freakishly good at math. Not just "kind of good," not just "better than average," but bizarrely and inhumanly capable at math.

The original post over at cosmic variance seemed to suggest that if you work hard enough and long enough and if you show enough grit and determination, you can slog your way through any subject.

In my experience this is just not true.

In many cases outside of physics, yes -- hard work does pay off. In a lot of areas of life, you can at least attain a modicum of competence. Foreign languages (which are primarily a matter of vocabulary), computer programming, history, philosophy, knitting...in many cases just put in enough hard work, and you can get reasonably good at doing something.

However, there are exceptions to the "true grit" notion of getting ahead in life. And genuinely advanced math is one of them. It is my experience that beyond a certain level, 99% of the human race bounces off math. I certainly did. As long as I can visualize a mathematical function, I'm okay. Green's Function is straightforward because it boils down to saying that if you shoot arrows into a watermelon, you'll get as many arrow heads inside the watermelon as you have arrow shafts puncturing the skin of the watermelon. That's obvious and easy to visualize. Parseval's Theorem is likewise sensible, as long as you recall that the power in an electrical circuit = I^2*R. Then it makes sense that the power of a signal is given by the square of the fourier cofficients (of course since they come in real + imaginary varieties, you must multiply the complex conjugates to avoid getting an imaginary number as the output). That stuff makes sense at an intuitive level. But there comes a time in physics when you hit math you can no longer visualize, and at that point, intuition fails. No matter how hard or how long I studied, I could never get anywhere at math when I could not longer visualize it. Never. Get. Anywhere.

Some people _can_ get past that point. I never could. The people who can get past that point make it into graduate school in physics. The rest of the human race, and that amounts to well over 99% of us, find ourselves forever barred from participation. It would be nice if that weren't the case, but here in America, education in the hard sciences is intensely mathematical. Extremely mathematical. Fanatically mathematical. Mathematical to the degree that no matter how superb your intuition about physics, no matter how excellent your grasp of experimental technique...here in America, if you can't do the math, you flunk.

That's life. We can argue over whether it should or shouldn't be that way here in the U.S. of A. Acquaintances have remarked that European graduate-level hard science educaiton tends to be significantly less math-oriented that American science grad schools. I don't know if that's so, but if it is, it bespeaks a difference in culture in how science education works in America vs. Europe. In any case, I don't live in Europe but in the U.S. of A., so it's a moot point.

It remains a fact, as far as I can tell, that if you want to do anything significant in the hard sciences, you had better be superb at math. Not just "okay." You have to be *outstanding.* Mutant-brainiac good at math. Eyebrow-raisingly good.

The reason isn't hard to discern. We have climbed an exponential curve over the last 200 years in terms of knowledge in the sciences. Once upon a time, you could do serious physics knowing nothing more than algebra. Then, by the 1700s, you could do serious physics knowing nothing more than derivatives and integrals. By the mid-19th century, you had to know differential and partial differential equations to do serious physics. But by the mid 20th century, you had to know one hell of a lot more than ODEs and PDEs to make any kind of serious contribution to the hard sciences, as far as I can tell. Even as an experimentalist, you damn well had to be able to read those incomprehensible equations and unscramble 'em just to know what to look for. If you couldn't untangle the scary-difficult math, even as an experimentalist, you were cooked, because you couldn't even figure out where to start.

If I had been born in the 1790s I probably would have been able to do serious physics research because I can handle differential equations. I cannot handle the math beyond that, no matter how hard I work and no matter how often I go back to it. Moreover, it's been my experience that most other people hit the same wall. There's a level of math beyond which, no matter how many times someone else explains it to you, you just can't get it.

Let me give a specific example: Calabi-Yau manifolds. Now, it's clear why we have to deal with manifolds if we're dealing with more than 4 space-time dimensions. A manifold is just a fancy way of saying "n-dimensional surface," so if you're talking about objects (like hypothetical strings) embedded in N-dimensional space, common sense tells you that you will have to talk about manifolds. Likewise, the compactification of Calabi-Yau manifolds is obvious and necessary because our everyday experience tells us that we perceive 3 dimensions of space and 1 dimension of time. Therefore common sense suggests that if there do exist more than 4 space-time dimensions (as string theory posits) then those extra dimensions must be curled up somehow so tightly that they aren't readily observable. "Compactification" is just a fancy way of saying that.

But why Calabi-Yau manifolds? I have no damn idea. Not a clue. Nothing. Nada. Zip. Dick. Diddly. Squat. Bupkiss. When it comes to answering that question, I am completely 100% lost. There exist many types of theoretical manifolds but for mathematical reasons only Calabi-Yau manifolds work. Only C-Y manifolds matheamtically fit with the rest of physics. I don't understand that mathematically. I've studied it and studied it, I've had folks explain it to me, and it does no good. I cannot grasp mathematically why only C-Y manifolds will work. It is not a matter of hard work. I lack the mathematical ability to understand what's going on.

Tha's an example of how you hit a brick wall, and no amount of hard work will get you beyond that brick wall.

The problem, of course, is that at each stage as physics has advanced, the number of people who can handle that level of math decreases, and tends to do so exponentially. I can handle linear homogeneous partial differential equations for only one reason -- because in most cases you can reduce 'em to ordinary differential equations, split up the variables, and then you've got simple straightforward ODEs that can be handled in familiar ways that we all learnt as undergrads. (Mind you, as for non-linear differential equations, I'm lost, and when it comes to inhomogeneous differential equations, it's real tough slogging for me. Only linear homogeneous ODEs are really something I can readily handle.)

But when it comes time to write down a partial differential equation to describe an _arbitrary_ system...a PDE which is likely to be nonlinear, and non-homogeneous...then I'm stuck. I can't get anwyhere. Can't visualize it, can't do it, can't see it, can't understand it. And no matter how hard or how many times someone tries to explain it to me, I _still_ can't do it. To this day I cannot visualize what the two parts of the string equation do. I can listen to someone describe what's going on, but mathematically, I just don't get it.

This is one of the basic things any research physicist must be able to do. If you observe a system, you had damn well better be able to write down a PDE to describe it. With all respect, I would claim that it's an ability that 99+% of the human race just doesn't have. I sure don't. And I watched people drop like flies around me in undergraduate phsyics courses because they couldn't handle the math. I barely could. But by the time I started taking graduate level physics courses, it became pellucidly clear that mathematically I was completely out of my depth.

That's just the way it goes. If you're not so freakishly good at math that you get the math by smell, that you just know it cold, that you just feel it in your bones, you are not going to get through grad school as a physics major, in my experience. You will certainly not get to the level required to do serious cutting-edge physics.

I think that people who _are_ freakishly good at math tend to greatly underestimate how impossible it is to actually do serious math. It's not just hard. It's like doing a 14-foot high jump -- you have to have something extra to get there, the average person can't get close. Not even close. Serious math isn't just hard. It's impossible for most of the human race. Calculus is hard, but doable. Dealing with n-dimensional manifolds is impossible for all but a tiny fraction of the human race. Moreover, this difficulty with genuinely hard math doesn't just apply to physics -- it applies to advanced signal processing in engineering. I found myself just as lost trying to figure out what was happening in, say, the Kalman Filter, or in Z transforms, as in alegbraic topology. FIR filters I get; IIR filters remain a mystery. What happens in an IIR filter isn't hard, it's _why_ it happens that's impossible. Never could do the math for IIR filters. And once again, this is baby math, speaking in terms of engineering. Engineering courses hit you with an onslaught of the Turing Halting Theorem and Cohen's Theorem and so on to the point where if you're not freakishly, superhumanly good with math, by-bye. You're gone. You will flunk right out of the course.

There exists a level of math that most of the human race is forever barred from participating in, just as there exists a level of physical skill that most of the human race can't hope to attain (the Olympics, professional ballet dancers, and so on).

Nowadays, math has become the sine qua non for participating in serious science, at least in the hard sciences. Touchy-feely doesn't get you very far in chemistry or neural networks or materials science or physics. You had better understand the mathematical foundation of Pauling's work on orbitals, or you had better have a rock-slid grasp of the math of quantum band gaps in semiconductors, etc.

It's nice to believe that since we live in a democracy everyone has equal value and if we all just clap our hands and close our eyes, Tinkerbell will come alive again. But physics is not a democracy. Some people are really good at it. The rest of us...well, we can muddle along up to a certain point, but beyond that, it just doesn't work, no matter how much time or effort or determination you put into it. Efforts to deny this basic reality run afoul of the way the world works. It's all very well to be encouraging and give stiff-upper-lip speeches to struggling physics students...but at the end of the day, if you can't do the math in graduate physics courses, you won't get the grades, and if you don't get the grades, you will _not_ get a masters or a doctorate in the hard sciences. That's life. It's not pretty. But it's the way things work.

Mind you, this isn't true of all areas of knowledge. Many professions reward continuous effort and determination. But a few don't. In a few cases, you get only so far with hard work, and beyond that, you hit a brick wall. I would contend physics is one of them.

mclaren,

If your point is that doing physics requires mathematical talents beyond the reach of a large proportion of the population, then I would agree with you. This is perhaps unfortunate, but there it is.

However, if your point is that in order to do "serious physics" one need to be able to master mathematics at the level of sophistication and abstraction of Calabi-Yau manifolds (and this strikes me as implicit in your argument) ... well, I think you're off your rocker.

"But by the mid 20th century, you had to know one hell of a lot more than ODEs and PDEs to make any kind of serious contribution to the hard sciences, as far as I can tell."

What does knowing more than PDEs buy you? What can't you describe in terms of a system of PDEs?