Tuesday, February 26, 2013 ... /////

Learning physics is futile without practicing

LHC news: CMS released two papers with 20/fb of the 8 TeV 2012 data. Remarkably enough, both the dijet paper (Figure 5) and the dilepton paper (Figure 5) show 2-sigma excesses for a possible new object of mass 1,750 GeV or so. They're much weaker signals than what is expected from the typical theories for the objects searched in these papers but the overlap of these excesses is still a bit interesting.
After a break, I answered a package of questions at the Physics Stack Exchange. It's sometimes fun and some of the questions are even interesting but there are also some omnipresent sources of frustration.

Let me mention some of them.

Pretty much every day, there is a question or two that tries to announce the discovery that quantum mechanics has been overthrown and/or may be employed to send faster-than-light signals, and so on. See e.g. user1247 yesterday. Or another question by the same user that tries to overthrow the postulates of quantum mechanics within quantum field theory "only". Or joshphysics who is convinced that observables can't be observable. And so on.

This is a topic that will probably never go away. A physics discussion forum that is open to the laymen must probably inevitably become a place for street rallies where protesters protest against what they hate about modern physics – and its very foundations, principles of quantum mechanics, are of course among the most hated aspects of modern physics.

Very rarely, one feels that they're starting to get the basic points which are really simple if you're impartial. All physically meaningful questions are questions about observables. All observables are represented by linear operators on the Hilbert space. The a priori allowed values of their measurements are given by the eigenvalues. Only the probabilities of each possible outcome may be predicted through a simple and universal formula, the Born rule – in one form or another. This is the framework for modern physics.

If a question can't be formulated in this way, as a question about probabilities of observables (linear operators), it doesn't belong to science. If an answer to a question uses different rules than the quantum-evolution-based and Born-rule-backed calculations of probabilities of propositions about observables (or some legitimately derived approximate theories that ultimately boil down to quantum mechanics), it's wrong. These claims have no exceptions, loopholes, and they don't allow any excuses. Everything else are "technical details" – what the observables are, how they commute or not commute with each other, how they act on various states and how they evolve – and one needs lots of experience. But the conceptual foundations may be summarized in the few principles fully captured by the few sentences above.

The foundations of quantum mechanics are almost never taught correctly, especially outside top universities. The teacher usually never understands modern physics himself or herself so even if the material is presented in some way, it's sold along with a completely wrong focus and "comments surrounding the material". It leads the students to reinforce their misconceptions. It leads them to believe that the foundations of quantum mechanics are a controversial optional luxury that may be replaced by something else at any point. It leads them to believe that $|\psi(x)|^2$ doesn't have to be interpreted as the probability density.

Perhaps it's "ugly" and it may also be the number of sex partners that a woman has had, a voltage in a vibrator, or anything else. They think that everything goes and anything goes. They don't care that the probabilistic interpretation of the objects is a directly observable experimental fact and any modification of it would instantly invalidate the whole theory. Virtually every question about quantum mechanics starts with some deeply emotional words proving the writer's discomfort with some very basic principles of quantum mechanics.

Sometimes I tell myself not to answer because it's completely unproductive – it's just beyond the people's abilities; the emotions sometimes look so deep that I decide that the question was asked by a psychiatrically ill person who should better not be told the truth because it may lead to dangerous situations. Sometimes I react differently when I tell myself: It's just not possible that the folks are this breathtakingly stupid and can't understand these simple points. In the latter case, I am usually destined to a journey towards a pedagogical failure. The people simply are breathtakingly stupid, prejudiced without any limitations, closed-minded, thoroughly incompatible with modern physics.

But that doesn't mean that everything is pleasing about the questions by those who shut up, who kind of correctly use the formalism of quantum mechanics, and who ask different questions. As the title of this blog entry indicates, this blog entry is about the practicing, exercises, particular examples, applications of abstract physical knowledge. Needless to say, this is ultimately what physics and science is good for. It's the bulk through which physics manifests itself in the human society.

Physicists have the working knowledge of all the things in the Universe – OK, I mean all the important things in the Universe – but they only have it if they can actually think. If they know not only the laws and principles of physics but if they also operationally mean what the laws and principles mean and if they have mastered a sufficient chunk of maths that is needed to connect the statements about physical phenomena with each other and with the mathematical expressions, structures, and propositions.

At the same moment, I would agree that too much exercising becomes useless, too. Physics – and research-level maths – is ultimately not about following some procedures and protocols infinitely many times. We want to find new ones. When we have mastered some old classes of problems, we understandably lose our interest in them. This is how physics differs from chess or recreational mathematics. We just don't want to play chess millions of times because it's ultimately almost the same thing. We don't want to solve billions of Sudokus. A person who has the curiosity of a physicist simply wants to learn new things that are qualitatively different from the things he has already learned.

So I want to say that it is indeed natural if a physicist doesn't want to spend too much time with practicing the same thing. Engineers or athletes spend much more time by doing the same things all the time – which may ultimately be a good idea for practical or financial reasons. A physicist wants to get as far as he can in his mastery of the Universe or its chosen part. On the other hand, and this is what the title is about, a certain amount of practicing is simply necessary even for the most exercise-hating physicists because it's needed to guarantee that the knowledge is genuine and usable.

When we study a course or a book, it doesn't mean that we must understand every letter of it. An analogous claim holds for research, too. We shouldn't get completely stuck and terminate all learning or further thinking when we encounter first equation or question that seems confusing to us. After some attempts to lift the fog, we must live with it, remember that the confusion was there, and try to continue, probably using the assumption that the answer obtained differently (or the claim by the author of a textbook or the instructor in a course) is right.

There are also gaps that may seem technically demanding and we may sometimes uncritically accept claims by the instructor, textbook, or fellow researchers that "if you calculate this and that, you obtain this". Because of assorted sociological criteria, such a claim may look plausible and we sometimes need to save the time. I would say that a good scientist should ultimately rely on almost no examples of "blind faith" of this kind. He or she should verify and/or "rediscover" everything that he or she wants to use as a starting point for further research or calculations.

However, in some cases, e.g. when you're learning a body of knowledge someone has packaged for you and you were not necessarily interested in every product in the package, it's natural to skip certain aspects, especially if the other material doesn't seem to depend on them (much). But the gaps in our knowledge that is created in this way should stay "under control". You must "almost know" that if you will need to learn some method or verify the proof of a claim or something of the sort, you will return "here or there" and spend roughly XY hours of time and you will be done.

I would argue that these gaps must be "mostly exceptions", perhaps like the holes in a Swiss cheese that still holds together. I don't say that the critical percentage is 50% but there is some rough percentage and if the number of holes is just too large, the Swiss cheese of your knowledge decays into hundreds of Swiss cheese balls that don't hold together.

When the amount of ignorance and the number of holes grow too large, you not only fail to know many particular things but you also lose the idea about how many things you're actually ignorant about and how to ask questions that would give you a chance to fill the holes, and so on. Your physics knowledge becomes unusable.

I had this feeling when user6818 asked four questions about Polchinski's textbook on string theory. How to derive the Green's functions on a sphere, disk, projective sphere, why this is canceled, and so on? The questions are based on Chapter 6, Volume I.

Superficially, they're legitimate, even high-brow questions. What's problematic about them is that every person – even person who knows no physics, not to mention string theory – could ask such questions. You pick an equation in a book and ask "Why is it true?" But in physics, statements such as equations for Green's functions don't have simple, generally understandable (by everyone), and self-explanatory one-sentence answers. They're derivations that may be compressed or need to be inflated depending on one's knowledge or experience or the lack of it, derivations that always depend on lots of other background.

When one asks why is [relatively straightforward] eqn. (6.2.17) right – without specifying any details about his or her confusion etc. – it suggests that he or she has made no attempt to derive the equation. And it seems that he or she probably can't solve or prove even similar, simpler equations. And although I don't have any rigorous proof, I would bet that these worries are justified by the facts – in this case and many others.

To meaningfully answer such questions and to have an idea how many details the answer should discuss, one needs to know how much the person who asks something actually understands. Does he understand why the logarithm solves the Poisson equation in two dimensions? Can he use substitutions while solving differential equations similar to the known ones? Does he know how to calculate Gaussian integrals? Does he understand why the holomorphic functions have a vanishing Laplacian? Does he know that the sphere is conformally equivalent to the plane, that the disk is conformally equivalent to the half-plane, that the projective sphere may be obtained by identifying opposite points on the sphere as well? Does he know what the boundary conditions at infinity must be for the plane to conformally represent the sphere? Does he really want to explain the blind mechanical calculation proving some independence on the conformal factor or the conceptual reasons? Does he understand the general concept of "solving differential equations", especially the fact that there exists no mechanical procedure that would lead to the solution of any equation?

There are tons of questions – the list above isn't complete. If the answer to some of the question(s) above is "No", the thing may be given a meaningful answer. But if someone just asks why is (6.2.17) and three more formulae right, it looks like the answers to all the questions above are "No". What I mean is that the person must be taught complex numbers, calculus, integrals, conformal transformations, symmetries in physics, two-dimensional Riemann manifolds and their relationships, and many other things pretty much from scratch, and in a more detailed way than Polchinski's book. (Of course, the equation (6.2.17) isn't the first equation where the ignorance about any of these topics should show up – but that is just another detail that makes the question about a seemingly random equation in the middle of the book slightly more surprising.) That's a big task, however, because Joe Polchinski spent a decade by writing his perfectionist textbook on string theory so you may need 50+ years to write the more detailed one.

But would it make sense? I don't think so. If someone really needed to explain all the things above – and I don't claim it's the case of user6818 – he or she simply shouldn't study Chapter 6 of Polchinski's book because he doesn't have the background for that. It's useless to learn some material if you can't use it. And if you can't understand how the material is derived – and the derivation is just an application of a "simpler" material you should have known before – then it shows that you probably can't apply this material (because you can't apply even simpler one) so it's useless to learn it.

The number of questions on Physics Stack Exchange where the answer could very well be 100 times longer than the question itself is significant. (A particular user named Anirbit has asked 60+ questions that mostly fall into this category: "Explain everything in a paper to me".) I am afraid that it is a waste of time to be answering these questions – questions of the type "explain every line of a paper or chapter to me" even though the paper or book have already been written with the assumption that they speak for themselves. A meaningful communication and explanation only occurs when the two sides are at least a little bit on the same frequency and/or if the "teacher" has some idea about the things that the "student" knows. Without such a context, it is impossible to teach physics. And if you teach physics to someone who will ask you "Why it is so?" almost after every claim he sees (and sometimes again – many times – after you already answer it), it's like training armless boys to become construction workers. A teacher may build a house out of bricks but it won't make any impact on the skills of the student simply because he can't do it. So if the "pedagogical house" is useless, you may better avoid this futile pedagogic exercise.

In this respect, physics differs from literature or many other subjects – that often include natural sciences – where the structure of the background isn't hierarchical or is much less hierarchical than in physics. In other words, you don't need much and you may immediately learn some isolated insight from advanced research. You may have never read a book but you may memorize two sentences from a play by Shakespeare and some naive people will instantly think that you're a cultural human being. And so on.

But in physics, this doesn't work. String theory is arguably the tip of a pyramid of knowledge that has almost as many floors as the Empire State Building, if I count if in a fine-grained way. Memorization of an isolated insight or rule is almost worthless in physics because the meaning and power only emerges when many prerequisites are understood.

Needless to say, this is why some people hate physics – and maths – at school. If I don't count gyms and similar things, almost all other subjects at school are about memorization, a universal method that requires lots of RAM (or hard disk space) and almost no CPU or GPU, if I compare the students to computers. At most, some subjects require that the students learn how to follow a relatively mechanical procedure or two to "derive something".

Paradoxically enough, it's the same people who don't like maths and physics for their "requirements of creativity and practicing" who most frequently complain that maths and physics are mechanical, dull, narrow-minded, isolated from practice, and that they reduce people to mindless mechanical machines. When you look rationally at the situation, you notice that the truth is exactly the opposite. These critics of maths and physics are the mindless unthinking machines that only do mechanical things and they hate maths and physics exactly because they can't be mastered in this way! ;-)

But I was thinking about folks who would never open Chapter 6 of Polchinski's textbook, of course. ;-)

Let me return back and say that to learn a subject such as quantum field theory or string theory, one needs to practice, rediscover, verify his own predictions (against well-known insights if not experiments), and think about the implications a lot. Ignorance about particular things is permissible if not inevitable in science. But even ignorance has to be tamed and brought "under some control". We must have an idea how many things we probably misunderstand, how many things about them may be known to other people, how or where to find the answers if necessary, how much time it may take to find the answers or verify some results, and – perhaps most importantly – we must roughly know to what extent the things we're ignorant about may affect the things we think we know (where are the boundaries of our ignorance).

The holes in the Swiss cheese should never break our knowledge into a large collection of disconnected marbles. If that happens, physics ceases to be physics. It ceases to be the lively mechanism to find and incorporate all important insights about everything in the Universe.

And that's the memo.

snail feedback (35) :

Another cool topic ! :) Quantum postulates may be very obscure for someone used to old school approach to physics, which consisted in getting intuitive Physical principles first, then the underlying mathematical structure of the theory ( e.g, Einstein special relativity & Lorentz transformations ).With quantum mechanics there is no such thing as "intuitive physical principle ", one is presented with the mathematical apparatus straight away ! :) We guess "Shut up & calculate " approach to physics is probably the best !

Wow, this text gives me something to think about too; maybe I should just shut up and read at most at physics SE ...

Concerning the Polchinski question examples, this is exactly why I am very annoyed about David Zaslavsky (and to some part about people not (yet?) really knowledgable about advanced topics in physics such as Manishearth, Crazy Budy, etc who just have the rep needed to closevote) closing every question asking about the mathematical prerequisits to study string theory or closely related topics. Asking about the mathematical prerequisits for other things such as QM, Relativity, Maxwell equations, etc is allowed and these questions are left allone.

Some mathematical prerequisits are obviously needed, and closing every corresponding question is like slapping folks, who seriously want to know what is needed and want to learn it before opening a Polchinski book, into the face for it !

More generally, I dont know what happend to David Zaslavski. Before the moderator elections, he used to be such a nice, wise, reasonable, and patient moderator and I really liked him. But now, he (and Manishearth and other high rep users) are only interested in policying, installing new rules, etc and he gives a damn about (the level of) the physics content or the usefulness of the site for people seriously interested in studying physics at a technical level. He moderates the site now in the same way as many bored bureaucrats to their job, caring not in the slightest about what their decisions do to the people who actually depend on them etc ...

What a superb article.

I am a 32 year-old liberal arts major, with no A-Level qualifications in maths or science, who has made a thoroughly implausible, plainly drastic decision to become a physicist.

Whether or not I can (or even should) succeed at gaining the right qualifications to proceed, your post has answered a number of questions I've been trying to ask of myself and others for a few months now, and has spelled out in fine detail that which I need to learn to succeed, as well as illustrating an attitude most conducive to that end.

In concert with a daily study of maths, I've begun to follow both Leonard Susskind's The Theoretical Minimum and Roger Penrose's Road to Reality, which I intend to study for as long as it takes to grasp each fundamental idea. It's my understanding that these texts are comprehensive explanations of the most fundamental principles of physics, and I am pausing at each step in order to give myself time to master each mathematical tool or explanation before I progress any further (needless to say, I'm still enjoying chapter one).

The idea of keeping track of what, despite my best efforts, I may still fail to understand is an idea from this article I'll attempt to make fullest use of. I don't simply wish to barrel ahead, and skip important concepts; I want to fill each swiss-cheese hole (or at least as many as I can) in my understanding before I proceed, and let mastery be the constant and the time that it takes be the variable.

If this all sounds a bit insane, then just know that at least one core message of your article has gotten through to someone. One must understand what it is they understand, and keep track of what they do not in a rigourous, accountable manner.

And, also, perhaps choose better introductory texts.

Bravo on the article.

Even at top universities the foundations of quantum mechanics may be taught incorrectly, Lubos. I first encountered QM in a course taught in 1956 at UC Berkeley by Bob Karplus, who simply did not get it. We used David Bohm’s textbook on the subject, written by a guy who didn’t get it either. I am very fortunate to have learned better after obtaining my doctorate. Most are not so lucky.

I would add that my pyramid of physics knowledge is pretty small compared to yours but it is still a coherent, stable pyramid, little as it is. Many, if not most, comments on TRF are written by folks with only a crumbled pile of debris to stand on. Until they accept this reality they cannot progress, can they?

What a superb article.

I am a 32 year-old liberal arts major, with no A-Level qualifications in maths or science, who has made a thoroughly implausible, plainly drastic decision to become a physicist.

Whether or not I can (or even should) succeed at gaining the right qualifications to proceed, your post has answered a number of questions I've been trying to ask of myself and others for a few months now, and has spelled out in fine detail that which I need to learn to succeed, as well as illustrating an attitude most conducive to that end.

In concert with a daily study of maths, I've begun to follow both Leonard Susskind's The Theoretical Minimum and Roger Penrose's Road to Reality, which I intend to study for as long as it takes to grasp each fundamental idea. It's my understanding that these texts are comprehensive explanations of the most fundamental principles of physics, and I am pausing at each step in order to give myself time to master each mathematical tool or explanation before I progress any further (needless to say, I'm still enjoying chapter one).

The idea of keeping track of what, despite my best efforts, I may still fail to understand is an idea from this article I'll attempt to make fullest use of. I don't simply wish to barrel ahead, and skip important concepts; I want to fill each swiss-cheese hole (or at least as many as I can) in my understanding before I proceed, and let mastery be the constant and the time that it takes be the variable.

If this all sounds a bit insane, then just know that at least one core message of your article has gotten through to someone. One must understand what it is they understand, and keep track of what they do not in a rigourous, accountable manner.

And, also, perhaps choose better introductory texts.

Bravo on the article.

"Let me return back and say that to learn a subject such as quantum field
theory or string theory, one needs to practice, rediscover, verify his
own predictions (against well-known insights if not experiments), and
think about the implications a lot"

Well, that sums it up nicely. And this is why it's almost impossible for even an intelligent amateur physicist to make a ground-breaking observation in physics that hasn't been dealt with already by professional physicists. This is also why even a world class physicist can embarrass themselves every now and again by answering questions outside of their field; they simply haven't practiced within that field over the years enough.

I may be asking for this, but was my comment disallowed?

Very interesting about Berkeley, Gene! I think that my QM instructors in Prague and at Rutgers were highly competent - they also avoided the most philosophical issues surrounding QM, however. And there were some shining big shots, e.g. Prof Bedřich Velický in Prague - a friend of lots of Western physicists, by the way, I learned from the old famous chaps in the U.S. - who was actually somewhat more achieved in research than most of the typical instructors and who was complaining about the teaching of QM - but he was really into various wrong interpretations etc.

Right, it's hard to build on debris because it's really a skyscraper that needs several firm floors under the ground and then the floors above the ground, too. One may try to build on the sand or debris but the resulting structure resembles the soft science - that build on piles of jellyfish meat - and even by the shape, it's a sort of caricature of the actual building of physics which does require some firm materials and perhaps even sharp corners.

It's fun to build even a lower skyscraper and it may be a source of happiness as well as pride. But some people just want to scrape the skies too soon, too quickly, and they believe that when they replace steel and concrete by a quickly grown bamboo network, it's equally good but this usually ain't so. So maybe some modesty and self-inspections of the construction engineering quality is needed at regular intervals, much like the good feelings and satisfaction following even more ordinary achievements that are however done well and firmly.

Thanks, Dilaton, and sorry if I downvoted some people's question(s) at SE even if you liked it. I wouldn't close questions just because they're about maths or adjacent disciplines to physics or unusual aspects, perhaps even metaphysical ones, mathematical ones, sociological ones, and so on.

On the other hand, certain questions may be predicted to lead to unconstructive threads and many other questions may be predicted to be useless for everyone except the author of the question. I understand if moderators are trying to suppress such things. And there are of course questions whose authors only want to solve a homework or exam and use the server to cheat - that's unfortunate and it's equally unfortunate when the problems are actually interesting but they just don't care about them except for the grade.

"Physicists have the working knowledge of all the things in the Universe – OK, I mean all the important things in the Universe . . ." Including economics? he he

Right but the intuition may also start to become useless when things are sufficiently mathematically complex - even in classical physics. That's why I feel that "shut up and calculate" may have been a constructive approach even in many situations in classical physics.

Moreover, I feel that conceptually transforming the foundations of physics has been one of the subject's most satisfactory subjects from the beginning - I mean from Galileo's time if not older ones. The transition from mechanics to fields or from non-relativistic spacetime to special and general relativity may have been more modest an advance - but the more radical transformation imposed upon us by the quantum revolution should make us even happier if we like to make progress.

One more thing. I don't even agree with the comment that quantum mechanics doesn't admit "intuition". I think that intuition is something that allows one to subconsciously guess the results of things even though he doesn't really see the full sequence of thoughts and steps that led to the results. People may have hardwired intuition about things they have spent lots of time with. That's true for questions of everyday life but when one works with quantum mechanics for a long enough time, I think that he or she does develop some intuition again. At least I think that the ability to guess the right analogies or even results from the beginning must count as intuition.

What quantum mechanics doesn't allow is a visualization of anything that behaves as classical objective reality in all respects. But this lack of classical visualizability is something else than incompatibility with intuition, I think, because intuition doesn't need to use classical images. Of course it depends on how one defines intuition...

"Needless to say, this is why some people hate physics – and maths – at school. If I don't count gyms and similar things, almost all other subjects at school are about memorization, a universal method that requires lots of RAM (or hard disk space) and almost no CPU or GPU, if I compare the students to computers."

If string theory is like the top floor of the Empire State Building then obviously it requires lots of RAM as well as CPU. A good short and long-term memory are essential along with logical power and imagination I would say. Maybe the difference is that logic is a powerful mnemonic device for people who are good at physics and maths: they remember things that are logically related, whereas people who are good at, say, chemistry can remember lots of facts that are not necessarily logically related.

All my doubts about basics of QM were answered in one 2 hour lecture on QM.

And it was simply because the professor actually had written thouse few axioms that the whole theory follows and than did some magic on 2x2 and 3x3 matricies... :D

And that was it... no idea why other people start with hydrogen atom and so on... without explaining the basic ideas.

Aristotle says somewhere that men are born with an innate desire to know. I don't think that's true of everyone but it is certainly true of some. Unfortunately we have to specialize in this world so for someone like myself who was certainly born with an innate desire to understand physics, but an even more powerful desire to understand people, the most I can hope for is a kind of qualitative understanding. You can certainly get that from Susskind's video lectures. From Lubos I get a qualitative understanding not of advanced theoretical physics so much (though a tiny little bit) as what the mental life of an advanced theoretical physicist is like. That's worth something too -- among other things it is awe inspiring.

Hi Kieran, new users outside a white list are premoderated and shown a silly - buggy - message I cannot change in any way. But I am confident that I have approved all your previous comments and now you're also on the white list. If you think that your future/next comment doesn't appear automatically, feel free to post another one or edited one with a complaint because this isn't the desired behavior from my side.

Isn't intuition only a subconscious deduction of repetitive lookalike Wolfram-like patterns that reaches consciousness through basic brain signals... those who perceive these signals are intuitive. The others are plugged on other signals because their subconscious is not used to recognize these patterns. (IMO)

Excellent to hear, Karle! And exactly. if finite-dimensional Hilbert spaces and observables as matrices were started with, people may simply get the full glory of quantum mechanics without caricatures from the beginning.

Courses often start with continuous wave functions and hydrogen atoms because that's so similar to classical field theory and other things people know. And that's the main problem of this approach, too.

LOL, if you don't know The Bang Theory in detail, it's funny because your question is almost the same question as one asked by Penny:

http://youtu.be/97a03nOINnQ?t=1m30s

I re-quoted myself from the sitcom, of course. ;-) Watch the whole clip above.

Dear Lumo,

I am not upset about downvotes, but about David Zaslavsky having become an unfair, biased, desinterested in the topic of the site bureaucrat, since the moderator elections.

I dont like homework questions either and I understand and agree that certain questions have to be closed, but the goal of the changed moderation style is obviously no longer to make the site useful for serious physicists and physics enthusiasts who love the subject. And to many users who are not knowledgable enough, in particular about advanced topics, have attaind now the neccessary 3000 rep to damage the site by bad closures etc.

As said, I am most upset that questions asking about what mathematical prerequisits are needed to study or understand string theory (or topological field theory and other advanced HEP-th topics) are closed immediately. About 10 questions are asked about this and not a single one did David allow to get an answer.
This is very bad for all people who want to learn the proper prerequisits for understanding ST before opening a Polchinski book for example. David is no longer a good moderator, he no longer cares about the physics content or level of the site or about making it a good place for people seriously interested in physics at any level. The only moderators I trust now are Qmechanic and Dmckee (even though I had some issues with him earlier too ;-)...)

Davids behavior looks as if he had a personal problem with people wanting to seriously learn string theory and related things. He tries to drive these users away by closing their questions, if they ask about prerequisits they need to learn concerning mathematics. If he succeeds, these people will not be there to ask interesting high level physics questions either.

He closes exclusively in a very biased manner questions asking about the math needed for ST, questions about the math needed for other physics topics he generally leaves alone.

I fear I have been unfair to Prof. Robert Karplus, who passed away many years ago. He was a fine teacher and I surely learned how to use QM to solve real problems. To his credit he did keep speculations about hidden variables etc. separate from the actual work. The only thing I missed was an appreciation of the universality and profound depth of QM but that, I think, comes mainly from experience. Karplus’ course made it possible for me to read Neville Mott’s textbook on solid-state physics and then to learn from Herb Kroemer, a true master of the art. I owe much to Bob Karplus for helping me get the ball rolling and i should always speak well of him.

I concur with many of the superlatives already bestowed on you by Luke Lea. I don't know whether Aristotle or someone else first said that some or all persons are born with an innate desire to know.

Most of us who have such desire will never climb that mountain of string theory, build any skyscraper or approach a real understanding of any cutting edge. We trust or not based on some knowledge and some instinct.

I have long had Road to Reality and enjoy Penrose. Penrose sends the reader forward and back, to and fro with references. More a maze than a Road and hope you would urge Kieran to review some of your blog entries before deciding which paths should have different references.

It is as if you expressed near enough exactly my position on my luck and privilege to be able to read so much of what Lubos writes!

Believe me! I have often googled like mad to try - not always successfully - to fill in holes. :-)

You describe us as being scarily machine-like! ;-)

I came across a nice video by Connes

He claims that our very notion of classical geometry is an illusion, we just create these pictures in our brains (from spectral information). E.g. there is no such thing as a point in physics. The "true" geometry is "quantum" (i.e. noncommutative). That's why we need math to "get it", some will never like (or accept) this ...

I think time studying a subject is extremely important. I would propose that intelligence is more related to speed of comprehension, (and to some extent retention of knowledge). So for instance, one could write a simple formula,

Intelligence = (Comprehension + retention) / Unit time spent on subject

where comprehension would be a measure relative to some statistical group of individuals in similar states of "preparation" in regards to some subject at the introduction of new material, and would measure the ability to grasp relationships between concepts, and the retention would just be shear memorization. Relative intelligence could then be normalized by the "ensemble" in question.

As it is often said, it takes 10000 hrs to becomes a master of a subject (or 4-5 years working full time...not surprising that most of our education levels are set within those same wickets).

Logically one would assume that an A student is more intelligent than a B student, however, I would caution that there is a question of optimization that enters the picture. If individuals have goals outside of academics (getting married, raising families, making friends, building networks, careers, etc) then there is a question of how one balances individual "mental" and "physical" resources in the aggregate. So it may very likely be the B student, when viewed in the full spectrum of life, is actually better than the A person in terms of optimizing resources, although the A person may certainly be better at focusing resources.

I think some of the issue here is that some people can not comprehend something enough to gauge whether what they are being told is true. This can be good or bad, but certainly there is "truth" when it comes to math and physics, particularly in the context of a particular theoretical framework. In physics, when one is told that experimental evidence has consistently supported a theory to several hundred or thousand of sigma, it is probably reasonable to accept the statements of the theory as being "true". Alternatively, if one identifies a situation where there is a contradiction between a statement in the theory and the experimental evidence, then there is something "false" about the theory, and then the theory has a definite bound.

So I think you would agree that there are two types of defects associated with people learning physics, the first defect is benign, it is simply a question of ignorance because of resources. This can reasonably be expect to be removed with time and practice. The second though is more nefarious, and it is a defect related to a certain level of arrogance and ignorance and distrust that stems from some other mental considerations. This is not to say such defects can not be overcome with time, but there is a limit for others in terms of resources available to correct such things. Good optimizers are simply not going to dedicate more time than what they can afford.

Interesting post . I'm not a physicist or a mathematician .I've not been taught even calculus at university (I'm a medical school student)but I wanted to understand quantum field theory so I've read chuncks of calculus , linear algebra and quantum mechanics and I was able to read srednicki book on QFT .I ended up reading textbooks on algebraic topology , smooth manifolds and complex analysis aimed at mathematicians because I'm very interested in mathematics and I want to make contributions to string theory . So far everything is not too difficult . I can follow discussions in those textbooks and fill in gaps but I wonder if this is sufficient . I want to join a graduate school but I doubt that many of these would accept some-one with no formal undergraduate training in physics or math .

Lol. Peter, we *are* machines, just very advanced. Our brain works like a flowchart from the second we wake up in the morning until we switch off at night ;-). Doesn't QM works a bit like this too ?

Regarding the question as to whether QM admits intuition, Lubos, I would go even farther than you. As a solid-state physicist I spent my entire professional career thinking quantum mechanically for the obvious reason that classical mechanics is useless in understanding electron behavior in solids.

I assure you and the TRF readers that, with sufficient experience, intuition and discovery are precisely the same process whether thinking classically or quantum mechanically. In fact, classical mechanics eventually becomes strange and less intuitive than QM because QM is how things really are.

Great post, Luboš.

Not particularly relevant to your theme but I've come across people who have a great desire to know, especially little things others don't know (almost exclusively for one-upmanship reasons and public preening — laughable stuff), but little or no desire to understand. Weird psychology. I don't understand it. They love data, isolated facts especially. Arbitrary lists are a favourite. Arts men to a man. Rote learners the lot.

Maybe slightly more relevant — a couple of my favourite data points, non-arbitrary ones of course: :)

A Vulgar Mechanic can practice what he has been taught or seen done, but if he is in an error he knows not how to find it out and correct it, and if you put him out of his road he is at a stand. Whereas he that is able to reason nimbly and judiciously about figure, force, and motion, is never at rest till he gets over every rub. [Newton]

There's physics — the rest is stamp collecting. [Rutherford, but not quite verbatim.]

Incidentally, I love the controlled aggression in the way you express your views. Combative! Spartan! Excellent!

Great way to put it - I wanted to say the same thing but you really did it.

I agree with you Gene, had the same experience in a different discipline, experimental particle physics. If you do not intuit quantum mechanically you are lost and cannot walk the next step in research.

Logically one would assume that an A student is more intelligent than a B student, however, I would caution that there is a question of optimization that enters the picture. If individuals have goals outside of academics
Liam Hudson had a little tale about that: B+ beats A. His idea was that big discoveries are made as much by perserverance as intelligence. He claimed that the probability of winning a Nobel for a scientist was equivalent for IQ ranges 130-180. Still only circa 2% of the population and numerous studies on creativity do highlight that one must be possessed of a certain passion to continue working so hard and long. So perhaps Nietzsche was onto something when he wrote:

"The essential thing, in heaven and on earth, is that there should be a long obedience in the same direction, there thereby results, and has always resulted in the long run, in something which made life worth living."

Nietzsche, Beyond Good & Evil.

Not for me though, life is too enjoyable today and I'm not waiting for some treasure at the end of the rainbow thank you.

You are being too kind. I'm interested in the ways one person can influence the behavior of another by means of speech, of words and ideas. For instance this very conversation we are having right now. I'm not sure such influences -- the interplay between speech and human behavior -- can be understood or predicted quantum mechanically or even classically, even in principle, though I suspect you may disagree. The wave function is not real. I wholly get that.

But what is the status of an idea? I don't mean in terms of its truth value or semantic content (assuming it has any) but considered purely as a physical object -- a physical object, moreover, that is capable of influencing other, similar physical objects in other people's heads and thence their behavior?

As each of us threads his or her way through through life, through a maze of possibilities and probabilities, bounded by necessities, impelled by opportunities, buffeted by the accidents of chance (including words other people say) . . . well, what I want to know is if by the end we can agree, many of us at least, that the sum total life history of that person's choices (say, the life of Abraham Lincoln) makes a certain kind of sense then can't we say that that person's life exhibits a kind of supra-natural order? Notice I say supra- (with an r and an a), not super- (with an e and an r), meaning within not outside the laws of nature).

In other words the choices we make within a world of possibilities can of themselves exhibit a kind of order which is outside science yet also wholly consistent with the laws of nature. These choices cannot even remotely be predicted by science yet they can be appreciated, understood, and to a certain degree even predicted by other human beings, even very stupid ones.

In German (correct me if I am wrong Dilaton) it is the distinction beween Verstehen (interpretive understanding) and erklären (scientific understanding in terms of physical causation), a distinction made by a 19th century sociologist whose name I forget (Dilthey?).

Anyway there is nothing supernatural about this idea whatsoever (this idea about ideas) and it certainly does not violate locality, Lorenz invariance, the laws of thermodynamics, etc.. Nor can I see it is an issue of getting the right time-and-distance scale for an effective physical theory.

It's not science, I agree, but it is important, this vast world of ideas.