Explaining is a difficult art. You can explain something so that your reader understands the words; [I try to] explain something so that the reader feels it in the marrow of his bones.

Richard Dawkins

 

My private school taught biology from the infamous creationist textbook Biology for Christian Schools, so my early understanding of evolution was a bit... confused. Lacking the curiosity to, say, check Altavista for a biologist’s explanation (faith is a virtue, don’t ya know), I remained confused about evolution for years.

Eventually I stumbled across an eloquent explanation of the fact that natural selection follows necessarily from heritability, variation, and selection.

Click. I got it.

Explaining is hard. Explainers need to pierce shields of misinformation (creationism), bridge vast inferential distances (probability theory), and cause readers to feel the truth of foreign concepts (quantum entanglement) in their bones. That isn’t easy. Those who do it well are rare and valuable.

Textbook writers are often skilled at explaining complex fields. That’s why I called on my fellow Less Wrongers to name their favorite textbooks (if they had read at least two other textbooks on those subjects). The Best Textbooks on Every Subject now gives 22 textbook recommendations, for fields as diverse as scientific self-help and representation theory.

Now I want to jump down a few levels in granularity. Let’s pool our knowledge to find great explanations for each important idea (in math, science, philosophy, etc.), whether or not there is equal value in the rest of the book or article in which each explanation is found.

Great explanations, in my meaning, have four traits:

  1. A great explanation does more than report facts; it uses analogy and rhetoric and other tools to make readers feel the target idea in their bones.

  2. A great explanation is not a single analogy nor a giant book. It is, roughly, between 2 and 100 pages in length.

  3. A great explanation is comprehensible at best to a young teenager, or at least to a 75th percentile college graduate. (There may be no way to seriously explain string theory to an average 13-year-old.)

  4. A great explanation is exciting to read.

By sharing great explanations we can more often experience that magical click.


List of Great Explanations

I’ve barely begun to assemble the list below. Please comment with your own additions!

(The list below is exclusive to written explanations, but feel free to share your favorite explanations from other media. My favorite explanation of BASIC programming is a piece of software from Interplay called Learn to Program BASIC, and of course many people love Khan Academy’s videos and The Teaching Company’s audio courses.)

Epistemology

Math and Logic

Physics

Biology

Psychology

Economics

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I see that the physics list consists of math-free pop-science books. I don't see how these can possibly count as "great explanations," since it's impossible to gain any real understanding of physics from such materials.

For example, a good explanation of relativity would first present the concepts of Minkowski spacetime and proper time, and then show how all those "relativistic effects" follow from these reasonably simple concepts. (This as opposed to a bad explanation of the traditional sort that confuses the reader with various "effects" and "paradoxes.") Then it would explain how a curved metric with a Minkowski signature can make the geodesic lines look like the free fall curves of Newtonian gravity, thus providing an intuitive grasp of the whole "gravity is geometry" business. Beyond that, you just have to get into the hairy tensor stuff, and even this minimum requires a solid knowledge of algebra, analytic geometry, and basic calculus. Anything less than that is just rambling that may have a lot of entertainment and signaling value, but won't move you one millimeter closer to any real insight.

Agreed. A good elementary exposition of relativity along these lines is Bob Geroch's General Relativity from A to B.

EDIT: Actually, I realize I'm only in partial agreement with Vladimir. While I do think that many pop-sci explanations of theoretical physics are fairly worthless and often actively misleading, I do not think that it is impossible to gain real insight into (say) the general theory of relativity without mastering differential geometry. Geroch's book presupposes only high school mathematics, but it provides a genuinely deep insight into relativity.

While I do think that many pop-sci explanations of theoretical physics are fairly worthless and often actively misleading, I do not think that it is impossible to gain real insight into (say) the general theory of relativity without mastering differential geometry. Geroch's book presupposes only high school mathematics, but it provides a genuinely deep insight into relativity.

I described the requirements as algebra, analytic geometry, and basic calculus, which is more or less within advanced high-school math. (Without calculus, I don't see how you could explain integrals along the world line, which are the very heart of the matter.)

However, note that just high-school algebra is already worlds apart from purely prose-based, general-audience pop-science. I would guess that for an average reader (let alone owner) of pop-science books, following a text using algebra would be far harder than it would be to figure out tensors for a reasonably math-savvy twelfth grade student.

I agree, and I wonder how much more widely it applies than the mathematically based subjects. Just because something is expressed in words rather than formally manipulable symbols does not, I suspect, make it any easier to arrive at sound understanding based on sound assessment of evidence and argument. However, it does make it easier to mistakenly think that you have done so.

It is easy to just go with the narrative flow, nodding along to it without asking "is this true?", "is this cherry-picked evidence?", "is the writer working to a hidden bottom line?", and so on, and even if one does, it's a lot more work to answer these questions in a subject such as history than in mathematics, where you can and should work out the proofs yourself, or in physics, where a short trip to Google will turn up reliable sources about the red shift of stars.

I agree -- that's why I sometimes point out that for people who imagine themselves to be so much more rational than average, a real test would be to try and make some sense of such fuzzy and controversial topics.

Also, in subjects outside of hard sciences, the danger isn't just that you may fall in with the flow of a well-written bad argument, but also that a valid argument might have such bad ideological or signaling implications that you'll desperately grasp for any excuse to reject it without due consideration.

Speaking as someone who has read a few popular treatments and none of the proper textbooks ... yeah, I agree. I have no insight into general relativity at all, and precious little into special relativity.

Vladimir,

Our discussion has been long. Let me reply to you here, and leave the tangents of our discussion hanging so I can return to the origin.

You say "it's impossible to gain any real understanding of physics from such materials [e.g. pop-sci books]."

I'm still not sure what you mean by "real understanding," but all I'm trying to claim in the post above is that readable but non-technical explanations of scientific concepts and theories from people like Brian Greene and Richard Dawkins can be helpful. They have helped me, for one. They have improved not just my ability to guess the teacher's password, but also my ability to have more accurate anticipations in ways that help me achieve my goals. Is that something you actually think is "impossible"?

If so, what is your response to the examples wedrifid and I gave?

If not, what is it that you do mean?

I'm still not sure what you mean by "real understanding," but all I'm trying to claim in the post above is that readable but non-technical explanations of scientific concepts and theories from people like Brian Greene and Richard Dawkins can be helpful.

My original point was only about physics, not about evolution, and I have already written that there is an important difference between the accessibility of these two for lay readers. So by dragging evolution into the discussion again, you are obscuring the issue.

They have improved not just my ability to guess the teacher's password, but also my ability to have more accurate anticipations in ways that help me achieve my goals. Is that something you actually think is "impossible"?

Yes, I do think that math-free popular books about modern physics (by which I mean QM, relativity, and the more advanced fields that use them) cannot give the reader any such ability.

A physicist with good writing skills could easily write a book full of completely nonsensical pop-scientific "explanations" of relativity and QM (let alone cosmology etc.), and there would be no way for non-expert readers to notice that something's wrong (unless they noticed that it's explicitly contradicting something they previously read elsewhere). In contrast, someone who studies with full understanding from a real textbook will notice the errors of the author purely from internal evidence, since these errors will stick out blatantly in the otherwise smooth and clear logical flow of exposition. There we see the fundamental difference between real explanations and fake explanations.

If so, what is your response to the examples wedrifid and I gave?

Some of these examples are about using classical physics to improve on folk-physical intuitions or gain insight that can be based on folk physics or simple Newtonian physics. Such insights can indeed be gained without much (or even any) math, so I admit my claims should be qualified to exempt such cases. Note however that the original context was about advanced non-classical fields of physics, which are the subject of the overwhelming majority of pop-scientific texts, some of which you originally cited as supposedly "great explanations."

As for the example of the limited speed of light, it's a completely isolated rule that stands outside of any systematic understanding. It's as if you claimed that you can understand Maxwell's electromagnetic theory without knowing vector calculus, and then supported it by arguing that your non-mathematical understanding enables you to predict that bad things will happen if you touch uninsulated high-voltage lines. Yes, it is a correct rule that makes correct and practically useful predictions, but it's completely isolated and learned by heart -- and there is no way to connect it with some more general framework for understanding physical phenomena without learning real (i.e mathy) physics. [*] Similarly, unless you have real mathy knowledge of relativity, your knowledge of this particular fact about the limited speed of light is just an isolated fact, which is not integrated with any broader and deeper understanding of physics.


[] -- *Edit: Thinking about this a bit more, I would say that there are in fact such ways. Some people manage to develop amazing intuitive understanding of electromagnetic phenomena without any math or formalism at all, and sometimes their intuitions will be more accurate than the products of laborious number-crunching by experts. However, such understanding is about hands-on technical practice, and it's radically different from anything that can result from reading pop-science.

My original point was only about physics, not about evolution, and I have already written that there is an important difference between the accessibility of these two for lay readers. So by dragging evolution into the discussion again, you are obscuring the issue.

Huh? I didn't bring up evolution again. I mentioned Richard Dawkins, but not evolution, and the 'great explanation' from Dawkins that I list above is in physics (rainbows), not biology.

BTW, are the physics ones the only ones you object to? Are you still mostly on board with the project of tracking down good, engaging explanations of, say, biological and psychological concepts and theories?

math-free popular books about modern physics... cannot give the reader any such ability [to have more accurate anticipations in ways that help them achieve their goals]

You've offered enough exemptions now for your claim (speed of light, classical physics, and probably others) that I now understand that we agree more than initially seemed to be the case. Still, I think there are examples of math-free popular explanations of modern physics that can give readers like me the ability to have more accurate anticipations in ways that help us achieve our goals.

I gave a few examples but you didn't accept them.

I'm tempted to drop the discussion for now — unless you strongly object?

Huh? I didn't bring up evolution again. I mentioned Richard Dawkins, but not evolution, and the 'great explanation' from Dawkins that I list above is in physics (rainbows), not biology.

My mistake -- I didn't realize you were alluding to Dawkins talking about physics.

BTW, are the physics ones the only ones you object to? Are you still mostly on board with the project of tracking down good, engaging explanations of, say, biological and psychological concepts and theories?

I think I have already explained clearly enough that it depends on the concrete topic in question. Some technical and scientific topics can be explained in a non-mathematical way that increases the understanding of smart lay readers. Others however can't, and attempts to do so will end up as sheer confusion and fake explanations. Modern physics just happens to be in the latter category.

You've offered enough exemptions now for your claim (speed of light, classical physics, and probably others) that I now understand that we agree more than initially seemed to be the case.

Then you understand wrongly. I haven't budged one millimeter about the worthlessness of pop-scientific "explanations" of modern physics of the sort you cited initially. There is a fundamental difference between, on one hand, direct improvements on folk-physical intuition and simple facts memorized in isolation, and on the other hand, real understanding of complex and non-intuitive theories of modern physics.

I'm tempted to drop the discussion for now — unless you strongly object?

I have no problem with that. I think what I've said so far should be clear enough.

I haven't budged one millimeter

At least there is one thing in this conversation with respect to which everyone is in agreement!

What in the above comment is worth downvoting? All I say here is: "Here's what I'm still confused about; could you clarify?"

Luke. Do you realize for what kind of trivial stuff EY gets downvoted down to like below -10?

People adjust their expectations accordingly in order to, the charitable explanation goes, preserve the signalling value of karma as a low investment form of feedback for your comments, or as the less charitable goes, raise their expectations to an insane level.

You have the third highest karma score on the site. You acquired this position very rapidly and are very prolific, probably everyone knows your user handle, while for example I still didn't know most of the people on the top 10 list for like a year after I started reading.

Take it as a compliment.

Sit down and pour yourself a glass of your favourite beverage and smile when something that gets most people a 0 or 1 gets you a -1 and something that gets others a -1 or -2 will get you -5. Also remember it is bad signalling to ask "why am I donwovted?" when you have several 10k karma, it dosen't make sense, but people respond better at that point if you ask basically the same thing without using the word "vote" and its variants or "karma". :)

Karma isn't the point. As you've said, I'm not lacking in karma. The point is to improve the success of my communication by learning in detail what kinds of things set some people off.

Which, now that I think about it, is something I could have included in my original comment asking why the comment above it was downvoted.

Vladimir,

I gave four criteria to explain what I meant by a "great explanation" and your example fails criterion #3 and probably #4. We're talking about different kinds of "great explanations."

My post is meant as a list of great explanations intended for a general audience. The explanations you describe are mathematical explanations like you would find in textbooks written for people who will do advanced work in that field. I don't need to be able to prove Aumann's agreement theorem for me to understand how it should inform my Bayesian updates, and I don't need to be a physics graduate to understand why many worlds has advantages over the Copenhagen interpretation. Eliezer's relatively non-technical explanation of Occam's razor does move people more than "one millimeter closer" to understanding many things.

My post is meant as a list of great explanations intended for a general audience.

As you note, some things are impossible to explain to an average 13-year-old. By the same principle, some things are impossible to explain while remaining at the "general audience" level. You can produce confusion or fake explanations -- pop-science typically offers some mixture of these two -- but nothing even close to real explanations, let alone actual "clicks."

The explanations you describe are mathematical explanations like you would find in textbooks written for people who will do advanced work in that field.

No, these explanations are far short of what one needs to do actual work in that field. Even the hairy tensor stuff I mentioned is just the beginning.

I wrote this based on my own experience trying to make some sense of relativity. What I sketched is the barest minimum that enabled me to gain anything resembling real insight instead of just confusion and fake explanations.

What I'm trying to say is this.

I do not have a technical understanding of biological evolution. I haven't looked up the equations in population genetics and elsewhere. But even a good pop-sci explanation of evolution from Richard Dawkins does make a positive impact in my understanding of the world. My non-technical understanding of evolution allows me to make more accurate predictions about the world than I would have otherwise. The trick is in making sure I do not over-estimate how well I understand evolution.

Do you actually disagree with this?

Edit: Also, just in case anybody is confused... the meaning of "explanation" in my post "great explanations" differs from the meaning of "explanation" in the phrase "fake explanations." They are two different words that happens to have the same spelling. A "great explanation" is a presentation of a topic that helps readers understand it. The word "explanation" in the sense of "fake explanation" refers to the type of explanation also used in the phrase "scientific explanation."

My non-technical understanding of evolution allows me to make more accurate predictions about the world than I would have otherwise.

A good understanding of evolution can be had without any math, unlike physics. There is no math in The Origin of Species, but it's impossible to rewrite any major work of physics since Galileo without math while preserving its essential points.

In any case, what exactly are these more accurate predictions about the world that pop-physics enables you to make? I would be very curious to hear some examples.

Also, just in case anybody is confused... the meaning of "explanation" in my post "great explanations" differs from the meaning of "explanation" in the phrase "fake explanations."

My comment about fake explanations applies to any reasonable definition of "explanation." In fact, the points from the "Fake Explanations" article apply perfectly here. If the material from some prominent pop-science book were rearranged into something written in a similar style but in fact completely wrong and nonsensical and signed by an equally high-status author, how many readers of these books would realize that something's wrong?

In any case, what exactly are these more accurate predictions about the world that pop-physics enables you to make? I would be very curious to hear some examples.

  • I will never get a ping time to American servers from my home here in Melbourne of less than the distance times two divided by c.

  • If I drop a really heavy rock and a somewhat lighter rock from a moderate height there will be only a slight difference in how long they take to fall to the ground.

  • If I find some stuff that is really, really heavy and leave it in my pocket I will probably die of cancer.

  • Cars traveling towards me will sound slightly higher in pitch than after they go past me.

  • If I buy bullets that are designed to travel slower than sound they will probably make less noise than the bullets that go faster than the speed of sound.

  • If you give me some charts that show how much light of various wavelengths there is coming from two different stars and it so happens that they look really, really similar except that one is kind of 'stretched out' over the 'wavelength' axis I can tell you that the stretched out one is farther away from us.

And, the critical one:

  • If I make a bet with someone that a survey done ten years from now of prominent physicists fewer will declare affiliation with the Copenhagen Interpretation than if an equivalent survey was done today then I am more likely to make money than to lose money.

You might believe that my lack of status as a mathematical physicist doesn't give me the right to make claims about Quantum Mechanics implications but the universe doesn't care. I can apply basic principles of rational thinking to filter large swathes of evidence from those who popularize physics and, particularly, the verbal, non-mathematical claims of physicists in order to work out whether or not a specific claim is likely to be correct.

I will never get a ping time to American servers from my home here in Melbourne of less than the distance times two divided by c.

Funny you should mention that. I spent years working in IT, and this knowledge was actually useful once. I tried to ping a DNS router in Europe (I forget where) from California, and it came back in 1ms and I thought "Ummmmmm... no. You lie." It turned out one of the smart switches on the local network was fucked up and was somehow returning all pings itself.

Behold! Even a pop-sci understanding of physics controlled my anticipations in a way that was useful for accomplishing goals in the world.

I see that this is getting upvoted, but your example sounds like someone who realizes that a device doesn't work because it's not plugged in, and then makes a self-satisfied comment that his knowledge of electromagnetic theory usefully controlled his anticipations in this situation.

In other words, it's about simple conventional nuts-and-bolts technical knowledge, not an improvement on such knowledge brought by more advanced understanding of anything. There's no way someone who works in network administration wouldn't know that a "<1ms" ping coming from around the world is anomalous.

it's about simple conventional nuts-and-bolts technical knowledge, not an improvement on such knowledge brought by more advanced understanding of anything. There's no way someone who works in network administration wouldn't know that a "<1ms" ping coming from around the world is anomalous.

Actually, yes: I think this is correct.

But, I think I would have noticed something was wrong even if I hadn't worked in IT before. But I'm not certain of that.

That is pretty awesome, but I also don't think it's necessary to think about light speed to solve that problem. Anyone who spends a lot of time debugging networking problems knows that 1ms is unreasonably fast for any communication with a machine more than a couple router hops away, even if it's physically nearby.

Fair enough. There are indeed many ways in which the folk physics intuition can be improved by internalizing a rule that's simple enough to explain without math. I admit that my question was too aggressive and snarky.

However, I don't think any such simple insights will move you any close to understanding either QM or relativity (let alone more advanced topics such as cosmology or the controversies over QM interpretations), which was the topic of the original dispute. I must also point out that your rules are either from classical physics (and thus reasonably close to the relevant folk physics intuitions) or in the form of entirely opaque rules for which you can't find any justification except for appeal to authority. And I'm certainly not saying that as someone who has "status as a mathematical physicist"; I'm a complete amateur in physics, as I pointed out in an earlier comment. (Also, if you've read my earlier comments on LW, you'll know that I don't put much inherent weight on the official credentials of expertise bestowed by the present academic system.)

Also, regarding the trends in the Copenhagen vs. MW debate, how much of your opinion is based on understanding of the issues involved, and how much on mere perception of the social dynamics in the field?

However, I don't think any such simple insights will move you any close to understanding either QM or relativity

Then, like Luke, I must observe that your definition of 'understanding' is completely incompatible with mine. Actually, no. I think we've moved beyond that now. You are just wrong.

Look at the first example for a start. That claim is based on understanding of (part of) relativity. But that example was only included coincidentally. May I remind you that the examples you demanded were for physics in general (and, for that matter, that the demand was in response to professed understanding of evolution.)? Space considerations stopped me before I also threw in predictions about "two really good clocks, one of which you fly around really fast!", the fellow who gets into a rocket and outlives his great grandchildren and what happens when you shoot not very many very small things through very small slits.

Question: If the world were wiped out tomorrow leaving a few people and you were the only survivor with any minimal scientific training, could you on your own reconstruct special relativity? If the answer to this is "no" then you don't understand it.

Note that by this standard I don't understand QM and I really don't understand GR and I'm a math grad student. It is ok to say that one doesn't understand things.

Question: If the world were wiped out tomorrow leaving a few people and you were the only survivor with any minimal scientific training, could you on your own reconstruct special relativity? If the answer to this is "no" then you don't understand it.

Yes. I already declared this, in a comment you replied to no less.

Additional claim: It has been too long since I studied physics for me to remember all of the mathematics of special relativity. Yet when I did the aforementioned study I also gained a solid grasp of the fundamental principles. With that understanding I could recreate the interesting mathematics from first principles. I am confident of this because I've done it before, just for kicks. Because memorizing the math wasn't enough for me and I wanted to really grasp the science in depth. The way you do that is by knowing the concepts well enough that you could work out the equations for yourself. Because just memorizing them is detail work. (I haven't got a chance in hell of doing this with GR.)

Look at the first example for a start. That claim is based on understanding of (part of) relativity.

As I pointed out in my reply to Luke, knowledge of that fact constitutes "understanding of relativity" to the same extent that knowledge of the fact that you can get hurt by sticking your fingers into electrical installations constitutes understanding of electromagnetic theory. It's just a single fact you know in complete isolation, not a fact that is a part of some broader framework for understanding the world.

To build on this particular example, consider that some things like shadows, reflections, etc., can indeed move faster than light. Or, if you just spin on an office chair, in your rest frame the celestial objects are spinning around you way faster than c. Unless you can explain why such motions are consistent with the "no faster than light" principle, you have nothing more than a literally memorized fact that if it might be useful if something could move faster than c, it can't happen. It's a true fact, to be sure, and even a potentially useful one, but still.

May I remind you that the examples you demanded were for physics in general (and, for that matter, that the demand was in response to professed understanding of evolution.)?

The context was about the pop-scientific treatments of modern physics. If you insist on full precision, I will gladly admit that my wording about physics in general was imprecise, since there are indeed simple topics in classical physics where insight can be gained without math, building only on a folk-physics intuition. So please read my statements in their original context.

As I pointed out in my reply to Luke, knowledge of that fact constitutes "understanding of relativity" to the same extent that knowledge of the fact that you can get hurt by sticking your fingers into electrical installations constitutes understanding of electromagnetic theory. It's just a single fact you know in complete isolation, not a fact that is a part of some broader framework for understanding the world.

We aren't talking about the memorization of a simple fact about pings. We are talking about all the understanding you can have about relativity without having memorized a mathematical equation. This can be used to make the prediction that ping times will never be lower than the aforementioned limit.

You asked for examples of predictions about the world that can be made based on understanding physics minus the math. It would be disingenuous in the extreme to then dismiss all examples of predictions about the world given because they are, in fact, mere predictions about the world and therefore could have been memorized without real understanding.

To build on this particular example, some things like shadows, reflections, etc., can indeed move faster than light. Or, if you just spin on an office chair, in your rest frame the celestial objects are spinning around you way faster than c. Unless you can explain why such motions are consistent with the "no faster than c" principle, you have nothing more than a literally memorized fact that if it might be useful if something could move faster than c, it can't happen. It's a true fact, to be sure, and even a potentially useful one, but still.

My first prediction: If you (who I believe professed understanding of at least this much understanding of math) and lukeprog were locked in rooms disconnected from the outside world and given the task of answering this question not only would Luke be able to give an explanation, his explanation would be better than yours. You both know enough about physics to answer and Luke is better at explaining things.

My second prediction: There are many students who, in their physics exams, get all the questions that require mathematics correct and who, when encountering a question like this one, can't give an answer. Because not only is knowing the math not strictly necessary to answer this question, it isn't even sufficient.

Additional claim: It has been too long since I studied physics for me to remember all of the mathematics of special relativity. Yet when I did the aforementioned study I also gained a solid grasp of the fundamental principles. With that understanding I could recreate the interesting mathematics from first principles. I am confident of this because I've done it before, just for kicks. Because memorizing the math wasn't enough for me and I wanted to really grasp the science in depth. The way you do that is by knowing the concepts well enough that you could work out the equations for yourself. Because just memorizing them is detail work. (I haven't got a chance in hell of doing this with GR.)

In the various responses you have been given your claim that you can't have any understanding of physics without math has been overwhelmingly refuted. All you are left with is "But that understanding isn't true understanding, true understanding means you remember the math!" To that I reply "No, not all Scotsmen like haggis. You can only tell a true Scotsman by the kilt they are wearing!"

It is time to retreat from a complete rejection of all non-mathematical understanding so that you can express an actually tenable position regarding the limits of how much you can know about physics sans math. Because there really are such limits and I would love to be able to support you in declaring them. But right now you've gone overboard and tried to reject even that understanding which can exist. And that brings you to the realm of the absurd and I just can't support a position which is just obviously factually incorrect.

You asked for examples of predictions about the world that can be made based on understanding physics minus the math. It would be disingenuous in the extreme to then dismiss all examples of predictions about the world given because they are, in fact, mere predictions about the world and therefore could have been memorized without real understanding.

However, there is indeed a difference between rules memorized in isolation without any additional understanding and, on the other hand, real understanding of physical theories and generating predictions based on them. Just like there's a difference between understanding electromagnetic theory and knowing a few common-knowledge technical facts on how to deal with electrical devices, there is also a difference between understanding relativity and knowing a few facts that follow from it in isolation.

You are now employing a rhetorical tactic where you try to make this obvious and relevant point look like weaseling, but in reality it is a pertinent and adequate response to your example.

In the various responses you have been given your claim that you can't have any understanding of physics without math has been overwhelmingly refuted. All you are left with is "But that understanding isn't true understanding, true understanding means you remember the math!" To that I reply "No, not all Scotsmen like haggis. You can only tell a true Scotsman by the kilt they are wearing!"

This is sheer rhetoric. You latch onto one point I made, completely ignoring the context and making the most extreme uncharitable interpretation of it (one that is in fact bordering on caricature), all for the greatest rhetorical effect. Instead, a rational approach would be to see if there may be some validity behind my point even if its original statement was imprecise, especially since I readily admitted this imprecision on first objection. Not to mention that your own example is largely irrelevant in the original context, which was about lengthy pop-science works purporting to explain whole physical theories to lay audiences, not about isolated examples such as yours.

In any case, if you think the distinction I outlined above is invalid, or that I am applying it incorrectly, please go ahead and explain why you believe that. If you're going to latch onto a caricature of what I wrote while treating the discussion as a rhetorical context, I have no further interest in continuing this exchange.

I think it might be helpful for you to taboo the word "real understanding". It seems like a lot of the disagreement stems from luke and wedrifid being unable to understand what you mean when you use that phrase. To be honest, while I agree with many of your points, I also don't think I understand what "real understanding" is supposed to mean. Perhaps you could restate your original point without use of the word "understanding"?

I think it might be helpful for you to taboo the word "real understanding".

Even understanding isn't well-defined. I at least don't know of any agreed upon conceptual or mathematical definition. Does Wolfram Alpha understand math? Does a lookup table understand anything?

I think that you really understand a subject if you are able to transfer, teach and artificially recreate the skill or heuristic that is necessary to make useful predictions about the subject. The skill or heuristic needs to be utilizable given limited resources. You also have to be able to abstract and generalize your knowledge about the subject to solve new problems that are not similar to problems previously encountered. Furthermore, you have to be able to prove fundamental mathematical statements and relationships about the subject.

An example would be the ability to teach the game of Go to other agents, transfer your knowledge by writing books on how to play it, and create a Go AI that can play the game by predicting the success of its strategies.

My second prediction: There are many students who, in their physics exams, get all the questions that require mathematics correct and who, when encountering a question like this one, can't give an answer. Because not only is knowing the math not strictly necessary to answer this question, it isn't even sufficient.

No one else is arguing that it is sufficient, but they are arguing that it is necessary. In this context, I'm going to make a counter prediction: if one did give a test on SR to physics students just learning about it, the ones who answer the chair question correctly will be a proper subset of the ones who can do the mathematical manipulation.

In the more restricted context of highschool calculus or more basic physics this is (from my experience both teaching and tutoring) very much the case. There are students who say things like "I can't do the math but I understand the concepts" and this just nonsense. The ones who answer the conceptual questions correctly are almost always those who can do the math. There may be kids who can do the formal manipulation and can't connect to it conceptually but there's almost no one who can't handle the symbol pushing who can answer the conceptual level questions. They are too interrelated.

There are students who say things like "I can't do the math but I understand the concepts" and this just nonsense. The ones who answer the conceptual questions correctly are almost always those who can do the math.

Why do the they believe what they believe? The simplest two explanations I can think of is that they are mistaken about their grasp on the concepts (when you ask them conceptual-level questions, they answer incorrectly), or they are mistaken about their inability to do the math (they feel insecure before quantitative tests, but score high). Is one of these the case?

There's a combination of both, but there seem to be far more in this first category. To some extent it seems like it is an example of the Dunning-Kruger effect. And I suspect that some of the people here who are convinced that one can understand physics without understanding the math may be also running into Dunning-Kruger issues.

Given that the Dunning-Kruger effect may be a combination of noisy estimates and hidden, different scales (how do we weight the 1500-2500 separate skills involved in driving when deciding who's in the top 10%), could we extend competence from both ends toward the middle?

What I mean is that an essentially correct but very fuzzy understanding may constrain your anticipations in many different circumstances, but may not tightly constrain them--anybody knows you're not going to get a <1ms ping from another continent, and a car coming towards you makes a higher pitched noise than it does going away from you, but most people couldn't diagnose the 500 mile email or calculate the speed of the car from the frequency change.

On the other side, many people just learning physics can carry out the calculus, as long as they have the formula in front of them and the problem's explicitly stated; but would have trouble generalizing the formula to all the real-life situations in which it's applicable.

As the intuition becomes more and more precise, it gets closer to math. As the math becomes more and more intuitive, the situations in which its applications are evident grow.

An engineer is never going to be successful with just intuition, because he needs very precise results. But, depending on your use case, generality may be more useful than precision. This seems more useful than trying to decide which one is "real" understanding.

No one else is arguing that it is sufficient, but they are arguing that it is necessary.

Did I leave out the part where students who couldn't do all the math answered it correctly? Pardon me.

In this context, I'm going to make a counter prediction: if one did give a test on SR to physics students just learning about it, the ones who answer the chair question correctly will be a proper subset of the ones who can do the mathematical manipulation.

Then I would make counterfactual money by betting against you.

My comment about fake explanations applies to any reasonable definition of "explanation."

No it doesn't. It only applies to reasonable definitions of the particular homonym you're referring to from within the group of two different words that are spelled E-X-P-L-A-N-A-T-I-O-N.

Suppose I said a particular hockey goal was a "fluke" (a stroke of luck). Then, you pointed me to a post that implied that hockey goals are not good examples of flukes (the fish). This would be a confusion caused by the fact that word "fluke" (a stroke of luck) has the same spelling and pronunciation as the word "fluke" (a type of flatfish).

In the same way, "explanation" has two meanings. The first meaning has to do with a reason, what Wikipedia calls "a set of statements constructed to describe a set of facts which clarifies the causes, context, and consequences of those facts." The second meaning has to do with a description: an illustration, exposition, or demonstration of some idea.

"Explanation" in Eliezer's post 'Fake Explanation' has the first meaning, about reason and cause.

"Explanation" in my post 'Great Explanations' has the second meaning, about illustration, exposition, or demonstration.

Suppose I said a particular hockey goal was a "fluke" (a stroke of luck). Then, you pointed me to a post that implied that hockey goals are not good examples of flukes (the fish). This would be a confusion caused by the fact that word "fluke" (a stroke of luck) has the same spelling and pronunciation as the word "fluke" (a type of flatfish).

Suppose you said "That bird is a fine example of a robin!" and I insisted it can't be, because American robins have a red breast. Then you clarified that you were thinking of European robins, a very different bird that happens to be called the same thing. I would point out that my reasoning works just as well in that case, as European robins also have a red breast.

This seems more like what Vladimir was doing than your fluke example. And note you didn't answer this question (though you thought you did):

how many readers of these books would realize that something's wrong?

I agree that some of Vladimir's comments can apply to both meanings of "explanation," I was just trying to avoid conflating the two. I also think that most readers of pop-sci books generally wouldn't be able to tell if something's wrong, because most readers don't check multiple sources and use Google Scholar to find the papers being cited.

What I'm disagreeing with is Vladimir's original claim that "it's impossible to gain any real understanding of physics from such materials."

Almost all of the Sequences consist mostly of helpful explanations of concepts without going into full mathematical detail. I gave additional examples of non-mathematical explanations that improved my understanding in ways that informed my actions here, and wedrifid gave his own examples here. I gave one more here.

Because these all seem to be instances of improving someone's understanding of some idea in a useful way without them acquiring full mathematical understanding of that idea, I wondered if perhaps Vladimir was using an unusual definition of "understanding" such that for him "understanding" or "real understanding" only referred to detailed mathematical understanding. But in that case his original claim that "it's impossible to gain any real understanding of physics from [materials that don't impart a detailed mathematical understanding]" is basically tautological, and doesn't undermine the rationale for the original post ('Great Explanations').

Vladimir also seemed to claim that I can't have evidence about whether Copenhagen is correct or incorrect without understanding all the equations involved. But that seems obviously false, so I asked if I had misunderstood him.

"Explanation" in Eliezer's post 'Fake Explanation' has the first meaning, about reason and cause.

Fake explanation is also an incorrect description, so both senses you listed qualify. Explanation of a reason for something is also a description of that reason.

"Explanation" in my post 'Great Explanations' has the second meaning, about illustration, exposition, or demonstration.

It still seems mostly impossible to give adequate explanations in this sense. For highly technical topics, a fair share of difficulty is in understanding what the concepts are, so you won't be able to see what the relevant facts mean without already going a long way towards knowing why they hold.

Suppose I said a particular hockey goal was a "fluke" (a stroke of luck). Then, you pointed me to a post that implied that hockey goals are not good examples of flukes (the fish). This would be a confusion caused by the fact that word "fluke" (a stroke of luck) has the same spelling and pronunciation as the word "fluke" (a type of flatfish).

I cannot find it online, but there is an essay called "Rural literalisms" or something to that effect written by a guy who moves to the countryside and is constantly being thrown by the way his neighbors use expressions literally. Among others occasions, he's very proud of himself when he catches his first fish, only to be told by his neighbor, "It's just a fluke." These aren't homonyms; it's one word being used metaphorically in other situations.

I'm curious about why this was downvoted. Is there something in this comment that is incorrect? Do people disagree with my statement that 'explanation' has two meanings, or something?

I didn't downvote that comment, but might have, if I followed the conversation live. My thinking when I read it was: "He can't possibly really think that it is a homonym! So, for the sake of the argument he arrogantly (because that all caps spelling does show off some arrogancy) distorts reality and expects us to accept it?!"

But, now I see that this is too much of a correspondence bias. You probably just wanted to show that "explanation" has two different meanings, but in the head of the discussion just found a very bad example for your argument. Because "explanation" does have two slightly different meanings and this is relevant here. But let's be clear, these two meanings are close and in no way homonyms (as opposed to what you stated and what you clearly tried to present with the "fluke" example).

So, I think this comment of yours is bad and the downvotings were valid.

Edit: I didn't read the wikipedia article you linked when I wrote the above. I only ever heard/saw "homonym" being used in the sense of "two identically spelled and pronounced words with different meanings and of unrelated origin"; what Wikipedia calls "true homonym". In the more general sense the two "explanations" might qualify as homonyms (I am definitely not a linguist). But, your "fluke" example strongly indicated the more narrow (and, I think, more common) meaning. So, my reasons still stand.

My best guess: the comment seemed generally rude and "talking-down".

While I can see the motivation for typing "E-X-P-L-A-N-A-T-I-O-N", it seems to me like it would seem to people as somehow mocking, especially on a cursory read.

If the material from some prominent pop-science book were rearranged into something written in a similar style but in fact completely wrong and nonsensical and signed by an equally high-status author, how many readers of these books would realize that something's wrong?

By checking multiple sources, like I always do.

In any case, what exactly are these more accurate predictions about the world that pop-physics enables you to make?

My statement was about pop-evolution, but for starters...

My (mostly) non-mathematical understanding of contemporary physics allows me to understand, for example, that the most popular expression of quantum physics — the Copenhagen interpretation — is probably incorrect. This allows me to avoid confusions that may result from taking Copenhagen seriously, for example thinking that probability is in the world.

My non-mathematical understanding of contemporary physics also allows me to see that the "same atoms" account of personal identity is incoherent, which in turn increases my probability that I could survive cryonic preservation and revival, which informs my decision of whether or not to sign up for cryonics.

My non-mathematical understanding of contemporary physics allows me to see how the majority of scientists can be wrong about something because they're carrying along outdated philosophical assumptions (verificationism) and don't see why Bayesianism and Solomonoff's lightsaber outperform the qualitative approach to scientific explanation.

I get these benefits and more because I read great explanations of contemporary physics, even though they didn't include all the math you would prefer to include.

I don't know how much actual understanding you have about these issues, but if you really believe you understand them in some "non-mathematical" way, you are fooling yourself. Considering that all these are prominent recurring themes from the LW sequences, if you have no independent knowledge of these areas as a solid foundation for your opinions about them, it is reasonable to conclude that you have let your enthusiasm for the underlying philosophy of these sequences lead you to an illusory "understanding" that is in reality sheer rationalization.

Now, I don't think one could even state a workable definition of the Copenhagen interpretation without a sizable mathematical background, so that your self-confident assertion that you "understand" that it's "probably incorrect" strikes me as absurd -- let alone your claim that your "non-mathematical understanding of contemporary physics allows [you] to see how the majority of scientists can be wrong" about these issues. (They may well be wrong, to be sure, but I don't think you have any real evidence either way.) And what are the "predictions about the world" that the supposed wrongness of Copenhagen enables you to make anyway?

As for your assertion about the implications of QM on the questions of personal identity, this looks even more as a belief that you've taken on faith, backed by sheer rationalizations. (Again, regardless of its actual merits when the real arguments are considered -- I'm not saying that it's incorrect, merely that you don't have any good reason to believe either way if your grasp of the issue is entirely non-mathematical.)

I should add that I have no formal background in physics, but I do have a decent background in math (nowadays sadly a bit rusty), and I have spent quite a bit of effort over the years trying to get an accurate basic understanding of the fundamental physical theories out of sheer intellectual curiosity. And while I have managed to get a basic grasp of relativity, I am still nowhere near having a clear intuitive understanding of the fundamentals of QM, despite having spent a lot of time trying to get it, and even though I can handle the math of Hilbert spaces, Schroedinger equations, etc. (And yes, among other things I have read the LW QM sequence too.) To me it seems inconceivable that someone could gain such understanding in a "non-mathematical" way, based only on pop-science books and the LW sequences.

As for your assertion about the implications of QM on the questions of personal identity, this looks even more as a belief that you've taken on faith, backed by sheer rationalizations. (Again, regardless of its actual merits when the real arguments are considered -- I'm not saying that it's incorrect, merely that you don't have any good reason to believe either way if your grasp of the issue is entirely non-mathematical.)

Leaving aside for now the question of understanding of Copenhagen validity but as for the specific claim about knowing enough contemporary physics to understand the implications to personal identity your rejection is just nonsense. You most certainly can gain enough knowledge to make conclusions about personal identity without knowing math.

Ask an impressive physicist:

"Dude are, like, atoms and combinations thereof in any way uniquely identified?"

He says "nah".

You say "kk"

From there you have some utterly trivial philosophizing to do to reject ideas of "same atoms for personal identity". This is a trivial question and basically relies on not being philosophically incompetent while also checking with a physicist just in case some relevant, surprising and bizarre phenomenon has appeared recently at incomprehensibly high levels of physics.

You most certainly can gain enough knowledge to make conclusions about personal identity without knowing math. Ask an impressive physicist: [...]

The critical step however involves asking a physicist and accepting his opinion on authority. Such evidence is by no means invalid, to be sure, but it's something quite different from the original context, which was about understanding things yourself. (Plus, the weight of this evidence should be discounted due to the fact that the question is, strictly speaking, outside of the physicist's immediate domain of expertise, and dependent on issues that raise significant controversy.)

if you have no independent knowledge of these areas as a solid foundation for your opinions about them, it is reasonable to conclude that you have let your enthusiasm for the underlying philosophy of these sequences lead you to an illusory "understanding" that is in reality sheer rationalization.

I haven't read the quantum physics sequence on Less Wrong. I got my physics from lots of other sources.

if you really believe you understand them in some "non-mathematical" way, you are fooling yourself... it seems inconceivable that someone could gain such understanding in a "non-mathematical" way

Are we just disagreeing about the meaning of "understand" or something? Are you using the word "understanding" in an unusual way, such that there is no such thing as non-mathematical understanding?

Also, at one point you seem to say that I can't have evidence about whether Copenhagen is correct or incorrect without understanding all the equations involved? That seems too obviously false; I assume I'm misunderstanding you?

For what it's worth, I agree with Vladimir_M. Eliezer has a nice post about this. One way to test your understanding of physics would be to google for problem sets on special relativity, or something, and see how many problems you can solve.

For what it's worth, I agree with Vladimir_M.

About what in particular? Just the vibe of "How dare you? Get off my lawn!" or is there something in the comment you are actually replying to which you would dispute? If the latter then I must affirm the observation "That seem too obviously false" and, like Luke, apply the benefit of the doubt. That would mean assuming that you aren't really responding to the grandparent at all and more just chiming in with a general sentiment.

One way to test your understanding of physics would be to google for problem sets on special relativity, or something, and see how many problems you can solve.

That does sound like fun. It's been too long!

Linking this back to the question of verbal and conceptual vs mathematical understanding I note that some of these could (and probably should) be verbal problems. Of the kind that use a broad basic understanding of the physics principles and not just the juggling about of numbers and manipulating some memorized equations. Because that really is useful knowledge (as well).

Some verbal problems on special relativity:

  1. Alice is running very fast holding a long pole. The pole is held parallel to the direction in which she is running. She's running into a barn with an open door. When the pole is stationary relative to the barn, it does not fit inside it completely. A tiny bit sticks out, preventing the door from being shut. However, since the pole is now in motion, Bob, who is standing by the barn door, sees its length contracted, allowing it to fit completely inside the barn. This means that Bob can wait for Alice to enter the barn and then shut the door. Let's say he shuts the door as soon as the leading edge of the pole makes contact with the barn wall opposite the door. But from Alice's perspective, it is the barn's length that has contracted, not the pole's. From her perspective, how do you account for the fact that she is able to run the pole completely into the barn so that Bob can shut the barn door?

  2. The usual response to the twin paradox is that the twins' situation is not symmetrical because the one who leaves Earth must undergo non-inertial motion in order to turn around and return. However, it is possible to reproduce the twin paradox without either twin accelerating, and without moving to curved spacetimes. Can you think of how one might construct this situation?

  3. We all know that according to special relativity it is impossible for a massive object to accelerate until it catches up with a photon. However, it is possible for a massive object to accelerate so that, in its rest frame, its distance from a photon always remains the same. What would the word-line of such an object look like?

  4. We have two spaceships A and B, initially flying at the same uniform velocity. A is flying behind B, and there is a taut string connecting the nose of A to the tail of B. The spaceships both accelerate so that an observer who starts out at rest relative to them will measure the distance between them as unchanging throughout the acceleration. When the ships stop accelerating they are moving relative to the observer. This means the observer should see the string as Lorentz contracted, but, by hypothesis, the distance between the spaceships unchanged. Does this mean the string breaks? Describe what happens from the perspective of the space-ships.

EDIT: Solutions here.

Solutions, or at least sketches of solutions:

  1. Sbe Nyvpr, gur cbyr'f raq uvggvat gur onpx jnyy naq Obo pybfvat gur qbbe jvyy abg or fvzhygnarbhf. Jura gur cbyr uvgf gur onpx raq bs gur jnyy, fbzr bs vg jvyy fgvyy or fgvpxvat bhg bs gur onea. Ohg gung ovg jvyy pbagvahr zbivat vagb gur onea, orpnhfr vg unf abg lrg unq gvzr gb erprvir gur vasbezngvba gung gur bgure raq bs gur cbyr unf uvg n jnyy. Gung vasbezngvba pnaabg cebcntngr snfgre guna gur fcrrq bs yvtug. Guvf zrnaf gurer jvyy or n crevbq qhevat juvpu gur onpx raq bs gur cbyr pbagvahrf gb zbir nybat jvgu Nyvpr juvyr gur sebag raq unf orra fgbccrq ol gur jnyy. Gur cbyr qrsbezf, nyybjvat vg gb svg vagb gur onea ol gur gvzr Obo fuhgf gur qbbe.

  2. Bar cbffvovyvgl vf gb unir n plyvaqevpny fcnprgvzr, jurer fcnpr vf xvaq bs yvxr Cnpzna. N crefba geniryyvat ng havsbez irybpvgl va nal qverpgvba jvyy neevir onpx gb jurer fur fgnegrq sebz. Guvf vf n syng fcnprgvzr. Va guvf pnfr, bar gjva pna sbyybj na varegvny cngu gung jvaqf nebhaq gur plyvaqre, juvyr gur bgure gjva sbyybjf n cngu gung tbrf fgenvtug hc gur plyvaqre. Gur nflzzrgel vf qhr gb gbcbybtvpny qvssreraprf orgjrra gur cnguf, abg nppryrengvba.

  3. Gur jbeqyvar bs guvf bowrpg jvyy or n ulcreobyn jvgu gur cubgba'f cngu nf nflzcgbgr.

  4. Gur fgevat jvyy oernx. Sebz gur crefcrpgvir bs gur fcnprfuvcf, gur qvfgnapr orgjrra gurz vapernfrf nf gurl nppryrengr, pnhfvat gur fgevat gb oernx. Vs gur qvfgnapr orgjrra gur fuvcf va gur erfg senzr fgnlf gur fnzr rira gubhtu gur fcnprfuvcf ner nppryrengvat, guvf zrnaf gur qvfgnapr orgjrra gur fuvcf va gurve bja senzr zhfg or vapernfvat (gb pbhagrenpg Yberagm pbagenpgvba). Guvf vf Oryy'f fcnprfuvc cnenqbk, qvfphffrq va terngre qrgnvy urer.

  1. Jung vs gur gjvaf ner zbivat nebhaq gur plyvaqre ng gur fnzr fcrrq va bccbfvgr qverpgvbaf? Gung'f flzzrgevp.

Gubfr gjb cnguf ner vaqrrq flzzrgevp, naq gubfr gjvaf jvyy or gur fnzr ntr jura gurl zrrg. Gur fcnprgvzr vagreiny gurl genirefr jvyy or rknpgyl gur fnzr.

This is exactly what I need about when I'm starting to need it. Thank you.

The math of special relativity helps not one whit in solving problems 1, 2 and 4. Problem 3 of course can be answered qualitatively without math, but to indicate with any specificity you will need math.

Moreover, you can be quite proficient with the math of SR and be floored by these. I knew an undergrad who could do acceleration problems yet couldn't work his way through problem 1, because the course had focused on gamma and neglected spacetime diagrams.

Spacetime diagrams are the math of special relativity. Doing algebra with gammas without insight on how it relates to the Minkowski spacetime is like the proverbial blind men grasping various parts of the elephant. (Why such godawful approaches are still foisted upon students is a mystery to me.)

I think someone without a detailed grasp of spacetime diagrams should be able to answer problem 1, as long as they know the following principles:

Gur eryngvivgl bs fvzhygnarvgl

Ab fhcreyhzvany fvtanyvat

Just the vibe of "How dare you? Get off my lawn!" or is there something in the comment you are actually replying to which you would dispute?

Why on Earth do you think it's "my lawn" in any sense? I have readily and repeatedly pointed out that I'm a complete amateur in physics, and I claim no status whatsoever as a physicist. I am making a purely negative and reactive point that pop-physics is about fake explanations perpetuated mostly for status-related reasons.

Pardon me. I mean "Get off my neighbor's lawn! You kids these days need to learn respect!"

(In case my message is insufficiently overt I am suggesting you have the 'status-related reasons' approximately backward.)

pop-physics is about fake explanations

Nonsense. Or, again, you are talking about an entirely different thing. Sources with bullshit fake explanations can be found and so can sources that give solid explanations with only limited mathematical detail, summaries and take home findings that have been worked out by others. It is the latter that lukeprog is referring to.

Pardon me. I mean "Get off my neighbor's lawn! You kids these days need to learn respect!"

Your speculations about my motives here are stupendously wrongheaded. If you look at my previous LW comments, you'll see that one of my recurring themes, on which I hammer incessantly, is that in many fields, people who get officially credentialed as experts under the present system are in fact naked emperors whose supposed expertise couldn't withstand scrutiny by a smart amateur who tries to make honest sense of it. (And also that excessive faith in credentialed expertise is a widespread and under-appreciated bias even among many LW contributors.) Assuming that I would act as a self-appointed guardian of a credentialist intellectual monopoly is absurd given the views I have expressed loudly and repeatedly.

In fact, if we're going to discuss the workings of the present system, the mainstream respectable view is that as an amateur, one should pay homage to credentialed experts in physics for their pop-science writings, extol these writings as a great source of enlightenment, and raise one's own status by being an owner and reader of such books. So rather than looking at my claims as upholding the intellectual monopoly of credentialed experts, you can view them as attacking the way these experts abuse their position for monetary and status gain by getting into these enterprises in pop-science. Such enterprises are supposedly educational and enlightening, but in fact entirely obscurantist and subservient to upholding the existing credentialist and bureaucratic hierarchy. (This even aside from the purely venal interest involved.)

Are we just disagreeing about the meaning of "understand" or something? Are you using the word "understanding" in an unusual way, such that there is no such thing as non-mathematical understanding?

No. Some things can be understood without mathematics, and some can't. I'm just claiming that QM and relativity (including the basic intuitive understanding of these fields) fall into the latter category, and by extension also various fields of modern physics that have them as prerequisites.

Also, at one point you seem to say that I can't have evidence about whether Copenhagen is correct or incorrect without understanding all the equations involved? That seems too obviously false; I assume I'm misunderstanding you?

I have no idea what you mean by "all the equations involved"; I certainly never mentioned any such thing. There are many different ways in which Copenhagen might be formulated, which may involve different mathematical concepts and equations used to express them -- but any formulation that rises above fuzzy and obscurantist fake-explanation talk must necessarily have a mathematical basis. I mean, if you're going to talk about "collapse of the wave function," you'd better have a solid understanding of what a "wave function" is and what exactly it collapses into.

Luke, do you agree there is no such thing as a non-mathy understanding of graph theory?

I think Vladimir is saying physics is like that. Because when you take away the math, you are no longer able to explain what is really going on.

Can such an explanation really be called "great"?

I think Vladimir is saying physics is like that. Because when you take away the math, you are no longer able to explain what is really going on.

Is that the right link? Because the that post, "Guessing the Teacher's Password", gives a purely verbal description of a object getting heated up and turned around. Explicit mathematics doesn't come into it either on the part of the successful student or the reader of the document. Basically it provides yet another example which reduces Vladimir's claims to absurdity.

When I was young, I read popular physics books such as Richard Feynman's QED: The Strange Theory of Light and Matter. I knew that light was waves, sound was waves, matter was waves. I took pride in my scientific literacy, when I was nine years old.

When I was older, and I began to read the Feynman Lectures on Physics, I ran across a gem called "the wave equation". I could follow the equation's derivation, but, looking back, I couldn't see its truth at a glance. So I thought about the wave equation for three days, on and off, until I saw that it was embarrassingly obvious. And when I finally understood, I realized that the whole time I had accepted the honest assurance of physicists that light was waves, sound was waves, matter was waves, I had not had the vaguest idea of what the word "wave" meant to a physicist.

(They may well be wrong, to be sure, but I don't think you have any real evidence either way.)

Sociological data about trends in opinions, the opinions of newly tenured people, about the opinions of people in the newest branches of the field, etc. don't count as evidence?

Sociological data about trends in opinions, the opinions of newly tenured people, about the opinions of people in the newest branches of the field, etc. don't count as evidence?

Yes, that certainly counts as evidence if you're asking a yes-no question about whether one of these statements is true. But I think it's clear from the context that we're talking about evidence from real understanding of the matter, not just indirect evidence based on judging of what sorts of people believe what. Even though, of course, the latter can be perfectly valid evidence.

If the material from some prominent pop-science book were rearranged into something written in a similar style but in fact completely wrong and nonsensical and signed by an equally high-status author, how many readers of these books would realize that something's wrong?

By checking multiple sources, like I always do.

This reply seems to make more sense if I assume that lukeprog cut and pasted the 'would' over the 'many' while reading. I suspect the answer to the actual question to be along the lines of "Probably relatively few, but lukeprog would be among them, he always checks multiple sources!".

Yvain's Parable of the Heartstone is by far the best explanation of metaethics that I have read. (I am actually surprised how much better I find it than other explanations. Does anyone know of something of similar quality that addresses the same things?)

That whole FAQ is one of my favorite documents ever. It's certainly the best explanation/defense of utilitarianism and consequentialism that I've ever seen.

I don't think Eliezer's Introduction to Bayes's Theorem should be on here. I seriously don't think it's that good. It labors its points, and after I read the whole thing I still didn't get that you could use it to judge between different hypotheses, which is pretty much the most amazing thing I've learned this year, incidentally.

However, his new version, which he's working on and I got to read when I volunteered as an illustrator briefly, is absolutely amazing. When he gets that one finished, it will deserve its place on this list.

I agree. I read Eliezer's Introduction to Bayes's Theorem three times and it didn't click. When I watched the Khan Academy video Probability (part 7) it clicked.

Lazy Functional Programming: Learn You a Haskell for Great Good by Miran Lipovača

It's a bit long to qualify under requirement #2, so in particular I'd like to highlight chapters 11-13 as a great explanation of functors and monads. (Albeit one that requires you already grok haskell, hence the recommendation for the whole book).

I highly recommend it because it has a cute blue elephant on the front cover and a bonus monkey on the back.

It's likely worth specifying that Haskell is not only functional, but also lazy.

Okay, I don't mean to be annoying, but I'm curious if anyone else ever thinks this way.

Right after I read this: "Those who do it well are rare and valuable", this is what I automatically thought: 'Okay, so he's setting himself up to, in the future, pursue a career in explaining, and this sentence/article functions as a tool of justification by making the value of the endeavor "objective" through writing it here'

That is, some part of me sometimes leaps away from the normal way of reading--which sees things as from the writer to yourself--and considers the writer as writing from himself to himself, unconsciously using writing as a selfish tool to shape the narrative of his/her life.

I think this has something to do with the fact that I am probably way too pessimistic about humans in general and think that much of what we say masquerades as something virtuous like reasonableness but is really functioning in a very self-oriented way to affirm and justify oneself and that one's life is a good (enough) life. Has anyone else's thoughts while reading anything ever flinched in this direction? [p.s. this has nothing to do with lukeprog, in fact he's probably my favorite writer here]

Hi, thank you for creating this great list!

Aumann’s agreement theorem: Landsburg, The Big Questions, chapter 8.

This link is broken, and I was not able to find this chapter separately anywhere. I would appreciate if someone would be able to update the link or re-upload the document (or recommend another good article on this topic). Thank you.

Does anyone know of a great explanation of the very early universe, preferably from the past 5 years?

It's not detailed, but the 12th edition of University Physics, written in 2007, has a decent introduction to it.

The Cosmos TV series did a pretty brilliant job at the topics it covered.

...as long as you don't mind listening to Sagan drone out "millions, and billions, and millions" for millions, and billions, and millions... basically number-novocaine delivered verbally.

And surely aliens are everywhere, we just haven't noticed them yet.

I tried watching Cosmos about a year ago, and quickly stopped. Is there a case to be made that it's worth soldiering through the awfulness?

My takeaway: Sometimes people don't behave in aggregate the way we think they should. By replacing their money with money\k and convincing them it's still just money, we can manipulate their behavior by jiggling k.*

And it apparently goes without saying that the coupon-issuer has a good way to distinguish "legitimate" reasons to cut back on going out. E.g., flu outbreak, new compelling indoor family activity, all the other stuff no one's even thought of yet, etc.

The Keynesian "key to enlightenment" is that we can cram a knob onto the economy and jack with it?

FORTH is one of the most interesting programming languages, seeing as an implementation can be built in an afternoon of coding, in somehting as low level as assembly language. FORTH is home to many interesting ideas of metaprogramming, and serves as a worthy counterpart to LISP, albiet in the other end of the abstraction scale.

If you are familiar with assembly languages in general, the GNU Assembler's syntax or the Intel x86 platform, you might want to give Jonesforth a read. It is a well written Literal Programming style implementation of a FORTH for the x86 platform.

Also worth reading: eForth overview (there's something very annoying going on with the page style that makes the text lines always be just a bit wider than my browser window, no idea what that's about). This is about how a Forth can get started from an absolutely minimal kernel of native assembly code, and how the rest of the necessary words are defined in Forth.

Pretty much any chapter of Mermin's It's About Time is the best explanation I've ever read of its content matter, and he is not shy of giving you the maths to back up the intuition (or even to motivate it in some cases). If I had to pick one chapter, it would, perhaps oddly, probably be the first one, which explains Galilean relativity extraordinarily well.

General applicability of Bayesian inference: Judea Pearl, "Probabilistic Reasoning in Intelligent Systems", chapter 2. (Definitely not an explanation suitable for a teenager, but for a college student interested in the topic it is very good, I think.)

The first chapter of Tim Maudlin's Quantum Non-Locality and Relativity is a great explanation of Bell's theorem.

I doubt Tim Maudlin's (2011) explanation of Bell's theorem is the best available.

Skimming his chapter, I was troubled by his incorrect assertion that the GHZ state hasn't been experimentally demonstrated(!?). This is the kind of error you should only witness if a book was published back when that was true (in the 1990s) or if the author was an aging professor who was divorced from the experimental physics literature for the past ten years. Maudlin's effort is the latter.

See:

2000: First experimental demonstration of GHZ state (with photons)

2010: More recent experiment (with fully characterized, physical qubits)

If you want to learn quantum physics, I recommend skipping all textbooks written by career physics professors. Go straight to: Quantum Computation and Quantum Information by Nielsen and Chuang. They are experimentalists who actually needed to build quantum computers... so they had to write their textbook to make sense since most quantum physics texts are variously incomprensible and/or wrong. As with most fields, quantum physics textbooks only got good once there was a terminal application beyond "teaching".

You're right about the GHZ thing. The first edition of the book was published in 1994 and it looks like the appendix on the GHZ state hasn't been updated since then.

But this oversight has little bearing on Maudlin's explanation of Bell's theorem, which is, after all, a purely theoretical result. It's an excellent explanation, sophisticated but also accessible. Nielsen and Chuang is a great textbook, but it clearly does not meet the criteria laid out by Luke. A layperson could not just pick up their discussion of Bell states and understand it.

I don't have it at hand, but I recall the explanation of entropy, temperature, and the Boltzmann factor presented in the first chapter of "Thermal Physics" by Kittel and Kroemer being particularly clear, elegant, and direct.

More recently I enjoyed "Probability: a Survey of the Mathematical Theory" by John Lamperti. It is great for understanding things like Kolmogorov's 0-1 Law.

I recommend Spacetime Physics by Taylor and Wheeler for special relativity. This is a mathematical textbook however it only requires basic algebra and is accessible to highschool students.

For an overview of the evidence for evolution and why it is true (but not how evolution itself works), I really like the first chapter of Neil Shubin's Your Inner Fish [PDF].