Sunday, May 25, 2008

Text + Equations = Clarity ?

I want to pick up a discussion that emerged out of my previous post "The Illusion of Knowledge" in which I mention that I dislike the fear of using equations in pop sci books, and that I also very much appreciate further references to follow up on a topic. Equations make a verbal explanation much more precise, and if possible I think one should at least provide the basic relations or definitions.

"I think that the average professional in the theoretical physics need at least five years to learn the math which I consider directly relevant to formulation of Relativistic QM (QFT). After that he needs to develop the necessary supplementary math on his own."

I explain that one can extract some knowledge from equations even without understanding the full context: First, there is the benefit of knowing how a system is quantified to begin with. But more importantly, there is some information that can be rather easily extracted, like what fields couple to which other fields, or how sources affect the field. General parameter dependence of solutions e.g. how the number of dimensions affects Newton's law. Questions of definition, like what is the cosmological constant, or what are the dimensionalities of quantities, scaling, and so on.

Let me call that 'reading maths', in contrast to 'speaking maths' in which you'd have to find a mathematical description for a system you want to investigate which is considerably harder. I think with a bit practice most people could learn to 'read' maths if they get used to it. If I open a newspaper I also wonder how many readers actually understand the details of the business part. But they are just used to the terms, and after a while they get some notion of what's going on. Using equations in the science part wouldn't be all that different.

Dany then argues that using equations and further references isn't nice for the reader because

"after first reading one remains with the feeling that it is interesting but only illusion of understanding and knowledge"

And yes, that's exactly the impression I want the reader to have. If you've had to argue with people who believe the Hawking effect is nonsense because in 'The Brief History of Time' it's written that it requires the existence of anti-gravitating particles, you'll understand what I mean. In fact, this was one of the aspects I meant to emphasize in that previous post: It has noticeable drawbacks to leave readers with the 'Illusion of Knowledge'. This drawback esp. in the science area is that readers come to believe it's all trivial and obvious and they don't even need to take a physics class to conclude that String Theory is nonsense, the LHC piles dogs, Special Relativity is wrong, and the tenth dimension is where the consciousness lives. After all, equations are unnecessary.

Example: I am still waiting for somebody to point out if you believe Lisa Randall's explanations for why mini-black holes at the LHC evaporate very fast: "just as a small drop of coffee evaporates more quickly than a big one" (Warped Passages, p.380, US Hardcover) then the temperature doesn't increase when black holes get smaller as is the case with Hawking's radiation. The problem isn't with the reader who just swallows this, but with the reader who takes such a metaphor literally and then tries to build upon it. I am not against using such verbal explanations, I would just wish the limitations of their usefulness would be made clear. (The actual mass-loss rate is the surface times the fourth power of the temperature, and the temperature grows with the inverse of the radius).

Dany further argues:
"I would not provide you any tools “to dig a bit deeper” since I honestly respect my reader"

If a book doesn't come with further references to dig into a topic this a) lowers the usefulness of a book dramatically, b) is an indication the author either doesn't know the literature well, or didn't take the time to scan it so limits his/her credibility and c) it's just unscientific. If you write about a topic without explaining all details you should at least provide a reference to look them up. If pop-sci books don't do that how can you expect millions of people on the internet to appropriately references their sources?

I agree with Andrew who mentions:
"You can have equations in a pop science book as long as you also include a written description so people can avoid the equations if they so wish. But people who want to dig a bit deeper could consider the maths. So it appeals to the widest audience. At the very least, they should have references to arXiv papers."

Yes, some references at least to arXiv papers or so are very useful - just that not every field (yet) has something like the arXiv. See, if I read a pop sci introduction into a field I don't know very well but I find interesting, I just want to have some recommendation for where to continue reading, preferably by somebody who has an overview on the literature and not by a Wikipedia article.

To add another aspect: when I was a kid I read several pop-sci books on Special and General Relativity. There is a huge abundance of literature that will tell you a lot about rockets and elevators and trains and twins and flying angels and signals and clocks and again the rockets, the rockets are everywhere. It's not that I totally didn't understand these explanations, I just didn't know what to make out of all the extra baggage. And some of these explanations are not very insightful.

I recall there was an explanation in a book which was a construction meant to 'proof' length contraction. Unfortunately, I assume now it was a construction done with the knowledge of what the outcome should be, but not meaningful in itself. So I went and 'proofed' it nonsense and showed it to my physics teacher hoping for some insight. All my teacher said was 'We know Special Relativity is correct'. Which didn't help me anything. However, today I can kind of understand his hesitation to take apart somebody's fancy construction of flying rockets and clocks and so on.

Special Relativity began to make sense to me the moment I learned what SO(3,1) is. No more rockets. What a relief. Okay, it is maybe a matter of taste but I found the maths underlying Special Relativity more insightful than all the thought experiments. It's not that I would want to throw verbal explanations out, just I'd prefer to use them as a motivation instead of a substitute. (And no, understanding what a orthonormal transformation is isn't all that hard and can nicely be explained with some pictures. The part with the Lorentzian signature is a bit more tricky. To the German readers I warmly recommend Ulrich Schröder's book on Special Relativity.)

I got a lot out of the magazine Spectrum (the German version of Scientific American) and Bild der Wissenschaft (which is a very similar German magazine), both of which usually provided at least one or the other equation and some references. (At this time however it was for me almost impossible to get a hand on the mentioned references. This would be considerably easier today.)

On the philosophical side it is quite interesting to which importance thought experiments (Gedankenexperimente) have grown since Einstein.

Anyway, what is your opinion on having equations in pop-sci books?

@ Dany: I didn't mean to pick around on your comments because they are so outrageous, they just made for a nice pro/con situation.

1. Well, yes, I agree with you entirely. I'll just repeat what I said earlier that I don't like just being "told" something in the general sense in pop science books, I'd like to know a bit about "how" they got to the conclusion. So some sort of derivation would be ideal, but if that's not possible then at least provide a list of further reading. And if people don't want the equations they can just skip them.

But, of course, there's a huge rule in pop science that "each equation results in 1,000 lost sales", or something like that. And some books boast of their lack of equations. Shame.

You also made a very good point in your earlier piece that so many pop science books are just endlessly repetitive. How many books on Schrodinger's Cat do we really need?

2. But, of course, there's a huge rule in pop science that "each equation results in 1,000 lost sales", or something like that. And some books boast of their lack of equations. Shame.

Yes, I've heard of that myth. I wonder if somebody actually has any reliable evidence for this. As I said above, I think it is mostly a matter of getting used to it. I know there are many people who will upon being faced with an equation say 'I was always bad at maths', irrespective of the fact that most of what you do at school isn't maths but calculus. E.g. the concept of what a differential equation is is so essential to physics and not that complicated to understand, how come this isn't school-book stuff?

Anyway, questions of whether or not equations are cool are strongly influenced by social trends. I hope the fear of equations will vanish soon. Maybe the backboard helps for this purpose ;-)

3. "just as a small drop of coffee evaporates more quickly than a big one"

That is twice true. Relative for surface to volume ratio. Absolute for smaller radii of curvature having higher vapor pressures.

"Every Child Left Beind" boldly divorces mathematics from observation and analysis. Make everybody equal by cutting them down to size. Public shcool curriculum is so knowledge-free that half the students, on the average, are not below average; 100% are.

Support evolution - shoot back.

4. Hi Uncle,

Yes. I didn't say it's a bad comparison. But it's a comparison and it has its limitations. E.g. as I said above when it comes to the temperature of the coffee drop. Best,

B.

5. Dear Bee,

I like the titles of your posts :-)... and in this case, once more, I completely concur with what you write.

Popular science texts which cumbersomely paraphrase in words what could be stated succinctly in one short equation can be an imposition to mathematically literate readers. It has happened often enough that I mentally wrote down the equation that to my understanding seemed to be described by a lengthy fomulation - but could I be sure if I got it right when it is not in the text?

The same for references, and for endnotes: I usually like to see where I could look up details, or at times to read about some subtleties disregarded in the main text.

I've come to regard books without any endnotes as the equivalent of TV entertainment - it may have it's purpose, but it shouldn't be taken too serious. I mean, the times of authoritative writers who can afford omitting any references - as did Heisenberg, Einstein, Planck and their peers in their popular writings - are over. On the contrary, in my opinion it shows respect for the reader to provide endnotes, as it means, "look, you needn't just swallow what I wrote, look it up if you want, and draw your own conclusions".

As for math and equations, the problem could be that many people just do not see them as a language that is extremely powerful to state scientific concepts and relations. I mean, you learn in school how to manipulate the symbols, but often not what they actually can mean and what they can express. I have a vague idea that one could handle math in school a bit like Latin: You are supposed to learn how to read and understand it, but not necessarily to actively use it (say, at least the more advanced parts) - that latter part is for "the nerds".

To come back to the pop-science books, I think a good compromise can be to put relevant equations, together with endnotes and references, at the end of the text. That's what has been done (quite short, though) in Lisa Randall's book, and Weinberg's The first three minutes comes with a 13 page mathematical appendix (in my German paperback edition), though not directly linked to the main text.

Anyway, questions of whether or not equations are cool are strongly influenced by social trends.

That could be right... let's hope so that the trend will come ;-)

BTW, are there popular books about music? How do they handle notation?

Cheers, Stefan

6. Hi Bee,

Bee:” Dany: I didn't mean to pick around on your comments because they are so outrageous, they just made for a nice pro/con situation.”

Never mind, but you may present your POV without them.

Bee:” Let me call that 'reading maths', in contrast to 'speaking maths' in which you'd have to find a mathematical description for a system you want to investigate which is considerably harder.”

You repeated your statement from the previous discussion. The repetitions of the same statement without addition explanation don’t add anything. Your statement however is logically inconsistent. The scientific method requires speaking only about what you know and not about what you will know God know when. That is the area of another kind of human intellectual activity called science fiction.

You mixed newspaper with the mathematical physics. We do speaking math but since it is impossible to speak with layman our languages we translate it into suitable human language (here it is English). The most brilliant example of that known to me is THE FEYNMAN LECTURES ON PHYSICS.

I try to do the same. Every statement made is directly supported by the math relation. You completely distorted my POV. You misinterpreted my position since my English is poor, but I am speaking nearly fluent about 20 math languages.

I didn’t mention S.Hawking, L.Randall, String Theory, LHC piles dogs, extra dimensions and mini-black holes (the truth is that I like mini, but at the women and get exited when it evaporates). I used two extreme and in my POV the opposite examples of popsci literature: “The Evolution of Physics” and “The first three minutes”. The facts are:
1) There is no single math equation or expression in “The Evolution of Physics”;
2) There is no single ref in “The Evolution of Physics”;
3) Besides A Mathematical Supplement there is one single math relation in “The first three minutes” (first ed., 1977):

7/4+7/4+1=9/2 (p.103);

4) There are QED Feynman Diagrams presented on Figure 10 (p.136);
5) There are refs in the “Suggestions for the Further Reading”; they contain the self refs on S. Weinberg three times: “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity”, “Unified Theories of Elementary Particle Interaction”, “Recent Progress in Gauge Theories of the Weak, Elecrtomagnetic, and Strong Interactions”; they contain single ref on A.Einstein et al. Et als are: H.Minkowski and H.Weyl.

Bee, you didn’t identify my POV. Pity. It is identical to N.Copernicus, G.Galilei, I. Newton, P.Fermat, W.R. Hamilton, J.C.Maxwell, A. Einstein, E. Schrödinger and C.N.Yang.

Regards, Dany.

P.S. The causality (the second relativity postulate; The Principal Physical Postulate: Rule of the Game) requires SO(1,3).

P.P.S. Stefan:” BTW, are there popular books about music? How do they handle notation?”

I suggest reading A.Schweizer, J.S.Bach; complete ref you easily may find using Wiki.

7. Hi Dany,

Huh? Yes, I repeated my previously made statement because I thought not everybody might have read it.

I did not say I consider your comments as outrageous. I certainly don't. I used them because they are interesting - that's what my endnote was meant to convey. I wrote a lengthy reply to your last comment on the previous thread but then thought it got to long and was already quite off the topic, so I put it into a new post.

If I did not capture your point of view I am genuinely sorry, but your comment above does not clarify what I misrepresented.

I never claimed you said anything about dogpiling at the LHC, and it was not my intention to raise this impression.

You repeated your statement from the previous discussion. The repetitions of the same statement without addition explanation don’t add anything. Your statement however is logically inconsistent. The scientific method requires speaking only about what you know and not about what you will know God know when. That is the area of another kind of human intellectual activity called science fiction.

The scientific method requires you to discard what you have talked about when proven wrong or shown to be unfalsifiable. It does not require you to speak only about what you know. If that was the case we'd never get anywhere. I am actually not sure what you are trying to say here as I don't see what this has to do with my statement that you are criticising.

What I am saying is it makes a difference if you

A) present a model, say, about the universe, including the relevant equations and explain the central quantities, their relations, and how it describes the observations that we have ('reading maths'). Or

B) present the observations and ask the reader to come up with a model of the universe to explain them. That's a much harder task that requires a great deal more of education ('speaking maths').

I hope that clarifies it.

Since you claim I did not identify your point of view, then maybe you could tell me what it is? Do you, or don't you prefer equations and references for further reading in pop sci books?

I understand you didn't like the way this was done in The First three Minutes? Well, I already told you I didn't read the book so can't say anything about it, but besides this I don't know what you are saying except that you didn't like the book. Are you saying this is a reason why one generally shouldn't do it?

Bee, you didn’t identify my POV. Pity. It is identical to N.Copernicus, G.Galilei, I. Newton, P.Fermat, W.R. Hamilton, J.C.Maxwell, A. Einstein, E. Schrödinger and C.N.Yang.

Again, this doesn't help me much to understand your POV that I allegedly didn't identify. Except for a quotation by Hilbert whose relevance for the discussion I couldn't decipher you didn't explain what you are referring to. Einstein certainly used equations e.g. in his book 'Über die Spezielle und die Allgemeine Relativitätstheorie'. Whether or not an equation is useful and or necessary depends of course on the scope of the book.

Besides this, times change. One would hope that over the course of a century the knowledge of the average reader one can build upon increases since it becomes part of the standard education.

Best,

B.

8. I wonder if there really is a disadvantage to including formulas in a popular science book. I suspect that the professional book people who advertise and sell these books are not necessarily the same as the audience who buys them. But I think you have to let the professionals make the decisions and in this case, the professionals are the book people.

9. Hi Bee,

A most interesting topic and certainly one on which I’d like to comment. As you know and as I have said many times, I am a physics novice, which is to admit that I am far from being an expert or an authority in any of it. I have been this novice from the time I was a young boy to the state I find myself now, many decades later. My formal education in such doesn’t extend much past high school and what little that I do understand I accumulated from a variety of sources, which include, pop science books and magazines, text books, scientific papers and personal correspondence with real physicists as yourself. In having experienced this journey I have come to realize that if one is to depend solely on pop science explanations to serve as the total of ones understanding, it won’t amount to much. There is no short cut that can be found that can avoid this.

As an example of this almost two years ago I purchased Roger Penrose’ book “The Road to Reality-A Complete Guide To The Laws of The Universe” which I am reading as of now for the third time. In the preface of this book Penrose addresses the issue you raise when he states the following:

“The reader will find that in this book I have not shied away fro presenting mathematical formulae, despite dire warnings of the severe reduction in readership that it will entail. I have thought seriously about these questions, and have come to the conclusion that what I have to say cannot reasonably conveyed without a certain amount of mathematical notation and the exploration of genuine mathematical concepts. The understanding that we have of the principles of the actually underlie the behavior of our physical world indeed depends upon some appreciation of mathematics. Some people might take this as a cause of despair, as the have formed the belief that they have no capacity for mathematics, no matter at how elementary a level. How could it be possible, they might argue, for them to comprehend the research going on at the cutting edge of physical theory if they can’t even master the manipulation of fractions? Well, I certainly see the difficulty.

Yet I am an optimist in matters of conveying understanding. Perhaps I am an incurable optimist. I wonder if those readers that cannot manipulate fractions – or those that claim that they cannot manipulate fractions--- are actually deluding themselves at least a little, and a good proportion of them actually have a potential in this direction they are not aware of. No doubt there are some who, when confronted with a line of mathematical symbols, however simply presented, can only see the stern face of a parent or teacher who tried to force into them a non comprehending parrot-like apparent competence—a duty, and a duty alone—and no hint of the magic or beauty of the subject might be allowed to come through. Perhaps for some it is to late; but, as I say I am an optimist and I believe that there are many out there, even among those who could master the manipulation of fractions, who have the capacity to catch some glimpse of a wonderful world that I believe must be, to a significant degree, genuinely accessible to them.”

To conclude what I can tell you is that I entirely agree with Penrose and yourself in this regard. It is true that I never expect to understand all, as for instance to seriously extend these concepts; and yet I can say that in what I have read and learned to understand, has given me at least some footing and thereby an ability to discuss them and more importantly to appreciate them.

Best,

Phil

P.S. I have over the years forced upon myself a strange rule; and that is I was never to put a book up in my bookshelf if I had not read it and more important convinced I mostly understand as well. After all the years many are up there and yet I would be surprised to find if Penrose's wonderful book would ever be among them. Do I find this cause for concern or despair? I think not.

10. Actually, there's another dimension of all this which is even more serious: the way that professional physicists are trained. Bee said very truly that she never understood special relativity until she learned about Minkowski space and its symmetries. But the subject is still taught in terms of "Einstein's postulates" and all that junk about lightning striking trains. The problem is that many physicists --- especially American ones --- develop a very powerful sentimental attachment to the things they were taught in their first year at university. The result in this case is an irrational attachment to Lorentz transformations, a failure to recognise that they are nothing but the group of symmetries of Minkowski spacetime, so that you must expect them to go away when spacetime is curved. If they had been taught Minkowski space from the start then a lot of people would not say nonsensical things about Lorentz invariance in cosmology etc etc etc. So you see that math-phobia is a problem for professionals too, not just readers of popularizations.

11. As a long time science tourist I find almost all of the popular physics books too dumbed down ('then the cat is either living or dead ...')

Penrose Road to Reality (mentioned above) is great. There's math but he starts with the 'easy' stuff and takes it step by step with back references. I can't comment on the String Theory / Quantum Loop Gravity / Twistor opinions at the end as my comprehension had fallen bellow a threshold but he got me over some hurdles I'd been up against for years.

I don't think the ideas can be presented without the math, at least I haven't read that book.

12. This comment has been removed by the author.

13. (Bee, I know you say you shouldn't be able to edit comments, but surely if your comment is the LAST comment then you should be able to edit it!!)

Penrose's "Road To Reality" is a real mixed bag. Don't worry if you don't understand it, Phil. Some sections of it is genuinely unintelligible, atrociously badly explained, and resemble unedited lecture notes which he has just submitted to the publisher. He's naturally very strong on advanced geometry, and finds it so easy that he makes precious little effort to simplify it for a general audience. But at its best it does reward the determined reader with almost a complete overview of the state of the art, together with tons of those sought-after (and genuinely useful equations!

I've just read chapter 32 on loop quantum gravity which is excellent and ties in with his own work on spin networks, though Smolin describes spin networks better. I've been reading the book for three years on and off, and I've only got string theory and his unusual theories of gravity to go.

Penrose is such a brilliant guy, considered rather a crank for a while but has come storming back into fashion as people realised his work on spin networks, gravity, entropy, twistors, mathematical platonism (30 years before Tegmark) were all ahead of their time.

(I must say, though, anyone wanting the "Road To Reality" in readable form, go to my site!)

14. Hi Andrew,

I never said I don’t understand Penrose, I implied that I probably will never totally understand him and along with it much of all of physics for that matter. I would also maintain there is much to be gathered here by even the professionals. As for your advice on the math chapters I personally have found them to be pretty good. I must say however geometry, trig, vectors and related spatial maths have always been my strong suit, while I have always struggled with the methodology and rules of algebra, as they relate to equations. I try to keep them in the framework of set theory as this can also be looked at from a spatial perspective.

Also, if you find that if you hold to my rule and are able to put Penrose’s Book up on the shelf, I applaud you.

Best,

Phil

15. Hi Bee,

Bee:” Are you saying this is a reason why one generally shouldn't do it?”

I never say to somebody else what he/she should do. I don’t argue with you. I consider exchange of our messages and comments as a private conversation. I don’t see any reason to stop conversation at “The Illusion of Knowledge” and I will answer you there. You are free to stop it any time you want and indeed anybody else is welcomed to participate in the discussion there.

I don’t like what you did here. You are kindly requested to allow me to decide whether I want to join the new discussion and if yes when. Here I quit.

Regards, Dany.

16. Hi Dany,

I didn't write this post to get a response from you, but from other readers. I don't regard blogs or comments on blogs as private communication. They are publicly available. I occasionally use comments in my posts, I have done that before, and I will continue doing so. Best,

B.

17. Hello Bee,
I think your posts on the ‘the illusion of knowledge’ are spot on, and a worry for many of us.

On the use of equations, budding pop science writers in Europe are told 'each equation loses a hundred more readers'. I too wonder where this received wisdom comes from, or how publishers can be so sure....

It's quite a serious issue in science communication. Surely a popular book on physics, without any equations, is a bit like a book on the history of art without any pictures?

It sems to me that equations are not just a means of expression, or an aid to understanding. They play a truly central role of our description of the world, As you know, viewing a single equation from different aspects often leads to different insights, definitions and predictions.

Consider the classic F=ma. Not only does it define force as an influence that changes the state of motion of a body - it also gives us a definition of what we mean by inertial mass (a=F/m). It's hard to see how anyone could truly get a handle on Newtonian mechanics without considering this equation. Indeed, to an observer unaware of the predictive power of equations, science must seem like a large collection of unrelated assumptions.

I've been wondering how to get around the problem for some time and it strikes me that one soluton might be 'equations-as-diagrams'. In other words, I might put the relevant equation in a small figure, rather than in the narrative text. That way, those who wish to deepen their understanding of the passage can do so by studying the figure in parallel to the text. As a bonus, there would be give plenty of room to clearly define the quantities being equated, often the biggest hurdle to including an equation in text.
After all, figures in traditional books play a similar role don't they? a sort of parallel explanation to accompany the text.....perhaps this has already been tried
Regards, Cormac

P.S. This is a very interesting weblog, with your permission I’d like to list it over on my weblog (http:// coraifeartaigh.wordpress.com )

18. SciAm has a related article
about things like story problems hindering math ed.

19. Why do we need equations to understand the univere? If the nature of the universe is mathmatics, then how are we going to learn anything by doing mathmatics? When you learn how to swim, you can't become the water. You just learn how to use the water, trust me you don't need to know a single equation to lap the universe. Other life forms will have already learnt how to swim, long before we even existed.

Modern Language & mathmatics is too complicated, for it to be any use to the genetic evolution of mankind. The use of words and numbers as forms of comunication is poor! We can do better than that! Text + Equations = Stagnation. Bring out a book that only 3 people will buy!

20. Hi Dany

I suggest reading A.Schweizer, J.S.Bach; complete ref you easily may find using Wiki.

thanks - Albert Schweitzer may indeed be a good starting point... Actually, I do not want to read about music, I was just musing that books about music may face a similar problem, that they have to deal with musical notation. Keep it, so that it emands that readers can read notes, or omit it? This seems to be not too different to the questions how to deal with maths.

I just do not know how this is done, or if there is a market of "pop music" books in the first place.

Best, Stefan

21. I'll vote for more equations. All too many books avoid mathematics completely, and it is a shame. A lot of people actually want to be guided through a few equations, and to learn a few new abstractions. Even better than just equations are a few examples. Wheeler did a nice job with his A Journey Into Gravity and Spacetime. He actually did a few calculations relating curvature and velocity, and it was great to see that this isn't all handwaving.

22. Hmmm. All the physicists think equations are good. What a shock!

It all depends on the audience.

It's really hard to write well about physics without them, but there are some good popular physics books - I nominate Kip Thorne's popular GR book (not the "black hole", the other one) for example. Pictures help a lot.

23. I'm a geometer so I second the motion for more pictures :)

24. Hi CapitalistImperialistPig,

“Hmmm. All the physicists think equations are good. What a shock!”

I don’t know if you read all the comments, for I stated I was not a physicist, yet also would like to see more equations with related explanation included in the pop science books.

“It all depends on the audience.”

As it relates to my own experience, I would have to agree; that is it cuts along the line dividing those that would truly like to know more about it, from those that only think they would like know more. I’m afraid that analogies and factoids just don’t have the ability to form to be a total substitute. This doesn’t mean however that they should be omitted, just that they shouldn’t lead the reader to believe they alone can form to be the explanation. Rubber sheets and bowling balls may get one closer, yet one has to be brought to understand that in terms of the utility that is demanded of science, such explanations serve to be practically worthless.

Best,

Phil

25. Bee,

Special Relativity began to make sense to me the moment I learned what SO(3,1) is

Whatever floats your boat, as they say, but it's interesting that Hilbert, Poincare, and Lorentz, who knew what SO(3,1) was, didn't manage to figure out special relativity, while Einstein, who had no clue about SO(3,1), did.

If the physics of special relativity were taught only to those who knew enough differential geometry and Lie group theory to understand SO(3,1), it would do a lot to relieve competition for physics jobs - but it would also eliminate the need for them.

26. Hi CIP,

I didn't mean to say one should introduce SR by saying Lorentz trafos are the elements of SO(3,1), but that one shouldn't stop at the rockets and the trains and the twins etc.

Best,

B.

27. Hi Stefan,

Yeah, putting the equations together with the references in the end of the text would be an option - if properly indicated in the text. I find it really annoying if references and footnotes are in the back by pagenumbers but are not marked in the text. You'll either end up looking every page if the author maybe had more to say, or read them all in the end when you have already forgotten what all that was about.

Best,

B.

28. I remember being a kid and examining the equations in popular physics books with great interest. At a young age, this is some of the only exposure one gets to the more advanced machinery in physics -- since advanced texts are out of reach, and early physics texts are too elementary. So I'm a big proponent of having equations in popular books. The reader might not be "just a layman," but might be a kid who's thinking about studying physics. For this reason, it's especially important to give some enticing and clear views into the real stuff, which only equations and diagrams can provide.

29. Hi Carl,

I wonder if there really is a disadvantage to including formulas in a popular science book. I suspect that the professional book people who advertise and sell these books are not necessarily the same as the audience who buys them. But I think you have to let the professionals make the decisions and in this case, the professionals are the book people.

Depends on whether you agree with their goals? If their aim is to sell a book as being totally easy to understand and advertise it as such because they think that's the way to go I'd just disagree. Why not try to sell a book by knowledge gain for once in a while, even if the reader has to think about it? Plus, there is always the factor that one should check commonplace knowledge every now and then, and interestingly professionals are often the ones hesitant to try something different (probably true in many fields).

Best,

B.

30. Hi Qubit,

If the nature of the universe is mathmatics, then how are we going to learn anything by doing mathmatics? When you learn how to swim, you can't become the water. You just learn how to use the water, trust me you don't need to know a single equation to lap the universe. Other life forms will have already learnt how to swim, long before we even existed.

I don't think the nature of the universe is mathematics. Rather, if the universe is the water, maths is the swimming, whereas the written word is paddling around with your arms wildly and occasionally drowning. Maybe there will come a time when we learn how to build a boat, but until we are there I guess the best we can do is swim.

Bring out a book that only 3 people will buy!

If the book is sold only three times it probably is a total failure in communication (or marketing). It is for progress in science completely useless if you have an insight but are unable to share it with others. Best,

B.

31. All the physicists think equations are good. What a shock!

It all depends on the audience.

It's really hard to write well about physics without them,

It's really hard to write well. Period.

32. Hi Cormac,

Sure, go ahead and list our blog.

Another option that sits somewhere between your suggestion of a diagram and equations is to use equations, but leave the quantities as text. Like

Force = Mass x Acceleration

It saves a lot of brain time, especially if the reader isn't used to the symbols. I mean, if I open a book or read a paper on quantum field theory or cosmology or whatever, I can roughly guess what the quantities are they use because many are just standard notations (consider a in Cosmology, A in QFT, g in cosmology, g in QFT, H in cosmology, H in QFT, psi is a scalar field, \delta is a perturbation, sigma a cross-section, epsilon is always small, and so on and so forth). But if one doesn't know that and has in addition to learn what the symbols mean and then try to put together the notiations in the equation that can be an additional hurdle.

Best,

B.

33. Let's try with the given equation:

|Text - Clarity| = |Equations|

Doesn't sound very promising :)

34. Good point, Arun. But how about if clarity^2 = text^2 + equations^2then from Euler:

clarity.e^(i.θ) = text + i.equations

(I think!)

35. Ok now you're just spamming.
spam != clarity

36. "I don't think the nature of the universe is mathematics. Rather, if the universe is the water, maths is the swimming, whereas the written word is paddling around with your arms wildly and occasionally drowning. Maybe there will come a time when we learn how to build a boat, but until we are there I guess the best we can do is swim." :)

I think i'll just hold my breath and breathe shallow.:)

"If the book is sold only three times it probably is a total failure in communication (or marketing). It is for progress in science completely useless if you have an insight but are unable to share it with others.

So it has to make sense? Science is progress, at the sacrifice of evolution. Don't you think that by 1000 & 1000 of years of mankind doing this, we now have to rely on science to evolve? Is that a good thing? Because it looks to me like we no longer have no choice!

Nobody would write a book if they thought it would sell only 3 copys, but that my whole point.

37. Dear Bee,

I find it really annoying if references and footnotes are in the back by pagenumbers but are not marked in the text.

True. And even if endnotes are labelled by numbers, really cruel things can happen, for example, if they are numbered chapterwise, but when you jump to the book endmatter, you cannot recognize to which chapter the notes belong because chapters are not properly labelled. I really like it if the page headers of the book endmatter contain the page numbers to which the notes refer, but that's not often done.

Cheers, Stefan

38. |Text - Clarity| = |Equations| Doesn't sound very promising :)

That's because you've set ? = 1. In general, you have to factor out the question mark in the text and the equations.

39. Bee-Somewhat biased as physics was one of my undergrad majors, but I do agree that popular writing at more than the newspaper/general interest magazine level should have some equations to the extent they are generally understandable by those with basic calc.

Keep in mind, understandable does not mean 'able to calculate'. I still read many physics texts (most at senior undergrad or lower graduate level) with the aim to be able to follow broadly what is going on. Sometimes this means getting a little dirty when omitted steps are not obvious to me. But my aim in these readings is not to be able to do all the problems or use it for work/research.

Likewise, the aim of most readers of popular science books is to get a gist of the topic at hand. Most times an equation can do this nicely, especially when followed by a sentence or two of explanation. The equation is like a picture of sorts - it can give that 'ah ha' moment of understanding, just sometimes they need a caption too.

You know Bee I'm so glad for this cold front you sent my way, it was getting entirely too warm!

40. "I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."Donald (H. S. M.) Coxeter

Who in their right mind would think the basis of reality is couched in mathematics?:)

It's as if they placed a piece of paper over the E8 image, and then coloured in certain points and then added triangles/upsidedown, squares circle and used colours to denote those same point as if they meant something?

Wow! way to "abstract a painter" for me.:)

41. Here's another layman's vote for equations. Those of us who like math get really frustrated when all we get is a bunch of hand-waving. Plus if you give us some equations, we can do some plots or simple analyses with them, which is really the best way to learn something.

42. I straddle the line somewhat between the novices and the pros, and on both sides, I think it's better to offer as many different access points for your ideas as you can. When you're talking about something as counterintuitive as relativity or QM, any foothold helps: text, equations, and most importantly, pictures pictures pictures. Like, Roger Penrose writes miserable prose, but is remarkably lucid in his drawings:

http://www.flickr.com/photos/ethanhein/sets/72157603018401540/

43. Bee:

I also wonder what you think of Penrose's book, I remember well the introduction where he validated what you are saying here.

It took me a year to work my way through the book, and I was, basically, already trained in everything in it.

I dunno if it actually retaught me that much but it was a reality check.