## Happenings – 2011 Aug 13

Earlier this year, as I said when it happened, someone asked me if there was a book about string theory at a popular level. I had thought I was going to have to recommend one fairly challenging book (Brian Greene’s “The Elegant Universe”) and one anti-string book (Lee Smolin’s “The Trouble with Physics”).

There is an elementary book. As far as I can tell, there is only one… but that could change.

And where would we look for such a book?

Stuff “for Dummies”.

There is a “String Theory for Dummies”, ISBN 978–0–470–46724–4. I hope people aren’t put off by the title, but it’s one of a series of introductory books about just about anything.

No equations… broken into bite-size chunks and thoroughly cross-referenced, so it’s a book you can dive into for a topic rather than have to read cover-to-cover… it appears to be well-written and accurate (as far I know, which isn’t much).

Further, it’s the only book I’ve seen that looks readable by an educated layman. Second best, for prerequisite knowledge, was “Quantum Mechanics for Poets” by Leon M. Lederman and Christopher T. Hill… and it wasn’t anywhere near as readable. Nor did it have much if anything about string theory!

I will remind you that I have talked briefly about more advanced books about string theory here and here.

(I could have sworn that I wrote more about Smolin and Woit, but apparently not.)

The primary author of the book is Andrew Zimmerman Jones, who is the “physics guide” for about.com . His website is here .

While I was checking him out, after reading his book, I came across this post of his . It seems that since it opened in 1938, the Bronx High School of Science has had 7 of its graduates go on to win Nobel prizes in physics.

Well, just who was the physics teacher?

It turns out that the graduates cite the school atmosphere rather than any one teacher. They were encouraged to ask questions.

On the other hand, a German Nobel-prize-winning chemist named Ostwald (his Nobel biography and Wikipedia) may have had three of his students go on to win Nobel prizes.

Ostwald shows up in the history of color theory. He was an artist, apparently, and after he retired professionally, he worked on color theory. I can’t seem to find a clear reference, but he may have championed the description of colors as hue-black-white. (I really should remember that, but I don’t)

Unfortunately, I don’t trust the reference which told me that he had three students who won Nobel prizes.

The reference in question is “Color for the Sciences” by Koenderink. The book was recommended in this comment. I bought it… I’m reading it… and I’m looking forward to finishing it, but I already have mixed reactions. It has a few typos and some poor English, but it is nevertheless clearly and well written, and it seems worth reading, so far.

But.

For now, let me focus on one small issue that has weakened my trust. On page 43, Koenderink says

“A Hilbert space is a separable space (given any two distinct points, there [exist two] disjunct open sets such that each set contains one of the points) that is complete (every Cauchy sequence has a limit) and has an inner product.”

There are two problems with that sentence.

One, for at least the past 45 years, a Hilbert space has not been required to be separable. Yes, the definition changed over time. Still, he could simply assume that he has a separable Hilbert space instead of a Hilbert space. Only the words change – except that then he no longer has a general Hilbert space.

A Hilbert space is just a complete inner product space.

Two, he has mis-defined “separable”. A space is said to be separable if it has a countable dense subset; the real numbers, for example, are separable because the rational numbers are a countable and dense subset. (Dense? The real numbers are the closure of the rationals.) More to the point, what he did define is the separation axiom named T2 or Hausdorff.

As it happens, this second mistake is in fact irrelevant: every inner product space is a metric space, and every metric space satisfies very strong separation axioms; in particular, every metric space is Hausdorff.

So. Koenderink is shaky on his math. Well, so am I sometimes. I still intend to read him.

(Look, I make mistakes. Most people do. And it may be that no mistake in this book will invalidate its criticisms of the decomposition of spectra – he may be right even if the details are wrong – but I’m now hesitant to assert that Ostwald had 3 Nobel-prize-winning students just on the basis of Koenderink’s statement.)

And in preparation for Koenderink’s arguments I’ve been looking at infinite-dimensional vector spaces again. (That’s “functional analysis”.) But that’s for technical posts.

Oh, so far there have been no magnitude 7 or higher earthquakes this month.

Posted in diary. 6 Comments »

### 6 Responses to “Happenings – 2011 Aug 13”

1. geppi Says:

Hello Rip,

it’s really astonishing how your discovery of a minor mathematical shakiness in Koenderinks book did raise your awareness for a point where he really seems to have published completely wrong information. In fact I wouldn’t call any of the three Nobel Laureates, Svante Arrhenius (1903), Jacobus van ‘t Hoff (1901) and Walter Nernst (1920) one of Ostwald’s “pupils”.

Arrhenius had already finished his dissertation with which he arose Ostwald’s interest and moved to the university of Riga to work with Ostwald, who held a professorship there between 1882 and 1887. Walter Nernst habilitated 1889 in Leipzig where Ostwald held a professorship from 1887 until 1906. During the time with Ostwald, Nernst developed his famous Nernst equation within the framework of his theory on osmotic pressure and electrochemistry. I would better call them coworkers like van’t Hoff about whom I couldn’t find any reference to have worked at an institution headed by Ostwald but who was definitely a friend of Ostwald with whom he published the magazine “Zeitschrift für physikalische Chemie”.

With respect to their achievements and scientific history I would call these 4 men the fathers of modern physical Chemistry.

However, as a connection to your remark about the Bronx High School of Science it looks like Wilhelm Ostwald created an open atmosphere where it was encouraged to ask questions and to develop your own ideas. Beside the fact that Ostwald was an important scientist he definitely was also a distinguished teacher who influenced hundreds of scientists.

Back to Koenderink. I also have mixed reactions and I recommended his book not because of its strictness but because of the different view he takes on color science.

The book does contain typos in places where it’s hard to forgive them like for example on page 365 where he lists the U, W and V matrices of the SVD of the transpose of the A matrix of his discrete model. They contain several positive and negative prefix errors. I would appreciate if he would publish the Mathematica code for his examples and calculations somewhere on a website.
This would also help to comprehend parts of the book like “8.4 The difference the Scalar Product Makes” where he himself declares on page 358 that “Many professionals with whom I have discussed this matter were puzzled to the extreme and actually doubted the computations.” The best way to resolve those doubts would be to present the details of these computations and make them reproducible.

Another point I dislike is the fact that in some of his graphs the labeling of the axis is missing or at least he doesn’t specify the units of those numbers labeled at the axis. This is normally a no-go for a scientific publication.

However, despite all this I still think it’s an interesting book. I just don’t have the mathematical skills to cope with some of his statements. Neither to prove and accept them nor to disprove and reject them. Therefore I’m happy that you bought the book and started to read it. I have high hopes in your analysis 😉

On my side I did follow your recommendation and bought Halmos and started to work through it. It really seems to be a very good book and I started to wonder again why a lot of those books on fundamental scientific concepts that focus not just on instructing the pure mechanics but also help to comprehend and intuitively understand the subject have been written already so long ago and why so many modern books are simply missing this point.

PS: One thing you should definitely forgive Koenderink is the “poor English”. He’s Dutch and therefore not a native speaker as I’m not too.

• rip Says:

Hi Geppi,

Sorry for the delay in responding, but it’s been a very tough work week.

Thank you for the detailed information about the other three Nobel Laureates. I certainly appreciated it. But did I miss something? How did you know those were the three Koenderink was speaking of?

In addition, you point out that Ostwald was apparently a very fine teacher – even if none of his actual students went on win Nobel prizes.

More to follow. I am deliberately putting out a few small comments rather than one large one.

• geppi Says:

Hello Rip,

on page 26 in the box with the biography of Wilhelm Ostwald he says at the end of the first paragraph:

“Ostwald was very successful both as a scientist and as a teacher: Among his pupils were Arrhenius (Nobel Prize 1920), Van’t Hoff (Nobel Prize 1901) and Nernst (Nobel Prize 1920).”

I thought you were referring to that passage in the text.

BTW the year of Arrhenius’ Nobel Prize he stated is wrong. It should read 1903.

• rip Says:

Thanks Geppi… I have no idea why I remembered “three” but not that he listed the names!

• rip Says:

Geppi said:

“… on page 365 where he lists the U, W and V matrices of the SVD of the transpose of the A matrix of his discrete model. They contain several positive and negative prefix errors.”

You want to recall that if x is an eigenvector, so is -x. We can change the sign of an entire column.

Actually, it appears that his only mistake is the $1/\sqrt{2}$ outside his matrix for V. If I multiply out his U’WV I get A’ divided by the square root of 2.

I would not, however, have written the SVD the way he did – no wonder it’s confusing! If we start from the SVD of A

A = u w v’

then

A’ = v w’ u’,

versus his

A’ = U’ W V.

His U’, then, would be v from the SVD of A. Or, if we start from the SVD of A’, Mathematica gives us

A’ = u w v’,

and now his U’ is our u.

(It’s these notational differences that prompted me to check his answers by computing U’ W V. He has also cut-down the w matrix and dropped the corresponding row of his V.)

• rip Says:

Geppi wrote

“This would also help to comprehend parts of the book like “8.4 The difference the Scalar Product Makes” where he himself declares on page 358 that “Many professionals with whom I have discussed this matter were puzzled to the extreme and actually doubted the computations.””

I’ve marked that page of the book so that when I get there I’ll be reminded of your comment.