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May 04, 2003

THE RIEMANN HYPOTHESIS....I finished my book on the Riemann Hypothesis, and unfortunately I didn't think much of it. It turns out that I've already read another book by the same author, and I didn't like that book too much either, so maybe I just don't like his books.

But it goes beyond that, I think, and the problem is one that infects an awful lot of popular books about math: most of them discuss things that require at least some familiarity with topics beyond basic arithmetic. In this case, for example, you really need to know what a complex number is.

So the author has two choices: (a) write the book for an audience that already knows what complex numbers are, which reduces your potential readership to a Very Small Number, or (b) explain complex numbers.

Most authors, including this one, choose option (b). This is annoying to me because I then have to wade through dozens of pages I don't need to read, something that has to be done with care since these pages invariably also contain details here and there that are important to other parts of the story.

What's more, I have my doubts that this works anyway. First of all, I wonder just how many non-mathophiles are going to read a book like this in the first place, and of the ones who do, I wonder how many actually end up understanding the kinds of tortured analogies that are usually used to explain difficult concepts. In this case the author analogizes complex numbers to streets and cross streets in New York, and then tries to convince us that the cross streets are somehow related to the square root of -1.

Fair enough, and I don't know that I could do any better, but does it actually help? Does anyone who didn't understand the concept in the first place understand it better after reading an explanation like this?

I honestly don't know, but I can't help but think that these efforts are doomed. The book has plenty of scary looking equations, and I'm pretty sure that simply saying "don't be scared!" doesn't do much to broaden your audience. It might be better to simply assume that anyone interested in a topic like this is already familiar with high school algebra, accept the fact that this will reduce your audience a bit, and be done with it.

POSTSCRIPT: Of course, a lot of this has to do with the skill of the author. For example, a very good book about math that I read a few years ago was The Mystery of the Aleph, by Amir Aczel, a short book about Georg Cantor and transfinite set theory. Aczel did a great job of explaining the math in an understandable way, weaving it seamlessly with interesting historical background and a biography of Cantor himself. My recollection is also that Aczel simply assumed, for example, that his readers knew what an exponent was and didn't waste time trying to teach basic algebra. It made for a much better book.

Posted by Kevin Drum at May 4, 2003 03:02 PM | TrackBack


Comments

>Does anyone who didn't understand the concept in the first place understand it better after reading an explanation like this?

I am not even sure what it means to understand complex numbers. I went through an electrical engineering education, and most of us just sort of accepted that sqrt(-1) has nice (if weird) properties, e.g. a Fourier transform can be written much more elegantly using complex numbers, or that complex eigenvalues in a linear system implies oscillatory behavior. In other words, I know how to formally manipulate complex numbers, but don't have a "gut feel" for how those formal manipulations came about in the first place.

Perhaps this is the difficulty with writing about math. An idea in, say, economics, can be illustrated with a few key examples -- one can look "under the hood" of the formulas and talk about the underlying concept. With "complex number" this does not seem possible. There seems no way to look "under the hood" of the formal operations (maybe a graduate-level math education is needed for that).

Posted by: Gautam Vallabha at May 4, 2003 04:08 PM | PERMALINK

I think the explanation of what are to some people basic concepts is for a very specific portion of the audience: those who like math, are interested in math, but haven't actually done anything with the types of math discussed in the book for several years. You know, a brief refresher course before you can attack the rest of the book.

I mean, I've been out of highschool for a relatively short amount of time compared to the general population, but since my field of study doesn't involve much higher-level math, some of the old concepts, while easily grasped back in the day, have accumulated a bit of dust in my skull.

A few pages devoted to teaching a relativley concept makes a book that would be frustrating to mathematical laymen suddenly accesible.

Posted by: The Mighty Reason Man at May 4, 2003 04:43 PM | PERMALINK

Gautam, I think the biggest difficulty is just getting people to realize that the number system they've used since kindergarten isn't adequate to represent everything. And that alone is hard because we're taught that numbers should have some sort of physical representation, which isn't possible with sqrt(-1) = i (or j for us EEs.) There simply is no real-world example of i apples.

You actually do run into the same sort of problem with negative numbers. Those are a little more tangible, though, since you can liken them to ideas such as debt.

I think I got around it by deciding that it's perfectly reasonable to want to take the square root of a negative value. But since the real number representation I'd known so well couldn't do that, it made sense to use a more general notion of a number.

Posted by: Jon M at May 4, 2003 04:43 PM | PERMALINK

Change that last sentence in that last post to "A few pages devoted to teaching a relatively basic concept can make a book that would be frustrating to a mathematical layman suddenly accessible."

I need more editors around here.

Posted by: The Mighty Reason Man at May 4, 2003 04:46 PM | PERMALINK

I haven't read this book, but this is a pervasive and very difficult problem in semi-popular writing about mathematics---the audience is very spotty in terms of what they know. It's partly a question of critical mass---there is much more science writing in biology, physics, chemistry, etc. than in math, despite the fact that most people "take" much more math than these other fields. Another issue: even when people think they are acquainted with complex numbers, how they understand them is highly variable. Once when I gave a presentation to some high school students, I was startled to learn that although they had "studied" complex numbers, they were purely formal symbols to manipulate---they had no idea that they could be represented as points in the plane, with easy geometric interpretations for the arithmetical operations. These kinds of issues are crucial. As an intellectual enterprise, mathematics is tall and interconnected---new ideas depend on old ideas, and they build into tall scaffoldings. If someone's mental scaffold is shaky at any lower level, the whole thing is in danger of collapse.

For what it's worth, I think there are ways to ameliorate the situation. First, mathematics is actually quite broad, there are a lot of different possible things to focus on when writing about it. Since there's a lot of choice, it's better to emphasize areas that are more novel to more people, so that there's not as big a mixture of background. Second, it's hard but usually possible to find ways to discard technical constructions and instead give informal explanations of almost anything, provided you understand it well enough yourself. A good informal explanation can communicate at different levels to a varied audience. In this regard, I'd like to shamelessly plug to the videos "Not Knot" and "Outside In" (with pamphlet-sized supplements that answer a lot of questions) available from the publisher, A.K. Peters http://akpeters.com/. They're both full of mathematical content at many levels, but research mathematicians as well as people totally unversed in mathematics often seem to get something from them. E.g. the Grateful Dead used parts of Not Knot in their shows for about a year. We tried very hard to find ways to communicate sophisticated ideas without getting too pedantic.

Of course, as the internet and hyperlinking becomes more an everyday thing, it becomes easier to refer people somewhere outside the main narrative to learn some subject such as complex numbers. In actual math books, this is often done using appendices or extended glossaries, and can work pretty well. In a semi-popular physical book it's not what people expect to encounter, but when people start to habitually read books electronically it'll probably become the norm.

Posted by: Bill Thurston at May 4, 2003 05:02 PM | PERMALINK

I suspect a large part of the problem is the question of "why is this book being written"?

I think there is (one day, but not quite yet) hope for books written by scientists to appeal to science-minded people of whatever ability is of interest to the author. What defeats such books right now is that they are routed through a publishing process that ruthlessly strips out anything "scary looking", meaning equations, graphs and suchlike. Future web-published material (ie material written for love, not money) has the potential to do better.

Of course books written by professional authors are doomed and pretty much always will be, since their primary interest is as large an audience as possible.

As for what to do today, my advice is simply don't waste your time buying what I call "baby-science" books. You're a smart guy, which means that whether it's a pop-science book about biology, chemistry, physics, math or astronomy, you probably know pretty much everything in it; and you'll just be disappointed yet again by the predictability and lack of content [much like whenever you pay to see a hollywood movie]. Do yourself a favor and buy textbooks --- either glossy 1st yr textbooks (second-hand bcs new they cost $100 a pop) to get an overview of a field --- or Dover paperbacks ($5-$25, and generally already screened as being classics in the field).

Finally remember how to read math/physics text books. The first time you read a field, you're going to get lost pretty much after the second chapter. That's fine, persist, skimming over the parts that really mean nothing like the proofs. The goal is to get a feel for the terminology and notation of the field, to see what its concerns are, where it is headed, what it does that's ultimately interesting and what it does that is merely scaffolding, necessary to get to the good stuff but not much interesting in and of itself.
The second book you read, you can now try to see what's happening, given that you know the route and the destination; and the third books you read you can fill in all the gaps and holes in your understanding that you noticed as you read the second book. And accept that, with almost every physics/math book, the last section or chapter will never seem as good as the earlier stuff. By that point you are tired, the author is tired, the spark is gone. Struggle through it as an introduction for when you decided to read a full-fledged book on the subject, but don't obsess too much about how it seems to make so much less sense than the middle of the book.

Posted by: Maynard Handley at May 4, 2003 07:07 PM | PERMALINK

Is assuming that (in order to describe some phenomena conveniently) it makes sense after all to regard -1 as having a square root really any more outlandish than assuming that you can multiply a number by itself a FRACTIONAL number of times? Or, for that matter, that there can be such a thing as "less than zero"?

Posted by: Bruce Moomaw at May 4, 2003 09:46 PM | PERMALINK

Maynard: what you said, except for the stuff about price. You can find all kinds of textbooks dirt cheap at thrift stores and garage sales, especially if you're not looking for anything particular.

Me, I'm a humanities kind of guy (actually, I did have an early affinity for math, but I must have burned away those particular brain cells in high school), but I've lately become interested in economics, so I picked up a copy of Baumol and Blinder's Macroeconomics: Principles and Policy (6th ed.: 1994) at the local Sally. Price: 37 cents. Good readin', with lots of pictures, even the examples weren't quite up to date, and it was, I think, subtly right-leaning (soft on Reagonamics, and nothing about stuff liken the greater tendency of poor people to spend rather than save).

Posted by: David at May 4, 2003 09:52 PM | PERMALINK

Gautam: "In other words, I know how to formally manipulate complex numbers, but don't have a "gut feel" for how those formal manipulations came about in the first place."

Did you have a gut feel for similar triangles when you studied geometry? The shape of a triangle is essentially the same thing as a complex number---many phenomena in plane geometry can be translated into complex numbers in a way that clarifies both. This is better illustrated with pictures, but a good way to think about complex numbers, especially on the "gut" level, is that they measure the shape of a triangle. The ancient Greeks thought of proportions of line segments, rather than real numbers--- essentially the same concept, just a very different mental image to imagine a ratio of line segments rather than a string of decimal digits. Complex numbers are really a 2-dimensional version of a ratio---i.e. a complex number describes the relationship of two sides of a triangle that meet at a vertex. To multiply two complex numbers, you put similar copies of the triangles together at the vertex. i = sqrt(-1) is basically the statement that if you turn left twice, you're going backwards.

Complex analysis is actually a very beautiful and intuitive geometric subject, just usually not taught very geometrically. Quite a lot of it is accessible on "from scratch" --- a lot can readily be absorbed and appreciated say by an interested 4th grader. It also has surprisingly many interesting and significant applications. If you're actually interested, I think Tristan Needham's book "Visual complex analysis" takes a pretty good approach---various people have told me they liked it, but I haven't actually taught from it or more than browsed it so I'm not completely sure.

Posted by: Bill Thurston at May 4, 2003 11:18 PM | PERMALINK

> i = sqrt(-1) is basically the statement that if you turn left twice, you're going backwards.

Thanks, Bill! This is really intriguing, and I going to take a look at the Needham book you recommended.

By the way, this notion of geometric intuition seems to bear on the relation topic of whether mathematics is symbolic thinking. We seem to be biologically wired for thinking in certain ways about spatiality and causality, and "gut feel" seems to relate to how well we can map a domain into these biologically wired abilities.

Posted by: Gautam Vallabha at May 5, 2003 07:43 AM | PERMALINK

Why not have an appendix where the mathematical concepts
are explained so as not to clog up the narrative? Although
if you've bought a book with the title "The Riemann
Hypothesis" I think you should really be prepared for
a bit of maths. The 'cross streets' analogy isn't a
bad one, but analogies need to be used with caution
and precision lest the reader get a wrong idea. Addition
and multiplication in the complex plane are really
very simple things, so it can't be that hard to explain!

Posted by: Thomas Dent at May 5, 2003 09:20 AM | PERMALINK

If you are interested in the physical properties of i, I would suggest the excellent book: An Imaginary Tale, by Paul J. Nahin. I was a math major for 2 years before I changed my field of study, and it wasn't until I read this book that I really understood the function of i in real-world terms.

Posted by: Sebastian Holsclaw at May 5, 2003 09:49 AM | PERMALINK

I would second the recommendation for Visual Complex Analysis by Needham. As a math teacher, I have found that knowing many different analogies and representations for an idea can be very help for both understanding and explaining mathematical ideas. There is no one correct way of thinking of math ideas, which is why I think (back to a previous discussion) math is more than just pushing symbols around: the connections, patterns and representations are HOW we understand the math. I would recommend the Lakoff and Nunez book, Where Mathematics Comes From, not for its repetitious style or its minor errors, but because it makes explicit how much of how we think mathematically comes from our analogies and metaphors, i.e., our representations.

Cheers

Posted by: Eddi Vulic at May 5, 2003 11:05 AM | PERMALINK

UK reader here. I've actually skimmed the book a few times (being a cheapskate I'm waiting for the paperback) and find it quite enjoyable. Certainly preferred it to Singh's book on FLT, tho' that's probably a matter of personal taste.

Some scattered thouhgts and responses:

"I wonder just how many non-mathophiles are going to read a book like this in the first place, and of the ones who do, I wonder how many actually end up understanding the kinds of tortured analogies that are usually used to explain difficult concepts."

Good point. On t'other hand one reason I'd suggest this book to non-math friends is that it gives an impression (to me at least) of what *doing pure maths* is like. Maybe you do well out of the broader USA system but over here most people are convinced early in life that "pure" maths is about doing lots of calculations with unfamiliar notation; the creative possibilities and the sense of inter-dependent structure don't get much of a mention, because it's so hard to communicate them in "popular" terms. The whole de Branges aspect - which is more about mathematicians as a social group than the actual operator theory - is also dealt with quite well.

Regarding the question "why this book?" -- again, I think the main reason is (if you're not a mathematician) to get some idea of what some mathematicians do.

"A few pages devoted to teaching a relatively basic concept can make a book that would be frustrating to a mathematical layman suddenly accessible."

Amen. Same goes for any other subject really; the reader needs a way in, or a foothold of some sort.

BTW Kevin, I'd be interested to know if you felt the book left you with a better (or worse) idea of the mathematics involved.

P.S. I'm sure loads of people have already pointed this out, but ... Bill Thurston? Wow. Glad to see some people believe as much in communicating about maths as in advancing research.

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