I ran across this journey to understand PoincarÃ© and thought I would pass it on. I am a big fan of the idea of popularizations, and am especially enamored with the “you too could have invented X”Â leitmotif that is statrting to emerge in that space (I read your version on CS Monads, sigfpe. it only made me like the form more). This link isn’t in that vein, but any effort is a worthwhile one in my book. It is a work in progress, so I am worried about commenting on it, but I am interested in people’s opinion of it. Is it off target for any particular audience other than the author? By that I mean the people who know the math will think too little is being said, while the ones who do not will be under the impression the trees are occluding the forest. More to the point, is any popularization doomed to such a critique?

ah well, I stumbled on this site following a link from Peter Woit’s NEW. That site has also led me to Luminous Lumo’s Wonderful Universe(s?), but better not go into that.

To answer your question (somewhat), I really don’t like this dichotomous reasoning along the lines of “those who know” and “those who don’t know”. BTW, long time ago I studied mathematical economics but left the field ’cause I feel math. economists abandoned this planet long time ago (or at least their minds did), and I’ve always had a layman’s interest in esp physics and also but to a lesser extent computer science and mathematics (hence my amazement to discover this blogspace).

My point being: lots of so-called “laypersons” are in a similar position: economists, engineers, biologists, dentists even, …, are not complete ignoramuses, we do know ‘some’ math, but what do we get ? Almost all popular science books seem invariably aimed at complete idiots, for whom every single equation is considered something straight from hell. The only alternative would be to work our way through numerous textbooks aimed at students who quite frankly have little else to do all day for the next 2, 3 or 4 years. We certainly won’t learn much from those arXiv papers given the fact that even professionals seem to be having a hard time trying to comprehend what it is all about.

In other words: why are there so few good books on physics (and mathematics) aimed at the “intermediate level” ? I know it’s impossible to turn laypersons into professional physicists or mathematicians in 400 pages or so (even tho Penrose seems to think so – “Road to Reality” being one of the weirdest books I’ve seen in years) – but quite a few of us do have some basic knowledge of say calculus, matrix algebra or differential geometry. So yes, I thinks it’s odd we – poor laypersons – are almost asked to write these ‘intermediate level’ books ourselves – for our own purposes (tho, in all honesty, I have to admit one does find a gem every now and than on the Internet).

Sorry for this rambling speech – I just had to get it off my chest I guess

Well, I am glad you were able to get that off your chest, but I think you may have misunderstood me (or maybe I just did a really crappy job of making myself understood). This “dichotomous reasoning” that you mentioned is pretty hard to refute. Some people have studied the math, and some people would look at you like you were from mars if you asked them if they knew what an atlas for a smooth manifold was. Or what a Topology was. More to the point what the PURPOSE of a Topology was. In fact, as I read your post again, I think we are in agreement. I think we both have the question: What is the best way to write a popularization for people in the middle ground. Reading the link that started this post, I was struck with the thought that someone who was an interested beginner would have no idea where things were going because they were overwhelmed by all the groundwork that needed to be laid (the trees). On the other hand someone in the interested middle might not note all the subtlties that are not being covered (which I think is what really made me ask the question in the first place). Lastly, someone who has already studied the material is not served by it. So I really would like to come to some conclusion on the best way to approach such an endevour.

Some of the best “you could have invented X” kind of articles I’ve read were written by Tim Gowers: http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html They emphasize how naturally some mathematical concepts arise in certain contexts.

michael:

in hindsight I realize I was ranting (quite) a bit (I certainly didn’t mean to offend anyone)

yes, I think we are in agreement. Revisiting the link it strikes me he hasn’t really got that very far, given that he probably started the project beginning 2005. Some of those things I learned when I was only 12 (when Bourbaki still ruled high in some European secondary schools), everything else is covered in more detail in the first chapter of a first year university course on topology I should imagine. Maybe this shouldn’t surprise us. Doesn’t it take (‘normal’ people) years to master the groundwork? No good trying to read Marcel Proust in French, let alone understand all the subtleties of his prose, if you can’t even order a bread in that language.

I do not know how best to approach laypersons in the interested middle, but being in the middle one should at least assume they have already covered some of the groundwork. It would be very tiresome to start every discussion with the basics of set theory again (my Bourbaki past keeps haunting me).

But maybe your question is more profound than that. I sometimes wonder: do even professional mathematicians or (theoretical) physicists always understand all the subtleties of what they (or their colleagues) are doing ? One of the (rather unpleasant) surprises of entering these blog spheres is the habitual name-calling one encounters, with one brilliant theorist calling the other a complete moron, and vice versa (OK, most are more subtle than that, but the unpleasantness is the same).

To end on a pessimistic note: maybe one really has to be thru & thru French to really appreciate Proust in the original language.

the habitual name-calling one encountersYou must have stepped into a ‘conversation’ between Lubos Motl and Peter Woit. I just noticed that their flamewar has even spilled over onto Amazon.

You won’t see mathematicians acting in such an uncivilised manner.

Heavens no!

Thanks for the link Michael. Note that I didn’t set out to teach people the background to the PoincarÃ© Conjecture. Rather, I started reading papers on Ricci flow, found some gaps in my knowledge that prevented me from understanding them and decided to go back to square one to make sure my own understanding and intuitions were solid. My posts are really just a diary of my own journey. If other people gain anything from them, that’s a wonderful bonus. Judging from the occasional emails I get about them, I think there is a certain class of person (perhaps, like myself, someone who once studied mathematics at an undergraduate level but hasn’t touched it for years) to whom it is well targeted. But that isn’t deliberate. At the moment it’s a selfish endeavour. Perhaps at the end of it, I’ll look back and be able to refactor it into some kind of guide to PoincarÃ© but, at the moment, I’ll just be happy when I can sit down and read Perelman’s papers.

The philosopher of science, Paul Feyerabend, decided to read Wittgenstein’s “Tractatus” as part of his PhD, and, like most people who encounter this book, could initially make little sense of its strange structure. So, he sat down for 6 months and figured out a different way through the book, purely for his own benefit. When he showed his notes to his supervisor, his sup. thought others might find them useful, so he advised Feyerabend to publish them as an article, which F. did.

Johan: I think Penrose intended “The Road to Reality” to be at that intermediate level. Would you say that he missed the mark?

sigfpe:

yes, I did stumble on the PW – Lumo war ! Of course mathematicians are much more civilized, they’re also much cleverer

Walt:

I’m not sure Penrose had an ‘intermediate level’ audience in mind – he may have missed the mark in trying to serve everyone. It’s easy to understand those things one already knows, but everything else IMO is simply too terse and requires additional reading.