The only swans around here – for thousands of miles in any direction – are black.

He should've called it a 'platypus event' – now those things are both fairly rare and strange beyond normal reckoning.

The Antipodeans were a group of Australian modern artists who asserted the importance of figurative art, and protested against abstract … (in my case the truth) and polar opposite.

Skippy…thanks for the chuckle…BTW I can't look up, bloody NH skirts and all.

"And by the way, Triana adds, there’s no such thing as implied volatility anyway, just supply and demand pushing prices around."

I find supply and demand arguments way too simplistic. As we have seen repeatedly in the last several years, speculation can push prices around with little or no reference to supply and demand. Look at the oil bubble last year and this year, for example.

Not that one shouldn't try to teach things in the right generality. Results that don't have to do with distributions being normal should be taught in appropriate generality.

I feel there are (at least) two different ways of discovering mathematical truth. One is to choose a result one wants and that one thinks one can prove and start proving like crazy anything one thinks might help give the result one wants, until one has shot what one is aiming at. Another is to understand mathematics one finds beautiful very carefully, and to always be on the lookout for an impression that these arguments can be combined and extended, which impressions, being mostly just impressions of similarity between previous arguments one has worked through and present possibilities, rely heavily on past experiences of rational thoughts and the logical patterns underpinning the latter. Counterexamples play different roles depending on the method chosen.

In aim-and-shoot mathematics, examples and counterexamples are essential. One doesn't want to aim to prove something that is wrong, and so knowing counterexamples is a great help in determining what exactly to aim for, and what generality one should choose in aiming for it. (But Godel's incompleteness theorem says that in any consistent mathematical theory strong enough for arithmetic, there will always be statements neither provable nor disprovable, so one shouldn't be too stubborn if adopting this approach, or one can waste one's life trying to prove an unprovable result that has no counterexample.)

Examples and counterexamples are much less useful otherwise. Examples are interesting if they are interesting in their own right (if the mathematics dealing with them is useful and elegant). And counterexamples are somewhat useful quite generally when one makes mistakes in logical arguments. When one makes a logical mistake (as is inevitable), it sometimes takes a while before one proves something ridiculous (at which point one realizes one has made a mistake), and this time can be reduced by having a supply of counterexamples.

Anway, though I have not read the book, I am skeptical of mathematicians that say what is needed is more knowledge of arcane statistical objects and counterexamples, since it's aim-and-shoot mathematicians who find these sorts of things most useful, and an aim-and-shoot approach to mathematics is not the best sort for teaching one how to think, which knowing how to think in the long-run probably actually gives better results than specific training.

Ben Ross – you are right, there is the fact that a distribution has fat tails doesn't of itself imply that it has infinite variance too. But Mandelbrot really does mean infinite variance. You can find a summary of his arguments & some tech references for infinite variance in "The Fractal Geometry of Nature" Chap 37(which I hope is still in print…).

Incidentally the normal distribution actually is a stable distribution, but a rather special one with some obliging mathematical characteristics. There is a handy little summary here which I've just noticed has a sentence or two on Mandelbrot.

http://en.wikipedia.org/wiki/Stable_distribution

I'll have a bit of a chew on your point about market participants.

You refer to a theorem for "stable" distributions. Stable distributions are not something I've ever worked with, but I see that they are limits that arise when you sum a lot of independent random processes. Surely, in the processes that led to the current financial crisis, the behavior of market participants was highly correlated. So I suspect that it would be a mistake to assume that price distributions must be "stable".

I've never read Mandelbrot on finance, but a long time ago I studied his fractals book very closely. His basic message was to avoid tempting assumptions about probability distributions – I understood the distributions he applied to data as counterexamples to the normal distribution, rather than replacements for it.

But I'm not sure if they were normally distributed.