Recently I had a conversation with a mathematician who had worked on the theory of Banach spaces early in his career, but had since left that particular subject. He explained that he had become disillusioned by the fact that “all the natural conjectures turned out to be false”; indeed, Banach spaces can have some “strange” properties, such as having uncomplemented subspaces, lacking a basis, admitting operators with no invariant subspaces, or admitting almost no operators at all. (Actually, in fact, they can even be unexpectedly well-behaved!) The last straw for this fellow, apparently, had been the Gowers-Maurey space (for which Gowers won the Fields Medal) , which has a whole bunch of “weird” properties.

Disparaging language is used with disturbing frequency by mathematicians to describe mathematical concepts. Examples are labeled “pathological”; objects are described as “badly behaved”; functions are called “nasty”; problems are said to be “ill-posed”. In a library once I encountered a book whose title actually was Differentiable Functions On Bad Domains — where “Bad” here is not the name of a mathematician, but the ordinary English word meaning the opposite of “good”.

You might think this is nothing but picturesque language — like calling a certain group “the Monster” — except that there are plenty of mathematicians who actually seem to think the way the labels suggest they do. The ex-Banach-spacer I mentioned above is only one example; spend some time among mathematicians and you will find many more. Indeed, such aversion to the unexpected has a distinguished historical pedigree, I am sorry to say. Who can forget the dismay with which Weierstrass‘s construction of a nondifferentiable continuous function was greeted? And even now there are still some people who are pissed off about Cantor’s discoveries, and who would sooner overthrow the standard axioms of mathematics than confront the “paradoxes” of the infinite. Even Hilbert, who had the good sense to regard infinitary mathematics as paradisal rather than paradoxical, nevertheless reacted with anger (!) to Gödel’s results on the limits of formalization, according to Constance Reid. One is reminded of the discoverer of the irrationality of , who, you will recall, was allegedly thrown overboard by his Pythagorean comrades.

I have never sympathized with this way of thinking. As far as I am concerned, unexpected “pathological” phenomena are a large part of what makes mathematics interesting in the first place. Indeed, this accords with the attitudes of other kinds of scientists with regard to their own fields. You generally don’t find physicists crying in agony about the discovery of black holes, or biologists resenting the existence of extremophiles. Why, then, do so many mathematicians insist on doing the equivalent?

Well, some physicists did cry with agony about quantum mechanics. That’s why there are still those who push for hidden-variables interpretations, which are seen as less ‘pathological’ than the prevailing (“Copenhagen”, or just “Shut Up and Calculate”) one.

I’d say that the hidden variables interpretations aren’t less pathological, but what people really want is a good way to think about the physics, which “Shut up and calculate” doesn’t give. As far as I’ve seen, it’s really more of a hope that figuring out how to think about quantum mechanics properly will open up new roads to quantum gravity, which we are sorely lacking at the moment.

As far as mathematical pathologies are concerned, I’m with you. I’m studying algebraic geometry and I’m always hearing about how important Resolution of Singularities is, and I get that it’s a good way to prove theorems, but I (and many other researchers, it seems) are interested in the singularities themselves, and focus on the “bad” points of things. I do think that a lot of mathematicians become fascinated by the pathologies, and start going after them with vigor, just like physicists do with black holes.

I had a feeling someone would mention quantum mechanics. Perhaps I’ve just been too seduced by Eliezer Yudkowsky’s recent series to be able to think of quantum mechanics as really difficult!

But, come to think of it, I do actually think there’s something about the kind of reactions to mathematical pathology that I have in mind (at least in modern times) that differs from the initial anxiety people had (and some still do) about QM. Namely: in physics, no matter how strange nature seems to be, there’s never much question that nature’s strangeness is worth studying. Whereas in mathematics, pathology tends to be viewed (by the kind of people I’m talking about) as an indication of some sort of cultural decadence — a sign that an area has become “too baroque”, or “removed from reality”. To see what I mean, contrast Einstein’s confident declaration that “God does not play dice” with Poincaré’s eye-rolling lament:

“In the old days when people invented a new function they had something useful in mind. Now, they invent them deliberately just to invalidate our ancestors’ reasoning, and that is all they are ever going to get out of them.”

That difference might have to do with the fact that pathologies in physics tend to be partly forced on physicists by experimental findings. That’s why there’s no question that it’s worth studying — if you get ‘strange’ measurements, you have to account for them! Whereas investigating pathological structures in mathematics is (rightly or not) seen as completely optional. There might be a hidden assumption that ‘nice’ structures in mathematics are more likely to apply to the physical world. Perhaps Poincare would have been less disdainful if he’d thought the invented new functions were obviously ‘useful’, although it’s not clear here what sense of ‘use’ he meant.

More often than not, I think disparaging language in mathematics is used to describe things which complicate proofs or computations. I also think that for a fixed example, “good” and “bad” are descriptors which change over time, both for an individual and for mathematicians on the whole.

I believe what Poincare was getting at in the quote James referenced is that a lot of these pathological examples often complicate the main idea and don’t sit well with the general intuition. For example, when proving the chain rule, one wants to just multiply the top and bottom of the difference quotient by the obvious thing, swap denominators, and so on. If that simple proof doesn’t work, then the thing which is going wrong is that one quantity must be zero an infinite number of times in any neighborhood of a point without destroying differentiability. For almost any calculus student, such a function might as well be called a monster. As our intuition matures, though, this function doesn’t seem nearly as bad.

The same is true for “bad” domains. One of the only incorrect proofs I’ve ever heard associated with Gauss has to do with his “proof” that the Dirichlet problem cannot be solved for an arbitrary simple closed curve as a boundary: there are some fairly simple pathological ways to construct curves which prevents a solution from existing (though they are common place by today’s standards). So you can imagine that if Gauss’s intuition wasn’t good enough to see them coming, they were definitely abominations in their era. But there is an entire field of study dedicated to studying differential equations and what conditions on the boundary are good (whatever that means).

As an analyst, my idea of bad spaces seem extremely reasonable. For example, not all Hilbert spaces are separable (unless that is also an axiom of what a Hilbert space is). Non-separable Hilbert spaces are bad in the sense that working with them is harder than with a separable space. And basically any Hilbert space you’ll encounter in real life is going to have a countable linear basis. That has at least been my experience. However plenty of people do analysis outside of L^2. Going further, the most natural Lebesgue space, L^1, is “bad” in the sense that it is not it’s own double dual. In fact, it turns out that in many contexts even the concept of function is “bad” and should instead be replaced by distributions.

The following aphorism (due to Max Planck, of quantum mechanics fame) sums my point up rather precisely:

“A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.”

Sorry, the comment about Gauss should be “[Gauss’s] “proof” that the Dirichlet problem CAN be solved for an arbitrary simple closed curve as a boundary”.

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