The Universe of Disco

Mathematical jargon failures

A while back I wrote an article about confusing and misleading technical jargon, drawing special attention to botanists’ indefensible misuse of the word “berry” and then to the word “henge”, which archaeologists use to describe a class of Stonehenge-like structures of which Stonehenge itself is not a member.

I included a discussion of mathematical jargon and generally gave it a good grade, saying:

Nobody hearing the term “cobordism” … will think for an instant that they have any idea what it means … they will be perfectly correct.

But conversely:

The non-mathematician's idea of “line”, “ball”, and “cube” is not in any way inconsistent with what the mathematician has in mind …

Today I find myself wondering if I gave mathematics too much credit. Some mathematical jargon is pretty bad. Often brought up as an example are the topological notions of “open” and “closed” sets. It sounds as if they should be exclusive and exhaustive — surely a set that is open is not closed, and vice versa? — but no, there are sets that are neither open nor closed and other sets that are both. Really the problem here is entirely with “open”. The use of “closed” is completely in line with other mathematical uses of “closed” and “closure”. A “closed” object is one that is a fixed point of a closure operator. Topological closure is an example of a closure operator, and topologically closed sets are its fixed points.

(Last month someone asked on Stack Exchange if there was a connection between topological closure and binary operation closure and I was astounded to see a consensus in the comments that there was no relation between them. But given a binary operation !!\oplus!!, we can define an associated closure operator !!\text{cl}_\oplus!! as follows: !!\text{cl}_\oplus(S)!! is the smallest set !!\bar S!! that contains !!S!! and for which !!x,y\in\bar S!! implies !!x\oplus y\in \bar S!!. Then the binary operation !!\oplus!! is said to be “closed on the set !!S!!” precisely if !!S!! is closed with respect to !!\text{cl}_\oplus!!; that is if !!\text{cl}_\oplus(S) = S!!. But I digress.)

Another example of poor nomenclature is “even” and “odd” functions. This is another case where it sounds like the terms ought to form a partition, as they do in the integers, but that is wrong; most functions are neither even nor odd, and there is one function that is both. I think what happened here is that first an “even” polynomial was defined to be a polynomial whose terms all have even exponents (such as !!x^4 - 10x^2 + 1!!) and similarly an “odd” polynomial. This already wasn't great, because most polynomials are neither even nor odd. But it was not too terrible. And at least the meaning is simple and easy to remember. (Also you might like the product of an even and an odd polynomial to be even, as it is for even and odd integers, but it isn't, it's always odd. As far as even-and-oddness is concerned the multiplication of the polynomials is analogous to addition of integers, and to get anything like multiplication you have to compose the polynomials instead.)

And once that step had been taken it was natural to extend the idea from polynomials to functions generally: odd polynomials have the property that !!p(-x) = -p(x)!!, so let's say that an odd function is one with that property. If an odd function is analytic, you can expand it as a Taylor series and the series will have only odd-degree terms even though it isn't a polynomial.

There were two parts to that journey, and each one made some sense by itself, but by the time we got to the end it wasn't so easy to see where we started from. Unfortunate.

I tried a web search for bad mathematics terminology and the top hit was this old blog article by my old friend Walt. (Not you, Walt, another Walt.) Walt suggests that

the worst terminology in all of mathematics may be that of !!G_\delta!! and !!F_\sigma!! sets…

I can certainly get behind that nomination. I have always hated those terms. Not only does it partake of the dubious open-closed terminology I complained of earlier (you'll see why in a moment), but all four letters are abbreviations for words in other languages, and not the same language. A !!G_\delta!! set is one that is a countable intersection of open sets. The !!G!! is short for Gebiet, which is German for an open neighborhood, and the !!\delta!! is for durchschnitt, which is German for set intersection. And on the other side of the Ruhr Valley, an !!F_\sigma!! set, which is a countable union of closed sets, is from French fermé (“closed”) and !!\sigma!! for somme (set union). And the terms themselves are completely opaque if you don't keep track of the ingredients of this unwholesome German-French-Greek stew.

This put me in mind of a similarly obscure pair that I always mix up, the type I and type II errors. One if them is when you fail to ignore something insignificant, and the other is when you fail to notice something significant, but I don't remember which is which and I doubt I ever will.

But the one I was thinking about today that kicked all this off is, I think, worse than any of these. It's really shameful, worthy to rank with cucumbers being berries and with Stonhenge not being a henge.

These are all examples of elliptic curves:

These are not:

That's right, ellipses are not elliptic curves, and elliptic curves are not elliptical. I don't know who was responsible for this idiocy, but if I ever meet them I'm going to kick them in the ass.

[ Addendum 20200510: Several people have earnestly explained to me how this terminological disaster came about. Please be assured that I am well aware of the history here. The situation is similar to the one that gave us “even” and “odd” functions: a long chain of steps each of which made some sense individually, but whose concatenation ended in a completely different place. This MathOverflow post has a good summary. ]

[ Addendum 20200510: Mark Badros has solved the “Type I / II” problem for me. They point out that in the story of the Boy Who Cried Wolf, there are two episodes. In the first episode, the boy and the villagers commit a Type I error by reacting to the presence of a wolf when there is none. In the second episode, they commit a Type II error by failing to react to the actual wolf. Thank you! ]

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