## Unfortunate Mathematical Names

It has long been one of my pet peeves that, as a discipline, physicists seem to be way better than mathematicians at giving things cool and useful names.  This disparity appears to have grown in the last few decades, as physicists have started naming quarks (charm, strange, flavor, red/blue/green) and dark matter (WIMPs and MACHOs).  I hang my head in shame when I realize that the average mathematician would probably have named dark matter particles ‘pseudo-massive quasi-particles’ and called it a day.  Mathematicians, of course, can’t even stop giving things the same name – bundle, sheaf, stack, gerbe (french for bundle, sheaf or stack) – or just tacking on more prefixes to an existing name… I’m looking at you, deformed pre-projective algebras.

What mathematical term has always bothered you, for the uselessness, obtuseness or unfortunateness of its name?  I’m hoping to see what has always rankled other people.

For me, I’ve always been bothered by the Lyusternik-Schnirelmann category, which is the smallest number of contractable open subsets that cover a given topological space.  Now, I believe in naming things after people, but why do it when the underlying concept is so straightforward?  The name hasn’t aged well, either; in modern times few mathematicians think of a positive integer when they hear ‘category’.  The term is particularly frustrating when compared to the nearby concept of cup-length, which you might reasonably guess is the length of the longest non-trivial word in the cup-product.

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### 31 Responses to “Unfortunate Mathematical Names”

1. John Armstrong Says:

bundle, sheaf, stack, gerbe (french for bundle, sheaf or stack)

I have to disagree here. Yes, they all have the connotation of “collection” (as do “group”, “ring”, “field”, …), but they have widely varying shades of meaning. And “gerbe” is French for “gerbe”. Just because it’s archaic doesn’t mean it’s not an English word.

2. Isabel Says:

Some speculation — physicists need to get grant money. Notice that the really good names that physicists have come up with — WIMPs, MACHOs, etc. — are the names of things which can only be discovered by using large equipment that accelerates particles to very high velocities, and costs a lot of money. And it’s easier to convince people to spend their tax dollars on finding WIMPs and MACHOs than “psuedo-massive quasi-particles”.

3. Jim Belk Says:

I recently read that spaces of fractional dimension only really captured people’s attention after Mandlebrot coined the term fractal — a nice example of a mathematician doing a good job.

Another example is John Conway, who invented the game of life, surreal numbers, the monster group and monstrous moonshine, sprouts, etc.

But in general I agree that mathematicians tend to do a terrible job of naming things. I’ve always thought that Lie groups and Lie algebras in particular were awful — call them differentiable groups and infinitesimal groups and suddenly you can talk about them with undergraduates.

I’ve heard that there’s a recent movement in logic circles to systematically replace “recursive” with “computable”, since the latter term is clearly more descriptive. Maybe that could be taken as evidence that the names will slowly get better over time. On the other hand, it’s been several thousand years now that pi has been misnamed — it should be called “pi over two”!

4. A.J. Tolland Says:

1) WIMPS is just an abbreviation for weakly interacting massive particle. This is a very dull name, so it’s wise that the use the abbreviation.

2) Physicists using clever names may have something to do with grant money, but I think it’s mostly just tradition and intra-physics marketing. The theorists who invent the names don’t get all that much grant money.

3) “Quark” and “scheme” are nearly perfect names.

4) You’ve hit on one of my major peeves: using (adjective) (thing) to denote a generalization of (thing). I find it frustrating to see terms like “n-stack” and “derived scheme” tossed around. Then again, I may be making an error of perspective: both of these objects are simply categorifications of pre-existing ideas.

But Grothendieck was wise not to call schemes “generalized varieties”. And I occasionally wish that Toen, Vezzosi, and Lurie had chosen to use “dreames” instead of “derived schemes”. Of course, that would have led to “dracks” and “drerbes” as well… 😦

5) Sometimes bad terminology has its own appeal. “Shtuka” is a great terminology, as long as you don’t speak enough Russian to be confused by its colloquial meaning.

5. Greg Muller Says:

Number 4 reminds me of maybe my all time favorite made-up word for a thing: “Qyzygy”, which stands for quiver syzygy. Its the group of 2-automorphisms of any arrow in the path 2-category of a quiver with relations. The concept hasn’t proved particularly useful, but if its nice to come up with a word so powerful in Scrabble it can melt the skin from your opponent’s bones.

6. g Says:

“Surreal numbers” isn’t Conway’s name, it’s Knuth’s. I think Conway just called them “numbers” in the hope that the world would come to see them as the right class of (real) numbers to work with.

7. John Armstrong Says:

I hope not, g. Conway seems sharper than to think any one model of the real-number-structure has any sort of precedence.

8. Greg Price Says:

In the first edition of ONAG, Conway wrote something like “in view of the generality of this Class, we shall simply call its elements ‘numbers’.”

In the second edition, he thanked Knuth very graciously for the much improved name!

He definitely also makes an argument that the surreal construction of the reals is much nicer than the usual construction of integers, then rationals, then reals, with negatives thrown in somewhere along the way. There’s even a diagram in that interlude between parts 0 and 1 to illustrate the complexity of the usual construction.

9. John Armstrong Says:

Nice, yes. “Right”? Impossible. There is no “right” model of the structure, since they’re all isomorphic.

10. John Baez Says:

Using adjectives to subtract properties from a concept is one of my pet peeves too. What especially irks me is how the only sensible definition of “nonassociative algebra” must allow the algebra to be associative.

Until A. J. Tolland just mentioned it, I’d never thought of the prefix “n-” as an example of an adjective that subtracts properties from the term it qualifies. I guess I don’t think of prefixes as adjectives! Anyway, by now I’m too addicted to running around sticking “2-” in front of all my favorite concepts to give it up. I actually think it’s important to let people know that all the math they know and love is sitting there eagerly waiting to be categorified. Or at least, lots of it.

The main advantage of the term “Lie algebras” is that you can tell gullible students it’s the topic in mathematical logic that you study after “truth tables”.

11. Dr. X Says:

It’s a bit off topic, but I hate, HATE, it when people say “twiddles” instead of tilde. This seems to be becoming more and more common among mathematics lecturers. Is anybody else as riled by it as I am, or am I alone?

12. John Armstrong Says:

It bugs me, Dr. X, but I choose to just accept it as a consequence of being so much smarter than everyone else 😀

13. Scott Carter Says:

Maybe I should have left graduate school when I threw down Jacobson’s notes that said, “Of course, every irreducible module is completely reducible.” I came close. Some wish I had.

There are many other oxymoronic terms: Let A be a semi-simple complex Lie Algebra. Let B denote a simplicial complex. Let G be an infinite finitely generated group. What happens when you have a finitely presented infinite group and you want to build its classifying space as a simplicial complex?

Of course in studying train tracks on surfaces you have to look a closed geodesics on closed surfaces, and compact subsets (closed and bounded) thereof. And you also might want to understand if a given lamination contains all of its limit points. Then a leaf may be closed, but not a closed curve. If the surface is a torus, then it has a group structure, so its multiplication is closed.

The worst terms of all are complex numbers and imaginary numbers. Then the real numbers comes up a close third. The real numbers are so bizzarre that we spend our careers studying them and various analogies. They are not the numbers of measurement. Measurements have errors. The reals might be called the linear numbers. Then the complexes could be called the planar numbers. Maybe the reals are the line of numbers and the complexes are the field of numbers, but then what is a field?

The first example of a ring that is given is the set of integers. I learned new math, don’t the integers fit on a line, not a ring. One ring to rule them all …

Needless to say, my favorite zany term is “quandle.” Rourke said he hated it because it was silly. Not nearly as manly as a “rack.” No, Rourke didn’t say rack was“manly term” I think he called it a “clever term.”

Terminology that is motivated by observations in the perceptual world should reflect those analogies. The problem must be that there is not quite enough in the perceptual world that fits the ideas that we are trying to express. But real leaps in mathematics can occur when terminology and notation are chosen that reflect the deep ideas that are being expressed.

14. John Armstrong Says:

Joyce came up with “quandle” precisely because he wanted a term that didn’t already mean something in plain language. “Rack”, on the other hand, is a modification of the original “wrack”, as in the wrack and ruin left after violently ripping away the composition from a group and leaving only the conjugation action.

A completely different term I came across tonight while talking with an old computer science lecturer of mine: “completely full binary tree”. A binary tree can be complete, and it can be full, but a completely full tree is not one which is both complete and full.

15. James Says:

But John, the surreal numbers aren’t isomorphic to the real numbers, right? See the Wikipedia article. Or maybe I’m not understanding what you’re saying…

16. John Armstrong Says:

James, of course they don’t. The surreal numbers fail the Archimedean property. They don’t satisfy all the axioms of the real number structure.

Now, if you stop at the $\omega$ day, then they do satisfy that property, and they are isomorphic.

17. Donna Dietz Says:

homogeneous 🙂

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