A lot of people complain about bad terminology in mathematics. I don't really consider myself to be one of those people, but I have to admit they have a point. The choice of a word we use to represent a concept greatly influences the way we read and perceive it. Here are examples of terminology I like and dislike, with some whimsical alternatives for fun. It turned out to be much harder than I expected to come up with good terminology. Leave a comment if you can think of something to add to these lists.

**Like**

*Adjoint*: This word seems to only exist in mathematics. It captures a general pattern that appears frequently in different contexts; several formally related concepts share this terminology. Appropriately, it suggests pairing or duality.*Club set*: Short for "closed and unbounded". This neat portmanteau helps emphasize that the conjunction of the two properties is somehow more the sum of its parts.*Compactness*: This is such a carefully nuanced word for a subtle and fundamental property which occurs in many different contexts in mathematics, although I think it can take some getting used to. It really captures the idea of some kind of finitary character, or a strong notion of "smallness".*Mouse, weasel*: Inner model theory words that give a mischievous flavor to a difficult subject.*Mixing*: from ergodic theory. It's a gerund, which helps describe this dynamical situation well.*Rational numbers*: This is a nice play on words. It makes $\mathbb{Q}$ feel very concrete and friendly and is probably derived from the representation of these numbers as ratios of integers.*Supercompact cardinal*: sometimes terminology doesn't totally make sense, it just sounds cool. I have to admit that hearing this word early made me curious about set theory.

**Dislike**

*Amenable group*: This seems to be a pun that only makes sense with British pronunciations. According to Wikipedia, the original name given by von Neumann was "messbar", or "measurable" in English. Of course, this name is even worse.*Alternatives:*Maybe something like "well-measured" would be better.*Antichain*: It's actually a catchy word, if only it didn't have two closely related but nonequivalent meanings. In order theory people use it to mean a collection of pairwise incomparable elements of a poset, while set theorists would say it is a collection of pairwise incompatible elements. Let the order theory people take back the outdated term "Sperner system".*Cardinal collapsing*:*Alternatives: "*decardinalizing", "cardinality decreasing".*Commutative diagram*: To commute just means to go from one place to another. This got twisted into the property that $ab=ba$; I guess I can kind of see that they are moving past each other. So we can also use this for functions $f,g:X\rightarrow X$ to say $fg=gf$ (a special kind of commutative square). But what about commutative triangles, some functions that satisfy $f=gh$?*Alternatives*: "freely composing", or maybe something made-up and descriptive like "ambicompositional".*Countable chain condition*(c.c.c.): This one from set theory is unpopular among some people I know, since it's a property of posets that means there are no*un*countable*anti*chains (and I don't mean Sperner systems here). I saw one paper (Abraham--Shelah: Forcing closed unbounded sets) which called this the c.a.c., for countable antichain condition, which seems like a reasonable alternative. I recall that there's actually a good historical reason for this terminology, though I'm not sure exactly what that was.*Maths*: OK, so it's not really in the spirit of the other things, and maybe its just my American crudeness, but this sounds totally wrong to me.*One-to-one*: I thought it was OK until someone pointed out to me that it sounds like each member of the domain maps to a unique member of the codomain (what some people call*single-valued*). This has been the source of some confusion in classes I've TA'ed.*Alternatives*: just use the already existing technical-sounding words "injective" and "surjective."*Proper forcing*: It's just a bit too vague and authoritarian for me.*Alternatives:*Maybe something like "internally generic forcing".