Here are some thoughts about the names of mathematical objects. I don’t make recommendations about how to name things; I am just analyzing some aspects of how names are given and used. I have written about some of these ideas in abstractmath.org, under Names and Semantic Contamination.

Some objects have names from Latin or Greek, such as “matrix” or “homomorphism”, that don’t give the reader a clue as to what they mean, unless the reader has a substantial vocabulary of Latin and Greek roots.

Some are named after people, such as “Riemann sum” and “Hausdorff space”. They don’t have suggestive names either. Well, they suggest that the person they are named after discovered it, but that is not always true; for example L’Hôpital’s rule was discovered by some Bernoulli or other.

You could call both types of names learnèd names.

Others concepts have names that are English words, such as “slope” or “group”. I will call them commonword names. Some of these suggest some aspect of their meaning; “slope” certainly does and so do “truth set” and “variable”. But “group” only suggests that it is a bunch of things; it does not suggest the primary group datum, namely the binary operation. Not only that, but too many commonword names suggest the wrong ideas, for example “real” and “imaginary”.

In contrast, learnèd names don’t usually suggest the wrong things, but they can and do intimidate people.

One upon a time, Roger Godement and Peter J. Huber came up with an important construction for adjunctions in category theory. They called it the standard construction. That commonword name communicates very little. They named it that because it kept coming up in their work. Well, derivatives and integrals are each more deserving of the name. Eilenberg and Moore renamed them triples, which suggests nothing useful except that the concept is given by three data. Well, so are rings. Saunders Mac Lane renamed them again, calling them monads, a learnèd name that suggests nothing except possibly an illusory connection with a certain philosophical concept.

Perhaps learnèd names are better, since they don’t suggest the wrong things. In that case “monad” is better than the other names, but I have a personal prejudice since I have co-authored two books that called them “triples”.

Some writers of popularizations of math and science avoid using the names of certain concepts that suggest the wrong things. In Symmetry and the Monster, by Mark Ronan, the author talks about “atoms of symmetry” instead of “simple groups”, on the grounds that “simple group” is misleading (the Monster Group is simple!) and doesn’t suggest the important property they have. He called involutions “mirror symmetries”, which is appopriately suggestive. Centralizers of involutions became “cross-sections”, which I don’t understand; it must be based on a way of thinking about them that I am not aware of. He doesn’t change the name of the Monster Group, though; that is a terrific name.

Frank Wilczek, in The Lightness of Being, used “core theory” for the theory in particle physics that is commonly called the “standard model”. I suppose that really is more suggestive of its current place in physics, since as far as I know all modern theories build on it.

Marcus du Sautoy, in The Music of the Primes (HarperCollins, 2003), also introduces new names for concepts. His description of the meanings of the many concepts he discusses uses some great metaphors that clearly communicate the ideas. He talks about the “landscape” of the zeta function, how Riemann “extended the landscape to the west”, and refers to its zeroes as its places “at sea level”. But he also calls them by their normal mathematical name “zeroes”. (I could have done without his reference to the “ley line of zeroes”.) He refers to modular arithmetic as “clock calculators” and in one parenthetical remark explains that modular arithmetic is what he means.

Summary

The problem with learnèd names is that they don’t give you a clue about the meaning, and for some students (co-intimidators) they induce anxiety.

The problems with commonword naming are that what a commonword name suggests can give you only one connotation and it is hard to find the best one, and almost any choice produces a metaphor that suggests some incorrect ideas. Furthermore, beginning abstract math students are way too likely to be stuck on one metaphor per mathematical object and commonword names only encourage this behavior. I have written about that here and here.

One problem with popular renaming is that the interested reader has a hard time searching the internet for more information about it, unless she noticed that one place in the book where the fact that it was not the standard name was mentioned.

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