Descriptivist Manifesto
Disclaimer: This article is partly tongue-in-cheek. It is also not to be interpreted as about philosophy, which I don't find important, but about pedagogy, which I do find important.
A spectre is haunting higher mathematics - the spectre of dead mathematicians.
It's my belief that theorems should not be named after their discoverer (or inventor, etc. etc.), but should, as much as possible, be descriptive. Of course not just theorems, but properties (Hausdorff space), constructions (Cantor set), structures (Lie group) or in general any concept which is referred to by name of its discoverer.
Example: Bolzano-Weierstrass is not descriptive, even if it sounds cool. The hairy ball theorem is descriptive. And this problem extends right back to the first introduction to proof for many, with Pythagoras' Theorem.
My main problem with the system of concepts being named after their discoverer, which I'll call the discoverer-named system, is that it gives few clues to the meaning of the concept. More generally this article is against non-descriptive names in general (e.g. groups, named by Galois), but discoverer-named concepts form a broad subset of these.
The discoverer-named system is only descriptive so far as it describes the discoverer. This is not zero information: you know that if something is 'Cauchy', then it is a concept in analysis, or in continuum mechanics. But it is not nearly as good as a descriptive name. Before seeing the definition of a Cauchy sequence, there's no suggestion that successive terms get closer together.
Another example: how one might picture a Hausdorff space without knowing the definition beforehand.
Then even though remembering Hausdorff spaces as 'Housed-off' spaces is used as a funny mnemonic, if they were named that way in the first place, it would allow you to guess the meaning before reading a definition, or at least make the definition feel natural.
Descriptive names can help in a couple of ways. They work as a mnemonic for the definition. They can help motivate how you should think about or picture a concept, especially if the definition is unintuitive at first (c.f. compactness). For theorems, they could work as a mnemonic for a tricky step or technique in the proof, or describe the result that the theorem is about.
I'm not arguing that this is the limiting factor in learning mathematical concepts. When Juliet says
"What's in a name? that which we call a rose
By any other name would smell as sweet"
of course she was referring to the fact that the abstract rose and its physical properties is independent of the label we give to it. Similarly, the concept is independent of the name we refer to it by, and most of our thinking still goes into understanding the abstract concepts themselves. But a good name can aid this understanding. It's like choosing the right basis in linear algebra: the basis is extra structure that we impose on the abstract vector space allowing us to work with the space, and the basis we choose should be fitted to the problem we are trying to solve. Going along these lines, a descriptive name need not be unique, and could be fitted to the context in which the concept is being used.
(Edit 2/22) Indeed we might think of taking different possible representatives (or 'gauges' to a physicist) of some class of ways to refer to a concept. The 'discoverer representative' is admittedly a useful one. Another is a 'short representative', which we do anyway whenever we abbreviate. But it is best to seek a 'pedagogical gauge', one that aids the way one ought to use or think about the concept.
We can compare to script-writing in coding, and in fact a stint in software engineering motivated this article. I wouldn't call a function I wrote 'rickys_function' because no-one would know what it does, although they would know who to shout at. In the same vein you shouldn't write scripts with generic variable names like 'x' or 'tmp' or 'var'. Even if the script works perfectly as planned, it is not good.
"Programs are meant to be read by humans and only incidentally for computers to execute" - Harold Abelson
"Proofs are meant to be read by humans and only incidentally to prove a result" (Paraphrased)
Further, the script's usefulness is not just in whether it works or not. At some point, someone (possibly yourself) is going to have to look at the script again. This might be because there is a useful function in the script that can be used elsewhere, but more likely its because something has become utterly borked and the problem was caused by the script. If the script is well written, with logical names, this massively reduces the time and energy spent to understand the script. Turning to maths, the script becomes a proof. In undergrad maths, you are never proving anything new, so you aren't writing the proof in order to prove the result, you are writing it for your own understanding, and for your supervisor or yourself to read. The better the naming is, the more understandable it will be when you come back to read the proof.
Bad naming causes documentation problems too. You might get duplication of effort; as an example, Bloch's theorem and (Lyapunov-)Floquet theory have the same underlying mathematics, but are taught under both names in the maths Tripos (in Applications of Quantum Mechanics and Dynanimal Systems respectively), and in neither discussion is the other mentioned. If Floquet theory, developed in 1883, had instead been named something like 'periodic normal form', it may have been easier for Bloch, or other physicists, to stumble across the right formulation for lattice wavefunctions. On the other hand, it gave Bloch the opportunity to get his name on a theorem.
Another 'documentation problem' is that bad naming can hide links between closely related concepts. For example Pauli matrices and the quaternions do not sound like they have anything to do with one another, but there is an isomorphism between the two which motivates well the link from Pauli matrices to rotations. Again, this is not discussed when Pauli matrices appear in the Tripos, but a better name could have fostered this link.
Devil's Advocate
Admittedly, sometimes it is just not possible to give a descriptive name to a concept without essentially giving the full definition. This defeats the purpose of defining the concept in the first place, as in that case we could just give the whole definition whenever we talk about the concept. The fact mathematics is necessarily abstract/general runs opposite to our goal of giving descriptions that fully capture the concept. This is a problem for properties which are a synthesis of other properties, like a Hilbert space, which is really just best captured as a complete inner-product space.
Here, another possible naming system is the acronym system, and so we could call a Hilbert space a CIP space. This is already done for TVS (Topological vector spaces). However, software engineering has slightly turned me against acronyms, since my first week was spent drowning in a sea of them. There were so many to the extent that there was a document full of acronyms for new starters, with one definition for TLA: Three-letter acronyms. And acronyms are not descriptive. But they are short, and this is a useful property of a name, especially if you have to copy it out many times in lectures.
So finding descriptive names can be hard, and you could argue that any descriptive name should come to mind naturally and quickly, but I would disagree. Further, in many cases there are concepts that could be given descriptive names, but aren't, and I find this to be particularly bad in mathematical physics, where by virtue of being applicable to physics there should be some less abstract interpretation for concepts.
Second counterpoint: the discoverer-named system is a nice celebration of the discoverer's achievement. I have never proven a result notable enough for it to be named after myself in the literature, but if I did, it would be quite nice to have the result or a concept within the theory I explored to be named after myself, and so have my name in history. But I don't, and so I can propose all the revisionism I want at no personal cost.
Another point for the discoverer-named system is that it is natural, short, and often unique. It is natural in that theorems have a person who first proved the result, or at least is accepted in the literature as having first proven the result, ignoring priority disputes and independent discovery. On the other hand, as discussed before, there may not exist a natural descriptive name. The discoverer-named system gives short names as names are generally only one word long (but there are counterexamples). The system is also often an injective map, with some counterexamples (Euler, Cauchy), but this is often solvable by considering the context or by appending some qualifying words (Cauchy's theorem in complex analysis).
However, this point runs into problems with correct attribution and in modern research, where results are often produced by large collaborations (see HOMFLYPT polynomial), or as results get generalised and more and more names need to be appended. See also Stigler's law, or as I would call it the misnaming law.
There might be something to be said here about how the discoverer-naming practice can make mathematics feel more colorful, and human, and fosters a nice tie with the history of mathematics. There might even be something to be said about censorship, revisionism or decolonisation and representation. I'm not sure what there is to be said though.
Applied descriptivism
My theory is that a descriptive naming system makes your life easier. In practice, we can test this by finding instances of discoverer-named concepts, for example the Bolzano-Weierstrass theorem, and trying to come up with a descriptive name. This is left as an exercise for the reader, as is thinking of more examples of discoverer-named concepts and remapping them to a descriptive name. Then we use the descriptive name, in lecture notes or example sheets. If you feel that it helps with the readability of your notes/proofs, success! (for the theory).
Quite apart from my theory, I have found it a useful exercise to try and come up with my own name for concepts which are not named descriptively, and it is definitely not always easy.
Mathematicians of the world, unite! You have nothing to lose but your names!
A bank of examples of mostly non-descriptive names and some descriptive names is found at Wikipedia's list of theorems.
Some examples from myself and friends found here.
A Nautilus article on the same topic.
Feedback/your thoughts can be sent to me via email. Criticism can be directed to /dev/null.
TL;DR: Descriptive good, discoverer-named bad. Test this by coming up with descriptive names, and using them.