I used to work on complex differential geometry, on families of Calabi-Yau manifolds (which are Kahler manifolds, a special class of Hermitian manifolds whose Chern connection coincides with the Levi-Civita connection of the underlying Riemannian manifold) and the Weil-Peterson metrics defined in part by the Hodge period relations and the Kodaira-Spencer morphism.
As far as I’m concerned, the article makes a good point.
I’ve definitely been annoyed at this too. Learning math is really seriously hampered by the useless names of every single concept.
Imagine learning math basic math for the first time, but instead of learning that addition adds things together, you have to learn that the Ossenheimer operator does a particular operation on two operands. That’s essentially how I feel learning math at a university level.
By now it’s possible to come up with more descriptive names for many of these things. A Riemannian manifold could be an inner product manifold, versus a normed (Finsler) manifold. The Levi-Civita connection could be the natural connection. The Riemann-Roch theorem maybe the characteristic theorem? Maybe a step in this direction would be to convince authors of new-ish books to include descriptive names in new editions. The arguments over the names could be had on Wikipedia.
On the other hand, quasi-polynomial vs. pseudo-polynomial time uses no proper names, but it is also a thing that can only be learned by heart (of course there is also weakly-polynomial time). All the short descriptive names are probably also already used, and sometimes for multiple things in different contexts.
I guess now people can reasonably prioritise what gets the shortest name for things introduced many decades ago, but I have a suspicion that picking names well inside a single decade after introduction of things is just hard.
But at least a descriptive name as a little chance of ringing the bell about something known in a close field, or a field related in any matter. Which is useful for mnemonics.
A Last Name, is much more clue-less. I think it is really rare that a Last Name should offer any clue mnemonics for new comers.
Reading the definition from Wikipedia, the reference/reminder to the word polynomial does seem helpful in my taste: from https://en.wikipedia.org/wiki/Quasi-polynomial
“A quasi-polynomial can be written as q ( k ) = c d ( k ) k ^ d + c d − 1 ( k ) k ^ d − 1 + ⋯ + c 0 ( k ) is a periodic function with integral period.”
For me, it resembles the structure of a polynomial in k (degree d), plus with those additional cn(k) coefficient functions terms. So quasi is good name.
Of course, like you said, the difference between Quasi and Pseudo and Weak… is then… well…
I agree broadly with almost all points in the article. The specific points about medicine and law are refuted by Wikipedia’s list of surgeons who invented procedures, people who inspired legislation, litigants who inspired case law, and so forth. The law has a Miller test, while computer science has the Miller-Rabin test.
There are two important reasons to know the names, though. First, by knowing that a single person worked on many things, we can imagine their worldview and the sorts of mathematical facts and scientific theories which influenced their insights. The article invites us to think like Euclid, for example, which means both forgetting about all of our cohomology and calculus, and also recalling what it is like to do geometry by drawing lines in sand. The article doesn’t mention them, but folks like Hamilton or Noether need to be understood as having worked on all of the different subfields that they touched not as many different parts of their life, but as one single coherent vision for mathematical possibilities. (This is not to exclude folks like Nash or Erdös whose work was so segmented.)
Second, and more importantly, knowing the name of a person is a gateway to knowing the rest of their life outside of mathematics. We have the story of Archimedes, who died because he would not put down mathematics. But, in a more historically confirmable and recent way, people like Galois fought and died for rights, and folks like Gentzen were killed. It’s not possible to study particle physics, rocket science, computer science, or any other modern applied mathematics without talking about wars of the 20th century. We don’t just use names to memorialize people, but to remember that they were characters in a real and relevant historical context.
In order to prevent everything from being named after Euler, the mathematics community names many things after the first person after Euler to have discovered them.
I’m only slightly exaggerating.
On the other side of this is more overloading of terms. It’s hard to name things. Look at how physicists named the strong force charges: red, green, blue. It works fine because nobody who is studying nuclear/quantum physics is confused enough to think those correspond to actual colors of things, but still.
I remember in undergrad that my friends were confused that I was taking “linear algebra”, “abstract algebra”, and “ real analysis “ toward the end of my degree. Those sound like high school subjects!
It’s no different from naming code symbols in a common namespace: Use unique names to avoid collisions and unintentional shadowing, name things plainly and specifically to be clear for new readers, and otherwise name concisely. Higher math seems to have passed on the first two of those lessons.
It’s probably good to move away from people’s names, just so there’s more variety in nomenclature, but I’m not sure any push for “better” names is going to be useful.
A lot of the names held up as good examples are references to relatively local cultural bits (such as the Hairy Ball theorem), which won’t help people from outside that culture - what if someone doesn’t know what a cowlick is? Others, such as “isosceles triangle” started out as good names but now are just names people have to memorize anyway. Others, such as “Monster” are easier to remember but so vague they don’t help you remember what the thing actually is.
Author spends a couple of short (and shallow) paragraphs on “but why” at the end, but fails to consider the effects of the incentive schemes set up by a citation-count driven culture of professional advancement.
It’s perhaps not so much that research mathematicians, laboring in poverty and obscurity throughout their careers, crave some lasting recognition from their peer community… and more that they need it just to survive their tenure tracks.