Gauss, you mean the guy who lived 1500 years later? Yeah, he definitely set a higher bar. But a “first-rate” restaurant does not mean “the best ever of all time.” Theon and Hypatia were probably some of the best mathematicians of their century. That is enough to be both “first rate” and “formidable” — and I’d be surprised if Gauss would talk smack about them.

You're talking about two people who don't seem to have contributed a single new theorem, proof, conjecture, or formula to mathematics. That bar is cleared by their contemporaries Pappus and Diophantus.

I don't think your idea that being "one of the best of the age" makes you "first-rate" holds any water. A reenactor at a local Renaissance Faire who has never worn a suit of plate armor or ridden a war horse is probably one of the best jousters of the current age, because it's not really pursued any more, but I would never call them a "first-rate, formidable jouster". The medieval scribes who wrote dry annals of what happened each year were among the best writers of their age, because writing was rare, but I wouldn't call them "first-rate, formidable writers". There is an objective component to this kind of language.

Is "first-rate" the top few mathematicians of all time or the top few mathematicians of their age? Or how many total "first-rate" mathematicians have their been? That ambiguity alone can explain this entire argument.

Nit: if it were top mathematicians of their age, then there wouldn’t be any ages without first-rate mathematicians, which is obviously wrong.

That said, the “of all time” version also doesn’t sound right.

"First rank" is the phrase from OP's link, not "first-rate", but in both cases it is not obvious to me that it is wrong for there to be mathematicians with that descriptor in every age. They sound like relative adjectives; "first rank" undoubtedly so, "first-rate" to a lesser extent.

"Who are the first-rate/first rank musicians of $YEAR" does not seem to me like it could be answered with "there were none in that year".

I don’t know about “first-rank”. To me it sounds like an RPG term (“first-rank mage”). But in the case of “first-rate” I see nothing weird about the answer “there were none this year”. When quality is lower, that’s what I’d expect, since I think about this term a bit like the A1, A2, B1, B2, C1, C2 etc language proficiency certificates. They’re not assigned based on quantiles in current population, but based on a set of requirements applied to skill in absolute terms, not relative to skills of other people.

I think you'll find that if you set an absolute bar on most skills you'll end up with zero people above the bar in the past and a growing percentage of people above it in the future. The first person to run a four-minute mile was Roger Bannister in 1954. Now, that achievement is basically table stakes for college-level competitors. Similar effects can be observed in chess, piano performance, and just about any other activity you can measure. Making "first-rate" an absolute bar results in all sorts of crazy answers to the questions "who is first-rate now" and "who was first-rate 70 years ago". If you set the bar such that only the top 10 players of this year cut it, neither Bobby Fischer nor Michael Jordan would be first-rate.

In maths though there was some fluctuation. Mathematics in Roman times saw a decline, and in medieval times a complete breakdown (at least in Europe). If we take Euclid to be the bar for a first-rate mathematician, then there were none in medieval times, but later there appeared Gauss, Newton, Leibniz, Euler, Galois etc. I’m also not sure if today anyone clears the bar for Hilbert-level genius. So I don’t think the bar for “top 10 today” keeps going up. From time to time it goes down.

I don't think the fact that our 90th percentile capabilities have been known to decrease at times in history refutes the idea that the broader trend is dramatically upward, in such a way that any absolute measure of skill is very likely to give you ridiculous results if you pick it based on the standards of year X and then try to use it in year X + 1000 or X - 1000. Was Pythagoras a first-rate mathematician? By what measure that wouldn't also capture millions of people--most of them not even self-described mathematicians--alive today?

I found this site, which seems to collect usage examples (unfortunately no citations): https://sentence.yourdictionary.com/first-rate

Here are a few showing that, when speaking about history, it is normal to reserve terms like "first rate" for the truly exceptional:

"Jonathan Edwards, a very stern Calvinist, is one of the few first-rate geniuses America has to boast in theology."

"For excepting Philae, which belongs as much to Egypt as to Ethiopia, Abu Simbel is the only temple which can be ranked among first rate products of Egyptian genius."

"Pericles likewise is responsible for the epoch-making splendour of Attic art in his time, for had he not so fully appreciated and given such free scope to the genius of Pheidias, Athens would hardly have witnessed the raising of the Parthenon and other glorious structures, and Attic art could not have boasted a legion of first-rate sculptors of whom Alcamenes, Agoracritus and Paeonius are only the chief names."

Definition arguments are by their nature pointless, these concepts have no universally agreed meaning, but it's even more silly in this example where you changed the term used from "first rank" to "first rate."

Plus, all the actual definitions given there are vague enough to not actually solve the ambiguity.

the whole argument is about whether a sentence like "Hypatia was a first-rank mathematician" is accurate. Obviously that hinges on the definition of "first-rank" since no one is disputing her actual accomplishments or claiming that they really compare to all-time greats like Gauss or Archimedes.

It was the person I'm replying to who used "first-rate" actually. I don't think there's a big difference in meaning between the two.

Usage examples show nuance in the meaning of a word that are not captured in the definition.

>Obviously that hinges on the definition of "first-rank"

When figuring out the definition for a term used, the best source is the context it is used. In this case, Hypatia is the benchmark given for what qualifies as "first rank."

You can then argue the author is misusing that term, but that's a different argument that the article doesn't make.

My understanding is that they're synonyms, more or less:

“By the early years of King Charles I's reign, these four groups had been renamed to a numerical sequence. The royal ships were now graded as first rank, the great ships as second rank, the middling ships as third rank, and the small ships as fourth rank. Soon afterwards, the structure was again modified, with the term rank now being replaced by rate, and the former small ships now being sub-divided into fourth, fifth and sixth rates” --wikipedia