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I'm beginning my study of $p$-adic numbers, so naturally I've come across the non-Archimidean property of absolute values, i.e. $$|x+y| \le \max\{|x|,|y|\}.$$

A metric space with the analogous property i.e. $$d(x,y) \le \max\{d(x,z),d(z,y)\}$$ is called an ultrametric space. I was just wondering as to the etymology of this term, especially why ultrametric was chosen over non-Archimidean. I understand that this is a stronger condition than the usual triangle inequality, so I guess ultra-metric as in stronger than-metric is the idea behind it, but non-Archimidean would keep terminology consistent between metrics and absolute values.

ikey
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    The earliest reference I can find to the term (in English) is https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/an-invariant-characterization-of-pseudovaluations-on-a-field/6E868CDF4CE8D9419961C6FDAF40AD27 . I do not have institutional access to the article, so I have no idea what it says. Some of the cited Bourbaki articles (in French) may have earlier uses of the term. – Xander Henderson Mar 05 '24 at 23:28
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    @GerryMyerson An ultrametric space does not satisfy the Archimedean property, as $|\varepsilon + \varepsilon| = |\varepsilon|$. No matter how many $\varepsilon$s you add together, you'll never get anything bigger. – Xander Henderson Mar 05 '24 at 23:30
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    I don't know the exact history, but I think "ultrametric was chosen over non-Archimedean" is at least morally the wrong way around. It happens that an absolute value satisfies the ultrametric inequality iff it is not Archimedean in this sense, so it makes sense to describe such absolute values as non-Archimedean. In a setting as general as metric spaces, "Archimedean metric" doesn't have a popular definition I know of in this sense, so it would be a strange name. – Izaak van Dongen Mar 05 '24 at 23:35

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I prefer to use the term "strong triangle inequality".

Anyway, the page

https://mathshistory.st-andrews.ac.uk/Miller/mathword/u/

says ultrametric is due to Krasner in 1944 "according to an Internet web page." I've never seen Jeff Miller's pages on words in mathematics use such a lazy reference before. Imagine someone said something happened "according to a book".

The OED page

https://www.oed.com/dictionary/ultrametric_adj?tl=true

says "The earliest known use of the word ultrametric is in the 1960s." That should be easily disproven (or maybe we're distinguishing between English and French?) by tracking down Krasner's paper in 1944.

KCd
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  • This is because of Krasner's paper Nombres semi-réels et espaces ultramétriques, Comptes Rendus de l'Académie des Sciences, Tome II, vol. 219, p. 433 available here: https://gallica.bnf.fr/ark:/12148/bpt6k31712/f433.item – Johanna Hirvonen Jun 10 '24 at 23:28