体 = 畑
The other day I came across a mathematics terminology in Spanish, which was a bit strange🤔 Let me shed light on machine translation for terminology especially in mathematics. The "signifié et signifiant" argument by Ferdinand de Saussure is not my interest, more precisely speaking, I am a perfect or complete or entire... just a hopeless idiot relative to structural linguistics, though I am deeply interested in mathematical structures.
For those who love beautiful but hard-to-translate words, I respectfully recommend the following superb article, which is in stark contrast to what I write below:
Complete / Perfect
Many Japanese may have heard of Perfect number thanks to the following novel:
It's called 完全数 in Japanese, Nombre parfait in French and Número perfecto in Spanish.
The Professor picked up a branch and began to scratch something in the dirt. There were numbers, and letters, and some mysterious symbols, all arranged in neat lines.
...
"The sum of the divisors of 28 is 28."
"Indeed ... ," he said. And there, next to his outline of the Artin conjecture, he wrote: 28 = 1 + 2 + 4 + 7 + 14. "A perfect number."
"Perfect number?" I murmured, savoring the sound of the words.
"The smallest perfect number is 6: 6 = 1 + 2 + 3."
"On the contrary, a number with this kind of perfection is rare indeed. After 28, the next one is 496: 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248. After that, you have 8,128; and the next one after that is 33,550,336. Then 8,589,869,056. The farther you go, the more difficult they are to find"--though he had easily followed the trail into the billions!
…
On the other hand, Wikipedia says:
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y is a closed map.
Here the corresponding Japanese terminology of "complete algebraic variety" is 完備代数多様体, not 完全. They are called variété algébrique complète in French and variedad algebraica completa in Spanish.
Completeness 完備性 as well as Completion 完備化 are very important concepts in mathematics.
We have both Perfect field 完全体 and Complete field 完備体. These are utterly different concepts in algebra.
Another English adjective "entire" means whole or complete, with nothing missing, nonetheless has apparently different connotation.
For example, "entire" in the following statement cannot be replaced by "complete" or "perfect":
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane.
We have the following software-based translation samples of the above sentence:
複素解析において、整関数 (積分関数とも呼ばれる) は、複素平面全体にわたって正則な複素数値関数です。
複素解析学では、全関数（積分関数とも呼ばれる）は複素平面上で正則な複素数値関数である。
The first one is correct in terms of "entire function" that is called 整関数 in Japanese, but the second one is incorrect since "entire function" is translated to 全関数.
Both examples translate "integral function" to 積分関数, which is wrong❌.
However, "integral equation" means 積分方程式, which suggests the adjective "integral" has at least two different meanings in English even when restricted to mathematical context.
Exact
Exact sequence is one of the most fundamental concepts in Homological algebra ホモロジー代数 and is called 完全系列 in Japanese.
Interestingly, Google Translation provides the following translation though Google Translation recognizes 完全系列 as a Chinese word😵👎🤔
完全系列 ➡️ Complete series
Exact sequence ➡️ 正確なシーケンス
In French, Spanish and Italian, it's called suite exacte, sucesión exacta and successione esatta respectively.
Field
ガロア理論 Galois theory is explained with the following introduction in English, French and Spanish respectively (all taken from Wikipedia):
And this is where I felt strange. I found someone used campo(s) instead of cuerpo(s) for a mathematical object which is called "field" in English, 体 in Japanese, in the context of Galois theory. In Italian, however, the word campo/campi is used.
Class Field Theory, for which Teiji Takagi 高木貞治, a prominent Japanese mathematician (1875-1960), is considered one of the major contributors, is called 類体論 in Japanese. It's a beautiful theory, not easy, but not that difficult compared to the Modularity Theorem, which is used to prove Fermat's Last Theorem.
The terminology in other languages corresponding to Class Field Theory in English are as follows:
(Deutsch) Klassenkörpertheorie
(Français) Théorie des corps de classes
(Español) Teoría de cuerpos de clases (castellano) /
Teoria de cossos de classes (català)
(Magyar) Osztálytestelmélet 🙃😮😶
Incidentally, "Classical Field Theory" 場の古典論 is a theory in physics, and has nothing to do with Class Field Theory.
The English terminology "field" is deemed to be the source of the problem. Here's an excerpt from an article in Mathematics StackExchange:
Fields are a bit funnier. It started with Dedekind using the word "Zahlenkörper" (body of numbers). In a supplement that he wrote to Dirichlet's Vorlesungenueber Zahlentheorie, he used that word (looking for an image for that) instead of 'rationally known quantities.' It was Moore who coined 'field' - in 1893, he wrote on Galois fields for the Bulletin of the New York Mathematical Society. However, he was always careful to mention 'fields of order' something, as field had the additional meaning at the time of neighborhood (as we understand it today)…
Most people who are decently educated but not fully contaminated by modern mathematics would find the terminology "field" in English odd.
On the other hand, Quantum Field Theory (QFT = 場の量子論 in Japanese) is called
(Deutsch) Quantenfeldtheorie
(Français) Théorie quantique des champs
(Español) Teoría cuántica de campos
(Italiano) Teoria quantistica dei campi
Here the word "campos" appears in Spanish terminology.
So we cannot mechanically translate "field" in English to other languages, which is apparent but we tend to forget.
If we mechanically translate a word from English to French (or Spanish) and then into Japanese, we might develop a funny terminology:
Class Field Theory 類体論
⬇️
Théorie des champs de classe / Teoría de campos de clases
⬇️
階級的場理論 / 階級的畑理論
However, "多面数複素関数論"😰, blasphemously chimerized (NOT invented) by two creators, would never be generated by machine translation, and there's no such word in dictionary as long as I know:
Order
This word has a non-typical meaning like the following:
An order of an algebraic number field K is a subring O of the ring of the algebraic integers of a number field K.
It is used in the following way:
The Hurwitz quaternion order is a specific order (a subring) in a quaternion algebra over a suitable number field. It is not confusing but odd…
And I came across the following explanation just a few days ago:
G⁺ exists because the rings of prolongations are R-orders in the separable K-algebra A₀, so are all contained in the maximal order m the integral closure of R in A₀…
One would find funny images by image search specifying
Scheme
I don't want to use the word scheme (or schéma) except in mathematics.
The following is a screen-shot at Merriam-Webster.com in which #2 applies to the above definition in algebraic geometry.
Singular
Another troubling terminology in mathematics is the word "singular".
Using DeepL, this sentence⬆️is translated to
『もうひとつ厄介なのは、単数（形）という言葉だ』
while I'm not trying to talk about singular or plural at all.
It's not the technological singularity (by Ray Kurzweil) which people have been so much excited about, but what I have in mind is the word which appears in the following paper of extreme importance by Heisuke Hironaka in Annals of Mathematics (1964):
"Singular point" in Algebraic Geometry is translated to 特異点 in Japanese.
Varieties without singular points are supposed to be "well-behaved" but are deliberately called with the adjective "non-singular" obviously for special emphasis on their "well-behavedness". They are not called regular or normal. These two adjectives regular and normal are often used in the theory of Commutative Algebra 可換環論. Normal scheme and Regular scheme are defined from the perspective of commutative algebra accordingly.
In the theory of Elliptic Curves 楕円曲線, one would find a supersingular elliptic curve. The adjective "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular🙃. There's a terminology hyperelliptic curve 超楕円曲線 which has nothing to do with supersingular elliptic curve, however, both "super" and "hyper" are expressed with 超 in Japanese.
A supersingular prime is a prime number that divides the order of the Monster group M, which is the largest sporadic simple group in moonshine theory.
We also have "Regular Singular Points"(確定特異点 in Japanese) in Ordinary differential equation 常微分方程式.
These would sound disturbingly confusing💫😵…
🐟Ghoti = Fish
This is nothing to do with mathematics.
In English, we have strange phenomena due to its chaotic pronunciation. Around 25 years ago, my British boss with German heritage told me that
"Exceptions are the rule in English unlike in German".
So we get lost without translation…😮😵😶🙃
Header Image Credit: Pinterest
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