# The etymology of 関手【かんしゅ】

In Japanese mathematics, the word ‘functor’ is translated as 関手【かんしゅ】. What is the etymology of this word?

I suspect that it is a pun on 関数 (function). This leads to two further questions:

1. Why 関手 and not 函手? According to Wikipedia 函数 was standard until the late 1950s. Category theory (圏論) was invented in the late 1940s, and was certainly known in Japan in the 1950s, c.f. Nobuo Yoneda's (米田信夫) work.
2. Why -手? It seems to be the same agentive morpheme -手 as in 運転手、歌手、選手 etc., but these are words referring to people. Moreover (to my knowledge) -手 is not as productive as, say, -者. Either way, it seems unnatural to calque -or like this.

To ward off any confusion, I do not mean 関手 as in せきて.

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This is a great question. I searched the Iwanami mathematical dictionary 『岩波数学辞典』 and Sasahara's 当て字 dictionary, 『当て字・当て読み 漢字表現辞典』, and did not find a definitive answer. Here's what I did find, though:

The word 函数 was invented in China, not Japan. The characters were chosen for phonetic value as well as meaning. It's possible that "box that numbers go into" was the intended interpretation, but we can't know for sure. What we can know is that it was the closest the translators involved could get to the "fun-" sound. [Edit: If I had bothered to read the Wikipedia article, I would know that it said that 函 was chosen as the /han/ sound for its meaning of "enclose, contain", and is apparently used in the sense that that the independent variable (x) is included within the dependent one (y). The "black box" interpretation is specifically rejected.]

The spelling 関数 was created in Japan once the postwar kanji and vocab lists (tōyō kanji, 学術用語集) were created without 函 on them. 関数 is ateji upon ateji. The two forms would have coexisted for a while before 関数 became dominant. As Tsuyoshi's answer says, this means that we can basically ignore the question of 関 vs 函; official kanji lists aren't able to cause sudden and discrete changes in everyone's usage, so it's no big deal if we see some inconsistency and overlap.

What follows is pure speculation.

• If "functor" was coined in the 1940s, then 関手 was probably coined in Japan, because by that period Western science was not absorbed via China.
• The English word is the root of "function" plus the Latinate suffix -or which means "(male) person who performs a task." Actor, director, operator, etc.
• So maybe whoever coined the word 関手 took the first half of 関数 and added the 手 of 運転手, 歌手, etc., by specific analogy with -or.

Basically I am in agreement with your theory and offering a reason why a suffix normally associated with people would have been chosen: because that's how the English breaks down, too. It may have been the very fact that 手 is no longer productive that made it the choice for calquing -or: as Tsuyoshi says, using a productive suffix like 者 might have carried too strong an implication of "(actual) person."

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A Japanese mathematics dictionary you say? I might look it up someday if I get the chance. Can you give a precise citation? –  Zhen Lin Aug 4 '11 at 11:33
Sure: 岩波数学辞典 (4th edition), ISBN 4000803093. I don't actually own a copy; I dropped by a friendly local bookstore. I should mention for the record that it is a dictionary for mathematicians, not linguists, so it actually wasn't helpful on this specific issue at all. But if I had wanted to know what a functor was, it would have been great. –  Matt Aug 4 '11 at 12:52
Ah. That's fine too. (I'm actually more a mathematician than a linguist.) I'd imagine an etymological dictionary of mathematical words is of too little interest to be commercially viable... –  Zhen Lin Aug 4 '11 at 12:57

I do not have a definitive answer to either of your questions, but let me post my thoughts anyway because definitive answers may be hard to obtain.

As for 1, kanji 函手 is also used, for example, in 圏論の基礎, the Japanese translation of Categories for the Working Mathematician written by Saunders Mac Lane and translated by 三好博之 and 高木理 (translation published from Springer in 2005). I do not know which of 関手 and 函手 is more common among mathematicians, or how it compares to 関数 vs 函数.

As for 2, I can only speculate. I feel that -者 is associated with the notion of physical people more closely than -手, and that using -者 to a non-human object is a clear personification, which sounds unprofessional. But I am not sure about this.

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ありがとうございます。ウィキペディアが一貫して「関手」を使うので全然「函手」を検索しませんでした。グーグルによると「関手」のほうが圧倒的に一般なようですね。英語‌​圏ではCategories for the Working Mathematicianは聖書っぽい存在ですが、日本では違うのでしょうかな。 –  Zhen Lin Aug 3 '11 at 12:53
@Zhen: You are welcome. According to web search, it seems to me that 圏論の基礎 is one of the standard textbooks in category theory in Japanese. (But I cannot say for sure because I have never studied category theory myself.) I agree that web search suggests that 関手 is far more common than 函手. I do not know why. The reason 関数 is more common than 函数 on the web is simply because 関数 is used in grade schools and high schools, but obviously that reason does not apply to 関手. –  Tsuyoshi Ito Aug 3 '11 at 13:00