I think the reason "between" isn't precise in English is that the intended meaning on a spoken "between" is almost always clear from context, so there isn't much motivation to have separate words. Circumlocution is used when necessary. If my conjecture is correct, then the equivalents of "between" in other spoken languages should be similarly ambiguous. Mathematics doesn't use context - at least within formulas and sequences of formulas, so it needs unambiguous terms.

That was a lot of work to get all the examples in different languages. Still it was the same numerical example, and the subjects were all mathematically educated. How about "Everything between New York and Boston is one big city." Does this include New York and Boston? Does the convention depend on language.

Computer languages are just as bad. "from m to n" now means [m..n] inclusive, although English was never so unambiguous prior to computer languages. Fortran never took a stand, as it simply used "1,N", and didn't use an English word. I have always advocated "from m upto n" for the non-inclusive [m..n), since this matches 0-origin indexing much better than the "to" form: "from 0 upto length(vector)". At 06:45 PM 7/2/2004, Edwin Clark wrote:

On Thu, 1 Jul 2004, John McCarthy wrote:

...

I think the reason "between" isn't precise in English is that the intended meaning on a spoken "between" is almost always clear from context, so there isn't much motivation to have separate words. Circumlocution is used when necessary. If my conjecture is correct, then the equivalents of "between" in other spoken languages should be similarly ambiguous.

Since there are so many languages represented by students and faculty here at the Math Dept at the University of South Florida, I decided to try a little poll of the department.(If I waited till next week I know I could get several more languages, but this will suffice.)

This is what I asked them to do:

Please translate the following question into your mother tongue: "How many integers are between 1 and 3?" and let me know what the correct answer is: -------------------------------------------------------------------------------- NOTES: Some answers below give the answer as "2" meaning the only integer between 1 and 3 is 2. Others give the answer 1 meaning that there is only one integer between 1 and 3.

I owe the subject title "Lost in Translation" to my colleague Mile Krajcevski and you may try to guess which two translations he provided.

A word of caution. Since I happen to speak a little Japanese I questioned two other Japanese in our department. One a secretary who is not a mathematician and one a research mathematician visiting from Japan. Both seemed to be less sure of the exclusivity of "betweenness" that the person whose answer I quote below (Japanese student who did undergraduate work in Japan). My point is that these answers all come from mathematicians or graduate students in mathematics and my feeling is that one might find a little less certainty among general speakers of these languages. But that's just a guess.

If I had it to do over I would try to compose a better sentence for translation. --------------------------------------------------------------------------------- Afaan Oromo: Translation in "Afaan Oromo" one of the largest African Languages. Mother tongue for about 40 million people and second language for about 5-10 million people. Mainly spoken in the horn of Africa. It is instructional and media language.

The translation is: "Laakobsa meeqatu 1 fi 3 jidu jira?"

The answer is 1. ----------------------------------------------------------------------------------- German and Serbo-Croatian: In German, the phrase "Wieviel ganze Zahlen gibt es zwischen eins und drei" would never include 1 and 3 and that the equivalent in Serbo-Croatian even more explicitly excludes the borders.(See below for more on Serbo-Croatian.) ------------------------------------------------------------------------------------ Japanese: If I translate the question as "1 to 3 no aida ni ikutsu no seisu ga arimasuka" (Note that "to" here means "and" in Japanese.) Then the answer would be one.

But I translate the question as "1 kara 3 no aida ni ikutusu no seisu ga arimasuka", (Note that "kara" means "from" in Japanese.) then the answer should be three. But then, for this one, if I translate it into English, it will be

"How many integers are there from 1 to 3?"

[But when I learned English, more confusing to me was the expression "more than". Because in daily conversation we only use "ijo", which is more than or equal to. And when I learned that expression, my Engligh teacher translated it as "ijo". Took me a while for my wrong notion to get corrected.] ----------------------------------------------------------------------------------------- Hungarian: In Hungarian it can be ambiguous, through the strict translation of between leaves only 2 (but in many cases we have to specify if we include the limits or not). ------------------------------------------------------------------------------------------ Fulfulde: In my mother tongue one of the most spoken in Central and West Africa, called "fulfulde" (central Africa) and "pular" (west Africa). the word between which means "hakkude" never includes boundaries. So the only integer in "hakkude gotel be tati" is 2. The question translates as: "Limde noy ngoni hakkude gotel be tati ?" ------------------------------------------------------------------------------------------ Guyarati and Hindi: In my mother tongue which is Gujarati spoken in the state of gujarat in India and many IndianAmericans up north, the word between means "vacche" and never include boundaries. so the answer would be 1. Also in Hindi, the national language of India, the word "between" means "beech mein" and it excludes the boundaries. So again the answer is 1. -------------------------------------------------------------------------------------------- Chinese: The Chinese translation is 1 he 3 zhi jian you ji ge zheng shu? Answer: 1. ------------------------------------------------------------------------------------------ Arabic and French:

I'll try: "How many integers are between 1 and 3?"

Arabic: "Kam Adad bayna wahed wa thalatha" French: "Combien y a t-il d'entiers entre un et trois".

The answer is one for both languages.

P.S. The Arabic translation is somehow hard because there are arabic symbols which do not have an anlogue in english(The only way I can explain it is to say it in front of you but I think you know what I mean some when you pronouce them they come from real inside the throat). ------------------------------------------------------------------------------- Russian: In Russian, "between" usually excludes the boundaries. For example, 'some place between Moscow and St. Petersburg' usually means that it is not in Moscow and not in St. Petersburg (and even not very close to either of them). From this point of view, I would say that there is 1 integer between 1 and 3. However, with integers it is not as clear as with cities, because talking about integers, we switch from the regular Russian language to mathematics, and in mathematics it is not that clear. However, I would say that in Russian mathematics, 'between' without clarification would rather mean an open interval; the same as the word 'interval' in Russian means an open interval, and 'closed intervals' in Russian are 'segments' and not intervals. If somebody wants to include 1 and 3, they would use such terms as 'from 1 to 3' rather than 'between'. ----------------------------------------------------------------------------- Polish: In Polish it will be:

"Ile liczb naturalnych jest pomiedzy 1 a 3?"

and here Polish is also ambigious...

Definitely I would ask for a clarification, and that's really problem since many times we have to rewrite math olympic problems to avoid this very ambigiousity... we would said "scisle pomiedzy" (which means strictly between) but saying "pomiedzy" (between) might be understood both as strictly between and "usual" between. ------------------------------------------------------------------------------ Bulgarian: In Bulgarian "between" would mean just one integer, namely 2! ------------------------------------------------------------------------------- Macedonian and Serbo-Croatian: Macedonian version ( in cyrilic, but with latin equivalents ):

" Kolku celi broevi ima pomegu 1 i 3?"

Serbo-Croatian version would be

" Koliko celih brojeva ima izmegju 1 i 3?"

Macedonian answer is "eden" and Serbo-Croatian answer is "jedan" , so in both cases the boundaries are not included. ---------------------------------------------------------------------------------- English (comment by an erudite colleague):

The problem appears to be the origin of the word "between," which descends from the Old English betweonum, which is itself a descendent or variant of the Gothic tweihnai (= "two each"), and refers to such terms as: sharing the work between them deciding the case between them Old English handled your question with the word betwux, also a variant of tweihnei (in Gothic, tei is 2) meaning "in an intermediate position," which became the archaic word betwixt, which later became "betwixt and between." I guess people didn't like betwixt, because later we got "in between" and the mathematical "strictly in between. --------------------------------------------------------------------------------

Reporting from warm and humid Tampa, Florida home of the winners of the 2004 Stanley Cup and the Devil Rays who have the best record in baseball for the past month,

Edwin Clark