The names ‘into’ and ‘onto’ functions are just about as imaginative as the names for an escalator that goes up and an escalator that goes down: an ‘up’ and ‘down’ escalator. Never mind the names though. The ideas is what counts.
The point here is so trivial that it hardly merits a mention: some people just can’t accept that there is no one-to-one correspondence between words in different languages. But when the same idea is drilled into heads in a maths class, the lesson is taken for what it stands, that is, except in all those cases when it just washes over the punters.
An ‘into’ function from X to Y is a function where for every element in X there is an element in Y to which it is mapped, but where it is not the case that for every element in Y there is an element in X that is mapped to it.
An ‘onto’ function from X to Y is a function where for every element in Y there is an element in X that is mapped to it.
A situation where one element in X is mapped to more than one element in Y is not a function, as it goes against the definition of a function whereby for every element in the domain there is one and only one element in the range. However, we say that X has the same size as Y if and only if there exists a one-to-one function that maps X onto Y.
Thus, it will be easy to find many ‘into’ functions between the word-sets in any two languages. I’d hazard a guess that mapping the closest equivalent of the English verb ‘to get’ in any European language to the set containing ‘get’ in English will leave many English ‘get’s’ unpaired. On the other hand, as far as I can ascertain, mapping the English verb ‘to get’ to the semantically closest set in Polish exhausts the list of ‘dostac’. It is an ‘onto’ function.
Are there any same-size word sets in any two languages? For that we would need a one-to-one and an ‘onto’ function. If there are any such sets, they are likely to lie at the peripheries of our communication: either very general or very technical and thus very narrow. It is worth noting, for example, that a lot of units of measurement, which we would expect to be prime candidates for one-to-one and onto, cannot be mapped whole (with all their semantic accretions) to the nearest set in a foreign language. Assuming we have found a match for the word ‘mile’, we may still have to look elsewhere for the equivalents of ‘to be miles ahead’ or ‘to be miles away’.
Are there any word-sets in any two languages where an element in one is mapped to two or more elements in the other (i.e. is not a function)? At first glance, there seem to be many (the example often quoted is that of the English ‘morning star’ and ‘evening star’, being two names for what in another language might well be one), but we may not realize that we are hemmed in by usage in such circumstances, which undermines the one-to-one pairing. It would be unusual to say ‘I saw the morning star in the evening and the evening star in the morning’, but not so unusual to say so in a language where we only have one word for Venus, say ‘X’: ‘I saw X both in the evening and in the morning’.
However, consider flipping it around – going from English into a language with just one word ‘X’ for the morning and the evening star. It is a function because one element in E(nglish) is assigned just one element in X, and it is ‘onto’ because for every element in X (and there is just one) there exists an element in E that is mapped onto it. So, this time the translation ‘I saw an evening star in the evening,’ and ‘I saw a morning start in the morning’ as ‘I saw X in the evening’ and ‘I saw X in the morning’ respectively is perfectly legitimate.
The point here is so trivial that it hardly merits a mention: some people just can’t accept that there is no one-to-one correspondence between words in different languages. But when the same idea is drilled into heads in a maths class, the lesson is taken for what it stands, that is, except in all those cases when it just washes over the punters.
An ‘into’ function from X to Y is a function where for every element in X there is an element in Y to which it is mapped, but where it is not the case that for every element in Y there is an element in X that is mapped to it.
An ‘onto’ function from X to Y is a function where for every element in Y there is an element in X that is mapped to it.
A situation where one element in X is mapped to more than one element in Y is not a function, as it goes against the definition of a function whereby for every element in the domain there is one and only one element in the range. However, we say that X has the same size as Y if and only if there exists a one-to-one function that maps X onto Y.
Thus, it will be easy to find many ‘into’ functions between the word-sets in any two languages. I’d hazard a guess that mapping the closest equivalent of the English verb ‘to get’ in any European language to the set containing ‘get’ in English will leave many English ‘get’s’ unpaired. On the other hand, as far as I can ascertain, mapping the English verb ‘to get’ to the semantically closest set in Polish exhausts the list of ‘dostac’. It is an ‘onto’ function.
Are there any same-size word sets in any two languages? For that we would need a one-to-one and an ‘onto’ function. If there are any such sets, they are likely to lie at the peripheries of our communication: either very general or very technical and thus very narrow. It is worth noting, for example, that a lot of units of measurement, which we would expect to be prime candidates for one-to-one and onto, cannot be mapped whole (with all their semantic accretions) to the nearest set in a foreign language. Assuming we have found a match for the word ‘mile’, we may still have to look elsewhere for the equivalents of ‘to be miles ahead’ or ‘to be miles away’.
Are there any word-sets in any two languages where an element in one is mapped to two or more elements in the other (i.e. is not a function)? At first glance, there seem to be many (the example often quoted is that of the English ‘morning star’ and ‘evening star’, being two names for what in another language might well be one), but we may not realize that we are hemmed in by usage in such circumstances, which undermines the one-to-one pairing. It would be unusual to say ‘I saw the morning star in the evening and the evening star in the morning’, but not so unusual to say so in a language where we only have one word for Venus, say ‘X’: ‘I saw X both in the evening and in the morning’.
However, consider flipping it around – going from English into a language with just one word ‘X’ for the morning and the evening star. It is a function because one element in E(nglish) is assigned just one element in X, and it is ‘onto’ because for every element in X (and there is just one) there exists an element in E that is mapped onto it. So, this time the translation ‘I saw an evening star in the evening,’ and ‘I saw a morning start in the morning’ as ‘I saw X in the evening’ and ‘I saw X in the morning’ respectively is perfectly legitimate.
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