Read Online Of Minds and Language by Pello Juan; Salaburu Massimo; Uriagereka Piattelli-Palmarini - Free Book Online
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rest with something like the successor function. Peano was adamant in stressing that there can only be one empty set. This is a truth of reason, an inescapable necessary truth, that there is only one empty set. So, you form the set that contains it, and then the set that contains the previous one, and so on. The successor function and the necessary uniqueness of the empty set give you the natural numbers system. It does not seem to me to be quite straightforward to do something similar in the case of language. The necessary uniqueness of the empty set would be missing.
C HOMSKY : Thatâs one way of doing it. If you want to generate it from set theory, thatâs a rich way of doing it. If you want to do it without set theory, what you have is one element, and then you have an operation that forms a successor, and itâs simply repeating it. Okay, thatâs the numbering system. Now this system you can get by taking one lexical item, and one way of doing it would be with a Merge system, which does use limited trivial set theory. The one item could be, for example, the set containing 0. And then if you use internal Merge, youâll get a set which consists of 0 and the set containing 0, and you can call that 1, if you like. And you can do that again, and you get 2, and if you throw in associativity, you can get addition, and thatâs basically the number system. You can get addition, subtraction, and multiplication in the familiar way. So it does needjust a trivial amount of set theory, just as Merge does, and in fact I donât know if you even need that; it might be possible to develop a Nelson Goodman-style nominalist alternative. 25 So thatâs one way of getting numbers, and there are others you can think of for just getting a numbering system by restricting language to the very narrowest sense.
H IGGINBOTHAM : Just to help clarify this. You know that in the mathematics of these things one studies semi-groups? You have groups (with a reciprocal operation) and semigroups, which are merely associative. The âfreeâ semigroups have certain special algebraic properties; and then, as they used to tell us at Columbia, the numbering system is just the free semigroup with one generator. Thatâs it.
C HOMSKY : Yes, thatâs basically what Iâm saying. Thatâs correct, it means that the numbering system might just be a trivial case of language, which would solve Wallaceâs Paradox. Wallace was worried about how it could be that everybody has this number system but itâs obviously never been selected; itâs not very useful.
R IZZI : I have a question on the division of labor between UG and third-factor principles. In a number of cases that come to mind, it looks as if there is a highly general loose concept which applies across cognitive domains. Take locality, for instance, a concept that seems to be relevant and operative in different cognitive domains in various forms. And then if you look at language, it is very sharp, very precise. It gets implemented in an extremely sharp manner, only certain things count as interveners, only certain categories determine impenetrability, etc. So the question, related to your short comment on the fact that minimal search may be a third-factor entity, is how much of that is in UG and how much of that is derivable from external general principles.
C HOMSKY : This looks ahead to Luigiâs talk in this conference, 26 so he is going to elaborate on this, but he mentions two principles that seem to be involved in these kinds of questions. One is something that comes out of sequential computation, which has strong computational reasons for it, and that could take care of some kinds of extralinguistic effects â though as an aside, I think there is good reason to suppose that computation of syntactic-semantic objects involves parallel computation as well. But there is another one, which he mentioned now and which is intervention effects,
C HOMSKY : Thatâs one way of doing it. If you want to generate it from set theory, thatâs a rich way of doing it. If you want to do it without set theory, what you have is one element, and then you have an operation that forms a successor, and itâs simply repeating it. Okay, thatâs the numbering system. Now this system you can get by taking one lexical item, and one way of doing it would be with a Merge system, which does use limited trivial set theory. The one item could be, for example, the set containing 0. And then if you use internal Merge, youâll get a set which consists of 0 and the set containing 0, and you can call that 1, if you like. And you can do that again, and you get 2, and if you throw in associativity, you can get addition, and thatâs basically the number system. You can get addition, subtraction, and multiplication in the familiar way. So it does needjust a trivial amount of set theory, just as Merge does, and in fact I donât know if you even need that; it might be possible to develop a Nelson Goodman-style nominalist alternative. 25 So thatâs one way of getting numbers, and there are others you can think of for just getting a numbering system by restricting language to the very narrowest sense.
H IGGINBOTHAM : Just to help clarify this. You know that in the mathematics of these things one studies semi-groups? You have groups (with a reciprocal operation) and semigroups, which are merely associative. The âfreeâ semigroups have certain special algebraic properties; and then, as they used to tell us at Columbia, the numbering system is just the free semigroup with one generator. Thatâs it.
C HOMSKY : Yes, thatâs basically what Iâm saying. Thatâs correct, it means that the numbering system might just be a trivial case of language, which would solve Wallaceâs Paradox. Wallace was worried about how it could be that everybody has this number system but itâs obviously never been selected; itâs not very useful.
R IZZI : I have a question on the division of labor between UG and third-factor principles. In a number of cases that come to mind, it looks as if there is a highly general loose concept which applies across cognitive domains. Take locality, for instance, a concept that seems to be relevant and operative in different cognitive domains in various forms. And then if you look at language, it is very sharp, very precise. It gets implemented in an extremely sharp manner, only certain things count as interveners, only certain categories determine impenetrability, etc. So the question, related to your short comment on the fact that minimal search may be a third-factor entity, is how much of that is in UG and how much of that is derivable from external general principles.
C HOMSKY : This looks ahead to Luigiâs talk in this conference, 26 so he is going to elaborate on this, but he mentions two principles that seem to be involved in these kinds of questions. One is something that comes out of sequential computation, which has strong computational reasons for it, and that could take care of some kinds of extralinguistic effects â though as an aside, I think there is good reason to suppose that computation of syntactic-semantic objects involves parallel computation as well. But there is another one, which he mentioned now and which is intervention effects,
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