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I would like to get some feedback from you regarding the mathematical structures which describe the objects and/or properties described in the paragraph below, which I take from the book Mathematical Structures in Language by Zellig Harris, from Chapter 2 ("Properties of language relevant to a mathematical formulation" p. 6-19, and more specifically p. 16-17)). I would like to get some insight pertaining which kind of space talking or writing is carried out at (since he means it is not measured) and what kind of contiguous operators work the way he talks about in the last 2 paragraphs.

2.4. Operations are contiguous

Talk or writing is not carried out with respect to some measured space. The only distance between any two words of a sentence is the sequence of other words between them. There is nothing in language corresponding to the bars in music, which make it possible, for example, to distinguish rests of different time-lengths. Hence, the only elementary relation between two words in a word sequence is that of being next neighbors. Any well-formedness for sentence structures must therefore require a contiguous sequence of objects, the only property that makes this sequence a format of the grammar being that the objects are not arbitrary words but words of particular classes (or particular classes of words), But the sequence has to be contiguous; it cannot be spread out with spaces in between, because there is no way of identifying or measuring the spaces.

By the same token, the effect of any operation that is defined in language structure, i.e., the change or addition which it brings to its operand, must be in or contiguous to its operand. No space or distance is defined between operator and operand, Of course, later operators on the resultant may intervene between the earlier operator and its operand, separating them. In the description of the final sentence such separation (i.e. the embedding of later operators) can be recognized. But in defining the action of the earlier operator on its operand this separation cannot be identified; the separation can only have been due to a later event.

If (sic) follows that if language can have a constructive grammar, then for language there must be available some characterization of its sentences which is based on purely contiguous relations. The contiguity of the successive words is related to this situation, but does not satisfy this requirement, because a sentence characterization cannot be made directly in terms of the successive words in the set of all words sequences. The sentence characterization will have to define well~formed subsequences or operators which will determine the word sequences that constitute sentences; but these subsequences or operators will have to operate contiguously.

1st Question: What kind of mathematical space (Hilbert space, compact space, whatever) charactertizes the space for writing or talking as Harris describes it (a non measured space, he says)?

2nd Question: How would you characterize the contiguity operand-operator he talks about?It is crucial to note that such relation should exclude displacement, that is, movement of the elements.

Javier Arias
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    That doesn't really sound like mathematics. Or, for that matter, like linguistics. He seems to be arguing that only ordering of parts of speech matters, and that the precise timing of them is only relevant insofar as it gives rise to an ordering. However this is directly contradicted by the fact that, for example, many languages use a distinction between long and short vowels to carry meaning -- so not only the ordering of sounds but also their lengths is important. – hmakholm left over Monica Dec 26 '14 at 16:23
  • @HenningMakholm I think Zelling only wishes to abandon 'spaces' between words and that he still makes a distinction between, say, ojisan and ojiisan. – Git Gud Dec 26 '14 at 16:28
  • Well, he is not primarily concerned about phonology in this particular paragraph. Even if he talks about operand-operator being contiguous, a friend of mine suggested me (without further elaboration) that rather than operator theory, it is something called operads what is at stake here. I have no clue, to be honest, whether that is true. – Javier Arias Dec 26 '14 at 16:29
  • @JavierArias Can you ask a (it may be more than one if they sufficiently related) specific question? As it is, the question is not a good fit for this site. – Git Gud Dec 26 '14 at 16:44
  • Well, I wanted to quote the text by Harris, in order to avoid being unfaithful to its spirit. As a matter of fact, I would say it is 2 questions which I would like to ask: – Javier Arias Dec 26 '14 at 16:49
  • 1st Question. What kind of mathematical space (Hilbert space, compact space, whatever) charactertizes the space for writing or talking as Harris describes it (a non measured space, he says)? – Javier Arias Dec 26 '14 at 16:50
  • 2nd Question: How would you characterize the contiguity operand-operator he talks about?It is crucial to note that such relation should exclude displacement, that is, movement of the elements. – Javier Arias Dec 26 '14 at 16:52
  • @JavierArias I'm writing this as a comment because I wouldn't be surprised if my interpretation is completely off. As I see it, Zellig is describing some characteristics that a formalized natural language should have. The 'space' he talks about is nothing fancy, it's just the concept of space one naturally thinks about, it's not a mathematical concept. On paper it would be the physical space between words, speech-wise it would be silence between words. – Git Gud Dec 26 '14 at 17:31
  • To exemplify, given the words $\text{brown, dog, fox, jumps,lazy, over, quick, the}$, one can create the statement $\text{The quick brown fox jumps over the lazy dog.}$ The spaces serve no real purpose other than facilitating a human reading of the sentence and in a formal language the spaces can be abandoned yielding $\text{Thequickbrownfoxjumpsoverthelazydog.}$ Luckily in this example and given the list of words I mentioned, there's only one way of reading this sentence, so there's no ambiguity. Ambiguity can arise if the language contains the right words, but that's another problem. – Git Gud Dec 26 '14 at 17:32
  • With the above, I'm hoping to have answered the first question. – Git Gud Dec 26 '14 at 17:38
  • @Git Gud. Well, I am not sure Harris is only concerned with the blank spaces in writing, or with silences in oral speech, for that matter.He is not so much concerned about formalizing natural language as much as he is with finding the properties of language (discreteness,enumerability and so on) which reflect mathematical nature in it. – Javier Arias Dec 26 '14 at 17:55
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    I do not know if any of you are familiar with Noam Chomsky's work, specially his so-called Minimalist Program, since if I could refer to it without the need to explain the basic concepts of his approach, maybe that should shed a lot of light on the kind of answer I am searching for or, in other words, on the type of argument I am sketching for my ongoing papers..... – Javier Arias Dec 26 '14 at 18:02

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To me in his "Mathematical Structures of Language" this is more or less the context for which he describes how operations are contiguous. Reading anymore into it would certainly be going down the rabbit hole a bit..

While he is describing characteristics of a Language, natural or otherwise, but he is actually speaking about is the grammar rules for a given language but switches between talking about properties of a language and properties of a grammar that define the language without ever giving a prompt to when he is switching context.

Grammars define languages and operators on the languages must specifically act on a sentence or word in the languages such that the grammar of the language is not violated. The grammar will define a linear set of symbols, together making words, together making sentences, and as a whole is a string which is a subset of the language. Given any string in the language, any operation on to a Word, Sentence, String in the language will be contiguous by merely obeying the rules of the grammar for the language.

Note that all strings in any language seem to linear, there is a prefix and suffix to any string of the language, where the contiguous nature of the operands that he speaks of is not speaking of the grammar itself but rather the any final string in the language after the operator has been applied.

Think of conjunction in English: I can take two well formed sentences and join them together with a "and" or "but" and the operators are "closed" under contiguousness if you will. Any contiguous string given to a valid operator in my languages will produce a contiguous string (whether that string is in my languages after applying the operator is a matter for the grammar to decide.) Remember that this isn't just natural languages in which Harris is speaking. It could be a binary language, or any arbitrary language of symbols defined by their own grammar and property.

Symbols "may" be moved given a a specific operator with respect to their position relative to the magnitude of the string, but their relative positions should not change given an operator. (Operator rules are defined by the grammar that defines the language so this may not always be the case.) Moving elements around may make the grammar ambiguous or make the given string not in the language (because the string does not satisfy the rules of the grammar for the language.)

As far as the "space" that Harris speaks of- It really isn't what you think. Simply the pairing to the natural numbers. Its linear, zero or greater, because the "space" is actually the magnitude of a symbol, word, sentence, or string in the language which can never be negative, and is always zero or more.

Now the "But the sequence has to be contiguous; it cannot be spread out with spaces in between, because there is no way of identifying or measuring the spaces." I will agree that the sequence is contiguous, and always contiguous for any string in an arbitrary language, and all sequences will uniformly be contiguous.

The idea is less about spaces between words or sentences in the language, where that can be measured, but more of spaces between instances of the languages itself where that generally cannot be measured. However, lets take an intractable example.. Given the English language and it's ambiguous grammar, generate all possible strings in English. Without any doubt of mind the book, "Moby Dick" will eventually arise a valid string in our language. Now eventually another valid string would arise as be book, "Pride and Prejudice", now what is the space between these strings? Put these two strings in the same book separated by a single space, what is the space between these strings? The strings themselves are obviously always contiguous, and for any substring where it was constructed using operators, they are contiguous as well, and the concatenation of the two strings, "books", are also contiguous. The immeasurable space is that because it can be infinite.

Harris is simply describing foundational properties of languages and grammars, and in this instance the foundational property that if an operator exists for a language then any application of the operator will produce a string where all elements of the string are contiguous, and the strings membership of the language is defined by a grammar.

Also he speaks of classes of words and is "seemingly" using English or another natural language as an example.. Classes of these words are simply Nouns, Pronouns, Verbs, Adverbs, Adjectives etc. For "Structure-dependency" that is simply another fancy way of saying "grammar". Yes a string can be a word in a language, but move the elements of the string around and is it still a string in the language? Is it still the same word? No, because strings in languages are generally structurally dependent, i.e. they a defined by a grammar.

seeksUnderstanding
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  • @ seeksUnderstanding Thanks a lot for your prolific answer. I will think about it thoroughly and try to answer you the best I can. I am afraid there is more to contiguity in Harris that you just mentioned. Particularly, his reflections somehow anticipate Chomsky's ideas on Feature or Operator movement (or ban thereof) and the whole trace theory, if you know what I mean..... – Javier Arias Dec 27 '14 at 00:30
  • @Javier: I believe there is more to be elaborated on but what Harris is describing is a foundational property of languages in general and many consequences are derived from that. Chomsky, Hopcroft, Ullman and even Frege develop on these. In fact the entire field of the theory of computation is a consequence. I think that a small subset of languages have properties that you are reaching for, but not in for languages in general. Mathematics is a formal language, with a formal grammar- it would be interesting to study contiguity and space in this context. – seeksUnderstanding Dec 27 '14 at 01:19
  • Well, contiguity is the way Harris has to, say, talk about the forces at stake in language. If you like the analogy, you could say only forces in contact act and react (interact), distance attraction being rendered some sort of mysterious magic one should exclude from the theory (it is interesting to note that Chomsky repeatedly mentions the bewildered astonishment which physicists experienced in the old days of Newton with regard to distance action, which he himself renders troublesome for pure thought and logic). Displacement (i.e. movement) is discarded in language. – Javier Arias Dec 27 '14 at 09:09
  • Displacement (i.e. movement) is discarded in language, at first. It only takes place under some specific provisions, all of which seek to avoid a Crash of the computation. See what Chomsky says in the following lines: – Javier Arias Dec 27 '14 at 09:10
  • . . . ‘‘strong’’ features are visible at PF and ‘‘weak’’ features invisible at PF. These features are not legitimate objects at PF; they are not proper components of phonetic matrices. Therefore, if a strong feature remains after Spell-Out, the derivation crashes. (1993:198) Alternatively, weak features are deleted in the PF component so that PF rules can apply to the phonological matrix that remains; strong features are not deleted so that PF rules do not apply, causing the derivation to crash at PF. (1993:216) – Javier Arias Dec 27 '14 at 09:11
  • A central tenet of Chomsky’s account is that movement is driven (or better said, forced) by Last Resort considerations (do that as last resort or else everything, that is, the computation, will collapse). Movement is triggered to check features which would otherwise be left stranded, making the whole crash. – Javier Arias Dec 27 '14 at 09:16
  • As he puts it "Move is driven by morphological considerations: the requirement that some feature F must be checked. The minimal operation, then, should raise just the feature F." Harris does not talk so extensively about computation and crashes, since he is not so influenced by programming languages and computer science as Chomsky is, but his contiguity requirement wants to principally ban movement the way Chomsky later does. Harris will also allow that ban be raised under some circumstances, as Chomsky does, but that is another issue. – Javier Arias Dec 27 '14 at 09:21
  • So I sort of see contiguity in Harris as the same requirement (or at least very similar) to the ban of distance action or Move in Chomsky (he sometimes calls the instruction underlying it Attract). But while Chomsky is clearly algebraic in his formulations, Harris at some points seems to be using geometric or topological concepts. And I am interested in the properties of the "spaces" he is playing around to define those properties of language. I hope you all know understand better what I mean. – Javier Arias Dec 27 '14 at 09:25
  • So, any idea about the nature of that ban on movement (even if it is later on put aside under special circumstances)? Is there something of that kind in algebra or in any field Chomsky might have found inspiration from? – Javier Arias Dec 27 '14 at 18:06
  • Excellent work! What does "immeasurable" mean here? – JJJohn Aug 07 '19 at 12:04
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To me it seems that you can forget about algebraic spaces, but it looks a bit like graph theory: the observation that a sentence is a path graph of words colored with word classes. The operands and the operators might refer to way different words or classes acts on each other, without a system of brackets that separates phrases from each other.

Lukasiewicz' polish notation handle sequences of operands and operators in logic, and perhaps Harris was inspired of such ideas?

There is a site on stack exchange about linguistic. https://linguistics.stackexchange.com/search?q=zellig+harris

It seems as I was wrong about the "spaces": https://en.wikipedia.org/wiki/Zellig_Harris#Operator_grammar

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Lehs
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  • Sure Harris was influenced by Polish notation. He explicitly mentions it. He certainly refers at some point in his work to the ability to anotate syntactic structures in an unambiguous way without resorting to bracketing. Those are valid points you make. – Javier Arias Dec 26 '14 at 21:39
  • But I still do not know what kind of mathematical object his relation operand-operator (in natural language) is. He was influenced by abstract algebra, and also by category theory, and often resorts to homomorphisms and the like.... – Javier Arias Dec 26 '14 at 21:40
  • The crucial point here (it becomes more obvious with Chomsky later on) is the imposibility of movement (not without leaving a trace anyway, which is a way of coyping information and having it twice in the flow of speech, once with a phonetic realization and once as a dummy or mute unit which is only "visible"because we can attest how it blocks some processes (in the morphology, junctures, and the like). – Javier Arias Dec 26 '14 at 21:40
  • @Javier Arias: Algebra and category theory seems adequate, but not what mathematicians call vector spaces or anything like that. When he focus on the word "space" it seems like he focus on the absence of scales, maybe as one do in topological spaces? He refers to a more abstract structure in the sentence. – Lehs Dec 26 '14 at 21:50
  • What he means is whan later became know as structure-dependency, namely, that language is not a mere linear code or so where elements could be defined through numerical distance of the sort "the phoneme X is three slots away from phoneme Y" or "boy occurs three instances (of a given unit) before "pie" in "The boy eats the apple pie". It is only via constituency(hierarchical syntactic or morphosyntactic constituency) that units in natural language are to be understood. That is what he means with no measured space. The question is. what mathematical object is such a space? – Javier Arias Dec 26 '14 at 21:58
  • @Javier Arias: AFter reading https://en.wikipedia.org/wiki/Zellig_Harris#Operator_grammar I suppose only your own knowledge in linear algebra can help you to understand the associations Harris did between linguistic and linear algebra. I just see superficial similarities, and believe that he was inspired by single ideas rather than by whole mathematical theories. – Lehs Dec 26 '14 at 22:10