Mathematical structures of language (Zellig Harris)

Question

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.

Answer 1

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

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

Answer 2

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.