I keep on bringing some interesting analogies (at least I hope they are) between the study of language by authors such as Zellig Harris or Noam Chomsky and some mathematical issues I have read a bit about as a layman in maths. I quote some paragraphs from the book by Zellg Harris I am working on, namely, Mathematical structures of language (the quotes are long, for the sake of exactitude,but not necessarily contiguous to one another; I am marking them with Roman numbers, to facilitate references):
(I) “Sentences are shown to have, aside from certain degeneracies, a unique and computable factorization into prime sentences; equivalently, there is a recursive method for obtaining all sentences from a finite elementary subset (of assertions) by means of a relatively small set of operators. The properties of each sentence are completely accounted for by the properties of its factorization. There are also effective methods for constructing various languagelike systems with predictable properties (including languages without ambiguity, without synonyms, etc.) and for specifying the relevant differences between natural language and logic or mathematics.”
(II)“We can describe each proposition in English (including the carrier sentences themselves) as uniquely decomposable, i.e., factorizable, into such carrier sentences and kernel sentences, partially ordered. These carrier and kernel sentences therefore constitute the primes of the set of sentences.”
(III)“One can investigate the properties of the decomposition into primes, for all sentences, or for those in distinguished subsets. It is of interest to see how the differences between decomposition into primes here and those in the set of natural numbers relate to the great differences between the set of sentences and the set of numbers, or to the differences between the algebraic structures that can be usefully defined on each of these sets. In the set of sentences, the number of primes is finite; neither the primes nor the sentences as a whole are ordered, in any relevant way that has been noted so far. Furthermore, the decompositions of sentences are partially ordered, and the requirement of matching the resultant and the argument of successive primes (operators) in a decomposition means that certain combinations of primes do not occur in any decomposition, i.e., do not make a sentence. Thus we have decompositions containing kernel primes and carrier primes, and carrier primes alone (as in I know that the latter is only because of the former) and one kernel prime alone (as in A boy walked).”
(IV) “But no sentence contains more than one kernel prime without also containing a carrier prime of the c type for each kernel prime after the first. These restrictions on the combinability of primes may be varied in interesting ways, or eliminated, for suitable subsets of the setoff sentences (e.g. all sentences containing only one kernel sentence), or for certain altered definitions of the primes. For an example of the latter, the carrier primes for c could be based on a unary rather than binary treatment of c. Then, instead of B, C, D above we would have B*, The same is young, as carrier for wh- plus any sentence of the form N is young.”
(V) “Various subsets of sentences can be defined in respect to decomposition. For example, we can consider the set of sentences modulo their first kernel prime: i.e., all sentences whose first prime is, say, A boy walked; then all whose first prime is A man walked; etc. Any two such subsets whose first kernel prime is of the same kernel type (e.g., NtV, NtVN of particular word subclasses, etc.) will contain the same decompositions; there will be an isomorphism of the first subset onto the second preserving carrier products.”
(VI)“We have seen that each sentence has a partially ordered decomposition into elements, which may be taken either as prime sentences or else as base operators and kernel sentences. The decomposition is unique for each proposition, if the analogic transformations are taken as single elements. There are certain restrictions on the combinations of primes, or of , that occur in a decomposition; certain combinations occur in no decomposition. With the elements taken as kernel sentences, unary and binary base operators, and analogic transformations, each proposition of the language can be written uniquely as a sequence of element symbols requiring no parentheses, for example in the manner of Polish notation in logic. Certain subsequences are commutative (representing elements unordered in respect to each other).”
(VII)“Every sentence, for each unambiguous grammatical meaning of it, has a unique decomposition, via these transformations, into elementary sentences. The transformations can also be looked upon as operators on the set of sentences into itself, or as a special set of prime sentences such that each sentence has a unique factorization into these and the other elementary sentences mentioned above. The transformations themselves are products of a set of base transformations.”
The thing is, I have read a bit about Ernst Kummer efforts on a unique factorization of numbers and wonder to what extend the analysis by Harris can be reconducted or interpreted as the instantiation in human language of Kummer rings. Comments are welcome.