Right Ideals of a Ring and Sublanguages of Science

📝 Abstract

Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring.

💡 Analysis

Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring.

📄 Content

RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro
Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/

Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators.
This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring.

Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone.

§1. Preliminary Word
In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement.

§2. Introduction

Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. Nonetheless, not all of them achieved equal resonance in the field. Thus, the theory of what he labeled “science sublanguages” is widely overlooked and hardly understood, let alone appreciated. There are, of course, some noble exceptions, as we will see later (§5.2), but, for the most part, linguists and theorists of language have been oblivious to that core idea in Harris’s methodology. This scenario becomes even more acute when leaving the English-speaking community. The topic has received only slight attention in French. My own endeavors in the Spanish- speaking world have panned out quite fruitless thus far. No work in this domain in German, Portuguese or Russian has come yet to my knowledge2.

1 Cf. Javier Arias, “Preámbulo a un análisis de la relevancia de los estudios de Ernst Kummer sobre factorización para la historia de las teorías del lenguaje”, Eikasia, 64, May 2015, p. 53-80. In particular, on page 78 the reader may find an advance of the core of the present study:
“Sí puede resultar de interés, sin embargo, que le adelantemos al lector, de cara a futuros trabajos  por si estos llegaren a escribirse , que al elaborar su teoría de los sublenguajes (que incluye, como caso egregio, los de la ciencia) Harris se apoya, precisamente, en el concepto de ‘ideal’ del álgebra abstracta. En concreto, define la relación de ciertos sublenguajes con el lenguaje en términos de un ideal a la derecha respecto a un anillo.” 2 In the crucial contributions regarding computational linguistics  machine learnability, definite clause grammars, semantic unification and so on  by Fernando Pereira (see, for instance, Pereira and Warren, 1980, Dalrymple, Shieber and Pereira 1991, or Pereira 2000) Portuguese plays a minor role and no real trace of an analysis of Portuguese sublanguages is to be found.
As to German, Lothar Hoffmann’s extensive work on sublanguages certainly has to be reckoned with (e.g., Hoffmann 1985). However, its nature is that of a theoretical Auseinandersetzung leading to applications in the standardization of specialized vocabularies or terminological nomenclatures, as well as in the sociolinguistics behind text typologies, all of which are domains quite remote from the stance adopted here. The present paper provides justification for Harris’s statement that the sublanguages of science stand in a particular algebraic relation with the language they are a subset of. Such relation may be defined as that of a right ideal in a ring.

§3. Sublanguages of Science As starting point for our inquiry it will be wise to consider the following caveat: “[…] the sublanguage grammar contains rules which the language violates and the language grammar contains rules which the sublanguage never meets. It follows that while the sentences of such science object-languages are included in the language as a whole, the grammar of these sublanguages intersects (rather than is included in) the grammar of the language as a whole.” 3

To put it graphically:

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