1) LINGUISTICS IN THE EIGHTIES

We cannot begin to talk about linguistics as a science without previously asking a fundamental question: what are linguists in fact talking about? The answer would appear to be rather easy, for human language constitutes the obvious object of such research. However, when contrasting the statements of current schools of linguistics on this subject, a surprising discordance immediately appears: as a matter of fact, linguists do not only disagree on the methodological approaches that they aim to apply, but on the status of the object of investigation itself as well.
Some schools of linguistics -the so-called "descriptivists"- have pointed out that human language consists of a set of utterances whose formal properties they intend to specify. Some others would prefer to establish the set of rules that human speakers supposedly use when performing these same utterances in everyday life -i.e., their innate competence, as stated by "generativists". There are also scholars -commonly referred to as "structuralists"- who are interested in neither utterances nor rules, but prefer to commit themselves to structures, that is, to the relationships held by some instances of discourse in a text or when placed in memory chunks.
It is worthwhile to notice that the above mentioned discordances do not solely arise as a result of reductional points of view. It would be a mistake to compare them with the disciplinary division of each science into several subsciences: there is a physical geography that studies the physical configurations on earth, and there is an economical one whose main point of interest is business. Likewise, linguistics can also distinguish phonemic branches alongside semantic ones no matter what kind of theory (descriptivist, generativist, or structuralist) we are dealing with.
Linguists have often characterized these differences -descriptivist vs. generativist vs. structuralist- as methodological ones, but such an epistemological reduction, although very convenient at first glance, proves to be false too. At the outset certainly such a distinction does work: descriptive linguistics, as stated in Bloomfield's  Language (1933), was conceived as a purely empirical description of language facts; on the contrary, Chomsky's Syntactic Structures (1957), that followed a well known rejection of positivism in his review of Skinner's Verbal Behavior, paid close attention to mental facts as expressed by deep structures, that is, it emphasized intuitive knowledge from an idealistic point of view; F. de Saussure's structuralist position, as adopted in his Cours de linguistique générale (1916), could be called a phenomenological approach due to his stressing of the oppositive, relative, and negative nature of linguistic events, that is, the fact that they are not understandable without the consciousness of other events that they replace. 
Nevertheless, I would like to point out that epistemological queries can be found in any linguistic theory. With regard to our three main classical schools it could be noted that each has developed from a strong epistemological position to a weaker stage that, in some sense, takes root in the opposite direction.
Generative grammar, for example, after having reached its more abstract and mysterious stage in the seventies -the generative semantics period- later became either a rather positivist hypothesis, when extended standard theory -Chomsky's 1972, Studies on Semantics in generative grammar, and later, 1981, Lectures on government and binding- recognized the contribution of surface marks to semantic interpretation, or even a phenomenological one as soon as Gestalt principles were incorporated to the model -Jackendoff's Semantics and cognition (1985).
From the structuralist point of view, the purely phenomenological position, which under the influence of F. de Saussure has been sustained by Jakobson's "Zeichen und System der Sprache" (1959), has led, curiously enough, sometimes to highly speculative grammars, such as Hjelmslev's Prolegomena to a theory of language (1961,1st 1943), and at others to descriptive studies on the grammar of natural languages, such as Togeby's Structure immanente de la langue française (1951) which is said to be supported by the former.
Descriptivism has also proved to be an ambiguous word: besides the full empiricist position of American Immediate Constituent Analysis, we should mention that tagmemics, one of its daughter-branches, when opposing emical and etical points of view, shows a strong commitment to phenomenology as it deals with events and their consciousness at the same time; and looking at recent developments of another branch of American descriptivism, it is not surprising to find that classical levels, as stated by Lamb's Outline of stratificational grammar (1966), have been implemented with abstract ones where our attention is drawn to pure mental facts -Makkai,  1972, Idiom structure in English.
Let's go a step further. In my opinion the differences that separate one linguistic theory from another lie exactly in the question we started this chapter with. Any linguist would agree that his/her science talks about codes and messages, but so far no balance has been attained between both notions:
a) From a descriptivist point of view, a linguistic code is only a set of major classes of items that can be obtained by analyzing a corpus of utterances and by observing resemblances among their linguistic behavior. As a result of this position the linguistic code always remains dependent on the corpus of utterances that supports it: any statement about this corpus will prove to be correct (if scientific method is performed in a convenient manner), but any statement will also be an oversimplification of the facts, because the number of utterances of a human language is infinite. In other words: a linguistic code, as used by the descriptivists, cannot convincingly explain other messages apart from those that formed the original corpus from which it arose, although it performs this task in an accurate and elegant way:
                                                                                                                                                      
                                            Æ?                                                                                                    (distributional classes)  code          messages (corpus)
                                             ¨
b) From a generativist point of view, things seem to happen in the opposite way. Now the code or, as is commonly said, the competence is a set of rules that generate not only a set of structural descriptions adequate to real utterances, but also a variety of descriptions that do not correspond to any message of the language that the code is describing at all -as is known, generative grammar developed by putting increasing restrictions on the power of rules. In other words:
                             Æ                                                                                                            (competence)  code          messages (perfomance)
                           ? ¨  
c) From the structuralist point of view, no inadequacies should arise between the code and its messages as the former is the consciousness of the latter, and both are described at the same time. However, no one has found a procedure to test whether a given code overlaps with a given set of messages, thus an inherent hiatus remains between them. In other words:
                            Æ?
(langue)  code          messages (parole)
                            ?¨
      
    Such a situation could not be solved without taking into account the code and the messages at the same time. But doing so proves to be impossible without previously choosing an adequate formalism which   would  incorporate their relationship.
It is known that modern science developed as a result of discovering convenient mathematical models capable of supporting the facts it intended to talk about. The notions of space, time, and speed, whose mutual relationship determines movement, were well known  from antiquity throughout the Middle Ages; however, before Galileo Galilei, no one had proposed a mathematical model capable of joining them together in a single formulism as differential equations turned out to be; since the discovery of such a statement, physics was not only allowed to explain some previously proven relationships of space, time, and speed, but also to predict new ones that could be obtained by substituting two of the variables for numerical values in order to obtain the remaining unknown factor.
Linguistics has often used mathematical apparatus to state the facts it aims to explain. Nevertheless, it is doubtful that the correct ones have been used. Descriptive linguistics, for example, has normally stated code relationhips in terms of set theory, or, on the logical side, in terms of logic of classes: a given element x is said to belong to class X, this class is included in another class X´, etc. Generative grammar on its own was conceived of as a Boolean algebra where combinations of a finite set of symbols should bring about an infinite set of symbolic chains, thus  constituting a logic of relations. On the other hand, when using a mathematical model as in glossematics, structuralism takes advantage of propositional logic and its mathematical background, e.g. the three main relationships that Hjelmslev used to distinguish interdependence, determination, and constellation could easily be proven to be based on the truth values of conjunctive, disjunctive and conditional connectives respectively.
No formalism has convincingly solved the problems it was supposed to. The reason being that, when the previous situation of the model is applied to its object of study, the theoretical mathematical models only account for the code from which messages are supposedly derived, but it never includes both the code and its messages at the same time. Whether the code is too weak, too powerful, or simply unrelated to its messages, descriptivism, generativism, and structuralism have shared a fundamental inadequacy in their respective classical periods, because set theory and its derivations constitute only an explanation of the code, and never an explanation of the code and of its messages simultaneously.
In the last decades linguistics has slowly been approaching a model that is simultaneously capable of dealing with a set of elements (messages) and a series of subsets of elements that intend to explain the former ones (a code). Such a formal apparatus, though derived from classical set theory, has lately acquired an autonomous entity called topological space.
It is worth pointing out that the need for topological formalism has arisen in all three schools of linguistics we are reviewing in this chapter. Although American descriptivism started by establishing very simple set theory relationships among elements, it soon surpassed such strict restrictions and, at the end of the fifties, began to make use of graph theory diagrams to represent syntactic relationships as well. This practice, that in Chomsky's Syntactic Structures is shown as an inheritance of his formative period, takes on epistemological significance in stratificational grammar where scholars insist on the fact that each message can be formalized -i.e. codified- either in an algebraic or a graphic spatial manner. Graph theory is simply a branch of topology, though descriptivists have not always been aware of this fact.
	
Nor did generativists realize that when raising the problem of establishing formal devices to support a set of messages and the need to select one of them, they were really posing a question that only topology could solve. Some scholars have nevertheless been aware of this fact -Kuroda, S.Y., 1973, "Généralisation de la notion d'équivalence de grammaires: une méthode topologique",The formal analysis of natural languages (Proceedings of the first international conference), ed. M.Gross, M.Halle, and M.P.Schützenberger, The Hague, Mouton, 362-72- and have claimed that the different grammars that prove to be adequate for a specific set of utterances constitute their explanation, and that comparisons should be done on topological grounds.
Structuralism has also surrounded itself with topological notions even from its very beginnings. No wonder R. Jakobson quoted the mathematics historian T. Bell when emphasizing the importance of topological relations -S.W., I, 637- as early as 1926; Husserl's 1882 dissertation on variation theory -a typical topological problem- had obviously preceded his disciple's own thoughts. However, the first studies of the Praguean school of linguistics show that a switch from mathematical exactness to rhetorical uncertainty occurs almost every time. In the technical sense of the word, topology has been present in structuralism only since R. Thom's 1972 Stabilité structurale et Morphogenése where dynamic linguistic processes -phonetic change, morphological development, etc- are explained by means of topological formalism. One has to regret that these scholars, having mainly a mathematical background rather than a linguistic one, have been unable to predict significant linguistic generalizations and have limited themselves to stating linguistic facts that were firmly established by linguists a long time ago. In any case the mathematical side of this attempt remains very impressive, and one should expect that collaboration between mathematicians and linguists in the catastrophe theory will go further.
In any case, it is remarkable that our three main schools of linguistics are not unfamiliar with topology as a method, even if this formalism has been consciously adopted only very recently. It appears that  topology may constitute the proper mathematics for linguistics because it permits the  simultaneous treatment of codes and messages as required above. Mathematics usually develops many years in advance to its practical applications as the history of physics demonstrates more than sufficiently. Topology was born in our century and has thus far shown very few applications of its own mathematical consequences. The use that linguists make of such a formalism may prove to be very productive in the future.
The reason lies in the fact that, as I have just mentioned, only linguists and their semiotic colleagues have to talk about reality and how reality is perceived at the same time, and only topology is able to describe a set of elements and the subsets of elements that limit or describe them all at once.
Let's explain it in a very simple manner. A topological space could be conceived of as a set of elements where some of them limit the others and are said to be open-sets, and the former and the latter together constitute the closed-sets. For example, when you consider the set of students that have taken the course you are teaching, there are always some of them whose examinations serve as points of reference to classify all other examinations -that is, the papers that have a straight A, B, C, etc, and make it possible to mark other papers as deserving an A+, B-, and so on: the former, (A, B, C, D, for example), belong to the open sets, and the two combined, (A+, A, A-, B+, B, B-, C+, C, C-, D+, D, D-), to the closed sets. Geometry constitutes a good source to provide examples in topology: every point in a circle belongs to the closed sets, but only the points of the surrounding circumference, that contain them and could be called their boundary, belong to the open sets.
 Linguistics, when approaching its object of study, is faced with a very similar situation: a language is an (infinite) set of elements -from utterances to minimal phonic items- where each is described by other surrounding elements, either in the chain or in the memory (syntagm vs. paradigm). It follows that linguistic elements could be considered closed when they are described, and open when they are describing. Topology knows such a situation where all elements are alternatively either closed or open as trivial  topology.
But when one aims to attain a deeper insight of the structure of language, it is recommendable to keep some elements as specialized describers -open sets- which put some limits on all the elements of the set -closed sets-, not only on the others but on themselves as well: such a methodological decision is at the foundations of grammar whose terms "adjective", "syllable", and so on,  are applied to other elements  -"red" is an adjective- and sometimes to the elements of their own set too -"adjective" has three syllables. In other words, we can conclude that the opposition "code vs. message" or, if you like, "metalanguage vs. object language" can be formally translated by "open sets vs. closed sets" and dealt with using the methodological apparatus of topology.
It should not be thought, however, that the process of convergence of the linguistic schools we are developing here happens only in a purely formal realm. Language always involves human life as a whole, and any hypothesis on its formal nature has psychological implications too. What are the psychological assumptions of such a topological approach to language?
In fact contemporary schools of linguistics were aware of this side of the problem long before they achieved the formal theory they needed to deal with it adequately. At the very beginnings of structuralism it was obvious to everybody that from a psychological point of view it was a Gestalt theory: E. F. K. Koerner -1978, "European Structuralism: the Beginnings", Current Trends in Linguistics 13: Historiography of Linguistics, The Hague, Mouton- has recently made clear that F. de Saussure was strongly influenced by the German gestaltist Ehrenfels; in the Praguean school Bühler's Sprachtheorie (Jena, 1934) explicitly states that linguistics is a Gestalt science, and R. Jakobson, who had attended a course on Koffka's theory as a student at the University of Moscow in 1915-16, explicitly related phonology to these modern psychological views in 1931 - "Prinzipien der historischen Phonologie",T.C.L.P.,IV-, and finally tried to continue the work of two Gestalt scholars, Köhler and Stumpf, in his studies on aphasic phenomena -1941, "Kindersprache, Aphasia und allgemeine Lautgesetze",S.W.,378ff. The Gestaltic roots of structuralism have later been ignored by many students, perhaps as a result of their dedication to specific studies instead of being concerned with the problem of linguistic foundations, but these roots should never be forgotten: after all structuralism and phenomenology were born at the same time and depend on the same set of assumptions, and both remain Gestaltic as far as Psychology is involved.
One may think it would be hard to find Gestalt antecedents in the case of descriptivism too. But I would like to mention here the fact that distributionalism, as a method, was philosophically promoted by Wittgenstein's Philosophical Investigations (1953) when he observed that the meaning of a word cannot be understood out of its environment; a word has a specific value if related to some other word, and sometimes a completely different one when related to another: this situation, Wittgenstein said, could be compared with certain pictures where one can recognize a "duck" or a "rabbit" according to which lines the individual observer considers most prominent. In other words, the Immediate Constituent Analysis cannot be performed without implicitly applying Gestalt principles.
      The Gestaltic nature of the generative grammar postulates, as they evolved during the eighties, is particularly interesting as this theory was born as a rejection of behaviourist psychological models. No mention of the Gestalt approach can be found in the classical works of 1957 and 1965 for they were highly mentalistic and they emphasized the innate knowledge of native speakers: but after Lakoff's 1977 "Linguistic Gestalts" (Papers from the 13th Regional Meeting of the Chicago Linguistic Society),  scholars got strongly involved with Gestalt psychology, not only in linguistics, but in other perceptual strategies, such as music as well ( see R.Jackendoff, 1983, Semantics and Cognition, The M.I.T. Press, and F.Lerdhal, R.Jackendoff, 1982, A Generative Theory of Tonal Music, The M.I.T. Press). Nowadays Gestalt psychology has almost completely taken over generative methodology.
Contemporary schools of linguistics have arrived at Gestalt convergence at the same time they joined the formal topological models. No one should be amazed by this fact. A Gestalt configuration is a set of stimuli where some are perceived as foreground simply because the others are left as background. In the picture :

we can see a couple of faces that are looking at each other if the central area is perceptually weakened, but we would see a kind of cup  if perceptual weakness were to affect the lateral areas.
 Properly speaking the foreground/background distinction of Gestalt psychology corresponds to the topological opposition of closed sets/open sets. Although there are situations where an element could belong to either the foreground or the background, we usually tend to specialize some stimuli as constitutively weaker than others. For this reason we can see constant objects in real life when they are outlined on the surrounding environment: people do not perceive a drawer in a desk as if it were a hole in the desk or they only very rarely do so. This situation has its topological equivalence in the fact that topological spaces normally exhibit some preferred topologies and not only the trivial topology: in linguistics, as I have mentioned above, it corresponds to human tendency to construct grammars whatever their formal character may be.
It would be interesting to underline that, if topological spaces allow the introduction of more than one topology, languages -i.e. sets of utterances belonging to the same speech community- can likewise be described by more than one grammar without lacking explanatory power. Similarly we can classify a set of students by either the boundary grades (A, B, C,...) of some students, or by the minimum athletic scores (jumping a height of 1.5 mts, running 100 mts in 15 seconds, etc.) that another set of pupils presents: in both cases the boundary grades/scores would thus allow us to classify the entire group of students/athletic team depending on whether our proposal was to constitute academic groups or athletic teams. 
 The history of linguistics has usually been pictured as a quarrel between schools and theories. Nevertheless, when it is viewed as a set of couples L-M1, L-M2, L-M3, ... L-Mn, where one and the same set of utterances -L- can be described (limited) by several different subsets emerging from their proper elements -M1, M2, M3, ... Mn-, the picture obtained is much closer to reality, whether one grammar is said to be "morphosyntactic" and another described as "semantic": in each case the point of view is a different one, and the results of investigation, obviously, differ too.