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  QUANTITATIVE MODELS IN COGNITIVE LINGUISTICSChuluundorj Begz 21 ABSTRACT Human verbal thinking is an object of multidisciplinary studies. Complexity of verbal cognition is often an integration of neurocognitive and psychological, bottom up and top-down processes. Anal ysis of Human verbal performance and language acquisition leads to combination of qualitative and quantitative approaches. Human mental states as constituents of mental continuum represent an infi nite sets of meanings. Verbal perception and interpretation of multiple meanings in mental continuum must be modeled by applying quantitative methods, particularly set theory and tensor methods.Notion of vector space in combination with fuzzy set theory presents a way for analysis of hu man semantic vocabulary and mental representations, rules of clustering and mapping. Number of meanings is not limited, but number of words and rules for building syntax structures are limited. For description of human semantic vocabulary and high order syntax need to apply also non-Euclidean metric spaces (Minkowski, Hausdorff, Levenstein etc.) In this connection concept of topological iso morphism/ homomorphism in synthesis with ideas of s6t and subsets must be applied to classification of words, syntax structures, grammatical categories.Author considered that different forms of verbal representation should be analysed in the light of tensor (vector) transformations, also manifolds. Change in semantics and structures are matter of analysis in 3D and other N-dimensional spaces.Above presented ideas is supported by empirical data from typologically different languages as a Mongolian, English and Russian.Author argues that quantitative approach to cognitive linguistics offers new opportunities to de velop alternative version of quantitative semantics and thus to extend theory of UG in new dimen sions. KEYWORDS: Verbal cognition, mental representation, vector space model, fuzzy set, non-Euclidean space,   isomorphism and homomorphism, quantitative semantics. Basic ideas of the presentation:1. New challenges to develop multidisciplinary study of human verbal thinking proposed more dynamic approach to cognitive research in linguistics based on human mental space. Modeling human mental spaces as non-Euclidean spaces has lead to concept of Hilbert spaces which extends the me thods of vector algebra and calculus from the two dimensional Euclidean plane and three dimensional space to space (Minkowski etc.) with any finite or infinite number of dimensions.The infinite cardinality of emergent propositions in a like-quantum semantics is closely con nected to quantum coherence: ^  = ai a2 : the properties and ^ 2  belong to ^ with de grees of membership and a 2  respectively. Differences in human mental clustering and mapping reflected in parts of speech and types of syntax structures in typologically different languages present special case for analysis in terms of like-quantum semantics and fuzzy sets. Fuzzy set operations (also properties of Boolean algebra-idempotence, commutativity, associativity, absorption, distributivity, complementarity) have a significance for interpretation of attributive and associative relations, tax onomic ranking which means classification of verbal structures expressing different dimensions and values of object (referent). This is important for analysis of mixed semantic and pragmatic interpreta tions of verbal structures.Cardinality of continuum and fuzzy sets with members connected by quantum coherence are framework for modeling verbal structures and mental states in non-Euclidean space time and this framework consistent with tensors analysis of human mental states. Comparative study of typologi- 21  University of the Humanities, Ulaanbaatar, Mongolia, chukab@hotmail.com 46  cally different languages conducted by us supports to an idea that scalar product of vectors is effective tool for interpretation of word meaning. In verbal communication each point on the sphere fixes a single interpretation of a given situation. Amplitude between the logical descriptions must be ex pressed by the angle between the two interpretations in terms of cross product and outer product in such analysis.In same way vector products is applied for interpretation of multiple meanings, dot product, cross product of vector - for interpretation of context dependent word meaning and pragmatic situa tion, cross product, outer product as akronecker product of tensor - for interpretation of sentence and discourse with synthesis of semantics and pragmatics.2. Language comprehension is both embodied and symbolic: symbolic-through interdependen cies of modal linguistic symbols, embodied-through references these symbols make to perceptual re presentations. In visual and verbal perception recognition ability to identify stimuli is important for object recognition. Multidimensional stimuli reflected in multidimensional scanning (MDS). Identifi cation of multidimensional stimuli has performance limit 7±2 (Miller, A. 1956) which has strong cor relation with stimulus structure.Intrinsic features of an object characterize the surface independently of any particular coordina- tization systems. Intrinsic features of space-time (curvature, metric tensor) are objectively real. Ex trinsic features are mere artifacts of the form of representation of subjective coordinatization, particu larly of verbal thinking spaces.Combination of intrinsic and extrinsic features has caused semantic changes, transformations, pragmatic interpretations. Human evaluational (or semantic) reactions are neurologically based on responses to symbols, and other events. Mapping this event in the brain (verbal mapping) is differing, varying. Space-time dimensions of verbal (visual) perception are varying. There are differences which are reflected in syntax structures, in grammatical categorization: SOV-SVO structures, manner of motion, causative and attributive constructions. In the light of human verbal mapping ambiguities in syntax, representational granularity, color recognition presents an interest for developing hypothesis of linguistic relativity (Sapir-Whorf)in terms of differential geometry and topology.3. The notion of divisibility and decomposition of an integer into a product of primes basic me taphor of arithmetics means that primitives of mental map should be described as a product of primes, thus proposition and concept - as the product of a unit.An experiment on universal syntax for verbal and musical, numerical production is best practic al application to above named notion. One of effective ways to analyze difference between musical and numerical processing, and syntactic processing is to use metrical stimuli for analysis of musical numerical and verbal structures. The subjects clearly answer faster if they have to decide that a num ber is bigger with their right hand, and smaller with their left hand. This spatial association depends only on the relative sizes of the numbers. “It is very interesting to see that primary colors are asso ciated with small numbers and that larger numbers are associated with more complex colors: black and white with 0 and 1or 8 and 10; yellow red and blue with small numbers such as 2, 3, and 4; and brown, purple, and grey with larger numbers such as 6, 7 and 8. Stanislas Dehaene suggests the fol lowing explanation for that association: “Because the number of neurons remains constant, the growth of the numerical network must occur at the expense of the surrounding cortical map, including those coding for color, form, and location” (Stanislas Dehaene, 1997).Primes in mathematics served as a basis for mental transformations, thus - for semantic trans formations in linguistics. Rules for mental transformations based on rules (mechanism) of perceptual spaces (modalities) serve as a basis for semantic transformations clustering, embedding and multidi mensional mapping.Idea of perfect number + 1)« *3 = 4(3) + 1 = 9 + 4 = 32 + 22jn combination with prime num bers presents revised vision of deep structures in linguistics leading to new version of generative grammar.4. Graph is effective tool for description of discourse structure. In the discourse a propositions (as a lines, vectors) have exactly one point in common and they intersect at this point. Presenting a discourse as a set of propositions (algebra) means that propositions have their directions (links) and 47  magnitude (intensity). So propositions should be modeled in terms of vectors. But Implicitness of propositions is object of analysis in Non-Euclidean geometry. So different graphs are different repre sentations of same semantic structure.Weighted graph is one of interesting models to a describe structure coherence between semantic units in a discourse. In graph model, particularly directed graph the degree = indegree(number of edges entering) + outdegree (number of edges leaving) is an indicator of cohesion and coherence. It means applying vector (tensor) models to discourse analysis in combination with graph models is powerful approach to discourse modeling. Knot theory provides different forms of representation of human discourse in Euclidean and non - Euclidean space. Knot is significant for modeling human mental structures and mental transformations.Knot has some advantage for embedding: Reduction of different structures of discourse to one basic (primitive) structure: Two knots are equivalent when one curve can be deformed into the same shape as the other. Knot polynomial as an invariant of the knot is used for modeling semantic struc ture of discourse, its transformations. Graph-based, knot-based comparative analysis of discourse ad dressed isomorphism problem which has to lead to typology of discourse.5. Discovery of mirror neurons supports the idea of cross modal organizational structures (COGs). Idea from neuroscience in the conceptual theory of metaphors also supports the idea that links between domains are creating a binding between the two domains. In conceptual blending the underlying cognitive mechanisms are bodily-grounded and not arbitrary. This ground is constrained by biological phenomena such as neuroanatomy, the human nervous system. Input spaces, mappings, and projections are realized by bodily-grounded experience such as thermic experience, visual per ception, spatial experience and so on. Image-schemas like container schema, source-path-goal schema, spatial relation schema (above, magnitude), complex relations (trajector-landmark relation) are exam ples of COGs. Image-schematic transformations as a basis for semantic transformations: mapping of the event components to linguistic structures in typologically different languages: motion paths, its manner, source and goal of motion, egocentric and retinocentric constructions.In conceptual blending natural numbers in one mental space in blend with proportions (of ob  jects) in the other space. Example: 6:3, 12:6, 500:250 are fused in blend with 2. The blend develops emergent structure using mechanisms of composition (integration the separate units), completion (background knowledge and meaning), elaboration (imagination) and also compression, recursion (Faunconnier, G & Turner, M. 2002).Blending can compose elements from input spaces to provide relation that do not exist in the separate units -(metaphor 1 -f-1 =£ 2 jCognitive process of creating metaphor is similar in mathematics and verbal thinking. A meta phor is the link between two domains a source domain, more concrete, and a target domain, more ab stract.Operation serving metaphor (homomorphism) building is embedding. Cross-language embed ding must be introduced to establishing an equivalence of structures and classes in languages related to different families, in that’s way - to analysis of no-linear verbal cognition. References David Landy, Naoh Silbert, Aleah Goldin. (2013). Estimating Large numbers. Cognitive Science 37(5), 775-780. Wiley Blackwell.Faunconnier, G., & Turner, M. (2002). The Way We Think: Conceptual blending and the mind’s hidden complexities. Basic books. NewYork, NY.James Robert Brown. (2008). Philosophy of mathematics, Routledge Taylor & Francis Group.Kurt W. A. J. H. Y. (2005). Manifolds, Lie Groups, Lie Algebras, with Applications.Spring.Miller, A. (1956). Magical number seven, plus or minus two: Some limits on our capacity for processing information. The Psychological Review.Stanislas Dehaene. (1997). The number sense. New York, NY: Oxford University Press. 48
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