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International Journal of Applied Linguistics & English Literature ISSN 2200-3592 (Print), ISSN 2200-3452 (Online) Vol. 2 No. 4; July 2013 Copyright © Australian International Academic Centre, Australia
Mathematical Approaches to Cognitive Linguistics
Chuluundorj Begz University of the Humanities, Sukhbaatar Square – 20/4, Baga Toiruu, Sukhbaatar District Ulaanbaatar City, Mongolia Phone: +976-9911-5670 Fax: +976-11-322 702 P.O. Box – 210646/53 E-mail: chukab@hotmail.com
Received: 13-04-2013 do

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International Journal of Applied Linguistics & English Literature
ISSN 2200-3592 (Print), ISSN 2200-3452 (Online)Vol. 2 No. 4; July 2013Copyright © Australian International Academic Centre, Australia
Mathematical Approaches to Cognitive Linguistics
Chuluundorj BegzUniversity of the Humanities, Sukhbaatar Square
–
20/4, Baga Toiruu, Sukhbaatar DistrictUlaanbaatar City, MongoliaPhone: +976-9911-5670Fax: +976-11-322 702P.O. Box
–
210646/53E-mail: chukab@hotmail.com
Received: 13-04-2013 Accepted: 24-05-2013 Published: 01-07-2013doi:10.7575/aiac.ijalel.v.2n.4p.192 URL: http://dx.doi.org/10.7575/aiac.ijalel.v.2n.4p.192
Abstract
Cognitive linguistics, neuro-cognitive and psychological analysis of human verbal cognition present important area of multidisciplinary research. Mathematical methods and models have been introduced in number of publications withincreasing attention to these theories. In this paper we have described some possible applications of mathematicalmethods to cognitive linguistics. Human verbal perception and verbal mapping deal with dissipative mental structuresand symmetric/asymmetric relationships between objects of perception and deep (also surface) structures of language.
In that’s way methods of tensor analysis are ambitious candidate to be applied to analysis of human verbal thin
king andmental space.
Keywords:
Mental mapping, verbal cognition, continuum hypothesis, Hilbert space, tensor transformations, vector space model, human mental lexicon, representational granularity, embodied and symbolic cognition, mental models.
1. Introduction
Cognitive linguistics, in close relation with neuro-psychological research of human verbal perception presenteddynamical approach to linguistics created new framework for multidisciplinary research. Object of research in cognitivelinguistics, complexity and multi-dimensionality of phenomena have required re-thinking a methodology of research byapplying more powerful theories and methods of mathematics. Researchers in USA, European countries, Russia andJapan have introduced mathematical concepts and models in psychological and neuro-cognitive studies of language.Human verbal perception, phenomena of embodied and symbolic cognition, metaphor are an object of multi-disciplinary research both in linear and non-linear frameworks. This framework provides sound basis to apply powerfulideas of quantitative, mathematical methods to cognitive linguistics.Tensor theory, Vector transformations in finite and non-finite continuum and in Hilbert space serve as a basis for modeling above named phenomena of hum
an verbal thinking and in that’s way offer new perspectives in extension of
the theory of Universal grammar. Application of mathematical models in cognitive linguistics lead to development of comparatively new area of multidisciplinary which should be called as a quantum semantics, or dynamic model of Universal grammar.Author of this paper presented some ideas in applying mathematical methods and ideas to cognitive research. Theseideas have been used in research in cognitive linguistics including empirical data from typologically different languageslike Mongolian, English and Russian.
1.1 Continuum (Cantor G) and Hilbert space in cognitive linguistics
In the light of cognitive research, mental states (in human brain) served as a basis for generating meanings which are producing high-order syntax structures, discourses.
In that’s way applying an idea of Continuum hypothesis to modeling mental states and meanings are useful to describe
human verbal thinking categories and primitives. Mental states are infinite (infinite sets) and the size of these sets, their cardinality, is infinite ).
Cantor’s theorem “the cardinality of the power set
(set of all subsets of S) is greater than the cardinality of
S:< (S)
has direct links to analysis of human mental space: power of subsets of propositions (meanings) indiscourse is greater than sum of propositions (meanings) in mental discourse. But words and rules (grammatical andtransformation rules) are finite sets.
In that’s way concept of cardinality has an significance for comparing infinite sets of mental concepts, meanings and
finite sets of vocabulary, grammatical rules and syntax transformations, bijection between these two kinds of sets.It means
than Cantor’s continuum hypothesis, particularly notion of set in combination with latest results of
psychological and neurolinguistics researches presented new opportunities to develop theories of generative syntax andsemantics, particularly ideas of UG in new dimensions (Chomsky Ch. 1993).Modeling mental states and meanings as an infinite sets is leading to concept of Hilbert space (Naohito Chino, 1998)which extends the methods of vector algebra from the two
–
dimensional Euclidean plane and three
–
dimensional spaceto spaces with any finite or infinite number of dimensions. Hilbert space as an abstract vector space served as an
IJALEL 2 (4):192-199, 2013
193
effective method to modeling macro (semantic) structure of mental discourse, its latent organization, mental force andcoherence, cohesion.Tensor (Vector) analysis in combination with ideas of continuum and Hilbert space of support an interpretation of storing words in a memory, their distribution and relationships capacity of Semantic memory, neurocognitivemechanism of semantic organization of vocabulary, syntax of high order mental structures.
2. Tensor analysis in cognitive linguistics
Formalization of Semantic space in terms of Hilbert space is background to apply tensor models to analysis of vocabulary and syntax structures in psychological and neurocognitive linguistics.
2.1
Measuring human mental lexicon
In
–
dimensional semantic spaces standard Euclidean distance function is used for measuring the similarity betweentwo words or concepts and the similarity is partially defined by the degree to which their features (sensory, motor,affective features such as shape, size, color, distance, location) overlap, the degree to which they share same contextsare basis to measure direction and magnitude (size, force) of relations between words in mental lexicon. Many patternrecognition techniques are based on similarity measures between objects (nearest neighbor classification, cluster analysis, multi-dimensional scaling).The example in which two-dimensional semantic (metric) space between representatives of subsets is measured byusing standard Euclidean distance function:Fig 1. Semantic space between wordsMeasuring semantic space between words (concepts) is not point of analysis. Metric space (Euclidean, Minkowski,Hausdorff etc.) is important starting point for interpretation of psycho-cognitive operations as an operations of mentalgrammar, rules of verbal thinking. The similarity between words and concepts in Human semantic space presentsinterest for analysis of the phenomenon of generalization and mutual exclusivity in terms of continuum and Hilbertspace. Infant extend labels on the basis of category membership (principle of categorical scope) to avoid second labelsfor objects (mutual exclusivity). So for infant multiple meaning is different words. In contrast to it, infant also used a principle of generalization. So in this case Euclidean distance is simplest variant of generalization and mutualexclusivity in human mental space.Extension of the principle of generalization and mutual exclusivity to an area of classes of words or concepts is leadingto ideas of fussy sets. Several authors have proposed similarity indexes for fuzzy sets can be viewed as generalizationsof the classical set-theoretic similarity functions. A systematic investigation of this notion was performed by Dubois andPrade.The membership function as a representation of magnitude and scalar cardinality of a fuzzy set must to serve as a basisfor classification of words a syntax structures. In his well- known paper entitled Features of Similarity (Tversky)similarity among objects is expressed as a linear combination of the measure of their common and distinct features.Each object in domain D is represented by a set of features or attributes, and A, B and C denote the set of featuresassociated with objects a, b, and c, respectively.Representation of two objects that each contains its own unique features also contains common features. An importantaspect of Tversky's model is that similarity depends not only on the proportion of features common to the two objects but also on their unique features.Euler diagram presents good example to supporting an idea of description attributive properties and values in the fuzzysets, subsets and on this basic for interpretation of degree of membership of words and concepts in human mentallexicon. /Khan H. 2012/
(1)
(2)
нөхөр
-friend,
амьтан
- animal,
нохой
- dog,
муур
-cat,
чоно
-wolf.
IJALEL 2 (4):192-199, 2013
194
Example:
Мах
(meat) : Set of all products animals that are meat.
Сүү
(milk) : Set of all animals that are milk.
Шинэ
(new) : Set of all fresh products.Fig 2. Intersection of members in mental subset Notion of scalar cardinality of fuzzy sets in supporting an analysis of Human mental lexicon and syntax in combinationwith an idea of metric space (Human Semantic space).
2.2 Modeling tectonics of syntax structures
The distances between words are also structural characteristic of sentences. More easy formula is the cosine of twovectors by using Euclidean dot product formula.Fig 3. Distance between words in a sentenceAnalysis of word distance based on vector (tensor) models has close relation to statistical similarity measures betweensentences. In this connection sentence similarity measure presented by Junsheng Zhang Yunchuan Sun, Huiling Wangand Yanging He will be developed with application of tensor ideas. (Junsheng Zhang, Yunchuan Sun, Huiling Wang,2011). Co-occurrence of words in sentence reflects relationships between words (word categories) in N-dimensionalspace presenting complexity of semantic and thus human mental regions.
Fig4.Co-occurrenceofwordsinsemantic(mental)spa
Measuring co-occurrence of words, distance similarity between words in a sentence by using vector (tensor) modelsinteresting application is must be applied to an analysis of tectonics of syntax structures and thus of word/ sentencerecognition patterns.One of phenomena at syntax level is a permutation and substitution which is related to combination of semantic andgrammatical categories. In this connection the Levenshtein distance between twostringshas some importance. Tensor- based permutation particularly special unitary group transformations, irreducible representations of
SU
(n) are orientedResearchers to rethinking a permutation in Linguistics as an act of changing the order of elements, transformations.
Би бэлэг аваад гайхсан.
I have been exhausted with receiving this gift.
(гайхах–
exhausted,
бэлэг –
gift)
(3)
Fig 4. Co-occurrence of words in semantic (mental) space
IJALEL 2 (4):192-199, 2013
195
Eigenvalues and Eigenstates in linguistics are basis for analysis of transformations as dynamic properties of humanmental states.Angle preserving similarity transformation and rigid transformation (distance preserving), affine transformation(parallelism preserving) has taken place in syntax of different languages. So these classes of transformations must beconsidered as properties of mental syntax generating linguistic universals.Transformations of syntax structures based on mental mapping and embedding depend on space-time coordinatizationin human mind. According to differential geometry, distance between two events in space
–
time is presented dependingon particular coordinatization system. Intrinsic features of an object in space
–
time coordinate characterize the surfaceindependently of any particular coordinatization systems. Extrinsic features are mere artifacts of the form of representation. In our opinion intrinsic and extrinsic features in combination have caused difference in syntax structures.Interpretation of intrinsic and extrinsic features in terms of representational granularity needed to consider followingneurocognitive processes of verbal thinking. Strong influence of extrinsic features on an object must srcinate adifference in color presentation in typologically different languages as a Mongolian, Russian, English and Chinese.Color is recognized faster than size because color is not a relative property. Head - noun referring, structure with twosatellites referring to color and motion (
хөөрсөн улаан бөмбөлөг, гүйж яваа өндөр хүү
), computational algorithms ondeterminism (minimally specified expressions. over specified expressions. The big red car. The red big car), GRZalgorithms (Generation of referring expressions-
“size” has great disc
riminative power) also present an example for analysis of human computational algorithms with vector application to in differential geometry. (Deemer, K. V. CS.2012. 36/5 P 830).Difference in object localization is an example of coordinatization dependent of perceptual granularity for which scalar product (scalar triple product) is basic interpretation.
Энд ширээ байна.Өрөөнд хивс дэвсээтэй байна.
(Mongolian)
Здесь стоит стол.В комнат
e
лежит ковер
. (Russian)There is a table.There is a carpet in the room. (English) Notion of representational granularity is very important for testing symbol interdependency hypothesis becauselanguage comprehension is both embodied and symbolic. Human algorithm on sequence fixation also must beinterpreted in terms of tensor (vector) models. Specific object of tensor analysis are differences in event mapping:SOV
–
Би ном авсан.
(Mongolian)SVO
–
The man opened the letter. (English)SVO -
Парень открыл дверь
(Russian)Tensor (Vector) in terms of their transformations presents effective way for describing mapping of the eventcomponents to linguistic structure, particularly mapping in argument structure an object, motion paths, its manner,source and goal of motion. In tensor-based mapping trajectory of motion, its speed manner are choremes. According toKlippel A., conceptual primitives combine movement choremes into chunks. Chunks (CMPs) are the basis for aconceptual grammar of movement patterns. (Klippel, A. Topics in CS.2011.3/4. P722).In tensor analysis choremes should be included as a proper of direction and magnitude. Chunks are considered as avector multiplication in this modeling. Differences in coordinatization are surface illustration of change in vector direction and magnitude in mental mapping. For example in Mongolian and English languages source /agents should bemore prominent than goal/ patients for vector transformation.Giving
Багш хүүд ном өгөв.
taking
Хүү багшаас ном авав
.Throwing The girl threw the ball to the boy.catching The girl caught the ball from the boy.Source and goal paths in conceptually different domains is an object of such transformations:
Дорж машинаас онгоцонд суув.Дорж ажилчинаас эзэн суув
.Similar examples in English.Brian went from the car to the store.Brian went from sad to happy.

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