This document discusses various models and concepts for analyzing systems, including: 1. It describes different types of systems including mechanistic, animate, social, and ecological systems. 2. It discusses open, closed, and semi-closed systems and how they interact with their environments. 3. Several mathematical and scientific concepts are proposed as models for linguistic and semantic analysis, such as group theory, topology, Hilbert spaces, and quantum mechanics. 4. The document suggests that these concepts from mathematics, physics, and other fields can provide frameworks for understanding semantic structures, mental representations, and cognitive processes.

1. Linguistics: models for
system analysis
Academician Chuluundorj. B
President, University of the Humanities
1

2. TYPES OF SYSTEMS
Universal categories
Space
Time
Mechanistic Energy
System +
Change
Entropy
Causation
2

3. Types of System Parts Whole Example
Model
Mechanistic No choice No choice Machines
Animate No choice Choice Persons
Social Choice Choice Corporations
Ecological Choice No choice Mature
3

4. TYPES OF SYSTEM
Closed system
is independent of its environment
Semi-closed system
A thermostat, circulatory system
Open system
organization systems
interact with the outside world
exchanging information, energy or material
4

5. Homeostatic Systems Or Dynamic Equilibrium:
Types of System Example of System
Homeostatic Human body
Closed A closed economy of a country
Semi-closed A circulatory system or heating
system
Open A business
Deterministic A computerized accounting
system
Probabilistic A football playing system
Cybernetic Most businesses
5

6. SYSTEMS THEORY
 Systems theory is the interdisciplinary study of systems:
- Bertalanffy‟s general system theory (GST)
- Action theory of Talcott Parsons
- Social systems of Niklas Luhmann
 Self-regulating systems:
- Physiological system of our body
- Human learning processes
Chaos Theory – The behavior of certain dynamical systems (as the butterfly
effect)
6

7. COMPLEX ADAPTIVE SYSTEM
Macroscopic collection‟ of relatively „similar and partially connected micro-
structures - to adapt to the changing environment - dynamic networks of
interactions, equilibrium conditions
7

8. DYNAMIC SYSTEMS THEORY
Dynamic systems theory is an area of mathemitics used to describe the
behavior of complex dynamical systems, usually by employing differential
equation(continuous dynamical systems) or difference equations(discrete
dynamical systems)
The Lorenz attractor is an example of a non-linear
dynamical system.
8

9. A dynamical system has a state determined by a collection of real
numbers, or more generally by a set of points in an appropriate state
space.
Small changes in the state of the system correspond to small changes in the
numbers.
The numbers are the coordinates of a geometrical space – a manifold.
Symbolic dynamics - a topological or smooth dynamical system by a discrete
space consisting of infinite sequences of abstract symbols, each of which
corresponds to a state of the system, with the dynamics (evolution) given
by the shift operator
9

10. Dynamical system theory – the neo-Piagetian theories of cognitive development
The learner‟s mind - A state of disequilibrium
The spontaneous creation of coherent forms
Newly formed macroscopic and microscopic structure support each
other, speeding up the process.
10

11. STRUCTURE
Structure – system is made of configuration of items, a collection of inter-
related components, network featuring many-to-many links.
11

12. Structural system
A structural system one-dimensional, three-dimensional depending on
the space dimension.
Real-world structure is strictly three-dimensional.
Verbal thinking is oriented to description of real world structure
most of mental models have reflected real world structure – three
dimensional structure.
12

13. FUNCTION
f(x) = … is the classic way of writing a function. And
there are other ways, as you will see
 The Input
 The Relationship
 The Output
Input Relationship Output
0 x2 0
1 x2 2
7 x2 14
10 x2 20
… … …
13

14. Injective, surjective, and bijective
A function is way of matching the members of a set “A” to a set “B”:
Multiplicative function: preserves the multiplication operation
Continuous function: in which preimages of open sets are open f(xy) = f(x)f(y)
Composite function: and be two functions. The composition of f
and g defines a function such that .
14

15. Semantics
General semantics (GS) - relations between the non-verbal and the verbal, including
our verbal and nonverbal transactions
GS - time binding - human engineering
GS - the meeting point of scientific-mathematical methods and daily life
The basic unit of study for general semantics - Human evaluational (or semantic)
reactions
Evaluational reactions neurologically based responses to words, symbols,
and other events
Meaning - ideas – mental representations
- truth conditions (T/F – Semantics and Pragmatics)
- semantic externalism (reference)
- determined by the consequences of its application (pragmatic theory) -
- prototype - radial structures
- in relation to other concepts and mental states (conceptual role semantics)
Semantic features. bachelor [+HUMAN, +MALE, +ADULT, +NEVER-MARRIED (?!)].
15

16. Semantics and Pragmatics
The infinite cardinality of emergent propositions in a like-quantum
semantics.
Hilbert space + Fuzzy theory – a set to a variable degree of membership;
a proposition and its variable relation to the true and false logical
constants. (New version of pragmatics T/F)
Quantum coherence: . The fuzzy interpretation the
properties and belong to with degrees of membership and
respectively. It means for complex systems: the Schrodinger’s cat can be
simultaneously both alive and dead!
16

17. Pragmatics
S - a finite set of signals
T - a finite set of types (information states) the sender might be in
- a prior probability distribution over
- a truth relation between and S
A - a set of actions that receiver may take
and - utility functions for sender and receiver to map triples from
to real numbers
17

18. Riemann’s sphere – a complex amplitude (Dirac, 1947) each point on the
sphere fixes a single interpretation of a given situation ,i.e. the assigning
of a coherent set of truth-values to a given proposition. Amplitude
between the logical description of the two worlds – is expressed by
(1+cos ), where is the angle between the two interpretations.
(Semantics/Pragmatics)
18

19. System analysis:
Ideas and models
Continuum Hypothesis (Cantor)
The natural numbers are in the set The size of this set, its cardinality, is
infinite
Thus /E/=N. The sets of real numbers - the continuum, “R”-the set of all real
numbers or the continuum.
Ideas: Finite set be applied to measuring number of meanings
Infinite set if words are finite set, meanings are infinite set.
Comments: mental states (бодол) – infinite set of sentences presenting these
states – infinite set of
words Finite set of
rules
19

20. …
Cantor’s theorem: The cardinality of the power set P(S) (set of all subsets
of S) is greater than the cardinality of S. In symbols │S│< │ (S) │.
Cantor’s theorem establishes a hierarchy of sets with infinite cardinalities:
…
Ideas: Notion of “set” be applied to classification of words.
Notion of “continuum” (N as a proper subset of R) be applied to
mental lexicon and thus to measuring capacity of semantic memory.
Sum of propositions in the discourse (set) and sum
of subsets of propositions in the discourse (set).
20

21. Russell’s paradox:
Comment: “X is Red” is equivalent to “X is a member of the set of red things”.
The set B of bananas is not itself a banana.
Idea: to be applied to the invariant theory (Boolean algebra, Cayley-Hamilton
theorems , Hilbert spaces) to invariance in semantics.
The characteristic polynomial of A is:
The Cayley-Hamilton theorem states that “substituting” the matrix A for in this
polynomial results in the zero matrix:
Hilbert spaces-extends the methods of vector algebra and calculus from the two
dimensional Euclidean plane and three dimensional space to space with any finite
or infinite number of dimensions.
Mental spaces-Semantic spaces-embedding
Non-Euclidean space
21

22. Comments:
Version I. The intermediate value
theorem Bolzano’s theorem states the
following: If f is a real-valued
continuous function on the interval
[a, b], and u is a number between f(a)
and f(b), then there is a c ∈ [a, b] such
that f(c) = u.
The intermediate value theorem
Ideas: Semantic space (mental space semantic space)words (or set
of words) in semantic space.
Invariance (invariant word) and variant words as elements of set (prototype
and radial structures) should be modeled in terms of continuous function
(Bolzano’s theorem).
22

23. Isomorphism: Two structures are isomorphic when:
They have the same number of elements or objects
The relations among elements of the one structure have same pattern as
the relations among the elements of the other.
Homomorphism is a weaker notion-requires the second condition, but not
the first to have different number of elements.
Isomorphism is structure preserving mapping.
Homomorphism is a topological isomorphism.
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24. Embedding is one instance of mathematical structure contained within another
instance such as group that is a subgroup.
Idea: to apply to the analysis of vocabulary in combination with Russell’s paradox +
Continuum hypothesis (Cantor) and Cantor’s theorem+ Bolzano’s theorem.
ХСЭ ном p 11-19
to reduce dual frame to a much smaller dimensional space through natural
embeddings(reduction of large spaces to finite dimensional Hilbert spaces).
Embedding: in algebra (structure preserving map in category theory-
morphism)
in topology (injective continuous map-“x” as a subspace of “y”-
homomorphism)
So: One space X is embedded in another space Y when the properties of Y restricted
to X are the same as the properties of X in terms of typologically different
languages.
Cross-language embedding-to establishing an equivalence of structures and classes.
Local isometry between Riemannian manifolds Riemannian symmetric spaces
Symmetry/assymmetry of Linguistic structures.
24

25. Differential geometry: distance between two events in space-time - its dependency on
particular coordinate system general relativity.
Intrinsic features characterize the surface independently of any particular coordinatization
systems.
Intrinsic features of space-time (curvature, metric tensor) are objectively real.
Extrinsic features are mere artifacts of the form of representation of subjective
coordinatization, particularly of verbal thinking spaces.
Idea: Intrinsic features are objective, but in terms of interpretation by (subject) may have some
influence. So these two factors have caused semantic changes, transformations, pragmatic
interpretations.
Relativity event structure is same for all.
Mapping this event in the brain (verbal mapping) is differing, varying.
to test: SOV, SVO- is a matter of extrinsic features (?).
Extrinsic features – color recognition (?)
Стол стoит
Intrinsic features are identic, but coordinatization system is ? - Ковер лежит
Idea: Sapir-Whorf hypothesis of linguistic relativity be renewed in terms of differential
geometry + topology.
25

26. A Structure is the abstract form of a system focusing on the
interrelationships among objects and ignoring any features of them.
Set theory axiom: two primitive concepts-set and member.
A set is a collection of objects. But set’s identity is wholly dependent of its
members-change the member and you change the set. By
contrast, groups very nicely fit the structuralist account.
Idea: Structure must be presented on the form of sets (set-based model for
structural analysis)
Group theory for structural analysis
26

27. In projective geometry: every pair of lines intersect at a point, the
exceptions are parallel lines with the introduction of a point of infinity,
even parallel lines can intersect .
Ideas: In the discourse a propositions (as a lines, vectors) have exactly one
point in common and they intersect at this point. (Super proposition?).
Discourse as a set of propositions (algebra).
Propositions have their directions (links) and magnitude (intensity). So
propositions should be modeled in terms of vectors.
Implicitness of propositions Non-Euclidean geometry.
be applied to a perception of music, to interpretation (verbal) of paintings.
27

28. Formalism in mathematics: Knot theory provides different forms of
representation .Knot-a closed, non-intersecting curve in space. Knot can
be transformed in various ways and properties which hold through such
deformations are an invariants.
Idea 2: Semantic structures at the sentence level-for a typology of syntax
structures.
Two knots are equivalent when one curve can be deformed into the same
shape as the other.
Applications:
Semantic transformations: invariant and variant (structures)
Identic structures and interpretations in varying pragmatic contexts.
28

29. In geometry: representational granularity.
Verbal text are different representational types – different inferences.
Diagram /depending on coordinatisation, surrounding topological
spaces.
Math Formula
Morning Star have same reference (Venus), but differ greatly in (mode
of representation)
Evening Star
The “2” has a sense (natural number which is the successor of “one”) and
has reference (the number two).
Энд ширээ байна.
There is a table. Брат приехал
Здесь стол стoит. Брат приежал
Representational similarity
Space-time dimensions of verbal (visual) perception. There are differences
which are reflected in syntax structures, in grammatical categorization.
Invariant of surrounding topological spaces intrinsic features of objects
invariant in linguistics.
29

30. Graph – as an ordered triple
Idea: Vertice point as a point for a coherence (logical) and cohesion (in three
dimensional space).
Next graph is a picture of the same graph as G in spite of their very different
appearances.
Vertex – as a point where straight lines
meet – lines of semantic force. to
analyze a coherence. Proposition as a
vector dimension, magnitude.
Idea: to apply to discourse
analysis, (comparison of semantic
structures. (?) Knot theory.
30

31. A weighted graph
•is a graph for which each edge has an associated weight, usually given by
a weight function w: E /discourse – set of proposition/
Idea: weight of proposition –in Vector model-magnitude (size) of
semantic force.
(Measure of the length of a route, capacity of a line, the energy
required to move between locations along a route).
31

32. Isomorphism Problem
 Determining whether two graphs are isomorphic
 Although these graphs look very different, they are isomorphic; one
isomorphism between them is
Comments: a h 1 2 g b 5 6 etc
i 4 c 8
Idea: Isomorphism between semantic structures of discourse – typology of
discourse
32

33. In quantum mechanics: Hilbert spaces are linear, vector spaces have infinite dimension = mental
spaces have infinite dimensions. States (state of an electron) are represented by a vector Ѱ in the
Hilbert space and properties (position, momentum, spin or energy of the electron) – by linear
operators.
Idea: Energy transmission (brain to brain) →mental spaces through which energy is transmitted
should be modeled as a Hilbert spaces or vector spaces of infinite dimension.
Structure component as a fermions (quarks, leptons)
Semantic component as a bozon (force carrier particles, guaze bosons, photons, gluons, W and Z
bosons)
Photons – carriers of electromagnetic field
W1Z – carriers of weak force
Gluons – carriers of strong force
33

34. Any prime of the form “4n+1” can be expressed as the sum of two perfect
squares in one and only one way.
13=4(3) + 1 = 9 + 4 = 32 + 22
Idea: Rules for semantic transformation.
To extend Chomsky UG - Rules for mental transformations (not logic
rules, but some ideas of logic be applied) based on rules (mechanism) of
perceptual spaces (modalities) – to serve as a basis for semantic
transformations.
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35. Bolzano-Weierstrass theorem: Every bounded infinite set S has at least one
cluster point. Q0 is divided into quarters. Q1 – into four quarters. Infinite
sequence of subsets: … Q3 Q2 Q1 Q0. Each has many points of S.
There is at least one point “P”, common to them all.
Idea: Discourse structure → super proposition
→ quarters as a part of discourse (in standard discourse) But: quarter is ?
→ In case of non-standard discourse, where coordinatization systems are
different or Hilbert spaces of infinite dimensions.
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36. Perfect number is one which is equal to the sum of its positive divisors. This “6” is
perfect number since it is divisible without remainder by 1, by 2, and by 3, and
1+2+3=6. Today about 45 (perfect number) are known.
21(22 - 1) = 6 to find some basic perfect numbers
Perfect
22(23 - 1) = 28 (structures) at morphological, syntax
number and discourse levels-structure
24(25 – 1) = 496
primitives to UG.
Idea: deep structures are limited, mental structures are not limited.
Idea: Math transformations should be applied to semantic (mental) transformations -
New model for structural analysis.
36

37. (-1, 1) (1, 1)
Small
sphere
(-1, -1) (1, -1)
Figure: with the small central sphere stay contained in the box?
Distance from the origin to the centre of any sphere is
Each sphere has radius 1; this radius of the central sphere is , for , there is
Idea: Visual (verbal) interpretation and non-Euclidean (Elementary application to
space propositions, space expressions) geometry.
(PoM. p 204)
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38. AoA hypothesis (age-of-acquisition)
Synaptic plasticity → (synaptic pruning, distribution of
neurotransmitter receptors, maturation of inhibition)
Age of acquisition
Cumulative frequency → (links between codes that are formed
of words
by the network become entrenched as a result of early
experience. Later experience in the word frequencies do little
to change them
The process of learning creates neurobiological changes that reduce plasticity. This is standard
view applied to learning of language (not mathematics) – When to start SL (FL)
Idea: Acquisition of word.
Zero and first order tensors should serve modeling an acquisition of words at different stages of
cognitive plasticity.
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39. Semantic relationships among words in Human mental space as a basis to form
syntactic structures – be modeled as a vector having magnitude and direction.
Scalar - words in mental space having only magnitude (size) – be modeled as a
component of mental vocabulary. But there is only isolated component, no
mention about a direction.
Idea: Human mental vocabulary should be presented in vector and scalar models.
Zero order tensor → Scalars weight (as a mass), electric charge (+ ; -)
I order tensor → Vectors direction of cohesion.
+ р 11,15,16,29
39

40. The inferior frontal gyrus (frontal operculum) was activated more strongly for semantically
unrelated words and for words created syntactic violations – this activation was mostly
bilateral, but stronger in left-hemisphere.
Hypothesis: Shared syntactic integration-combination of discrete structural elements (words or
musical tones) into sequences-perceiving complex acoustic, non-verbal structures (symbol?).
New hypothesis: lack of musical priming in patients with Broca’s aphasia who have difficulties in
Linguistic syntax processing.
Холбох р 6,11,12,20,26
Idea: acoustic structures –wave +particle.
Verbal structures wave-particle (listening)
particle (reading)
X? carrier of semantic force (quant ?)
40

41. - Arithmetic: musical processing-numerical processing
Violation in number series
S in sequential regularities-aphasia
Violation in syntactic structures
Violation in musical structures
Musical structures (tone)
Metrical stimuli:
Syntactic structures (morphemes etc).
Musical processing – perceiving waves (quantification of musical structures) + tones
as a particles.
Dynamic attention – Cognitive sequencing.
Idea: Experiment on universal syntax for verbal and musical production.
Холбох р 25
41

42. Inferential relations between: Sample-population
Example-prototype Distance between “these…”
Member-class
Instance between member 1-class
Member N-class
Idea: Inferential relations and distance between invariant and variants (prototype –
example etc) in Hilbert spaces (finite dimensional Hilbert space) + Bolzano’s
theorem
Idea: Inductive inference – isomorphism, homomorphism
Холбох р 24
(least square–for measuring a dispersion caused by an inference).
42

43. Cognitive maps are mental representations.
Visual
Perception of spatial information Verbal
Semantic relations between cities in case when spatial information (about
locations – distance between cities) are presented:
Idea: Symbol interdependency hypothesis - Language comprehension is both
embodied and symbolic.
Symbol-through interdependencies of a modal linguistic symbols.
Embodied-through references these symbols make to perceptual
representations
Visual and verbal perception / recognition.
In verbal for (discourse-text based)
In visual form (geographic map-map based)
Idea: - Projective geometry, representational granularity.
-Vectors.
43

44. Ability to identify stimuli: Types of stimuli (verbal, non-verbal etc).
Features (simple, combined).
Multidimensional stimuli – multidimensional scanning (MDS).
Identification of multidimensional stimuli-performance limit 7± 2 (Miller) – MDS.
Relationships between identification performance and structure
(stimulus structure) Strong
Psychological representations (simple and complex)
Idea: Object recognition + projective geometry – representational granularity: set
theory, graph theory.
44

45. Conceptual metaphor (Cantor’s metaphor) “same number as is pairability”.
Conceptual blending – BMI (the basic metaphor of infinity)
Transfinite cardinals are the result of a combination of conceptual metaphor and
conceptual blending (done by the creative mind of George Cantor).
The underlying cognitive mechanisms are bodily-grounded and not arbitrary. This
ground is constrained by biological phenomena such as neuroanatomy, the
human nervous system.
Source, input spaces, mappings, and projections are realized by bodily-grounded
experience such as thermic experience, visual perception, spatial experience and
so on.
In the case of transfinite numbers these constraints are provided by container
schemas.
Idea: (be extended in difference versions) – human perceptual activity → verbal
presentation and interpretation → mapping and blending in verbal forms
(sentences etc)
- associative mechanism for building a metaphor – is grounded (fox, head etc)
45

46. Container schema: If A is in B and B is in C, then A is in C.
Systems of mirror and canonical neurons points to joint action – perception circuitry
– binding circuits - two primary metaphors form a complex metaphor – to
interpret processes on the basis of container schema, inference by applying an
algebra.
Inference – аn operation serving metaphor building – embedding
Life needs a container with an
Should be modeled in terms of topological, interior, a boundary, and an
non-Euclidean geometrical notions exterior.
Container schema – is a spatial-
relations concept, a gestalt +
multimodality + embedding.
46

47. Georg Cantor’s fundamental conceptual metaphor Same Number As is Pairability.
This simple but ingenious metaphor is at the core of transfinite numbers and
modern set/theory.
Source domain Target domain
Mappings Numeration
Set A and Set B can be put Set A and Set B have the
into 1-1 correspondence same number of elements.
Set A and Set B can't be put Set B is larger than Set A. It
in 1-1 correspondence, and has more elements than Set
Set A is a proper subset of B. A.
Isomorphism (Set theory)
Homomorphism
Idea: Cantor’s continuum hypothesis p 2, 4
47