The Time Operator of Reals
Miloš Milovanović
1 a
and Srđan Vukmirović
2 b
1
Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia
2
Mathematical Faculty, University of Belgrade, Belgrade, Serbia
Keywords: Intuitionism, Continuum, Real Numbers, Measurement Problem, Multiresolution, Self-organization.
Abstract: The purpose of the paper is to establish the continuum in terms of the complex systems physics. It is based
upon the intuitionistic mathematics of Brouwer, implying a processual definition of real numbers that
concerns the measurement problem. The Brouwerian continuum is proved to be a categorical skeleton of
complex systems whose determining feature is the existence of time operator. Acting on continuous signals,
the time operator represents multiresolution hierarchy of the measurement process. The wavelet domain
hidden Markov model, that recapitulates statistical properties of the hierarchy, is applicable to a wide range
of signals having experimental verification. It indicates a novel method that has already been proved
tremendously useful in applied mathematics.
1 INTRODUCTION
Announcing decline and fall of reductionism in
mathematics, Gregory Chaitin (1995, pp.156-159)
considers randomness in arithmetic elucidated by
some results of the computation theory. In the
conclusion concerning experimental mathematics, he
highlights an impact of the computer that has so vast
mathematical experience people are forced to proceed
in a more pragmatic fashion. In that manner,
mathematicians go ahead proofs postulating some
hypothesis based on the results of computer
experiments. Chaitin particularly points out a relation
to contemporary physics wherein the randomness is
regarded to be a crucial agent, which is the core of an
emergent paradigm in science. He ends by a remark
that the question of how one should actually do
mathematics requires at least another generation of
work.
The occurrence of randomness in a formal theory,
set upon deterministic assumptions, corresponds to
the complex systems physics that considers systems
for which the best method of their description is not
clear a priori (Sambrook et al., 1997, p.203). The
statistical complexity suggested by Grassberger
(1986), that concerns stochastic computing in terms
of the Bernoulli-Turing machine, is an analog of the
a
https://orcid.org/0000-0002-2909-451X
b
https://orcid.org/0000-0002-5135-869X
deterministic one designed upon the Turing machine
(Crutchfield et al., 1990). Within stochastic
computation theory, deterministic and random
behaviors are considered to be elemental extremes
deprived of a vital component since they share
common failure to support emergent properties.
Being an amalgam of both, the complex patterns have
an inherent tendency towards hierarchical
organization (Sambrook et al., 1997, p.200).
The hierarchy has substantial implications
concerning cognition, since observation and
comprehension are related to the neural architecture
whose structure is a reflection of the cognitive
complexity (Sambrook et al., 1997, pp.204-206). It
corresponds to an evolution in the hierarchical
manner (Simon, 1962, pp.470-477) indicating a
concept of time operated not only physically or
biologically, but in terms of organization theory. In
that respect, time represents a primordial intuition
being the very base of conscious life – as stated by
Brouwer in his attempt to found the continuum upon
such an intuitionism (Tasić, 2001, pp.36-45). He
considers mathematics to be an intellection of
increasingly complex features, meaning actually self-
organization of complex systems due to the definition
originated by Shalizi et al. (2004) that concerns the
increase of complexity over time. The issue requires
MilovanoviÄ
˘
G, M. and VukmiroviÄ
˘
G, S.
The Time Operator of Reals.
DOI: 10.5220/0007747000750084
In Proceedings of the 4th International Conference on Complexity, Future Information Systems and Risk (COMPLEXIS 2019), pages 75-84
ISBN: 978-989-758-366-7
Copyright
c
2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
75
a formulation of complex systems by the time
operator acting on the space of continuous signals
(Prigogine, 1980). It is regarded a straightforward
generalization of multiresolution that relates the
measurement process (Antoniou et al., 2003, p.107).
The paper is aimed to establish the Brouwerian
continuum in terms of the complex systems physics.
In the preliminary section, the intuitionistic
mathematics is briefly outlined, as well as the concept
of the time operator. The next section elaborates the
continuum structure that concerns the measurement
problem, defining the time operator of
multiresolution hierarchy in the space of continuous
signals. The last section contains some concluding
remarks.
2 PRELIMINARIES
2.1 The Intuitionistic Mathematics
Brouwer`s contribution to the foundations of
mathematics should be regarded in a context of the
XIX and XX centuries whose significant persons
were Hilbert, Russell and Whitehead. Russell`s and
Whitehead`s view were based upon logic,
1
stating
that it represents the fundament of mathematical
thought and Hilbert`s considered a formal language to
be the design of mathematics. All of them were
labelled by Brouwer to be the platonic ones, since
implying timeless conceptions. Elucidated in physical
terms, they have been the deterministic theories.
Brouwer claims irreducibility of mathematics to a
language, aiming to separate it both from formalism
and logicism.
2
According to his opinion, a basis of
consciousness is the time continuum transcending
any language in order to provide an original creation.
It represents a continual activity of the creative
subject, that is not formally determined.
Consequently, for mathematics there is no certain
language (Brouwer, 1929). To elucidate such a
conception, he uses the term choice sequence,
diverging intuitionism also from constructive
1
The logicism had actually come from Gottlob Frege
whose heirs Russell and Whitehead were.
2
Interpreting mathematics as being based on logic is like
considering the human body to be an application of the
science of anatomy (Brouwer, 1907).
3
The Principle of Excluded Middle has only a scholastic
and heuristic value and therefore the theorems that in their
proof cannot avoid the use of this principle lack any
mathematical content (Brouwer, 1928).
4
Einstein`s intension to reach the timeless world of
supreme rationality was definitely manifested in the
mathematics based upon deterministic decisions
(Tasić, 2001, pp.38-40).
Considering the intuitionistic logic, one should
mention its deviation from the laws of excluded
middle and double negation. Brouwer regards a
structure to be discrete if the law of excluded middle
holds in the form , which is not
generally applicable.
3
Intuitionism is therefore a logic
of the continuum, unlike formal logic that is actually
the discrete one. Hereupon, the double negation law
is also violated allowing the
existence of infinitesimals since a negation of
diversity from zero is not equivalent to the
identity (Bell, 1998). Consequently, the
continuum does not reduce to a pointwise structure,
considering that the primordial intuition is not
dependent on a formal spatiality. However, the sense
of time has been seriously damaged by treating it as
an additional dimension of formal space that the
modern science peddles for an ultimate reality (Tasić,
2001, pp.36-37).
Brouwer actually refers to the elimination of time
stated by Meyerson (1908) through his definition of
the modern science history in terms of progressive
realizing the fundamental bias in human reason,
which is about reducing of difference and change to
identity and constancy. As early as the XVIII century,
d`Alambert (1754) noted that one could regard the
duration to be a fourth dimension supplementing the
common three-dimensional design and Lagrange
(1796, p.223), more than hundred years before
Einstein and Minkowski, went so far to term it the
four-dimensional geometry. The climax of such a
historical trail was Albert Einstein in categorical
rejecting the existence of change that was considered
by him to be a mere illusion (Prigogine, 1980, pp.201-
203).
4
In that respect, contemporary restoration of
time indicates a postmodernism in science, whose
forerunner Brouwer has been (Tasić, 2001, p.49).
celebrated Einstein-Bohr debate on the foundations of
quantum theory. A core of the debate was about the
fundamental role of randomness in specifying a system`s
state, that was emphatically denied by Einstein who was
supporting the objective epistemology of science. Although
he overturned the Newtonian mechanics, Einstein was
firmly holding the Cartesian intent of reducing physics to
geometry in terms of the formal spatiality that was but a
deterministic assumption.
COMPLEXIS 2019 - 4th International Conference on Complexity, Future Information Systems and Risk
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2.2 The Time Operator
To be a conception of the postmodern science, one
considers the quantum physics whose origins date
back to the beginning of the XX century (Toulmin,
1985). The statistical formulation, concerning an
evolution of probabilities, has given rise to the
operator mechanics by Koopman and von Neumann.
In terms of the theory, a system corresponds to group
of transformations
acting on the Hilbert space
through
, whereat a dynamics of
the phase space is represented by one-parameter
family
that preserves probability measure
. Due to the measure preservation,
is unitary and,
according to Stone`s theorem, the group has an
infinitesimal generator termed the Liouvillian
which is a self-adjoint operator defined on the dense
subspace (Koopman, 1931).
The uncertainty principle concerns a pair of
conjugate observables, whose main example is
manifested by the position and the
momentum operator
.
5
Formulated in terms of the commutator
, the uncertainty principle implies
relation
. In the same manner, Prigogine
(1980) defined the time operator to satisfy the
uncertainty relation
,
(1)
considering the Liouvillian .
The variable whose domain is evolves by
action of the group
, whilst the distribution
density is acted by adjoint operators
implying the Liouville equation
Consequently, the Liouvillian
5
Operators act on functions defined in the phase space
having a coordinate . In the quantum theory, the position
is related to multiplication and the
momentum
to the Hermitian
derivation by a coordinate.
6
The value , however, is not a coordinate of the phase
space and there is no guarantee of the time operator
existence. If it does exist, the system is termed to be
complex.
7
is an operator function in the sense of operational
calculus.
and the time operator correspond to conjugate
observables similarly to the position and the
momentum in the quantum theory
6
. In terms of the
group action, the relation (1) is equivalent to
(2)
which comes down to
, supposing a cyclic
group generated by
.
The existence of the time operator in the system
induces a change of representation
,
7
transforming with no loss of information the group to
a semigroup action (Misra et al., 1979). The
semigroup
(3)
corresponds to an irreversible evolution of the
complex system,
8
conjugated to the reversible one of
the group. It addresses a stochastic process
irreducible to the deterministic description, whose
existence is analog of Gödel's incompleteness
theorem (Misra et al., 1983). In the modern science
based upon an elimination of time, the irreversibility
is cognized through the measurement problem that
demands a departure from determinism in favor of the
statistical causality (Prigogine, 1980, pp.65-67).
9
3 THE TIME CONTINUUM
3.1 The Continuum of Reals
The concept of continuum corresponds to real
numbers and the measurement process, originating
from the geometrical algebra of Euclid. In the V book
of the Elements, he elaborates the doctrine of
proportion that concerns commensuration of
magnitudes. According to the Euclidean algorithm,
magnitudes and measure each other in the form
8
Although there are also the operators
, they are not positivity preserving
and therefore not corresponded to system`s evolution that
maps one distribution to another (Misra et al., 1979, p.23).
9
In the quantum theory, the measurement corresponds to
reduction of the wave packet, which is a nonunitary
transformation. Von Neumann (1955) has expressed its
difference from a unitary evolution by the Schrödinger
equation in terms of the entropy increase, invoking a
substantial role of the observer. Therefore, the problem of
irreversibility appears at the very core of physics.
The Time Operator of Reals
77
(4)
that is termed the continued fraction having the
spectrum
10
The proportion
,
indicated by matching of the respective terms in both
spectrums, induces identity on the continuum of
reals.
11
Consequently, the real number corresponds to
the fraction expansion (4) implying the measurement
process taken place step by step in a manner of time.
The time evolution is represented by the Ford
(1938) diagram of circles, whose intersections with a
vertical line correspond to the sequence
(5)
identifying a real number . Its elements
are termed the Diophantine approximations of ,
being the most approxiate to the real number in regard
to the fractions
with denominators not greater than that of
.
Denominators and numerators of the sequence are
given by recurrence equations of the form
(6)
considering the initial conditions
and
. The difference of successive
members is
10
One assumes that , i.e.,
. If not, the
representation applies to
.
11
Defining proportion in terms of the continued fraction
spectrum is not done by Euclid, but by Omar Khayyam in
A Commentary on the Difficulties Concerning the
Postulates of Euclid's Elements. The conception presented
in the Elements, that originates from Eudoxus of Cnidus,
means that the proportion
(7)
supposing
which implies
(8)
and, having in mind
, one
gets
, i.e.,
(9)
Accordingly, the continued fraction concerning a real
number takes the form of an alternating series
(10)
that is a sparse representation (Mallat, 2009)
composed of terms from the redundant dictionary
The equation (10) corresponds to a binary code
wherein 0 is assigned to terms of the dictionary that
do not participate in the series and 1 to those that do
participate, having a sign which alters in front of
them. Such a record of the real number is, however,
highly redundant since the entire dictionary cannot be
involved in a series. Therefore, one should eliminate
excess zeros, which is achieved by coding the
spectrum
A binary code like that is
composed of alternative values different from
zero at the positions
, which
gives rise to the Minkowski question mark function
(11)
holds if each of propositions
is equivalent to the respective one of
. Displeased with such a philosophy,
Khayyam redefined it by the use of continuous fractions,
where the concept of real number was established.
Eudoxus` definition was employed by Richard Dedekind,
concerning his construction of the real line in terms of
rational cuts.
COMPLEXIS 2019 - 4th International Conference on Complexity, Future Information Systems and Risk
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transforming the continued fraction to the binary
code. It is an automorphism of the continuum,
mapping a real number
to the analog value
.
Under the continuum, one implies a skeletal category
being specified up to an isomorphism. In that regard,
the transformation is considered to be
automorphism of the structure, since isomorphism
(12) hereinafter is also an identity.
The Ford diagram is structured by a hierarchy of
scales, each corresponding to the insertion of circles
tangent to two of them at the previous scales, as well
as to the number line. In such a hierarchy, each circle
is attributed to an irreducible fraction that represents
its contact to the line.
Assuming designators of the circles to be
and
,
an inserted circle between them corresponds to the
fraction
.
One denotes it
,
which defines an operation termed the mediant or the
Farey sum.
12
The question mark function maps the
mediant to the arithmetic mean (Minkowski, 1905)
(12)
being isomorphism of the topological quasigroups,
whose action turns circles of the Ford diagram into
squares. The diagram of squares has a binary tree
structure whereby the nodes, coordinated by and ,
correspond to paracomplex number ,
forming an algebra of segments
(Warmus, 1956). Branching of the segments
12
It is due to John Farey (1816) who noticed that the
successive fractions , whose denominators in the
reduced form are up to a given value, relate by .
For example, the fractions up to the denominator value 5
form the order
designates bit by bit of a real number and, in that
regard, hierarchy of continuum is related to the binary
coding. The Renyi map
(13)
that is a shift in terms of binary digits, represents self-
similarity of the structure mapping both left and right
subtree to the entire one.
3.2 Wavelets and Multiresolution
Aiming to construct the time operator corresponded
to an evolution of continuum, the Hilbert space of
continuous signals is considered. In that respect, one
suggests the space
consisted of square
integrable functions on the domain . Crucial
for the issue are hierarchical bases consentient to the
continuum structure, which are the wavelet bases on
the interval. According to a wavelet base of the
support
, a signal
is decomposed
in the form
(14)
wherein indexes the dyadic scale and the spatial
position of an element
(Daubechies, 1992,
pp.304-307). The Haar base
(15)
is obtained by translations and dilatations of the
mother wavelet
.
In the decomposition sum (14),
is the average
value which is a projection onto the subspace of
constant signals . The hierarchical structure of a
wavelet base is reflected to the detail coefficients
forming the binary tree. Each node at a scale of the
tree has two successors at the next one sharing
its position in the hierarchy. The succession is related
to the measurement process whose steps correspond
whereby each successive threesome is related by the Farey
sum.
The Time Operator of Reals
79
to the scales of hierarchy. In that respect, the time
operator concerning a wavelet base is given by
(16)
wherein the eigenvalues correspond to the scales of
eigenvectors
, increased by a unit in order it to be
the invertible operator the dense subset of
.
The operator domain includes finite sums
whose components
constitute detail subspaces
wandering by the
unilateral shift
(17)
induced by the Renyi map (13).
13
It generates the
time succession of the space
, establishing
a multiresolution analysis whose basic axiom is the
shift property
(18)
Presented in the form
(19)
whereby each projector
(20)
corresponds to the details at a resolution scale, the
time operator appears to be a straightforward
generalization of multiresolution (Antoniou et al.,
2003, p.107). The basic axiom (18) is equivalent to
the uncertainty relation
(21)
that concerns the definition of time in complex
systems (2).
In order to establish a measure preserving group,
the Renyi map should be extended naturally to the
baker transform
13
Mutually orthogonal subspaces
are termed to be
wandering if the shift property
holds.
They are termed generating if the sequence is a cover of the
space (Antoniou et al., 2000, p.446).
14
A problem occurs concerining preservation of positivity,
since the approximation operators
does not
preserve it any more (Gustafson, 2007, pp. 16-18). It is
reflected to the positivity preserving of and
, that
(22)
inducing the bilateral shift (Antoniou et
al., 2003, pp.35-38). According to that, the signal
space
is embedded into
by the rule
, i.e.,
. The relation
(23)
that holds on the space
, gives rise on
to
(24)
meaning that the projectors onto detail subspaces
interrelate by conjugation. The time operator
of
the baker map has been explicitly constructed
(Antoniou et al., 2003, pp. 47-60), whose natural
projection onto
corresponds to the Haar
multiresolution (15). However, any wavelet
mulitiresolution could be obtained likewise, since the
change of base
does
not violate the shift property (18) and therefore the
operator
also satisfies the uncertainty
relation (21).
14
The time operator induces a change of
representation transforming the group
evolution, generated by
, to the semigroup one (3)
by
(25)
whose adjoint operator corresponds to the Markov
process (Misra et al., 1979, p.9)
(26)
The semigroup action signifies a blurring of the
signal, related to an extension of the spatial domain
due to an action of the operator on
.
3.3 Complexity and Self-organization
In the poem On the Nature of Reality, Lucretius
describes not only how things vanish at a distance –
but also how they appear to change (Lucretius, 1948,
should not correspond to an evolution of probability.
However, it is all about a mere change of base conjugating
the approximation operators to those of the Haar
multiresolution, that still preserve positivity. In order to
resolve the concerns, one requires a base independent
definition that presumably affects the concept of probability
distribution. The authors do not discuss it in details.
COMPLEXIS 2019 - 4th International Conference on Complexity, Future Information Systems and Risk
80
p.156).
15
The effect concerns blurring of the signal,
wherewith the details are successively suppressed. On
the other hand, the emergence of details unfolding the
time of a system is termed self-organization that is the
increase of complexity (Shalizi et al., 2004). The
concept originates from Grassberger (1986) who
defined the statistical complexity to be minimal
information required for an optimal prediction.
Crutchfield et al. (1990) elaborated the conception by
accurate definitions of the optimal predictor and its
state. In that manner, the causal structure has been
established corresponding to the intrinsic
computability of a process in terms of the Bernoulli-
Turing machine.
The detail coefficients of a signal
are
considered to be distorted measurements of a causal
variable evolving in a stochastic manner. It is
factorized into local variables
related to each
node of the tree. The Markovian tree
contains all correlations in a signal, that occur only
along branches linking the local states due to a
structure of the continuum. The wavelet domain
hidden Markov model established like that has been
proved tremendously useful in a variety of
applications, including speech recognition and
artificial intelligence (Crouse et al., 1998, p.887).
Elaborating a statistical model of the wavelet
transform, a signal and its coefficients are regarded to
be random realizations. In that respect, one requires
the space of distributed signals
whose
constituent
is a variable of the domain
implying Lebegue`s probability measure, that
varies over the codomain
. The coefficient
distributions are given by
(27)
since from (24) it follows that details at a resolution
scale are
(28)
considering
.
15
For instance, distant square towers look rounded. A pair
of distant islands appear to merge into single one. When
distance is increased, details generalize and distinctions
merge or vanish (Koenderink, 1997, p.xv).
16
Statistical stationarity of the system, that concerns
translational invariance of a signal distribution, enables
reduction of the model parameters. The practice is known
as tying in the hidden Markov model literature (Rabiner,
1989), aiming to estimate the parameters robustly by a use
of the Baum-Welch algorithm given an observation from
According to that, the detail coefficients at a common
scale are regarded to be equally distributed
(29)
which reflects to the causal variable
whose
distribution is independent of the position index .
16
The information contained in local variables
(30)
dependent on the scale only, is termed the local
complexity whose increase in the temporal domain
represents self-organiozation. One implies the
Shannon entropy
, being the
extensive measure of a random variable.
The time is unfolding in a manner of the
complexity increase, and so it is significant to find an
optimal base of the signal wherein self-organization
is the most prominent. The complexity
(31)
termed the global one, is proven to be a measure of
the optimal representation (Milovanović et al., 2013).
The signal information is decomposed through the
canonical equation
(32)
wherein
corresponds to the complexity and the
extensive term represents an irreducible
randomness that remains even after all correlations
are given. Adding white noise to the signal,
17
only the
randomness should increase while the complexity
remains unchanged. Thereby the optimal base
performs superior denoising, since it best respects
self-organization of a system corresponded to the
time operator. A multiresolution it provides
temporally decomposes the signal, specifying its
significance by a complexity insight. In that regard,
the multiscale pyramids are proposed to be likely
models of the visual perception (Koenderink, 1997,
p.xx).
An incisive phenomenology of the fact has been
presented by John Ruskin (1844, p.174):
Go to the top of Highgate Hill on a clear
summer morning at five o'clock, and look
at Westminster Abbey. You will receive
the signal space
. It is about sharing statistical
information between related variables at certain scales,
whose distribution parameters are tied to a common value.
The algorithm usually converges in as few as ten iterations
supposing a locally two state causal structure (Crouse et al.,
1998, p.893).
17
The term white noise means uncorrelated Gaussian noise
independent of the signal it is added to.
The Time Operator of Reals
81
an impression of a building enriched
with multitudinous vertical lines. Try to
distinguish one of these lines all the way
down from the one next to it: You cannot.
Try to count them: You cannot. Try to
make out the beginning or end of any of
them: You cannot. Look at it generally,
and it is all symmetry and arrangement.
Look at it in its parts, and it is all
inextricable confusion.
Yet Ruskin adamantly insists that the draughtsman
should render the confusion veridically, meaning that
the complexity is optimally represented. Such a
rendering is however done in a hierarchical manner,
since it describes a complex object (Simon, 1962,
p.477).
Koendreink (1997, pp.xvii-xx) indicates that one
is faced with a fundamental and important, though
unfortunately ill understood, aspect of perception.
Having taken a first look at the subject, he admitted a
shock by the fact that there existed essentially no
science on the topic. The only discipline that carried
about such phenomena turned out to be cartography
(Greenwood, 1964). Although there is certainly a lot
of science in cartography, its arguably the most
important aspect has always remained an art
conducted largely on intuition. It corresponds to an
aesthetical criterion relating truth to the original
creation (Milovanović et al., 2016), that has been
termed by Gaston Bachelard (1961) the poetics of
space.
Concerning physical reality, Koenderink
concludes the same as Mandelbrot (1983) in terms of
the fractal geometry that a complex description of
nature is required. The conception is concisely
exposed in the book Powers of Ten, giving to the
number a significance corresponded to
multiresolution (Morrison et al., 1982). A link
between the number of ten and multiscaling is about
the continuum structure designed by the measurement
process. According to the Lochs (1964) theorem, the
number of terms in continued fraction requisite for
determining a decimal digit tends to be
,
meaning that each step of measurement roughly
designates digit by digit almost certainly. A
significance of the decimal system in coding numbers
is based therefore on the structure of continuum.
18
The extreme concerns deterministic computation based
upon the Turing machine that is a reduction of the stochastic
one using the Bernoulli-Turing machine. In that manner, the
concept of statistical complexity is reduced to the
algorithmic one (Crutchfield et al., 1990).
Since the continuous signals are equally distributed
with no dependence on a horizontal position of the
hierarchy (29), a real number in the tree
representation corresponds to a choice sequence
unfolding in time from the top downwards.
In that manner, the time continuum appears to be
a model of intuitionistic logic wherein the excluded
middle does not hold, considering a
dynamical identity unfolded by choice (Milovanović,
2018). However, the law of excluded middle is valid
concerning the diversity since it holds
(33)
Respecting the intuitionism, a negation of identity
is diversity , but its negation is undiversity that is
a discrete relation. A negation of undiversity is
diversity , and thus the law (33) takes the form
(34)
indicating a discrete structure corresponded to the
formal logic. It is obtained through a negative
translation of the intuitionistic one (Glivenko, 1929),
meaning that formalism is a form of the intuitionistic
continuum reduced to a discrete method.
18
The discretization due to the double negation of
identity makes a pointwise structure based upon the
undiversity of elements.
19
In terms of the signal
space, it gives rise to a point operator that is required
to be sufficiently smooth in order to transfer the
concepts of continuity and differentiation onto the
discrete functions (Florack, 1997, pp.57-65).
20
Regarding the time continuum, however, all functions
on the domain have been considered continuous since
representing morphisms of the structure (Brouwer,
1924).
4 CONCLUSION
Elaborating relation between wavelets and stochastic
processes, Antoniou et al. (1999, p.96) asserted that
wavelets had not been motivated by any underlying
dynamics of the phase space. He concludes that the
ergodic theory is richer than the wavelet one, since
the former involves fundamentally an underlying
dynamical system of point trajectories. However, it
has been demonstrated that wavelets involve the
baker transform, acting in the domain of
distributed signals. Due to the existence of the time
19
It also emerges in JavaScript, being the legendary cast-
to-bool operator written in a form of the double negation
(!!).
20
The point operator represents a measurement process
corresponded to blurring of the signal (26).
COMPLEXIS 2019 - 4th International Conference on Complexity, Future Information Systems and Risk
82
operator, the group comes down to the semigroup
action generated by the Renyi map. The complex
systems physics is therefore implied by the very
concept of real numbers, that addresses the
measurement problem.
According to Brouwer`s view, time is a primordial
intuition being the base of conscious life.
Mathematics is regarded to be the paradigm of self-
organization, i.e., an intellection of increasingly
complex features. In that respect, the basic structure
is the time continuum that is a categorical skeleton of
complex systems. The dynamical identity it implies is
unfolded by choice, similarly to one of Jungian
psychology whereby the natural number emerges to
be a timestamp (von Franz, 1974).
A complex description of nature following the
evolution of continuum is designed by fractal
geometry, wherewith time is established in terms of
multiresolution. Considering the statistics of
continuous signals, the wavelet domain hidden
Markov model has been proved tremendously useful
in a variety of applications (Crouse et al., 1998). It is
obtained in a manner of experimental mathematics,
elucidating the complex systems physics to be the
paradigmatic framework for such an activity.
Referring to Chaitin (1995, pp.156-159), one
concludes that another generation of mathematics has
come – aligned to the Brouwerian method.
ACKNOWLEDGEMENT
The authors acknowledge support by the Ministry of
Education, Science and Technological Development
of the Republic of Serbia through the projects OI
174014, OI 174012 and III 44006, also by the Joint
Japan-Serbia Center for the Promotion of Science and
Technology of the University in Belgrade and the Ito
Foundation for International Education Exchange.
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