P H I L O S O P H Y P A T H W A Y S ISSN 2043-0728
Issue number 167
28th November 2011
I. 'Symbolism: From Supernatural to Science: an Epistemological
Inquiry' by John Nwanegbo-Ben
II. 'On Concepts: With or Without Perceptions' by Ali Yousefi Heris
III. 'Generating Stable Knowledge via Reduction in Entropy' by
Georgios Constantine Pentzaropoulos
For this issue of Philosophy Pathways we return to the topic of
Epistemology (see Issue number 164 29th July 2011). How do symbols,
for example the symbols in religion or science, embody human
knowledge? What does it mean to say that human beings possess the
'innate capacity' to organize raw perceptions into a meaningful form?
Under what circumstances is the information thus gained sufficiently
ordered or organized to qualify as knowledge?
John Nwanegbo-Ben invites the reader to consider symbols in their
historical aspect as well as their contemporary use. Symbols
'represent a state of affairs which has been relatively
universalized', that is to say, they embody the accepted knowledge of
the time. The key question to ask, however, is how we decide whether
the proposed introduction of a symbol into the language is
acceptable. The use of symbols enables us to increase our knowledge,
by facilitating thought. However, what is the case, the facts, should
not be seen to depend on our arbitrary decision whether to use a
particular symbol or not.
Ali Yousefi Heris looks at the innateness debate, in the light of
genetics, for example the genetics of inherited diseases or
disabilities. It is an uncontested observation that the very same
gene or combination of genes can produce different effects in
different environments. In that case, how does one disentangle the
combined effects of environment and genetic makeup? If that question
proves intractable, then it is hardly surprising that there will be
no simple answer to the innateness question, for example, the
question raised by Chomsky to what extent our knowledge of the
grammar of our language is 'innate'.
Georgios Constantine Pentzaropoulos in his second article for
Philosophy Pathways offers an interpretation of Plato's account of
knowledge in the dialogues Meno and Theaetetus, according to which
the 'account' which is required in order to convert true belief into
knowledge can be understood in terms of the physical concept of
entropy, or 'disorder'. When our beliefs are organized into an
ordered system, they gain the stability necessary for knowledge;
which also explains why, as Plato observed, knowledge has much
greater utility than mere true belief. This line of inquiry could be
fruitfully extended to explore Edmund Gettier's challenge to the
traditional definition of knowledge as 'justified true belief'.
I. 'SYMBOLISM: FROM SUPERNATURAL TO SCIENCE: AN EPISTEMOLOGICAL
INQUIRY' BY JOHN NWANEGBO-BEN
Symbolism as a theoretical entity is as old as human knowledge. Its
major expression appears to be within the confines of supernaturalism
exemplified in various religions of the world. Symbols represent
ideals and acts as the mirror of the unseen. In formal and empirical
sciences symbols are codified and universalized for general
acceptability. The universal acceptability has a double entendre and
this is of vital importance in epistemology. The importance of
symbolism cannot be over emphasized. However, how reliable are
symbols as mirrors to reality? To what extent do we say that a symbol
is a true representation of the ideal or absolute knowledge? The
purpose of this paper is to analyze the epistemological character of
symbolism in human knowledge.
Human knowledge grows by accretion through problem solving. Knowledge
of both physical and mental realities most times is represented
symbolically. Different forms and levels of experience and
relationship to reality both within the confines of the sacred and
profane are linked together with the concept of symbols, signs and
The secularization or universalization of symbols for objective
interpretation to boost human knowledge is of vital importance.
Religious symbolism has played major roles both in the African
traditional religion, Christianity and eastern religious lives. It
has aided a standard perception of ideals of each of these religious
lives within their socio-cultural milieu.
Our emphasis in this paper is to identify the import of symbolism
both in the supernatural and science and in human knowledge. In this
we hope to discover or explicate the significance of symbolism in
human knowledge and how symbols standardized and make intelligible
our understanding of reality whether in the confines of the
supernatural or science.
Meaning and Functions of Symbolism
A symbol may be anything, objects, words, colours, or patterns; their
defining characteristic is that they stand for something other than
their intrinsic property. In aesthetics, symbolism represents an
object which apart from its own immediate and proper significance,
suggests also another, especially a more ideal content which it
cannot perfectly embody. Symbols may be either natural: as light is a
symbol of truth; or traditional and conventional: as the cross is a
symbol of sacrifice. Another example is while there is nothing
intrinsically dangerous about the colour red; it has become a symbol
for danger in almost all societies.
Symbols are theoretical or mental entities that are subject to
verification. Though these symbols are subject to verification, yet
they are of practical value to thought or mental acts and science in
general. In African religion, ethics, agriculture, and even in
communal political life, symbols are of practical value. As a result
of the necessity to exhaust all frontiers of knowledge, there is now
a resumption of series of inquiry into the metaphysics and
epistemology of symbolism as a category of human knowledge. This
resumption is predicated on the intent to standardize knowledge as
objectively as possible.
In the 1960s anthropologists delved into the study of symbolism to
investigate what symbols stand for and the processes by which meaning
is attributed to them. The common agreement between various research
schools on this issue 'is the need to explain the universal presence
of symbols in all societies and to explain their importance. In human
history, symbols have been used to express ideas, communicate
meanings, and convey emotions'.
Different languages of the world have developed different scripts as
symbols of meaning. Hieroglyphics from the ancient Egyptian
civilization are among the earliest in Africa and the world. The
function whereby a mental result primarily referring to one set of
object is transferred to another set of objects; first set is said to
be symbolic of the second. Symbols are frequently used in a very wide
sense as equivalent to any kind of sign. But it seems desirable to
limit its application in psychology to cases in which the sign is
provisionally substituted for the thing symbolized. In examining the
relationship between 'sign' and 'symbol', D. Okeke states '...sign
indicates, symbols represents'.
Symbols are used as a kind of key to convey religious concepts,
ideas, visual, auditory and even kinetic representation of events.
Nature is replicate with symbols. Other non-religious types of
symbols have achieved increasing significance in the 19th and 20th
centuries, especially those dealing with man's relationship to and
conceptualization of the material world.
The function of symbol is to represent a reality or a truth and to
reveal them either instantaneously or gradually. The symbol is
sometimes identified with the reality that it represents and
sometimes regarded as a pure transparency of it. As a 'sign' or
'picture' the representation of the experience of and relationship to
reality has either a denotative or a truly representative meaning.
Scholars are now appreciating the theoretical and scientific
functions of symbolism. African philosophers have noticed that within
the content of both traditional and contemporary life of Africans,
symbolism has played very vital roles. Symbolism has been the
integral part of African medical practice, religious worship, culture
Symbolism in the Supernatural
Symbolism began with the emergence of religion and religion is as old
as mankind. The reason being predicated on the conception that man is
naturally a worshiping being. And the only way for man to identify
the supernatural being that he believes is responsible for his being
is to use symbolic representation. This in other words shows that man
really knows nothing about his beliefs or the supernatural but only
that which appears.
The supernatural is simply an act or condition which transcends
explanation within the confines of physical or corporal existence.
Terry Davidson's description of the supernatural is, 'The
Supernatural posits sentient agents who are somehow 'above' or
'beyond' nature (or whatever), who have the ability to alter the very
ways in which nature would ordinarily work had these agents instead
chosen simply to sit back and watch.'
Issues within these areas have to do with faith and this is premised
on the belief on the possibility of suspension or violation of
natural laws. Religion is based on supernaturalism, the world of the
mysterious, or the unknowable that cannot readily be comprehended by
Symbols can be traced back to prehistoric times, where they were
totally linked with ritual practices of their religions. A group of
so called 'Venus' figures provides one of the earliest examples. The
most notable of these is the Venus of Willendorf, a small, stone
carving of a faceless pregnant woman, which dates from 30,000-24,000
In Egypt, the use of animal symbols was clearer, since most of the
gods were associated with creatures. In many cases, they were
portrayed with animal heads and human bodies. Unlike the Egyptians,
the Greeks preferred to portray their gods as human beings, using a
variety of symbolic props to identify them.
In the west, symbols developed after Christianity was proclaimed the
official religion in AD 313. The symbols were simple: the cross, a
dove etc. Religious images were placed inside the church for
educational purposes, both for the preacher and the congregation. The
crucifixion, for instance, could be symbolized by the instruments of
the passion (the crown of thorns, the Lance, the nails, the harmer).
In a similar way, an apple and a snake could be used to symbolize the
temptation of Adam and Eve.
The oldest Buddhist image represented Buddha through a series of
symbolic objects. These include among others the Buddhist tree under
which he was sitting when he attained a state of Enlightenment.
African religion, culture and even political life are embedded with
symbolism. Most African communities created their gods and have
symbolic representation of these gods in shrines prepared by them.
The ontology of African symbols emphasizes that symbols cannot be an
isolated compartment but could be seen as a synthesis of his
cultural, social and philosophical thinking. Thus in every aspect
of the life of an African that has to do with medical practice or
divination of any type, symbols play a direct link. Just as the
stethoscope is a symbol of the orthodox medical practice, so do 'the
Afa seeds make up the diviners paraphernalia'.
Symbols or symbolic relationships are tied or linked to culture. It
could be identified in dressing, dance and even musical instruments;
these are basic to our understanding of the consciousness of a given
people. The issue of the supernatural or African medicine (whether
divination or curative) can be identified with symbols such as the
mask, shrine, red, white or black cloths. Each of these symbolic
representations identifies spiritual forces, each having a particular
connotation. As stated above, red in most societies may connote
danger, black may connote death, while knives, mirrors, skulls of
animals and birds are symbols representing contact points to unseen
forces in nature which the medicine man (dibia) uses both for
diagnosis and treatment of diseases and ailments that are
supernatural in character.
Herbs as another symbol represent various aspects of forces and
unseen and unquantifiable energies in nature. The herbs are used for
the preparation of drugs or medicine for both curative and even
protection for men against evil forces. Beyond the realm of herbs are
more mysterious symbols like the effigy as practiced in Haiti as
'Voodoo'. The effigy can be used as a direct contact to the person
involved. Through incantations and invocations, it is believed that
if the effigy is stabbed, the stab will be felt by the person
involved. In most cases, if the intention was to kill, the person
dies. There has however been no scientific interpretation to this
claim. And to cap it all, without symbolic representations we cannot
rightly talk of magic, mystical experiences or miracles in any social
milieu. Thus, there is always and element of symbolism in anything
related to the supernatural. The reason being that it acts as a
reference point for the unseen or that which is hidden.
Scientific Character of Symbolism
Symbolism is as old as knowledge itself hence was not confined to the
perceived knowledge of the supernatural alone; symbols also featured
significantly in secular themes. Modern day organized science
utilizes symbols for general acceptability. Concepts which may appear
ambiguous due to difference in Languages and methods of interpretation
use symbols. Symbols in scientific knowledge are necessary for
objective communication by scientists.
The scientific import of symbolism in human knowledge cannot be over
emphasized. The reason is that rational, scientific-technical symbols
have assumed an ever increasing importance in modern science and
technology. These symbols serve partly to codify and indicate,
abbreviate and make intelligible the various mathematical and other
scientific and technical relationships and functions. Examples of
these are mathematical or logical symbols like <-> Bi-conditional or
Equivalence, = equal to, > greater than, -> implication, < less than,
Â negation. These are some symbolic representations of theoretical
entities in mathematical and symbolic logic.
In the physical and biological sciences symbols such the symbol (~)
for alternating current, or the symbols for male and female
respectively are symbols that have been universalized. It should be
noted that these type of universalized or secularized symbols are
also rooted to a great degree in the realms of religious life like
the example of red, white or black cloths. Scientific symbolism
functions in a manner similar to that of the religious symbols which
associates a particular meaning with a particular sign.
Symbolic representations are usually depicted in diagrammatic or
ideographic modes as signs, abbreviations, images, and objects of all
kinds that indicate a larger context. In this category belong the
simplified or abstract forms, as well as colours, letters and
numbers. The circle, the disk, square, the cross, Star of David,
Pentagram etc. may symbolize, the sun, universe, stars, the earth,
eternity, the flow of time or even a magical spell.
Ordinary formal logic has from the earlier times, substituted symbols
(via, the letters of the alphabet) for significant terms, and has thus
added much to the facility with which the validity of arguments can be
tested. Symbolic logic goes a step further, and adds symbols to stand
for combinations of terms, or functions of terms, and statements of
relations between terms. Symbols, pictures do mirror the world just
like the picture theory of logical atomism of Wittgenstein as
documented in the Tractatus Logico-Philosophicus. If we wish to
identify a particular element, chemical or compound, we can do so by
a particular symbolic representation. The perfect language or
statement pictures the structure of reality. An ideal statement gives
us the structure of facts, and since facts are composed of objects and
their properties, we can say that they are symbolic. The structure of
the world corresponds to the structure of language and is pictured
for us by language. This picture of language, like any other kind of
picture or symbol 'is a model of reality.'
Every model of reality has its symbolic representation. We all
(assumption) within the discipline of philosophy, political science
etc always have a particular ideology. An example is when the picture
of Karl Marx is seen, socialism is imagined hence Marx's picture is a
model representation or symbol of socialism.
Why should symbolism be important in human knowledge? Of what
significance is it? What we should note is that symbolism has a
scientific character. It is this characteristic that makes it
significant in human knowledge. Science, etymologically is an upshot
of the Latin word Scientia which literally means knowledge. But the
knowledge we are addressing now is the contemporary definition and
understanding of science which is 'knowledge arranged in an organized
or orderly manner, especially knowledge obtained by observation and
experimentation. Thus, scientific knowledge is proven knowledge or
systematic knowledge based on evidence'.
Symbolism is based on the theoretical conception of the
correspondence theory of truth and the relativist theory. Symbols
represent a state of affairs which has been 'relatively universalized'
and accepted by community of scientists, individuals or people. It is
this general acceptability as a representation of the real state of
affairs that makes it scientific in character. The symbol > greater
than has been generally accepted within the scientific world to
represent the statement 'greater than', or the symbole < for 'less
than'. The colour red or a red cloth has been universally perceived
to connote danger.
This universal acceptability depends on the point of view from which
it is perceived. For example, we have identified the symbol (~) as
representing alternating current in the physical sciences; this same
symbol also represents negation in symbolic logic. In contrast to
natural language, symbolic logic is artificially contrived by
logicians. Its rules and principles are arbitrarily fixed by some
sort of fiat that requires logicians to work within their framework,
thus, the symbol (~) though accepted generally, but still relative
epistemologically. The same can be said of the symbol for a
conditional statement (if... then...) that is usually expressed
symbolically -> [arrow]. Irvin Copi uses É [horse shoe symbol] and
this is accepted generally by logicians.
Knowledge presupposes a subject and an object that is, the knower and
the known. The subject is the human mind, the seat of perception, the
object of knowledge may not be the things of the material world, but
the ideas or symbolic representation in the world of ideas. Thus
symbols are seen as the representation of the 'ideals'. An object of
knowledge, for example water could be represented symbolically in
form of a picture, drawing or by a better scientific identification
H2O. The symbol H2O can be understood universally by all scientists
just as <-> [biconditional] is to all logicians. Symbols can be seen
as universals or essences or the general idea of a thing formed in
the mind through the process of abstraction. Thus, just as the symbol
of the cross signifies sacrifice and as salvation for mankind within
the Christian faith and understood generally by all as thus, (even if
its meaning is not accepted as a belief by some) so also are other
There is however an element of the principle of Occam's razor in
symbolism. This principle states that one should not make more
assumptions than the minimum needed. Occam's razor helps us to shelve
those concepts, variables or constructs that are not really needed to
explain a phenomenon. Symbols are like short cuts to realities that
may look ambiguous to express. The issue is that symbolic
representation as a means of comprehending reality does not
adequately give us a true knowledge of the 'thing in itself'. The
crucifix, picture of God or Buddha tree does not adequately give us
the true nature of Jesus, God or the Buddha. Names and symbols are
ambiguous. The name John (if it is mine) does not represent my
personality adequately nor does it represent my body as a corporeal
What should be noted is that one of the characteristics of a
scientific statement is its general acceptability. The fact that
Aristotelian cosmology was dethroned despite its general
acceptability did not negate it as a scientific knowledge. Symbolism
though assumed to be an ideal representation of reality by all does
not constitute an absolute means of representing the sacred or
supernatural nor the secular or science.
1. Dictionary of Philosophy and Psychology. Edited by James Mark
Baldwin. Vol. II Gloucester: Peter Smith. 1960. p. 640.
2. Mukulika BanerJee, Symbolism (Anthropology) Microsoft. Encarta
2006. Microsoft Corporation.
3. David. C. Okeke. Foundations of Religious Studies. Enugu: Raboni.
Nig. Ltd. 2006, P. 90.
4.Terry Davidson. Paranormal, Supernatural, Ghost, Science, and
Atheism http://www.positiveatheism.org/mail/eml18958.htm Retrieved
5. C.I.C. Mbamara 'Creation of Deity in African Clinical and Social
Settings: An Epistemological inquiry' in Essence:
Interdisciplinary-International Journal of Philosophy. No. 2, 2005.
6. Paul Ogugua 'Understanding Igbo-African Medicine (A Metaphysical,
Epistemological Study' in Essence: Interdisciplinary-International
Journal of Philosophy. No. 3. 2006. p.50.
7. M. F. Asiegbu 'Dibia Ogwu: An Indispensable Agent in the Search
for Ultimate reality and Purpose in Igbo World' in Essence. Op. cit.
8. For these examples I am Indebted to the New Encyclopaedia
Britannica Vol. 17. 15th Edition. 1982. P. 900.
9. Ludwig Wittgenstein, Tractatus Logico-Philosophicus London:
Routledge and Kegan Paul. 1961. para 2.12.
10. John Nwanegbo-Ben. Philosophy of Physics: The Growth of Man's
ideas on the Nature of Matter. Owerri: Advanced Graphic 2008. p.40.
11. Kwesi Wiredu. Philosophy and African Culture. Cambridge:
Cambridge University Press. 1980. p.115.
(c) John Nwanegbo-Ben 2011
Department of Philosophy of Science & Technology
Federal University of Technology
II. 'ON CONCEPTS: WITH OR WITHOUT PERCEPTIONS' BY ALI YOUSEFI HERIS
Innateness, though a widely admitted view in cognitive science, is a
vague and confusing hypothesis. Conceptually analyzed, the notion of
innateness either leads to the nonsensical view that we are born with
some actual innate stuff, or to the truism that we are born in some
sense with certain capacities and dispositions that develop through
interaction with perceptual and environmental factors. The first
conclusion is empty and unacceptable even by nativists. The second
leaves nativists with an acceptable but vague and confusing face of
On concepts: with or without perceptions
I had a project to do on 'Expert systems: knowledge acquisition' for
which I developed an ontology using the Protege platform. Ontology is
a formal representation of the knowledge of a particular domain,
expressed as a set of concepts, individuals and the relations between
them. This formal and explicit specification captures and models the
structure of the domain (conceptualization) with possible logical
restrictions, and can be used for automatic processing in computers
to reason about the relations between those concepts and individuals.
Part of the work was such that for every individual in the domain, I
had to define a concept to which that individual belongs. In other
words, defining a domain individual required subsuming that
individual under a concept.
Having finished the project, I had to present it to the professor and
other students in the class. But when I was making slides to talk
about the same individuals and their relations in the domain, I
didn't need to define every single individual by classifying that
individual under a concept. The audiences in the class already had
those concepts. Intuitively, I knew that they would be able to make
sense of the tokens and individuals in the domain exactly in the same
way that I do. Indeed that is what we humans automatically do in
almost all situations in our everyday behavior. We recognize things
around us as belonging to certain classes and this recognition
retrieves information about those things, which enables us to predict
their behavior, draw inferences about them, and consequently we can
make decisions to control our behavior. In fact all the agents,
either biological or electromechanical, that are somehow engaged in
activities such as language production and understanding,
categorization, decision-making and reasoning have access to a
repository of concepts. In this way, we can say that concepts are the
building blocks of our mental life.
Discussions on our mysterious ability to organize the world into
objects, to experience the world as a layout of physical entities
that endure through time, and the origin and nature of concepts have
been pursued traditionally within the empiricism-nativism framework.
The most influential hypothesis is the empiricist model in which we
construct the world by first detecting the sensible properties of a
scene, and by learning how to use words to denote things in the world
(Spelke, 1998a). On the empiricist side, it is generally supposed that
we initially interact with the world via fleeting sensations, which
they gradually organize into more structured entities. The
empiricists believe that our knowledge about the world, and a
fortiori all concepts, are derived from sense experiences. The
nativists or rationalists, on the other hand, claim that mere sense
perceptions are not enough to explain our knowledge. They believe
that there is some core knowledge that is not acquired, but develops
and interacts with later-acquired knowledge. This non-acquired
knowledge is commonly called innate knowledge. Like concepts in
Protege, we might like to regard it as a kind of predefined
knowledge. The nativists argue that the domain-neutral empiricist
model is not sufficient enough to explain all our cognition and hence
the rationale for innatism.
Cognitive psychology and developmental studies provide a good source
of evidence about the origin of concepts. Some studies point to a
close link between our perceptual experience and our cognitive
capacities. Results from a recent study (Domahs, Moeller, Huber,
Willmes, & Nuerk, 2010) on mental representations of numerosity
suggest that there is a strong relationship between number processing
and finger counting. The study shows that even in the most elementary
numerical tasks with hearing educated adults, there is no exclusively
abstract magnitude processing. Rather, it suggests that the process is
always functionally influenced by persisting counting experiences. The
study not only confirms other findings in developmental, behavioral
and neurocognitive researches, but also goes beyond a general
relation between finger counting and number processing, and implies
that the way number magnitude is processed is determined by the
specific structure of finger counting system. In this way the
presented data dismisses the idea that our numerosity representation
and number processing is purely abstract in nature.
The above study on the relation between cognition and perception goes
in the empiricists favor and provides evidence that even seemingly
abstract cognition turns out to be rooted in persisting bodily
experiences. Contrary to this result, there are other studies in
support of innatism suggesting that at least part of our knowledge is
independent from perceptual experience. Typical instances are the idea
of Universal Grammar as an innate and domain-specific body of mentally
represented knowledge (Chomskian module), and studies that advocate
initial knowledge during infancy (Leslie & Keeble, 1987; Spelke,
1988; Spelke, Breinlinger, Macomber, & Jacobson, 1992; Wynn, 1992)
which consider, for instance, solidity, substance and causality in
young children as innate principles of arithmetic, physics and
It is the early existence of such cognitive capacities in infancy
that leads the above studies to the postulation of the innateness
hypothesis. Experiments on infants' ability to compute the numerical
results of arithmetical operations lead Wynn (1992) to conclude that
the existence of such abilities so early in infancy suggests that we
innately possess the ability to perform some arithmetical
calculation. This and the other studies mentioned above show that
researchers normally draw innateness with respect to the contribution
of perceptual experience in the emergence of cognitive abilities. In
other words, innateness is understood as conception independent of
perception. This is actually one of the reasons that these
experiments are mostly performed on infants. Compared to adults,
infants obviously have the minimum perceptual experience possible.
This conception-perception dichotomy is best illustrated in the
following passage from Spelke (1988a):
Our research suggests that these views are wrong for a
common reason. All assume that objects are perceived: that
humans come to know about an object's unity, boundaries,
the persistence in ways like those by which we come to know
about its brightness, color, or distance. I suggest, in
contrast, that objects are conceived.
This apparently suggests that there are two aspects to our
experience: perceptual and conceptual aspects. Perceptual aspects
include our experience of color and distance while our experience of
unity and boundaries of objects is conceptual. But what is it to have
perception or conception of something? What is it to have either of
conception or perception with or without the other one?
Perceptual experiences, one might say, are based on purely sensory
data while conceptions do not depend on sensory data, or in other
words, conceptions do not depend on perceptions. In this sense, our
concepts of color or brightness of an object are derived from purely
sensory data while our very concept of object, that is, the
conditions under which something is recognized as an object, or its
unity over time, are independent from sensory data. This means that
we come to know about an object's unity in the same way that we come
to know about, say, an object's market value. Studies show that
infants who cannot yet reach for objects or talk about them
nevertheless organize the visual world into unitary bodies (Spelke,
1988a). Then it might be concluded that the organized world is not
constructed from a sensory tableau, but from a conceptual system,
which is independent from perceptions. The idea of being equipped
with such an independent cognitive repository sounds to be the core
of innateness hypothesis.
The idea of conception or cognition independent of perception is of
course far from clear and demands for clarification. It is plausible
to suppose, at least as a first approximation, that we are born with
bodies of information or principles whose development is quite
independent from perceptual experiences. This doesn't seem to be
promising as the idea of present at birth imposes a kind of temporal
constraint, which is neither sufficient nor necessary for innateness.
As Samuels (2004) points, the condition is not sufficient because the
prenatal learning is possible. In addition, traits that
paradigmatically are not innate can be present at birth. Present at
birth is not a necessary condition as there are typically innate
traits that are not present at birth and appear later in development.
Typical examples are the growth of pubic hair and sexual
characteristics. Therefore, the idea that we are born with some
actual innate stuff seems unacceptable. On the other hand, if innate
traits somehow instantiate late in development, then they enter into
interaction with environment and perceptual data. With respect to
this situation, one might suggest that cognitive innate stuff is not
quite independent from perceptions as we envisaged before. Rather,
they are influenced by perceptual experiences and the influence is
limited only to some transitory stages of development.
This conception of innateness suggests a kind of interactionist view
according to which innate cognitive characteristics are the result of
interaction with perceptual data. In other words, they are the result
of both internal and environmental factors. They are like
dispositions that can be developed, but the development requires the
trigger of perceptual experience. A classical and interesting example
of this conception of innateness is illustrated in Descartes' analogy
between innate traits and congenital diseases:
In the same sense we say that in some families generosity
is innate, in others certain diseases like gout or gravel,
not that on this account the babes of these families suffer
from these diseases in their mother's womb, but because they
are born with a certain disposition or propensity for
contracting them (1648, p. 304).
As babes, according to Descartes, do not suffer from disease in their
mother's womb, the analogy dismisses the notion of innateness as
present at birth. In addition, since inborn diseases are influenced
by environment or perceptual factors, it also instantiates an
interactionist approach to innateness. This is close to the
biological conception of innateness in which innate traits, although
influenced by perceptual experiences, are developmentally stable.
What would be Descartes' answer, if we ask him what is it for a
concept or trait to be innate, or in other words, what are the
conditions under which a trait is innate? He would probably refer to
his analogy and reasonably answer that the innateness conditions are
the same as the conditions under which a disease is inborn or
congenital. This answer immediately leads us to the question of the
conditions under which a disease is inborn. Obviously diseases do not
develop in vacuo, and any answer to this question certainly does not
ignore the role of environmental factors. The situation is also in
line with our second conception of innateness in which the
development of innate cognitive perceptions is not away from
environment, but influenced by perceptions. In this way, the
conditions of being an innate ability and being an inborn disease are
identified, and therefore characterization of inborn diseases leads to
a clarification of the concept of innateness.
One might propose that to say that an organism has an inborn disease
is to say that if it comes to an appropriate age, some specific
symptoms appear that indicates he has been afflicted with certain
disease. It is noticeable that we are looking for conditions that are
characteristic of inborn diseases. Since the conditions of innate
traits and inborn diseases are identified, the proposed solution is
useful in understanding characteristic features of innate traits, if
the proposed conditions exclude non-inborn diseases. But this is
exactly the problem with the above solution since both inborn and
non-inborn diseases satisfy this condition. Human diseases are
classified in a number of ways, from those caused by nutritional
factors, to those caused by infectious agents and also those with a
genetic origin. Cytomegalovirus, for instance, is a viral genus
infection, which has the characteristic ability to remain latent
within the body over long periods, and tends to reactivate
intermittently without symptoms. Once a person becomes infected, the
virus latently persists in the body for the whole life and can
exhaust the immune system in old age (Scott, Gibbs, Karlan, & Haney,
2003 p. 192). This simply shows that there are some infectious
diseases that satisfy the above condition, which was supposed to
belong specifically to inborn diseases.
It may be argued in response that Descartes' analogy identifies
innate traits with genetic traits. In other words, by comparing
innate traits with inborn diseases, Descartes is characterizing
innateness as part of our latent biological endowment in the same
sense that biologists consider genetic traits as part of our
biological endowment. In fact some biologists have put forward
similar proposals. Lorenz, for example, declares 'an innate trait is
one that is genetically transmitted as opposed to acquired by
cultural transmission or individual learning' (as cited in Ariew,
1996). But how are we going to understand the notion of a disease
genetically transmitted? What is it exactly to associate innateness
with a process that is 'in the genes'?
This requires an analysis of the relation between genes and
characters. Indeed it is customary in biology to use shorthand such
as 'gene for brown hair' or 'brown hair gene'. It also seems correct
to say that my hair or eye colour, all the processes in my body from
small-scale like cell division to reproduction and my capability to
survive, even my learning abilities, are all genetically encoded. But
it is an oversimplification if we treat these statements and
terminology as suggesting a direct causal relationship between a
particular gene and the production of a process or a character such
as brown hair. At least in the case of hair colour, the gene involved
in making hair brown is primarily involved in pigment formation
throughout the body, not just in hair colour.
There is also a more serious problem with the above proposal in that
it is based on a sharp contrast between genes and environment. It
supposes that a trait is either acquired or genetically transmitted.
But even genetically transmitted traits are not developed independent
of perceptual and environmental factors. Thus, this proposal doesn't
seem to be promising for the following two reasons: first, there is
no direct causal relationship between a particular gene and the
production of a process or a character and second, by the truism that
phenotype always requires an interaction between genotype and
environment, that is, phenotype is the sum of the combined action of
our genes and the environment in which we grow and develop.
It is worth considering that our characters are influenced either by
a single gene or involve two or more genes, which are called
multifactorial characters. Although the development of all these
characters requires interacting with environment, for multifactorial
characters there is a strong interaction between genotype and
environment (Metcalfe, Hirst & Saunders, 2001). The following cases
illustrate the relative influence of genetic and
Siamese cats: In order to appreciate the effect of the environment on
the phenotype, let's consider the temperature and coat colours in
Siamese cats. In these cats, the pattern of brown extremities --
feet, face, ears, and tail-and cream coloured body is transmitted to
their descendents. The kittens are all cream coloured at birth; some
days later, pigment appears in their new fair, first along the margin
of the ears and gradually over their extremities. If the kittens grow
and develop in a warm environment, the amount of brown fur is less
than the amount they gain if they develop in cold temperature.
Although it appears that the brown and cream pattern is itself
inherited, in fact what is really inherited is the capacity of the
fur to form brown pigment, depending on the particular temperature at
the time of growth. So a single genotype may produce different
phenotypes, depending on the environment in which the organism
develops (Metcalfe et al. 2001, p.52).
Identical twins and multifactorial characters: Another interesting
way of considering the relative influence of environmental or genetic
factors in the development of characters is the study of the identical
twins. In this study, identical twins reared together are compared
with those in which members of each twin-pair are separated at birth
and raised in different households. The two individuals in each
twin-pair are genetically identical because both are formed from a
single fertilized egg, which splits early in development to form two
distinct embryos with identical genes. Thus identical twins separated
at birth can be studied to examine the effect of different
environments and perceptual factors on the same genes. Studies have
revealed that although the twins have identical genotypes, they have
different multifactorial characters (Metcalfe et al. 2001, p. 55).
These cases illustrate that we cannot provide a successful account of
innateness in terms of gene-environment dichotomy. There are
alternative biological accounts that explain innateness as a matter
of degree rather than gene-environment dichotomy. Sober (1998), for
example, proposes that 'a phenotypic trait is innate for a given
genotype if and only if that phenotype will emerge in all of a range
of developmental environments', and Ariew (1996) suggests that 'the
degree to which a biological trait is innate for a genotype is the
degree to which a developmental pathway for individuals possessing an
instance of that genotype is canalised'. None of these ideas seems to
be clear and helpful as they lack a precise account of the relation
between genes, characters and the role of perceptual and
The notion of environment, with respect to the development of
phenotypic characters, has a very broad sense, ranging from cytoplasm
that surrounds nucleus within a cell, to specific environments such as
uterus, or even to more general factors such as human culture or
earth's climate. It includes any social or physical factor, other
than genes, which can interact with the genotype. This means that the
above accounts not only require providing a dynamic interaction
between genes and perceptual factors, but they also require
explaining this interaction with respect to such a broad sense of
environment. The concept of environment is so broad, inclusive and
indeterminate that any attempt to account for innateness in this way
is condemned to vagueness. This implies some kind of insufficiency in
the innateness hypothesis.
Almost all nativists wholeheartedly accept the interactionist view
that innate characters are caused jointly by both internal and
perceptual factors. But none of them give an account that provides
the necessary and sufficient conditions that characterize nativism.
The situation needs to be clarified by the advocates of nativism.
Otherwise, innateness sounds like a negative hypothesis, established
and interpreted vaguely as nothing but the negation of Empiricism.
1. Such as blood groups or some disorders like Huntington.
2. Such as height, weight or coronary heart disease.
3. There are exceptions such as Jeffry Elman suggesting that a trait
is innate if and only if it is the product of interactions internal
to the organism. (1996)
Ariew, A. (1996) Innateness as Canalization. Philosophy of Science,
63 (Proceedings), pp.S19-S27.
Descartes, R. (1648). Comments on a Certain Broadsheet, in
Cottingham, J., Stoothoff, R., & Murdoch, D. (1985). The
Philosophical Writing of Descartes, volume 1. Cambridge University
Domahs, F., Moeller, K., Huber, S., Willmes, K., & Nuerk, H. (2010)
Embodied numerosity: Implicit hand-based representations influence
symbolic number processing across cultures. Cognition, 116, 251-266.
Elman, J. et al. (1996) Rethinking Innateness: a Connectionist
Perspective on Development, MIT Press.
Lepore, E., & Pylyshyn, Z. (ed.), (1999) What is Cognitive Science?
Metcalfe, J., Hirst, M., & Saunders, R. (2001) Human Genetics and
Health Issues, The
Murphy, G. (2002) The big book of concepts. The MIT press.
Samuels, R. (2004) Innateness in cognitive science. Cognitive
Science, Vol.8 No.3
Sober, E. (1998) Innate knowledge. In Routledge Encyclopedia of
Spelke, E. (1998a) Where perceiving ends and thinking begins: the
apprehension of objects in infancy. In A. Yonas (ed.), Perceptual
Development in Infancy, Hillsdale, NJ: Erlbaum, 197-234.
Spelke, E. (1998b) Nativism, Empiricism and the Origin of Knowledge.
Infant Behavior & Development, 21 (2), 181-200.
Scott, J., Gibbs, R., Karlan, B., & Haney, A. (2003) Danforth's
Obstetrics and Gynecology. Lippincott Williams & Wilkins; Ninth
(c) Ali Yousefi Heris 2011
III. 'GENERATING STABLE KNOWLEDGE VIA REDUCTION IN ENTROPY' BY
GEORGIOS CONSTANTINE PENTZAROPOULOS
All men by nature desire to know
Abstract. The purpose of this work is to examine the conditions under
which reduction in uncertainty contributes to knowledge stability. Two
of Plato's works, Theaetetus and Meno, are used to illustrate the
difficulties. Links with information theory are established by virtue
of a computer metaphor. Results are expressed in a series of four
statements. According to the final statement, by reducing entropy,
uncertainty is also reduced, and the flow of information approaches
steady state. In that state, knowledge is always stable and valuable.
Keywords: Epistemology; knowledge acquisition; cognitive sciences;
computer metaphor; stability; entropy; information philosophy.
Knowledge acquisition is a central subject in classical epistemology,
theoretical computer science, and the cognitive sciences. Given that
much of our knowledge is gained via interaction with the environment,
it follows that the study of knowledge requires the examination of all
possible feedback mechanisms between the human brain and its
surrounding physical space. Memory is an important part of this
feedback process as it involves brain's neuronal circuitry.
Leading experts in neuroscience have for some time been trying to
find a link between short-term memory and the creation of a long-term
memory (Menzel, 2005). EU-funded research has also shown that the
hippocampus inside the brain's limbic system plays a major role in
memorizing recent events (Moser, 2005). Recently, it has been
suggested that the human brain could be simulated as an artificial
neural network with a two-level memory hierarchy Pentzaropoulos
(2011). By virtue of this computer metaphor, the short- and long-term
memories are connected via a feedback mechanism which allows the
short-term memory to be regularly renewed in order to store all
recently used information. Then, this renewal process guarantees
brain's responsiveness to intellectual challenges such as making new
decisions. Finally, it was noted there that any knowledge gained is
valuable only if the information from which it is derived reduces
The purpose of this work -- a follow-up of our research above -- is
to examine the conditions under which reduction in entropy may
contribute to knowledge stability. Our approach involves aspects from
classical epistemology, computer science, and information philosophy.
Part 1: Nature of Knowledge
Epistemologists often make a distinction between two types of
knowledge: ability or know-how, and propositional knowledge. Here we
consider both types. However, we retain the view that much of our
knowledge is gained via perception following the tradition elaborated
by Democritus. Therefore, in this paper, the senses (aestheses) are
considered more important than intuition (noesis). We also retain the
view expressed by the British empiricist, John Locke, that the mind at
birth is more or less like a blank table or slate, i.e. a tabula rasa
(Blackburn, 2008). However, we give to this term an
information-theory definition as explained below.
At its initial state, the mind is assumed to have no information.
However, given the mind's constant exposure to the outside world, it
becomes clear that the tabula will eventually become full unless we
assume that it has infinite capacity. Our earlier work
(Pentzaropoulos, 2011) suggests that memory can be very large but not
infinite despite brain's extensive neuronal circuitry. Therefore, we
consider the tabula to be like an erasable tablet, a term which is
consistent with the origin of the Latin word (verb) radere which
means 'erase' (Morwood, 2003). Human memory has the property of
writing and overwriting information, much like computer memory.
Roughly speaking, human short-term memory is analogous to the
(volatile) computer main memory, while long-term memory can be
associated with the (more stable) computer virtual memory. The latter
could be seen as a very large tabula rasa such as a database
containing billions of records. Such databases form the
infrastructure of today's digital libraries; and, of course, digital
libraries are sources of knowledge.
The problem of defining knowledge in acceptable terms began with
Plato's view in the Theaetetus that knowledge is true belief plus a
logos, or else certification by reason (Blackburn, 2008). In this
dialogue, which takes place in the presence of the mathematician
Theodorus, Socrates announces that he has no idea what knowledge is
like; and that he is seeking some form of a definition. Theaetetus
says that he cannot think of any definition of knowledge. Thereafter,
Socrates and Theaetetus engage in a lengthy conversation both trying
to arrive at an acceptable definition. Three possible definitions are
examined in this dialogue:
(i) knowledge as pure perception;
(ii) knowledge as true judgment; and
(iii) knowledge as true judgment with an account.
Although none of the three definitions proved to be completely
satisfactory, the last definition seems to be somehow better than the
other two, prompting the young Theaetetus to note the word 'account'
(logos). Then, Socrates comes to the point that, in his opinion,
things without an account are unknowable, while things with an
account are knowable. The final conclusion of the Theaetetus is that
we do not really know how to define knowledge. But this is not to say
that we have not learned anything about what knowledge is like. As
Theaetetus admits, he has given birth to far more than he had in him
In computer science, knowledge and memory are defined in somewhat
different terms than those used in epistemology. Let us look at this
definition: 'Knowledge: the objects, concepts and relationships that
are assumed to exist in some area of interest'
Knowledge is different from (raw) data or information, two key
concepts in computer terminology, in the sense that new knowledge may
be created from existing knowledge by logical inference. Information
is the result of applying some kind of processing to this (raw) data,
giving it meaning in a particular context.
Wisdom has been excluded as it concerns a higher state of mind along
with other human properties. Wisdom may never be achieved, at least
according to Pythagoras, who thought of himself not as wise but only
a friend of wisdom (friend, philos plus wisdom, sophia equals friend
of wisdom, i.e. philosopher). Nevertheless, the above computer
metaphor illustrates that, if knowledge is a prerequisite to wisdom,
then we will always want more data and information. But, knowledge
gained through the senses can sometimes be deceiving; therefore,
information gained through data processing should always be held up
for inspection. Following the above discussion, we can write our
S1. Data (raw) fi Processing fi Information fi Knowledge.
If we accept that knowledge is true judgment with an account, as
noted above, then we must also note that -- because of S1 --
knowledge cannot really be separated from data or information. The
last two usually come from experiments or measurements performed in
the framework of physical or technical systems. Therefore, knowledge
is affected by the process of gathering data or transforming data
Part 2: Reliability and Stability
The observations made above lead to our second statement:
S2. Knowledge inherently contains uncertainty, which comes from the
methods of obtaining data and information. Such uncertainty can be
The second part of S2 is not immediately obvious, but it will become
clearer later when we will be discussing the problem of stability of
knowledge and its relation to information entropy. The latter is a
well-known index of uncertainty in both physical and technological
systems. For now we examine stability.
The following passage in Plato's Theaetetus -- known as the wax
tablet paradigm -- makes a clear distinction between perception and
thought. The objects of perception are a series of
constantly-changing immediate awarenesses. The objects of thought are
those objects of perception to which we have chosen to give a measure
of stability by imprinting them on the wax tablets in our minds
Plato's Theaetetus [191c-191e] (http://www.perseus.tufts.edu)
Socrates. Please assume, then, for the sake of argument,
that there is in our souls a block of wax, in one case
larger, in another smaller, in one case the wax is purer,
in another more impure and harder, in some cases softer,
and in some of proper quality.
Theaetetus.I assume all that.
Socrates. Let us, then, say that this is the gift of
Memory, the mother of the Muses, and that whenever we wish
to remember anything we see or hear or think of in our own
minds, we hold this wax under the perceptions and thoughts
and imprint them upon it, just as we make impressions from
seal rings; and whatever is imprinted we remember and know
as long as its image lasts, but whatever is rubbed out or
cannot be imprinted we forget and do not know.
Theaetetus. Let us assume that.
The most remarkable thing here is Socrates' view that human memory is
actually split into parts, i.e. it is a non-homogeneous structure in
computer terminology. Moreover, Socrates actually insists that human
memory has the property of writing and overwriting its 'images'
(icons): this is, of course, a well-known property of all kinds of
modern computer memories, e.g. electromagnetical, optical, and
semiconductor memories. Anything (data) that is not stored in memory
cannot be used in a subsequent processing activity; similarly, as
Socrates points out, whatever is rubbed out or cannot be imprinted
Returning to the problem of the stability of knowledge,
epistemologists generally agree that true beliefs are better than
false beliefs in the sense that the former have greater instrumental
value. However, true belief can never be as good as knowledge
(Pritchard, 2008). The fleeting nature of true belief was early
identified by Plato in his work Meno. There, it is argued that
knowledge is more stable than belief (opinion) because knowledge is
not easily lost. In the course of the dialogue Socrates refers to an
old legend (mythos) according to which the ancient sculptor Daedalus
had inserted a mechanism in his statues by which they could move.
Plato's Meno [97c-98a] (http://www.perseus tufts.edu)
Meno. I wonder, Socrates, this being the case, that
knowledge should ever be more prized than right opinion,
and why they should be two distinct and separate things.
Socrates. Well, do you know why it is that you wonder, or
shall I tell you?
Meno. Please tell me.
Socrates. It is because you have not observed with
attention the images of Daedalus. But perhaps there are
none in your country.
Meno. What is the point of your remark?
Socrates. That if they are not fastened up they play truant
and run away; but, if fastened, they stay where they are.
Meno. Well, what of that?
Socrates. To possess one of his works which is let loose
does not count for much in value; it will not stay with you
any more than a runaway slave: but when fastened up it is
worth a great deal, for his productions are very fine
things. And to what am I referring in all this? To true
opinion. For these, so long as they stay with us, are a
fine possession, and effect all that is good; but they do
not care to stay for long, and run away out of the human
soul, and thus are of no great value until one makes them
fast with causal reasoning. And this process, friend Meno,
is recollection, as in our previous talk we have agreed.
But when once they are fastened, in the first place they
turn into knowledge, and in the second, are abiding. And
this is why knowledge is more prized than right opinion:
the one transcends the other by its trammels.
Meno. Upon my word, Socrates, it seems to be very much as
The passage cited above confirms this problem: right opinion (or mere
true belief), unlike knowledge, can be unstable. True beliefs are
still useful, but they cannot stay in their place unless they are
tethered to the ground by what Socrates calls a logical explanation
(aetias logismos). The distinction between knowledge and true belief
forms the basis of the definition of knowledge as justified true
Pritchard (2008), who also quotes the same paradigm (The Statues of
Daedalus), links stability of knowledge to reliability of information
which underpins knowledge. This author offers the following example.
Suppose one is trying to find his way to the nearest restaurant in an
unfamiliar city. Suppose, further, that the particular map is
unreliable. Eventually, he would notice that none of the landmarks
corresponded to the where they ought to be. He would quickly realize
that the map is unreliable, thereby preventing him from getting to
Let us now recall our statement S1 according to which information and
data are prerequisites to knowledge. Unreliable information will
eventually lead to trouble as in the example cited above. Of course,
one might still reach his goal, even if the map was unreliable,
entirely by chance; but epistemologists generally agree that one
cannot gain knowledge by chance. This leads into our third statement:
S3. Knowledge gained through reliable information has greater
instrumental value than true belief. Further, knowledge is always
more stable than true belief. Thus, knowledge increases the
probability that one achieves his objectives.
Part 3: Entropy in Relation to Stable Knowledge
According to our last statement (S3), reliable information is a
requirement for achieving genuine knowledge. Knowledge is always
stable. In everyday life, information is commonly associated to a
sense of order. Any lack of order brings about uncertainty, and
sometimes chaos. In physics, as well as in information theory, the
entropy of a system is defined as a measure of the system's intrinsic
uncertainty (Soanes and Stevenson, 2005). The presence of a feedback
mechanism often observed in closed systems contributes to greater
entropy. That is the reason why in many system performance evaluation
studies entropy is described by an objective function that deserves
Entropy comes from an ancient Greek word (entropia), which is formed
from the prefix en- plus trope. Any logical structure e.g. a memory
containing information can be thought of as a 'system'. Entropy
distorts information content, and this quite often leads to
incomplete knowledge. The amount of information stored in memory, as
well as its internal organization, gives us an index of how well we
can understand that information. Such an understanding always reduces
uncertainty in systems as explained below.
Entropy was given a new meaning -- away from thermodynamics -- thanks
to the pioneering works of Claude Shannon and Norbert Wiener. Shannon
introduced the complex term 'information entropy' when he was
developing his mathematical theory of communication, which is now the
basis for modern information theory. And Wiener, the founder of
Cybernetics, regarded information as the negative quantity of
entropy. Therefore, any decrease in entropy results in more
information. Probability and statistics are both key elements of
Closed systems gradually become less and less organized with the
passage of time. Wiener believed that communication of information
has a tendency to resist entropy. Therefore, an increase of
information will increase organization within the system, whether
communicated by a living being or a machine (Wiener, 1950). The
ancient Greek word 'homeostasis' was interpreted by Wiener as a kind
of steady state applicable to both organisms and machines. Life
itself continues by maintaining an internal balance of factors.
According to remarks made by several present-day philosophers of
information, especially Doyle
(http://www.informationphilosopher.com), the notion of the so-called
'information philosophy' quantifies knowledge as actionable
information. In line with this new method of enquiry, when
information is stored in any structure, two fundamental physical
processes occur. First is a collapse of a quantum mechanical wave
function. Second is a decrease in the system's entropy corresponding
to the increase in information. Entropy greater than that must be
transferred away from the system in order to satisfy the laws of
The above discussion naturally leads into our fourth and final
S4. Reliable information leads to greater stability of knowledge.
Stability requires some form of internal organization as regards
information, which can be achieved at the expense of uncertainty. The
latter can be described as a manifestation of entropy. Thus, by
reducing entropy, uncertainty is also reduced, and the flow of
information approaches steady state. In that state, knowledge is
always stable and valuable.
Blackburn, S. (Editor) (2008).Oxford Dictionary of Philosophy, Oxford
University Press. See 'epistemology' and 'tabula rasa'.
Chappell, T. (2009). Plato on Knowledge in the Theaetetus. Stanford
Encyclopedia of Philosophy,
Free Online Dictionary of Computing (www.foldoc.org). Imperial
College, University of London. See 'knowledge' and 'information'.
Information Philosopher (www.informationphilosopher.com). See
Menzel, R. (2005). RTD Info, Special Issue on Science and Memory.
Brussels: European Commission, p. 25,
Morwood, J. (Editor) (2003). Pocket Oxford Latin Dictionary. Oxford
Moser, E. (2005). RTD Info, Special Issue on Science and Memory.
Brussels: European Commission, pp. 28-29,
Pentzaropoulos, G.C. (2011). Knowledge acquisition as a memory
renewal process. Philosophy Pathways, Issue 159, Part I.
Perseus Digital Library (www.perseus.tufts.edu). Tufts University.
See under Plato's Theaetetus [191c -- 191e] and Plato's Meno [97c --
Pritchard, D. (2008). What is this Thing Called Knowledge? London:
Soanes, C. and Stevenson, A. (Editors) (2005). Oxford Dictionary of
English. Oxford University Press. See 'entropy'.
Wiener, Norbert (1950). The Human Use of Human Beings. The Riverside
Press (Houghton Mifflin Co.).
This work has been partially financed by grant no. 70/4/4733 awarded
by the Research Committee of the University of Athens, Greece.
(c) Georgios Constantine Pentzaropoulos 2011
Georgios Constantine Pentzaropoulos
Associate Professor, Information and
Mathematics and Information Technology Unit
Department of Economics
University of Athens
8 Pesmazoglou Street
105 59 Athens
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