A
MODEL OF GENERALIZED SYSTEMS
This paper is intended to be a brief note on a mathematical
formalization of the following methodology that we term as interactive,
integrative and evolutionary in terms of its knowledge-centric order. We will
then briefly indicate how such a complex model can be applied to specific
problems of economics and social contract. Briefly we examine the chain:
Stock
of Complete Knowledge (treated as primal topology), T. T is a universal
topology in the sense that it is defined by the Completeness of knowledge
called the Stock of Knowledge. Its cardinality, a concept required to generate
explainable mappings, is called the supercardinality, SC(T), of T. T thus
contains all classes of mathematical union and intersection of its subclasses
which are indefinitely many sets. Thus T contains the sum-total in the
supercardinal sense of all truth statements and its mathematical
complementation comprising false statements. By this mathematical
complementation property, T contains the null set and the universal set which
is defined by the complementation of the null set. The null set is defined by f = {F}Ç{F’}, where, {F} denotes the complete class of all truth statements;
{F’} denotes the complete class of all false statements. In
terms of the uniquely universal laws, as of the Unity of God and the cosmic
order, the world-systems, self and the other, abstract and experiential
knowledge, {F} and {F’} would denote Divine Laws explaining the totality of
Truth and the totality of Falsehood. We also call the latter disjoint
categories as Knowledge Stock and De-Knoeledge Stock, because of their
completeness in the supercardinal sense corresponding to their own disjoint
spaces. In the end we note that T is fully disjoint between {F} and {F’} while it remains unity in explaining both of these
by a singular set of universal laws. {F} is explained by T by Divine Laws premised on Truth;
{F’} is explained by the
mathematical complementation of the set of same Divine Laws, which because of
the universal topological property of T, is also contained in T. Hence, {F}Ì T; {F’}Ì T.
We
also note that {F} and {F’} form topologies in their own: For any set of elementary knowledge
encompassing the supercardinality, sc of {F}, denoted by {q}sc, sc’ of {F’}, the usual properties of a topology are satisfied.
It
remains to be explained that {q}scÇ{q’}sc’ = fsc e {q}sc, for {q}sc e {F}; {q’}sc’ e {F’}. Also, fsc’ e {q’}sc’. Here the
null sets, fsc and fsc’ have been differentiated in order to show their
candidacy in the corresponding sets as mentioned above. It is easy to note that
if fscÏ{q}sc and if fsc’Ï{q’}sc’, then some element of{q’}sc’ would belong
to{q}sc. Consequently, {q’}sc’ e {F}, which would yield {F}Ç{F’}, a contradiction to the topological property of T,
as mentioned earlier. The same inference holds if fsc’ Ï{q’}sc’. We refer to {q}sc and {q’}sc’ knowledge and de-kinowledge flows
from the Stocks {F and{F’}, respectively, all emanating from T. Thus by virtue of its
singularly complete nature with respect to the emanation of all flows including
the Knowledge Stock and De-Knowledge Stock, T denotes the unity of knowledge in
the sense of its completeness, uniqueness and thus absoluteness. The last
attribute is established by noting that it would be tautology to have any other
primordial inclusion of T in another similar set. If that were possible that T
would acquire the position of {q}sc È{q’}sc’, which would be a subset of T.
The
same attributes of completeness, uniqueness and absoluteness of T are
established by noting that for sequences if elements, say {q} e {q}sc, lim{q}e{q}sc e {F} (completeness of {F}). Likewise for {F’}. Hence the attribute of completeness of T is
established from the property of topology of T in terms of {F} and {F’}. For reasons of integration and consensus in the
system, for two sequential values, {q} and {q}*, |{q} - {q}*| < |({q} - lim{q}) - ( {q}* - lim{q}*)| < |({q} - lim{q})| + |{q}* - lim{q}*)| < indefinitely small value. This proves the uniqueness property
of T through its topological property in terms of {F} and {F’}. For the absoluteness property of T we note that
since T uniquely explains both {F} and {F’}, it is universal in the primordial sense of all
sets of {q}sc and
{q}sc’. Hence,
if there exists T*Í T, then T assumes the role of {q}sc and {q}sc’, and T* must become primal, etc. Thus, T*Í T is a tautology.
Construction of
knowledge-centered unified world-system
Furthermore,
the creative essence of T extends {F} and {F’} to the domain of objectification. Such a domain
comprises the observed domain and the abstract domain. Hence, as much as the
anthropic world-system is spanned by {F} so also is the relational order through its
interactions with the anthropic order.
We
will denote the spaces of cognitive entities by {X({q})} and {X’({q’})}. These have their topological properties as well,
for {q} ® f X: X({q}) = f({q}). For the entire class of such functionals we
obtain, {X({q})} as a monotonic
transformation of {q}in real-world relations via a sequence of mappings {f}. But we note
that by virtue of the topological property of {q} (hence of {q’}) and the monotonic transformation of X in terms of
{q}, that {X{q}}also forms a topology. Likewise is the case of {X’{q’}}in the de-knowledge domain. Now all the properties
relating to relationships between {X{q}} and {X’{q’}} will apply. We will not repeat the similar
formalization here.
However,
the distinction and characterization of the nature of f-functionals must be
noted. There are two stages of such functionals here. First, the derivation of
knowledge flows from the primal Knowledge Stock at the moment of establishing
abiding moral laws is based on immutable attributes which we denote by the
symbol A. Thus, ‘A’ augments both {F} as well as {q} values as {F}[A] and {q}[A], respectively. We will thus denote the derivation,
{F}[A] ® f*{F*[A]}: {F*[A]} = f*({F}[A]). Since f*({F}[A]) is a monotonic function of {F}, therefore, f*({F}) Ì {F}Ì T, after depressing A as being implied in these
functionals. The nature of A is such that it remains immutable and primordial
in the essence of T. Hence such an essence would not change in the induction of
the world-system by the knowledge-flows, {q[A]}.
However,
the second level of deduction from the primordial functional relationship via
f* is characterized as follows: {F*[A]} ®f1 {q[A]}: f1{F*[A]}= {q[A]}. f1 denotes the mundane understanding and progressive mechanism of
deriving worldly laws premised on the unity of knowledge T through the primal
mapping, f*. f1 is thus the result of discourse, consensus and further evolution
premised on the immutability of T in knowledge formation. Besides, since T
establishes the unity of knowledge in the Stock, therefore, both f* and f1 are
derivations that carry the same essence of unity in world-system. The nature of
recreated unity in world-system is depicted by complementarity among the
agents, variables and their relations in various sub-systems of the
world-system. Among these sub-systems are society, economy, science, community,
nations, subnations, the global order, self and the other. Broadly speaking the
sub-systems along with their various elements are pervasively interactive
within and across themselves. They thus form a cosmic order in the sense of the
grand nexus of socio-scientific commons. The discourse that takes place in the
essence of pervasive interactions leads to consensus in the sense of
integration among the agents, variables and their relations. Systemic
interactions leading to integration is the reflection of ‘unification’ of
knowledge flows in the light of unity of knowledge in the primal epistemology
of T. We call this universal principle of unification premised on the
fundamental unity of knowledge as the principle of universal complementarity.
Knowledge
flows are pervasive inter-systemically and intra-systemically in order to carry
the nature of unification across all changing nexuses of knowledge-induced
systems. Hence an evolutionary phenomenon is intrinsic in the evolving
world-system. But this stage of knowledge induction needs explanation.
Evolution from one level to another level of dynamic world-system comes
about by a complementary combination of knowledge flows and their
knowledge-induced forms. The latter are our X(q) and the tuplet is (q,X(q)). Evolution to higher
levels of knowledge flows appears through post-evaluation and confirmation of
certain well-defined criterion of well-being. One form of such a criterion is
the social well-being function denoted by W(q,X(q)). It is clear that each of the variables
and hence the evaluation of this criterion takes place in terms of the
immutable attributes, A.
We
now have a chain of interactive, integrative and evolutionary stages that
together define the process as follows:
T ®F {F} ®f* {F*} ®f1 {q} ®f2 {X({q})} ®¯®f3 New {q} ® continuity
® T=H (1)
W(q,X(q)) in repeated
processes
Primal ® Derivation® Process of ® Post-evaluation®Evolution®Continuity®Closure
Stock
of of primal deriving of similar in
the
Knowledge
knowledge knowledge processes very
flows flows by large
discursion scale
universe
H
here denotes the end of the cumulative process of all flows within and across
nexuses of world-systems. The appearance of the attributes which is pervasive
in all of the processes as the defining conditions of the nature of primal
Stock of Knowledge and the derived knowledge flows has been suppressed.
The process of de-knowledge flows
Unlike the knowledge flows, de-knowledge claims a
multitude of T to give rise to pluralistic knowledge flows with differentiated
pre-conditions of such pluralistic orders. Consequently, although the nature of
the evolutionary processes of de-knowledge flows is similar to knowledge flows,
their interactive, integrative and evolutionary conditions are distinct. In
de-knowledge flows these are defined by pluralism and dualism of interactions
lading to individuation and differentiation, conflict and independence from
each other. Thus, systemic interactions, integration and evolution devolve to
individual sub-systems but do not interact across systems to gain integration,
or unification of knowledge. Thus, universal complementarity is systematically
replaced by marginalism, trade-off, conflict and substitution in the
de-knowledge world-system. This is a pervasive fact of its socio-scientific
perspective and explanation. Because of the random nature of its
epistemological premise that remains disjointly distributed across sub-systems,
their variables and relations, the essence of rationalism fully characterizes
such sub-systems. Rationalism is defined as the epistemology of pluralism of
thought without a premise that inter-systemically unifies knowledge according to
certain unique laws. The premise of rationalism is sheer anthropic. The premise
of unity of knowledge is the set of unique laws that remain immutable and that
are then humanly comprehended and applied by reason to world-systems.
To formalize, let {T1,T2 ,…} denote the indefinitely
large number of competing epistemes from which individually arise relational
chains of the type shown by (1). Because of the axiom of systemic independence
either of the following conditions will apply to {Ti}, i =1,2,….: (1) Çi {Ti} = f (methodological individualism); (2) {Ti}=aj.{Tj}, i,j = 1,2,….
Consequently, each of the corresponding kinds of
entities in the chains shown in (1) for the case of de-knowledge, remains
mutually independent. Thereby, the beginning and endpoints of the chains in the
very large scale universe are not unified across systems. In such universes
equilibria can exist within the disjoint sub-systems but they do not exist
across systems. The concept of interactions leading to integration is lost.
Evolution of the disjoint systems result in infinitely many competing
sub-systems. We refer to such individuated sub-systems as being characterized
by endogeneity of de-knowledge flows within sub-systems but exogeneity of such
flows across sub-systems. Contrarily, in the knowledge-induced world-system
there is endogeneity of knowledge flows both across and within sub-systems of
the world-system.
A generalized model of knowledge
Because of the pervasively relational essence of
knowledge centered world system we will consider the following expression:
F1 Û F2 (2)
F1: T®F {F} ®f* {F*}®f11{q1}®f21 {X1({q1})}®¯®f31 New {q1} ®continuity ® T=H (3)
W(q1,X1(q1)) in repeated
processes
F2: T
®F {F} ®f* {F*}®f21{q2}®f22{X2({q2})} ®¯®f32 New {q2}®continuity ® T=H (4)
W(q2,X2(q2)) in repeated
processes
Between (2), (3) and (4) we obtain the following system
of interrelationships:
® {q1}®f21 {X1({q1})}®¯®f31 New {q1} ®continuity
®
W(q1,X1(q1))
¯
T ; ; ; ; ; T=H ….. (5)
¯
® {q2}®f22{X2({q2})} ®¯®f32 New {q2} ®continuity ®
W(q2,X2(q2))
By an elementary disaggregation of relations we note
that,
{q1}®f21 {X1({q1})}®
f31New{q1}
; X ; X ; X ; X ; …
(6)
{q2}®f22{X2({q2})}®
f32New{q2}
where,
X denotes cross-wise interactions. Such interactions are extensive as can be
worked out even from this simple disaggregation when extended to second and
higher number of processes (not shown). In terms of the functional mappings the
extensive interactions cause compound functionals to arise.
The
well-being criterion function resulting from the pervasive interactions across
the interactive, integrative and evolutionary branches (3) and (4) is the
non-linear aggregation of the separate well-being functions. One such non-linear
form would be the product function with indexed coefficients of elasticities of
the individual variables to the aggregate well-being function. The resulting
non-linear aggregate well-being function is a cardinal measure of the
complementarity among the various variables and their relations. Among the
variables are also policy and institutional ones that imply the complementary
role of agents in the underlying decision-making. Through the joint aspects of
interactions that lead to the compound form of the branches, functionals and
now well-being functions and the representation in the resulting well-being
function of the complementary role of agents, variables and their relations,
the essence of integration is introduced. Finally, because of the continuously
dynamic nature of knowledge flows affecting decision-making, variables and
their relations, evolutionary processes become the natural consequence.
The
evolutionary nature of the interactive and integrative processes at each stage
conveys the importance of simulative method of quantitative analysis in this
interactive, integrative and evolutionary system (IIE). It also points to the
replacement of all steady-state equilibrium points by a multiple evolutionary
knowledge-induced equilibria. Optimization as a method is totally rejected as a
method in the IIE system, as their cannot be any attained position of the
system except in the instantaneous case of the variables, their relations and
the underlying decision-making. The instantaneous case is not a sustainable
perspective in knowledge-induced evolutionary models. Besides, the presence of
unification of knowledge in the principle of universal complementarity by
negating the contrary idea of marginal substitution, as in the case of
neoclassical resource allocation, also rejects the idea of scarcity and
constriction in resource supply. The circular causation and continuity model if
unified reality in the IIE-system makes risk-diversification,
product-diversification, institutional development and participation among the
agents in the work place to constantly reduce the unit cost of production. The
principle of universal complementarity and diversity of methods as signified by
the branches of (5) followed by creative evolution are intrinsic in realizing
reduction in unit cost due to risk and scarcity.
The systems
methodology of de-knowledge
In
the de-knowledge system there will be independent branches of the type shown in
(1) with respect to each of the competing {Ti}, i =
1,2,… There will be a plethora of such emerging independent branches from any
given branch in as far rationalism rules the individuating configuration of
human thought and organization. Ultimately in such a process, all perspectives
of convergence will be dispensed with giving way to utter randomness. Along
each of these independent branches the condition of equilibrium is once again
of the evolutionary type but optimality of the objective criterion is a precept
independently of separate branches of the evolutionary process. Continuity and
interrelationships across branches not being present, the optimal conditions of
each of these branches are also unrelated to each other; so too are the
equilibria of the evolutionary branches.
To
formalize we let (xi*,Wi*(xi*)), i = 1,2,… denote the optimal values of the
variable, xi and the objective criterion function, Wi(xi). From our
formalization in (6) we note that in the generalized case a joint objective
criterion function is given by,
W((q, x(q)) = Si Wi(qi, xi(qi)),
because
of independence among the (qI, xi(qi))-tuplets, i = 1,2,…
The
optimal value of W((q, x(q)) with respect to (qi, xi(qi))-tuplets, i = 1,2,…yields,
dW/dx = Si (¶Wi/¶xi).(dxi/dx)= 0 ..
(7)
identically
in the terms, because each ¶Wi/¶xi = 0,
i
= 1,2,.. for reasons of optimality along the individual branches.
Likewise,
dW/dq = Si [(¶Wi/¶qi).(dqi/dq) + (¶Wi/¶xi).(¶xi/¶qi).(dqi/dq)] = 0 ..(8)
identically
in the terms, because each ¶Wi/¶qi = 0, i =
1,2,… for the same reason.
These
results clearly indicate that in a state of optimal trade-off must be
maintained among at least two of the xi(qi)-variables, i = 1,2,… The determination of such variables
by means of the trade-off is shown by dxi/dxj < 0 and dqi/dqj < 0. These results are consistent with the given
optimal value of Wi(.), ii = 1,2,... along the independent branches.
Aggregation of the knowledge flows and the variables being laterally additive
implying independence among the variables, this aspect does not convey the
meaning of interactions.
In
the case of the IIE-system we would have the non-linear result corresponding to
the expressions (7) and (8):
dW/dx = Si (¶Wi/¶xi).(dxi/dx)> 0 ..
(9)
dW/dq = Si [(¶Wi/¶qi).(dqi/dq) + (¶Wi/¶xi).(¶xi/¶qi).(dqi/dq)] > 0 ..
(10)
corresponding
to, dW/dq >
0, implying that dWi/dqI > 0, dWi/dxI > 0, dxi/dqi > 0, i = 1,2.., due to the positive induction of the complementary
variables by knowledge flows in the IIE-system.These identically positive
nature of the knowledge-induced variables and well-being function imply that
optimization fails to be an acceptable method for use in the IIE-system, except
in the instantaneous case of theq-value, when an attained consensus does not progress
dynamically to higher levels. Consequently, the corresponding x-values and
W-value also remain unchanging. We now relapse from the knowledge-induced forms
to the neoclassical case of independence and inter-systemic exogeneity from interactions.
This last case we will now prove.
Inter-systemic
generalization of knowledge-induced model
Expressions
(9) and (10) can be now extended to the inter-systemic case in accordance with
expression (6). The evolution of the interactive and integrative sequences of
knowledge flows and their knowledge-variables would now yield the following
well-being system, where i denotes the number of interactions, k denotes a
given numbered system; l denotes a numbered system, with k ¹l (=1,2,…):
Simulate
{qikl}W(qikl, xikl(qikl)),
Subject
to, } ..
(11)
qikl = f1(Èi{qik}Ç{qil}, x-ikl(q-ikl), W-(..)),
xikl(qikl) = f2(Èi{xik(qik)}Ç{xil(qil)}, qikl, W-(..))
-
symbol denoting
recursively lagged values.
Underlying
this simulation model is the IIE-epistemological circular causation and
continuity chain,
T ®F {F} ®f* {F*} ®f1 [qikl] ®[f(ikl)] [X ikl({qikl})] ®¯®[f ‘(ikl)] New[qikl] ® T=H
.. (12)
[W([qikl],[X ikl(qikl)]]
The square brackets indicate the matrix of variables, relations and
well-being function corresponding to the ([qikl],[X ikl(qikl)])-entries across
(k,l)-systems for given numbers of interactions (i). The same matrix meaning
applies to the functionals.
The variables shown in (11) and (12) can be simplified by taking
limiting values of the knowledge flows over given number of interactions, as
was explained earlier. In this way the values of i would be assigned with
respect to the number of interactions undergone to arrive at the limits of the
([qikl],[X ikl(qikl)])-values.
With such limit values of ([qikl],[X ikl(qikl)]), say ([qkl*],[X kl*(qkl*)]), the simulation path of
the well-being function and variables in expression (12) can be depicted in
figure 1.
Figure 1: Simulation path of the well-being function and variables
The arrows in the diagram show the simulative direction of convergence
of the knowledge flows, the knowledge-induced variables and the resulting
simulated well-being function. This whole convergence marks one given process
in the IIE-system. In the simplification of the expressions (11) and (12) we
would substitute (q*1kl, X*1kl) for stage 1 in the
interactions. The resulting simulated well-being function would be W*(q*1kl, X*1kl) for process 1 when the
second process commences, and so on.
The expanding knowledge-inducing random field of interactive events, Çkl(q*1kl, X*1kl) that can be made to converge
into integration represents an aggregate topological domain. This is shown by,
W(q*1, X*1) = òq*e {q*1kl} òX*e {X*1kl} W(Çkl(q*1kl, X*1kl)d X*1kldq*1kl. (13).
Furthermore, in the case of random field the variables are
probabilistic. We then have to replace the integrand by the expected value of
W(.). One is then led to determine the type of probability distribution of W(.)
in the random field of (Çkl(q*1kl, X*1kl). However, as knowledge flows
and the knowledge-induced variables progress towards their limiting values
greater certainty is gained. Consequently, it would be safe to assume a
multivariate normal distribution in this random field.
In the complex case of conditional probabilities of a q occurring, subject Bayesian probability
distribution of (Çkl(q*1kl, X*1kl) needs to be used. That is,
in terms of the inter-systemic interaction and integration of intra-systemic
events during any given range of interactions denoted by 1, such as, Zk
= (Çk(q*1k, X*1k)) and Zl = (Çl(q*1l, X*1l)), the Prob.(ZkÇZl) = Prob(Zk|Zl).Prob(Zl), where,
Prob(Zk|Zl) denotes the
conditional probability of Zl subject to the lagged occurrence of Zk.
But since there exists causal interrelationship between the events of
systems k and l, therefore, the simultaneous occurrence of these events will
have a probability of Prob(Zl|Zk).Prob(Zk)
or a linear function of this. But now, Prob(Zk|Zl).Prob(Zl) = Prob(Zl|Zk).Prob(Zk), or a
linear relationship of the two. Hence the probabilistic condition for the
simultaneous occurrence of events in
knowledge-induced random fields under the principle of universal
complementarity is given by the proportionality between the conditional
probability of occurrence to the probabilities of actual occurrence of the
events or a linear relationship thereof. That is,
Prob(Zk|Zl) / Prob(Zl|Zk) =.Prob(Zk) / Prob(Zl) (14)
Generalization to an expanding field of knowledge-induced random fields
can be given by,
W(q*1, X*1) = Convolution of òÇkl(q*1kl, X*1kl)W(Çkl(q*1kl, X*1kl)d(Çkl(q*1kl, X*1kl) (15)
k,l = 1,2,…
Pk,l=1N [Prob(Zk|Zl)].Prob(ZN) = Pk,l=1N [Prob(Zl|Zk)].Prob(ZN’) (16)
The proportionality condition as above is given by,
Pk,l=1N [Prob(Zk|Zl)] / Pk,l=1N [Prob(Zl|Zk)] = Prob(ZN’) /
Prob(ZN), (17)
where, k¹l, N ¹N’, but, k,l = 1,..N.
Diagrammatic explanation of evolutionary knowledge and de-knowledge
nexuses
The knowledge-induced nexus
In figure 1 we show the generalized region A as the intersection
meaning interaction and integration of the three systems 1, 2 and 3. The region
BBB shows the expansion or evolution of region A under the impact of knowledge
flows. Consequently, the properties of the regions A and B under the impact of
random field in which knowledge flows according to IIE are given by,
A = Ç k=13([qk*],[X k*(qk*)])
B = Ç k=13([qk**],[X k**(qk**)]),
with expanding limit values ([qk**],[X k**(qk**)]).
Aggregation of the well-being function is given by,
W(q**, X**(q**)) = òÇk(q**k, X**k)=B òÇk(q*k, X*k)=AE{W(Çk(q*k,X*k)d(Çk(q*k,X*k))} (18)
k = 1,2,3.
E{W(Çk(q*k,X*k) is the expected value of W(Çk(q*k,X*k) according to the
conditional probabilities explained earlier, now taken for the three systems k
= 1,2,3.
Expansion of the systems indicated by the arrows show the induction of
these interactive and integrative systems by means of knowledge flows. If any
of these three systems becomes disjoint with the other system, then interaction
of the disjoint system with the remaining system cannot be maintained. The
entire system then collapses into two competing system and disjointness
continues between these competing systems. The extensive interactive nature of
the systems on the other hand is the result of the complementarity between both
{q*k} and {X*k(q*k)}, k = 1,2,3. Consequently, Çk(q*k,X*k) ¹ f, Çk(q*k,X*k) ¹ f. The systems can be of the
most generalized kind ranging from scientific systems and global systems, to
systems of variables and their relations in the socio-scientific order and
between self and the other. We call any phase of such IIE-systems for a given
range of interactions to be the nexus. Thus, (Çk(q*k,X*k)) ¹ f, Çk(q*k,X*k) ¹ f are nexuses corresponding to
the range interactions given. The totality of such nexuses across various
phases of interactions is given by, Èi (Çk(q*k,X*k) ¹ f, È i Çk(q*k,X*k) ¹ f, i denotes sequences of
interactions.
Figure 2: Generalized knowledge flows in random fields
The nexus induced by de-knowledge
Figure 3 shows the methodological independence and disjointness of the
three systems. The direction of the arrows indicate increasing disjointness of
the systems as de-knowledge flows increase. The principle of complementarity is
thus replaced increasingly by marginalist substitution or trade-off between
alternatives. Contrary to the knowledge-induced generalized system, the
generalized system of de-knowledge is shown by,
A = Ç k=13([qk*’],[X k*’(qk*’)]) = f
B = Ç k=13([qk**’,[X k**’(qk**’)]) = f,
across systems. Within such systems there are interactions.
The primed values stand for de-knowledge, with expanding limit values
([qk**’],[X k**’(qk**’)]), independently across
the three systems, k = 1,2,3.
Aggregation of the well-being function is given by,
W(q’, X’(q’)) = ò(q*’3, X*’3) ò(q*’2, X*’2) ò(q*’1, X*’1)E{W(q*’k,X’*k)d(q*’k,X*’k)} (19)
The probabilities are now additive indicating independence among the events
,
(q*’k,X*’ k):
Sk=13 Prob(q*’k,X*’k) = 1.
Figure 3: Systemic generalization in de-knowledge flows
Legacy of the knowledge and
de-knowledge decision-making processes
Unity of knowledge has been the time honoured quest for scientific
explanation of reality. Yet it is also the scientific project that remains the
most evasive. This problem of lack of unity in knowledge across and within
disciplines has posed the greatest problem of unification in theories of
everything in present times. Yet this interdiscplinary insulation in the name
of efficacy and scientific rigour within self-same disciplines has marked the
nature of modern scientific inquiry. The post-modern inquiry into scientific
inquiry remains unhappy within such a dichotomous perspective in the sciences
by rejecting foundationalism, but at the same time a new barrier is raised by
the utter randomness of pluralistic thinking in post-modernism. In the words of
Foucault, “From one end of experience to the other, finitude answers itself; it
is the identity and the difference of the positivities, and of their foundation
…. “(Order of Things 315-16).
Yet unity of knowledge appears to be evasive because of its
anthropomorphic origins primarily. The change of this anthropic primacy to the
Unity of God returns the sciences and human designs to the axiomatic foundation
of fundamental unity as the root of all world-systems. The world is then
constructed on the basis of this singular, unique, complete and absolute
primacy of unity. The world in its details is recreated by cause-effect
relationalism in the midst of this fundamental Unity of God.
While
all religions have shared in this primal contest of Divine Unity as a
fundamental way to explain reality, yet the context of the emanating laws
enabling the organization of various aspects of life and thought has not been
uniform. In Islam this ordering of both the primal epistemology and ontology of
Divine Unity and its expression in terms of Divine Laws in precise ways for
externalizing to the world-system, is firmly established. The rumbling of
fundamentalism in the Islamic scholarly legacy of all ages is precisely this
causal relationship between the Divine Laws of Unity, their derivation,
understanding and application to the world-system. In the Qur’an the
fundamental Unity of God is termed as Tawhid.
The Divine Laws emanating from this Divine episteme are termed as Sunnat
al-Allah.The concrete way of deriving rules of life and thought from this
epistemic origin is termed as Sunnat al-Rasul also called Sunnah,
as the guidance of the Prophet Muhammad in their most authentic form. In the
context of these fundamentals the epistemological domain of Divine Unity (Tawhid
= T) interrelating Sunnat al-Allah with Sunnat al-Rasul, is given
by the phase, T ®F {F} ®f* {F*}. This primal chain is further is further composed
of T ®F {F} (Sunnat al-Allah) and {F} ®f* {F*}, which is Sunnat al-Rasul. The combination of these two
sub-chains is the foundation of the ‘core’ of Islamic Law (Shariah).
The derivation
of rules (Ahkam) premised on the episteme of Tawhid is
given by the extending mapping, ®f1 {q}. From the discourse or interactions (Ijtihad) arise
the limiting values of {q} or consensus within and across the interactive systems
(Ijma) that provide rules to guide the ordering of issues and problems
of world-system. Hence arises the Qur’anic
meaning of Signs of God (Ayath al-Allah) in the perceived world. This is
signified by ®f2 {X({q})}. From a combination of knowledge flows (Ahkam) and the ordering
of the world on the basis of these knowledge flows (Muamalat) is
determined the post-evaluation of the applications of laws to issues of
world-system and their reconstruction. This entire process is called the process
of Islamic discursive institution (Shura) called the Shuratic
Process. The Shuratic Process is pervasive and embryonic across and
within world-systems in all its details. It is shown by the chain,
[T
®F {F} ®f* {F*}®f1]0 [{q} ®f2 {X({q})} ®¯®f3 New {q}]1, (20) W(q,X(q))
where,
the subscript 0 indicates the complete exogeneity and immanence of the episteme
of Divine Unity in all matters. The worldly process of unification of knowledge
premised on Divine Unity is shown by
the endogenous knowledge-centered process subscripted by 1. The emergence of
new knowledge flows through the interactive (Ijtihad) and integrative (Ijma)
phases regenerates a new and similar Shuratic Process in which the exogeneity
of the primal chain remains the cause and endogeneity of knowledge flows derived
from the Unity episteme are regenerated. Such a discursive method of continuing
the Islamic process of knowledge regeneration is the essence of the ever
evolving nature of Islamic Laws (Shari’ah). The next round of the
Shuratic Process is given by the chain,
[T
®F {F} ®f* {F*}®f1]0 [New{q} ®f2’ New{X({q})} ® ¯®f3’ Fresh{q}]2, (21) New[W(q,X(q))]
This
kind of regeneration of the knowledge forming process in its cause and effect
circular causation interrelationships in a knowledge-centered world-system is
referred in the Qur’an as Khalq in-Jadid.
The
attributes ‘A’ that establish and sustain the uniqueness of the primal episteme
are referred in the Qur’an as Asma al-Husna (the beautiful attributes of
Allah). Because of the infinitely many diversity of the Asmas we reduce
this to the enabling vector of attributes, {Justice (‘Adl), Purpose (Maqasid),
Certainty (Haqq al-Yaqin), Well-Being (Falah & Tazkiyyah),
Re-origination (Khalq in-Jadid}. Such a derivation and ordering of the
attributes is not unique as many of the Asmas can be inducted into ‘A’ in
accordance to how we can comprehend and use them in systemic world-system
studies.
Since
(20) and (21) are uniquely premised on the same episteme of Unity, these must
be diverse expressions of unity across and within systems in all matters of
details. Hence we have the meaning of unification of knowledge by endogeneity
within and across systems. We call such extended methodology of unification
arising from Unity as the circular causation and continuity model of unified
reality.
The
endogenous knowledge inducing processual methodology just explained remains
perpatually evolving and discourse oriented in the IIE perspective. However, in
the very large scale perspective of the universal system, the cumulative
supercardinality of T is attained. This can be the only point of Optimum and
Final Equilibrium. Being so the End characterization of the topology of T, i.e.
Hereafter H, is equivalent to the Beginning T in this supercardinal sense. In
the Qur’an it is said that God will reveal Himself in Hereafter. This Qur’anic
mention is equivalent to saying that the total Stock of Knowledge (Lauh
Mahfuz) will be revealed.
Both
the Islamic epistemologists and rationalists in different ways though,
developed the epistemic worldview of Tawhid at great length. In the pure
science of epistemology are the immortal works of Ibn al-Arabi, Imam Ghazzali,
Shurawardi and Fakhruddin Razi. In the worldy matters applying the tenets
Fakhruddin Razi was an original thinker on human fulfilment and social
well-being and he made this idea epistemologically revolve around the precept
of obedience to God. Thus he thought early in those years of the precept of
self-actualization through a combination of self-reformation and appropriate
socioeconomic choices. Ibn Taimiyyah and Imam Shatibi were great exponents of
the social organization of well-being and market guidance. They considered
issues such as of money, finance, preference formation under guidance of public
policy, selection of means to attain public purpose, social contractarian and the
market order, the proper structuring of social institutions and control of
inflation. In recent years, Malik Ben Nabi wrote on the precept of Qur’anic
historicism in scientific endeavour. He called upon Muslims to discover science
in the Qur’an and take that scientific methodology to the highest level of
analytical discourse and investigation. Thus in all we have amply significant
historical legacy to confirm the place of the epistemology of Divine Unity that
is at the same time backed up by pragmatism through the guidance of the Prophet
Muhammad, what came to be known as the Sunnah.
Conclusion
The
analytical and historical basis of the epistemological of Divine Unity and its
crystallization in real world-system has left an abiding legacy for generations
to look up to and discover new and fresh paths to the answer of unification of
knowledge. Increasingly as the scientific endeavour marches on in answering the
initial conditions of the universe, it has become inextricably gripped in the
research project of theories of everything. In social theory as in scientific
paradigm, the praxis of unifying the premise of markets with institutions or
polity. The endogenous theory of institutions is notable in this area of
economic research of institutionalism. However, to date the research has either
taken up a purely historistic perspective or has been entrenched in
neoclassical roots of new institutionalism and social choice theory. This has
not helped in the critical study of discourse within market interactions. The
invisible hands of the market order has remained hidden in explaining the
preference formation of institutions, agents and markets.
The
theory of unity of knowledge premised on the Unity of God and its
externalization through concrete laws in the real world-system. This appears
not to be a problem of religious fundamentalism. Rather it is a matter of deep
analytical reasoning on which a substantive theory of science and society can
be premised. Thus when we write on the need for unification of knowledge we
treat the relevance of unity of knowledge on solid topological domains of
functional relationism of Unity of Knowledge with unification of knowledge in
endogenous world-systems. This precept of knowledge-induced endogenous
world-system and its extensively complementary explanation holds a prospect of
great significance in the scientific research program of theories of
everything.
References
Pollock,
J.L. 1990. Nomic Probability and the Foundations of Induction (Oxford,
Eng: Oxford University Press) Chap. 2.
Vanmarcke,
E. 1988. Random Fields: Analysis and Synthesis (Cambridge, MA: The MIT
Press) Chap. 2.