1
In: Sulis, W., and Trofimova, I. (2001). Nonlinear Dynamics in the Life and Social Sciences, pp. 133-152.
Amsterdam: IOS Press (NATO Science Series, Vol. 320).
Emergence, Radical Novelty, and the
Philosophy of Mathematics
Jeffrey Goldstein, Ph.D.
Adelphi University
Garden City, NY 11530 USA
goldstei@adelphi.edu
Introduction: Emergence Eluding Traditional Constructs
Complex systems research has sparked a resurgence of interest in emergence-- a
resurgence since the movement called Emergent Evolutionism in the 1920’s and 30’s
considered emergence as a fundamental principle of nature that could explain puzzling
aspects of evolution while steering a mid-course between mechanism and vitalism [1,3].
Although this movement died-out within two decades of its birth, the idea of emergence
occasionally found its way into the philosophies of biology and neuroscience [2]. Now, as a
result of research in Artificial Life, Neural Nets, Genetic Programming, and similar areas,
emergence has re-emerged, but this time within a more rigorous scientific and mathematical
setting [4-10]. Several avenues of current research, notably models from Nonlinear
Dynamical Systems Theory (NDS) and computational simulations from Artificial Life, have
decidedly enriched our understanding of emergence. Yet, emergence has remained an
elusive concept, primarily due to the lack of suitable constructs for investigating structure
and patterns [4, 6, Anderson cited in 11]. Not at all helping this situation is the fact that
emergence carries conceptual baggage having to do with causality, determinism,
reductionism, and so on [12].
Consequently, I think the time is right for a fresh look at emergence. We will be
developing a model of emergence by taking very seriously the claims made for emergent
phenomena, backing-in, so to speak, from the characteristics of emergents to consider what
kind of processes are capable of bringing them about. To ensure the freshness of our
perspective, we will need to uncover the assumptions of the constructs dominating current
research. Hence, we’ll be making forays into the foundations of the constructs, an area of
inquiry usually included under the rubric of the Philosophy of Mathematics. At the same,
time I will show how the study of emergence, in turn, may offer new insights in the
Philosophy of Mathematics as well.
A Black Box Model of Emergence
To provide a terminological and notational underpinning for our inquiry, let's start
with a simple black box diagram that illustrates claims made by proponents of emergentism.
Idiosyncratic elements in the notation will be explained below. Also note that this diagram
is only meant to focus on one “unit” of emergence, a somewhat arbitrary demarcation, but at
least a place to begin. And, of course, because the diagram is only presenting the bare bones
of this one unit, it shouldn’t be taken too literally.
2
The Emergent Bases are the lower-(micro) level components {’s} out of which
Emergent Phenomena {’s} arise. The superscript i represents the lower level, whereas the
superscript i+1 denotes the new emergent level [13].
Black Box of Emergence
Emergent Bases
Emergent Phenomena
( , ,... ), t 
i
1
i
2
i
n
Properties
( P , P
i
1
i
2
,...P ni ), t k
(
i 1
1
k


(P
,  2i 1 ,... ni1 ),t l 
Properties
i 1
i 1
i 1
, P
,...P
), t
1
2
n
l


The {’s} have their own Properties {P’s}at level i, whereas the {’s} have a
different set of Properties {P’s} at the new level i+1. The level distinction also represents
the coherence/correlation characteristic of the {’s} since a coherent pattern implies a
higher class than the objects {’s} which are cohering. Furthermore, tk and tl are dynamical
in the sense of evolving over time. Hence, emergence is distinct from pre-given “wholes”
such as universals, "gestalts", or Aristotle's "whole before the parts" [14].
We need to clarify which “emergence” our black box model is meant to represent.
First, we're not including "emergent properties" in the sense of unanticipated effects arising
from the interaction of factors, e.g., new effects resulting from the interaction of medicines.
There are two insufficiencies about this type of emergence: it leaves out {’s} and {’s};
and, I believe "emergent" in this sense is only a sign for the lack of a detailed enough
taxonomy of factors and a thorough enough statistical analysis. Moreover, our emergence is
more than simply a systemic property arising out of the collective behavior of elements, e.g.,
temperature arising from the collective motion of molecules or collective modes of

excitation like sound waves [15]. ( P i  1, P i  1,...P i  1), t
1
2
n
l
 is more than what happens
when {’s} are somehow together in a collective . The emergence in our black box will
need to include, therefore, both of Luc Steele's [16] first and second order emergence: that
which is not explicitly programmed in plus behavior that confers additional functionality on
a system. Also, our emergence includes both the weak (simulations of A-Life) and strong
(early emergentist views) forms of emergence posited by Bedau [17], a subject to which
we’ll return later.
The notation  is the “radical novelty operator” indicating the claim that emergent
phenomena and their properties are radically novel with respect to the properties of their
emergent bases. That is,  points to the processes taking place inside the black box that
enables the radical novelty to come forth. The Greek letter  (nu) is used since it is a
homonym to the English “new”; the arrow-like  indicates several things: first, the temporal
directionality of emergence; second, the related allegation for the irreversibility of
emergence; and third, the associated claim that {’s} are irreducible to {’s}. I am
explicitly using this unknown notation “” since I don't want to bias our understanding of
what happens inside the black box by associating it with such mathematical constructs as
functions, mappings, binary operations, relations, and so forth. As Humphries [13] has
pointed out, whatever it is that processes of emergence may consist in, they will not be
merely logical or mathematical operations like set formation, conjunction or disjunction
although such operations may be descriptive or used in parts of the mathematical
3
formulation. We can however, use the notation of mapping functions to describe the
multiple realizability property of emergence, i.e., the idea that the same emergent
phenomena may be realized with different emergent bases: f :    , where f is surjective
but not injective.
Radical Novelty
Since {’s} emerge out of {’s}, they must share certain properties, e.g., both the
{’s} and the {’s} of cellular automata consist of cells. Yet, the “life” suffix of Artificial
Life refers to the radically novel capacity of {’s} at tl for autonomy, self-replication, and
integration with other life-like entities not found in {’s} at , tk . This “radical novelty”,
then, means a difference in kind, not just degree. This requires more explication.
Novelty versus Radical Novelty: The radical novelty of emergence is distinct from
the newness accompanying any kind of change. In his critique of early emergentistism,
Baylis (cited in [1]) discussed how the mere moving of a book from one shelf to another
introduces both an integrational and a disintegrational novelty depending on point of view: a
new integration for the new pattern of books created, or a new disintegration for the
disarrangement of the previous pattern. Since Baylis thought it was arbitrary as to which
type of change emergent novelty was to be assigned, the whole idea of novelty in relation to
emergence was suspect. Similarly, Stace [18] argued that every time there was a causal
sequence, focus could be either on the way the effect is novel or on the way the effect is still
like the cause. Emergence then becomes an arbitrary designation: emergentists being those
who chose to emphasize the novelty of emergents-- "There is nothing new in novelty!" [19,
270].
The criticisms of Baylis and Stace, however, can be turned around to erect a hurdle
that emergent novelty must cross in order to be considered radically novel. If novelty can
cross the hurdle, it is emergent, radical novelty. Examples of what we could use for these
hurdles include Kilmister’s [20] “discrimination operation” and Crutchfield’s [4] “inductive
leaps” into novel classes of automata, both of which we’ll be discussing later on. The
difference between ordinary and radical novelty is also shown in the arising of the new,

emergent, macro level i+1. The claim is that ( P i  1 , P i  1 ,...P i  1 ), t
1
2
n
l
 are so radically
novel they demand a new set of laws, principles, and constructs appropriate to that level.
Inside the Black Box?
Because of the claim for radical novelty of emergents, emergentists are faced with the
dilemma: how can the radically novel at a higher level be generated out of components at a
lower level? In respect to our black box model, this dilemma means that whatever is taking
place inside the box must account for the radical transformation in qualities taking place in
the system. The dilemma then becomes the question:
EmQ: What must processes of emergence consist in for them to possess the kind of
potency that can bring forth the radically new properties of emergent phenomena?
This is really the question of what  is.
Stridently anti-reductionist proponents of emergence reject this question outright on
the grounds that the very nature of emergence precludes an answer to it [21]. However, I
believe that headway can be made on this question without falling prey to the pitfalls of
crude reductions by following three principles:
4

1.The more we know about ( P i  1, P i  1,...P i  1), t
1
2
n
l
 particularly as they differ
from ( P1i , P 2i ,...P ni ), tk , the more we can learn about the capabilities required of the
black box to bring about the former from the latter. This principle is similar to what guides
reverse engineering-- we will use emergent properties to “back-in” to the contents of the
black box;
(P
i 1
i 1
i 1
, P
,...P
), t
1
2
n
l

2. The more we learn about
,
the more we can
characterize the ways and means of the black box, e.g., by ruling out certain candidates. I
call this second principle the “homeoental” principle meaning “sharing the same elements”


since it suggests that the ( P i  1, P i  1,...P i  1), t reflect in some measure what has to be
1
2
n
l
happening inside the black box. Our strategy here is analogous to Tononi’s and Edelman’s
[22] reframing of the relation between consciousness and brain activity. Instead of arguing
whether a particular brain area or group of neurons contributes to consciousness, they
characterize the kinds of neural properties that might account for key properties of
conscious experience and then look for neural processes that could generate these
properties;
3. The more we can disentangle dilemmas about emergence, the more we can learn


about the precise nature of ( P i  1, P i  1,...P i  1), t . Here we will try to turn dilemmas
1
2
n
l
into lemmas.
With our black box model in hand, let’s begin examining the major current contenders in
the study of emergence.
Emergence and Nonlinear Dynamical Systems
The Dynamical Prototype of Emergence
The prototypical example used in the Nonlinear Dynamical Systems Theory (NDS)
model of emergence is the well-known logistic map whose evolution exhibits the following
facets of the model [23]:
1. The dynamical nature of emergence can be understood in terms of the sequence of
bifurcations into new attractor(s);
2. The novel properties of emergent phenomena can be represented by the shift in the
qualitative dynamics of new attractors in phase space;
3. The new emergent level can be seen in the “macro” status of attractors in relation
to the lower level of the solutions of the equation themselves;
4. The black box itself, i.e., the processes leading from {’s} to {’s}, can be
understood in terms of three factors: criticalization of the values of the control parameters;
instability of attractors at bifurcation points; and the mathematical operation of functional
iteration notated by Feigenbaum [24] as:
x n  f ( f (... f ( f (x 0 ))... ))  f n (x o ), where n is the total number of applications of f
and fn(x0) is not the nth power of f(x), it is the nth iterate of f.
5
The NDS model has, indeed, several things going for it. First, the discontinuity
between {’s} and {’s} is represented by bifurcations which seems a better way for
understanding the discontinuous nature of radical novelty than “older” functions exhibiting
discontinuity, e.g., Dirichlet’s famous function:
f  (x ,y), x  R,[0,1], {
y 1 when x Q
y  0 w hen x Q
Dirichlet's function has discontinuity "programmed" into it whereas the discontinuity
shown in NDS models emerges. Second, functional iteration involves recursiveness which
for example, corresponds to the “updating” procedure of cellular automata. Third, the very
simplicity of the model aids in computational simulations.
Limitations of the NDS Model
This very simplicity of the NDS model, though, in my opinion, betrays the questionbegging circularity of the model: the model equates emergence with the evolution of a
dynamical system via a bifurcation into new attractors: then it simply examines the
evolution of a dynamical system for insight into emergence. This is a ploy (it hoodwinked
me too) that, borrowing a phrase from Bertrand Russell, has all the advantages of theft over
honest toil! For in the wake of NDS’ simple model we are left hanging with critical
questions about emergence: What can NDS tell us about the structural and pattern
transformation seen in emergence? What are the cognates of the NDS model to cases of
emergence other than the evolution of a purely mathematical dynamical system? What does
the “qualitative” in qualitative dynamics really amount to when it comes to radical novelty?
What conceptual commitments are connected to the control parameter-driving picture of
emergence? What exactly would functional iteration correspond to in a naturally occurring
case of emergence? There is nothing in NDS itself that can answer these questions. But
that's only the beginning of its problems.
Bifurcation, Discontinuity, and Quantity into Quality:
The NDS model specifically pinpoints the discontinuity of emergence with its
construct of critical parameter(s) values where a bifurcation occurs. There is a problem,
though, with this very specificity. Imagine the control parameter of the logistic map, , able
to be slowly cranked-up by a dial. With a dial calibrated at a resolution of 0.2, as we turn
change  from 3.5 to 3.7, we see rapid period-doubling bifurcations and then chaos. But
what if we could increasingly fine tune the dial to 0.02, then 0.002, then 0.0002, and so on?
Since the parameter value is an irrational number we would indefinitely find it harder and
harder to discern a discontinuity between before and after "pictures" at each turn of the dial.
The more fine-tuned the dial, the less sudden and discontinuous emergence would be. The
observation of the discontinuity of radical novelty then would be a function of the coarsegraining in how the parameter is “tuned”. I’m not trying to introduce Zeno-like paradoxes to
the study of emergence, I’m simply trying to point out how, from an NDS vantage point, the
recognition of emergence is a matter of how fine tuned our ability is in observing the
system. If, however, we operationally adopt a rule of thumb based on Ernst Nagel's quip that
just because there’s no line down the side of your head doesn't mean there’s no difference
between your face and the back of your head, we are departing from the parameter-driven
picture of emergence offered by the NDS model. Furthermore, the specificity of emergent
discontinuity in the NDS model ignores the transient dynamics that are a crucial aspect of
6
emergence [25]. Indeed, since NDS typically ignores transients (they are the homeless of
NDS, Sulis [10] has developed a formulation for transient dynamics.
Related to the above problems is how NDS understands the novelty of emergence in
terms of the new qualitative dynamics of the new attractor(s). (Engels would indeed be
delighted by such a validation of his concept of the transformation of quantity into quality.)
In my opinion, this is an exceedingly paltry notion of novelty, e.g., consider a bifurcation
from a period-2 attractor to a period-4. This is an example of NDS’ qualitative change, yet
how much novelty is there really here? Of course, instead of period-doubling we could turn
to bifurcation into chaos as a model of radical novelty. But in this case, I can bring up the
question again of where exactly does this bifurcation into chaos happen? Only after an
infinite number of period-doublings! And it’s also the case that the kinds of emergence seen
in A-Life that have prompted so much interest in emergence do not appear particularly wellsuited to chaos.
Phase Space and Emergence
Fontana and Buss [26] admonish the NDS model for neglecting structural change by
the tracking in phase space only the magnitudes of properties abstracted from objects, e.g.,
velocity or concentration (of a chemical). The rates of change of these properties are
modeled as differential equations so that phase space displays portraits of the evolution of
these equations. I want to go out on a limb here and question the whole construct of phase
space as being that helpful to emergence. But we need a bit of history first.
One of the first important uses of phase space goes back to Gibb's kinetic theory of
gases [27] where he used it to collapse the immense number of degrees of freedom needed
to study gases. He proceeded indirectly by comparing phase portraits of similar gases
resulting in a theorem on the “incompressibility” of phase space volumes that enabled him
to feel justified with a statistical approach. Right from its inception, then, we can see that
phase space was used to depict the evolution of something with the least structure of matter,
a gas!
As is well known, Poincare created a global analysis of phase space to investigate
entire families of differential equations for stability properties. This was the birth of both
modern NDS as well as topology. Smale [28], for instance, defined a phase portrait as a
topological equivalence class of differential equations on an n-dimensional manifold. The
global study of these phase portraits is what is now termed qualitative dynamics. The place
of topology here, however, is very telling since topology is concerned with notions like
sidedness which it can only get to by necessarily leaving out most of the rich structure of
what it’s studying.
Fontana and Buss [26] attempt to go beyond these limitation of NDS’ phase space by
positing an “object space” instead which would depict construction relations formalized via
the lambda calculus, even extending their constructional "artificial chemistry" by employing
proof theory (indicating the need to go deeper into the foundations of mathematics). Action
between the molecules, forming their “chemistry”, is "parametrized" by structure, a
"derivative" in object space then providing information about change of object action
resulting from change in object structure. They admit, however, that serious questions
remain, about their "object space": What is the motion in "object space" induced by object
constructors? Is there a meaningful concept of trajectory in "object space"? Is there a useful
definition of "distance" between "attractors" (in their case, algebraic structures in lambda
space)? In my opinion, these are major issues whose lack of answers cast doubt on the
whole idea of substituting “object” for “phase” space.
7
Computational Models of Emergence
Crutchfield’s “Calculi of Emergence”
Crutchfield and company [4,5] have been developing a "computational mechanics" to
provide a "calculi” of emergence. In this approach, emergence is seen as endowing a
complex system with an "intrinsic computational capacity", the intrinsic quality of which is
supposed to ensure against subjective bias in detecting emergent structure. Emergent
phenomena are identified by reconstructing the least complex type of finite automata, “machines”, which adequately captures the complexity of emergent phenomena. Somewhat
analogous to the taxonomy of attractor types in dynamical systems, -machines are arranged
in a quasi-hierarchy of increasing complexity (“quasi” because there are only partial
orderings). Emergence is recognized by these machine reconstructions as a process of
"inductive" leaps of innovation from one level of complexity to a higher one. Moving up the
hierarchy happens when regularities are detected in a series of increasingly accurate models
that the machine makes of the data stream from the emergent phenomenon. An “inductive”
jump is occasioned when those regularities are taken as new representations, i.e., a new
model is formed. The key step of innovating a novel model class is the discovery of new
equivalence relations through the detection of new structure by grouping lower-level states
into equivalence classes of the same future “morph”.
Computational mechanics measures the complexity of emergent structure by
"statistical complexity" which Crutchfield sees remedying the domination of the usual
Kolmogorov-Chaitin measure by randomness, not structure. Not surprisingly, given its NDS
foundation, -machine reconstruction is tried-out on identifying emergence in the period
doubling sequence to chaos of the logistic map. The machine reconstruction shows the
emergence of chaos requires a switch to a more complex automaton, specifically, a “oneway nondeterministic stack automaton.”
Computational mechanics, in my estimation, exhibits an egregious circularity
reminiscent to that of NDS: emergent phenomena are observed in computer simulations;
this emergence is then conceived as bestowing an additional computational functionality on
the system; this additional computational capacity is next identified by computational
reconstructions; these reconstructions are interpreted according to their place in a quasihierarchy of computational devices; finally, the specific -machine required is supposed to
reveal significant structure in the simulation. As Wolpert and Macready [29, p. 626] have
pointed out, "...before a model-driven approach can be used to assign a complexity to a
system, one must already fully understand the system (to the point that the system is
formally encapsulated in terms of one's model class). So only once most of the work in
analyzing the system has already been done can one investigate that system using these
proposed measures of complexity". If everything is computational, then, of course, the
model class can encapsulate the emergence. But, one consequence would be that types of
emergent phenomena are limited to types of automata, which seems a very strange
conclusion. We’ll be returning to more problems with the computational mechanics model
of emergence later. For now, I want to go back to radical novelty and its place in the
philosophy of mathematics in order to provide a way to think about emergence that may be
able to surmount some of the difficulties with both the NDS and computational mechanics
models.
8
The Generation of Radical Novelty
The Bias Against Novelty
The emergentist claim for the possibility of irreducible novelty has placed it in the
path of a powerful current in Western thought which I am calling the Bias Against Novelty
(BAN) for its repudiation of the possibility of any natural process having the capacity for
radical novelty generation. This BAN has an ancient, even holy history. Thus, we can see it
expressed in two forms in the Bible. The radical novelty involved in the creation of the
world, an account brimming over with increasing complexity and emergence, is the result of
Divine action alone, definitely not a natural process. Then, there is the desparing utterances
of Ecclesiastes, "Is there a thing of which it is said, 'See, this is new?' It has been already, in
the ages before us" (verses 9, 10). The philosophical tradition in ancient Greece didn't fare
much better with novelty. Anaxagoras [30], for example, postulated that every new thing
was in fact already present in that out of which it arose in the form of "seeds" that are merely
rearranged. Anaxagorian tendencies can be found not only in the 17th Century doctrine of
preformationism but also in today's strict genetic determinism. According to Bunge [31],
mechanistic philosophies did not invent the theory of change without novelty, they just
adopted, reinforced, and rationalized the picture of change as a circulation of a limited stock
of forms. Mechanistic approaches can be seen as always flirting with BAN.
Scholastic thought incorporated the BAN into several precepts concerning causality,
e.g., the Thomist precepts "causa aequat effectum" (effects are equal to their causes) and "ex
nihilo nihil fit" (out of nothing, nothing can come) were combined to express the general
conviction there could be nothing in the effect that had not already been in the cause [31].
Such ideas were accepted as self-evident by two of the luminaries of modern mathematics,
Descartes and Leibniz. According to Bunge [31], this viewpoint provided a framework for
the laws of conservation. Indeed, the great mathematician of conservative systems, William
Rowan Hamilton, wrote that there was a tautology of cause to effect and that, "We think the
causes to contain all that is contained in the effect, the effect to contain nothing which was
not contained in the cause...that all that we at present know as an effect must previously
have existed in its causes" (quoted in [31, p. 210). Lest we think that the 20th century finally
discarded the BAN, I present a final example in Bertrand Russell's [61] principle of a
"structural invariance" between cause and effect. In fact, the extreme reductionist drive to
reduce everything to one fundamental level has been so decidedly anti-novelty, it provoked
Philip Anderson's [32] well-known Constructionist Hypothesis as a retort: the ability to
reduce everything to simple fundamental laws does not imply the ability to start from those
laws and reconstruct the universe.
A Method for Generating Radical Novelty
In this section I want to take on the BAN directly by presenting a method that does
just what BAN says can't happen: bring forth radical novelty. To be sure, this method is a
purely mathematical one: Cantor's anti-diagonal construction to prove the uncountability of
the set of real numbers (i.e., the lack of a one-to-one isomorphism between the natural or
counting numbers and the real numbers). But its transparent nature can provide hints as to
what's needed, in general, for the generation of radical novelty.
In his famous proof, Cantor [33] first assumed that any arbitrary infinite set of
numbers was denumerable, i.e., isomorphic to the natural numbers. Next, he showed how to
list the numbers of this set in a countable fashion. From certain simple operations performed
on this list, Cantor then constructed a new number that could not, by the rules of his
9
construction, be included on the original list. The existence of such a number, therefore,
contradicted the original premise that any set of reals was countable.
Since Cantor’s proof relied on a way to generate a radical novel number, let’s look at
his method more closely to see what it reveals about generating novelty. First, he introduced
the premise:
Let E1, E2, ..., Ev,... be any countable set of elements such as
E1  ( a1,1 , a1, 2 ,... a1,v ...),
E 2  ( a 2,1 , a 2, 2 ,... a 2,v ...),
E u  ( a u,1 , a u, 2 ,... a u,v ...).
The subscripts indicate how the elements of a set E are to be listed, i.e., denumerated.
In typical graphical illustrations of Cantor's proof, the subscript u refers to the index number
listing the elements of E listed in a vertical descent, whereas v refers to the horizontal index
of the decimal expansion of the number. Cantor then constructed a new number sequence,
b1, b2,… bv,… by making bv different than av,v at each step of the constructed sequence.
Again in the typical illustration of Cantor's proof, this new number b1, b2,… bv,… is
constructed as the diagonal sequence, where the vertical and horizontal index numbers are
the same. Hofstadter [34] pointed to the self-referential character of this mapping since the
same integer is used on two different levels (u and v or horizontal and vertical). This
diagonal sequence is then made into an “anti-diagonal” by changing each number digit by
digit.
Cantor's construction, therefore, included an ongoing twofold operation: first map the
u and v onto one another to generate a diagonal; next follow the diagonal and change (i.e.,
negate) each instantiation of this diagonal sequence. It is these operations which make the
new number b1, b2,… bv,… radically unique in relation to the countable list E . The coup de
grace of Cantor’s proof is that this new number cannot be included in any of the sequences
making up {E....} since it was constructed to always differ from any number in the original
list. Thus, the original premise, that any set of infinite numbers could be counted, is not
true—there are uncountable sets.
Here, I'm not interested in the transfinite implications of Cantor's method but how his
relatively simple method could be so powerful in generating radical novelty (Hellerstein
[35] sarcastically quips: has anything more ever been gotten by anything less?). Even critics
of Cantor's transfinite hierarchy like Poincare and Wittgenstein accepted that Cantor’s
method had shown how to generate a new number that was not able to be included in the
original list. Thus, we can consider Cantor's method a species of a  radical novelty
generation operator, one that was explicitly constructed to be just that.
Radical Novelty, Self-transcending Constructions, and -Structionings
Cantor's anti-diagonal method was not universally embraced. One later critic, Felix
Kauffman [36], a philosopher associated with the influential Vienna Circle, considered
Cantor's method an example of "self-transcending constructions" which should be barred
from mathematics since "...no construction can ever lead beyond the domain determined by
the principle underlying it" (p. 136). What Kauffman the strict finitist was really after was
Cantor’s transfinite sets which he could discount by denigrating the method that Cantor used
as a portal into the uncountable. I think, however, Kauffman’s term is particularly apt for
our purposes since it describes very well the self-transcending, i.e., radical novelty
10
generating capability of . Also, the term “self-transcending construction” points to
another pertinent feature of Cantor’s anti-diagonal construction: it can be applied repeatedly
like functional iteration, e.g., a new list which now includes a set of all countable numbers
with the addition of Cantor’s new number can itself be first diagonalized and then negated
resulting in yet another radically new number.
It is important to recognize that Cantor’s constructions were not piecemeal or step-bystep operations, a fact recognized by Paul Cohen [37], famous for his work on Cantor’s
Continuum Hypothesis. Cantor was, in fact, driven to such self-transcending methods when
he realized the new number class he was seeking could not be achieved by the continual,
stepwise formation of Cauchy sequences which would never lead outside the denumerable
[38].
To denote his transfinite sets Cantor used the Hebrew letter  (aleph), presumably
because both the Greek alphabet had been pretty much exhausted by mathematicians as well
as the connotations of  having to do with his mystical speculations about infinity.
Following on Cantor’s lead and to mark an association with his method, I am suggesting
that we denote “self-transcending” with the Hebrew letter  (Nun) because of its
connotations of productiveness, growth, transformation, and variation. I also want to append
 with the suffix “Structionings” to indicate the constructional nature of self-transcending
constructions. I am using this neologism “structionings” and not “structure” or
“construction” to indicate three things: one is the ongoing nature of the activity whereas
”structure” is typically static; two is the fact that “constructions” per se are not typically selftranscending; and, three, the plural form of the term alludes to the fact that Cantor’s method
is just one example of such self-transcending constructions. Other self-transcending
methods can be found in the emergence-generating “rules” used in Artificial Life, what
Langton [8] terms the “recursively generated” procedures of Holland's genetic operators,
Lindenmayer systems, Tierra’s evolving ecology, and so forth. But it needs to be pointed out
that self-transcending constructions must involve something in addition to recursion: they
require an element of negation that allows a space for novelty in the recursively generated
objects. Holland’s crossover does this through explicit genetic recombination and mutation.
We’ll come back to these links between self-transcending constructions and emergence very
shortly.
-Structionings and the Philosophy of Mathematics
We can use Rene Thom’s [39] lay-out to show where -Structionings might be
located in modern mathematics. On the vertical axis, Thom places “generativity” going from
“free” through “bound” to “constrained”; the horizontal axis goes from “discrete” to
“continuous”. The formation of sets falls under a “discrete” “free generativity”-- objects
within this region are usually of little direct interest although they are indispensable for
other fields. “Continuous, bound generativity” would correspond to pre-existing algebraic or
formal relations, e.g. differential equations, whereas, “continuous constrained generativity”
points to such mathematical objects as partial differential equations solved through the use
of boundary or initial conditions.
Thom then distinguishes between “beautiful” and “ugly” mathematical objects: finite
simple groups are “beautiful” since the mathematical structure itself imposes its own
limiting conditions as to what can be generated. “Ugly” objects are where very little can be
said and where singularities, accidents, the unforeseeable and the undecidable reign, e.g. the
continuum as well as compact topological manifolds of large dimension. For Thom, the
future growth of mathematics will be in the frontier branch where the beautiful meet the
ugly.
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It is here, at this frontier, where, for four reasons, I would place self-transcending
constructions. First, Cantor’s anti-diagonal, self-transcending constructional method was
instrumental in the foundational work on undecidability, inconsistency, and noncomputability by Hilbert, Godel, Turing, Kleene, and others. In that way self-transcending
constructions touch on the verge of Thom’s “ugly” objects. It is critical to note here that
while Rosen [21] and others have used an “argument from Godel” to claim emergence is not
formalizable, Webb [40] has persuasively argued how the foundational work of Godel and
Turing actually supports a formalization of the self-transcending construction of
diagonalization. Therefore, by placing -structionings at this frontier we are claiming
emergence via -structionings is formalizable but not formalizable in the way
formalizability itself has formerly been conceived.
Second, -Structionings introduce unforeseeable, radical novelty requiring the
establishment of new laws, principles, and implications. Indeed, -structionings thrust right
up against the BAN, again skirting Thom’s “ugly” region. Third, -Structionings in the
form of cellular automata penetrate into the domain of accidents and the unforeseeable since
they may utilize randomization and require simulations to observe their evolution. Fourth, I
want to include the particularly salient constructional feature of -Structionings, that,
although not formally in Thom’s considerations, speaks to them nevertheless. This
constructional feature, since it focuses on the construction process itself and not on
implications concerning the ontological status of the resulting construtced objects, allows Structionings to cut across Platonism, Formalism, Constructivism, Intuitionism, Realism,
Anti-realism, Structuralism or whatever other school in the philosophy of mathematics. Structionings take us to a strange paradoxical world where mechanism and creativity meet
[41], a more organic, evolving type of mathematics that emerges where Thom’s “beautiful”
objects meets “the” ugly ones. It is at this edge where research into emergence may provide
inspiration for future mathematics: e.g., the melding of group theory for emergent
invariances with measures of complexity and randomness emerging from the unfathomable
richness and novelty of the continuum!
Inside the Black Box: -Structionings and Emergence
If self-transcending constructions could accomplish all they did for abstract
magnitudes like cardinality, then, how much more could they do for emergence which
exhibits a far greater richness than magnitude alone. What I’m suggesting is that Structionings are what goes inside the black box as the radical novelty operator . But it
must be asked what is accomplished by doing this beyond merely substituting a
Hebrew/English expression for a Greek/symbolic notation that was used for an English
phrase? Here, we don’t want to fall into the inanity expressed by one of Moliere’s physican
characters who explained the efficacy of sleep medicines by appealing to their dormative
powers! So, let’s explore where the construct of -Structionings takes in relation to
emergence.
-Structionings Are More than Increasing Abstraction
First of all, -Structionings take us beyond conceiving the black box as processes
which create novelty through increasing abstraction. This notion of emergence brought
about through some sort of increasing abstraction is related to the idea of hierarchical levels
associated with emergence. In fact, according to Herbert Simon [42], complex systems that
are not hierarchical and decomposable largely escape our observation and understanding.
Yet, it is also true that looking at a system in a hierarchical fashion in large measure depends
12
on the observer's interest in the system [43]. One interest that seems to be common among
hierarchically-tinged theories is to think of emergence as a kind of cumulative stratification
of abstraction: emergence is supposed to be a process of going from objects… to patterns of
objects… to objects made-up of patterns of objects…. to patterns of objects made-up of
patterns of objects…and onward and upward. These increasing levels of abstraction parallel
Theories of Logical Types which erect increasingly abstract hierarchies of sets and classes to
avoid paradoxes like that of sets that have themselves as member. But they accomplish this
avoidance only too well since they thereby eliminate the “strange loops” or self-referential
entanglements that Hofstadter [34] has described. Since this self-referential entanglement is,
as we’ll discuss a little later, a crucial characteristic of emergence, understanding emergence
within the framework of increasing levels of abstraction which shun level entanglement is
destructive of what is being investigated.
This notion of emergence occurring via processes of increasing abstraction can also
be found in Crutchfield’s [4] quasi-hierarchy of -machines, this hierarchy reflecting the
levels formed by emergent "inductive leaps" of innovation. A shift upwards in the hierarchy
of -machines occurs when new equivalency classes are discovered in increasingly accurate
models. These recognized regularities are then transferred into the elements of a new model
class. “Inductive” is the appropriate word if, indeed, the basis of -machine reconstruction is
the recognition of regularities.
But induction through the recognition of regularities tends to leave behind a great
deal of structural richness, a fact recognized at least as far back as John Herschel [44]. The
impoverishment of induction for innovative scientific theorizing was also noticed by C.S.
Pierce [44] who felt the need to come-up with an alternative construct, “abduction”, to refer
to the creative formation of new hypotheses. Advances in the sciences and mathematics are
simply not the result of mere induction. Thom [39], thus, points out that mathematical
progress has consisted of the construction of new objects through a liberation from that
"which intuition had previously suppressed". It is unclear to me how induction alone could
transcend this suppression. In addition, Hintikka [45] has indicated that it is only the
deductive side of mathematics as found in proofs which corresponds to climbing up a
hierarchy of abstraction.
The role of increasing abstraction by way of induction is also found in computational
mechanics’ “inductive leaps” taking place at the discovery of new equivalency relations via
the recognition of regularities. Kilmister’s [20] model also uses equivalency classes to
determine novelty: values of the function for elements equivalent to already generated
elements lie in a fixed set which is disjoint from the set of values for truly new elements. It
seems to me, however, that the use of equivalency classes is problematic. Consider, e.g., the
equivalency relation: aRb (a  b mod 4; a  A; A  Z ) (a is congruent to b modulo 4). The
equivalency classes for this relation are {...-8, -4, 0, 4, 8, ...}, {...-7, -3, 1, 5, 9, ....}, {...-6, 2, 2, 6, 10....}, {...-5, -1, 3, 7, 11, ..}. Since equivalency is a synonym for sameness which is
the antithesis of novelty, these sets are not novel in relation to each other. Of course, what
the equivalence relation does, in fact, is to define in particular situations what is to count as
the same and, and, by extrapolition, what is novel. By grouping together what the
equivalence relation establishes as the same, regularities are recognized and put into a
higher-level class. This is more of the same inductive process of increasing abstraction. The
recognition of equivalency classes is a form of induction which might allow for the
discovery of a new class of -machine, but, as Crutchfield himself admits, this new class is
only a model of emergence, it doesn’t take us to emergence itself. In contrast, Structionings are about emergence itself, and here, at least, induction is not up to snuff.
13
-Structioning as Creative Processes: Emergence and Novelty
If emergence is about radical novelty generation, induction is not going to get you
there since it lacks the creative potency for generating radical novelty. If emergence is a
feature of nature, the profusion of radical novelty throughout nature calls for something very
different and more powerful than increasing abstraction as the basis of emergence. I am
proposing that we call this more powerful generative capacity “creativity” since it is of the
essence of creativity to bring about novelty. In fact, the field of creativity studies is the only
field where novelty generation itself has been the central concern. We are following two
principles here. First, human creativity cannot be less than nature's creativity since humans
are a part of nature. Consequently, from studies of radical novelty generation in human
creativity we can learn at least a little about nature's creative processes of emergence.
Second, as we’ll soon see from research, creative outcomes necessitate creative processes,
so that an investigation of creativity should reveal something about the processes leading to
more original creative outcomes, and we can apply aspects of these processes to our black
box. This is the “homoental” principle we mentioned above: creative outcomes demand
creative processes.
Emergent Cognitive Creativity: One of the most bountiful areas for studying creative
processes can be found in research into human cognition. Intriguingly, this area of research
has introduced the term “emergent” creativity to describe creative processes in which
novelty emerges along the way of the creative process. For example, imagining
combinations of images can result in the exhibition of emergent novel features not present
in the images by themselves [46]: imagine a creature with the heads and legs of an ostrich
but the body of a lion. Imaginatively exploring how this fantasy creature might walk, jump,
eat, hunt, or mate brings-out new and unanticipated properties. Such imaginative exercises
are essential for the generation of originality: "Because our images can only be formed
according to information that we have already acquired, how could they ever engender new
insights?" [46]. Emergent features can also arise not just through combination, but also
through the discarding of fantasized features, e.g., imaginatively removing a couple of legs
of a spider and visualizing the resulting creature and how it could function. In another
experiment [47], subjects were asked to visualize the letter "F", then imagine the letter "b"
hanging straight down off the vertical stem of the "F" so that the top of the "b" formed a
continuous line with the lower stem of the "F." Next, they were asked to visualize flipping
the loop of the "b" around to the other side. The resulting image is a musical note, but it is
emergent since the subjects did not anticipate the features through inspecting the imagined
parts alone. In yet another experiment, students were given a list of three figures to combine
in anyway without distortion of shape. The resulting combinations were then judged for
creativity. Outcomes included structures resembling a jellyfish, a speedometer, a bottle
rocket, a bubble, and a logo.
These novel outcomes were emergent in the sense that the students did not predict
them from just looking at the parts. It's as if all the possibilities of parts don't even come into
existence until they are combined with other parts in varied relations. In this way the whole
brings out more in the parts than the parts alone. This is another way to understand the
often-heard remark that emergent wholes are greater than the sum of their parts: they’re
greater since they help to bring out potencies of the parts that were not manifest before. This
could be taken for a version of macro-determination and feedback between levels, a subject
that we’ll soon be discussing.
A similar study was conducted dealing with the design of new tools using a randomly
selected set of parts [46]. An interesting finding was that when restrictions on the use of
parts were loosened, the results were rated as more, not less conventional. Restrictions
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enforce a kind of container for the creative processes that tends to intensify them—we’ll be
coming back shortly to this notion of the containment of novelty production. Although
creativity works best by some restriction on the elements to be used to generate the novel
outcome, premature categorization can impede the creative process. When students were not
given categories for their emergent patterns, they actually came-up with more inventive uses
since they were not constrained by conventional notions inherent in the categories. Instead
of function following form, here it is form following function [46]. Similarly, Wertheimer
[46] showed that setting subjects up with problems where past experience could be used
hindered their creative abilities in new situations..
Another creative process is to take an existing design, add a random feature to it and
then probe the potential implications. Bricolage and Ernst Mayr’s principle of “proximate
logic” for evolutionary advances come to mind here. For novelty to result, the combination
in the structure of the image must allow for unexpected features and relations and what
better way to do that than the serendipitous taking advantage of the random event. This
takes the idea of “what’s at hand” even further that what is immediately present to include
what becomes present during the process of creativity.
Homospatial Creativity: Getting back to the idea of fostering creativity through some kind
of containment, Rothenberg [48] compared various ways to visually stimulate artists, e.g.,
using slide projections of two scenes either next to one another on the screen or “laminated”
one on the other, the latter termed “homospatial” since the two slides occupied the same
space. He found that the most original art works were stimulated by very disparate, even
jarring, images shown homospatially, i.e, contained in the same space. Creative outcomes
were most original, i.e, most radically novel in our terms, when there were two things
present: homospatiality of the images and the most drastic disjunction between the images
in terms of content, mood, colors, pattern, theme, etc.
His research led Rothenberg to conclude that creative processes leading to originality
were not adequately understood in terms of a “synthesizing” operation in which some
essence is abstracted from two or more separate realms to generate a new, third one—that is,
thesis, antithesis, and synthesis. The jarring nature of the slides used in juxtapositon
mitigated against tendencies towards synthesis with the result that the artistic creations
could not be seen as syntheses of the contrasting slides. Synthesis is too much like induction
and increasing abstraction, whereas creativity evokes an originality that surpasses whatever
is already present, however abstractly, in the imagery of the slides.
-Structionings and Emergence
So what have we learned about emergence by our forays into radical novelty
generating self-transcending constructions, BAN and the philosophy of mathematics, the
limitations of abstraction, and the originality producing potency of creative processes? To
summarize, here’s what I think we’ve accomplished by considering emergence in terms of
-Structionings.
First, we’ve shown what’s either not in the black box at all, or is there in only a slight
degree: emergence is not dynamical bifurcations capturable in phase space, is not composed
of processes of increasing abstraction like induction or the recognition of new equivalency
classes, and is not adequately understood in terms of a process of synthesis. Instead,
emergence is constructional in which the emphasizes is on morpho-genesis and morphomaintenance and this constructional activity is self-transcending in its ability to generate
radical novelty. New levels arise not as higher classes of lower level regularities but as new
creative integrations with properties radically novel with respect to their components. I
15
believe that much further light can be shed on emergence by studying creative processes
than by conceptualizing emergence as the forming of more and more abstract higher classes.
The radical novelty of emergence comes about through creative processes that recombine,
alter relations between wholes and parts, take advantage of randomness serendipitously,
intensify under some kind of containment combination, and include some element of
negation operations that open a space for radical novelty.
Moreover, by focusing on processes of constructing, we can look at emergence across
the board, thus going beyond the artificial/natural distinction in relation to emergence, e.g.,
A-life versus Biology or Engineering versus Science. Langton [8] has classified A-Life
according to a movement from the artificial, i.e., Genetic Algorithms, to the natural, i.e.,
Tierra. To be sure, classifying Tierra as “natural” is certainly a stretch but I want to stay
within that same spirit by recognizing that “-Structionings” are self-transcending
constructions whether constructed by mathematicians, computer scientists, or nature. They
take current structures and transform them in a way that generates radical novelty. This can
be seen, for example, in Holland’s [6] theory of emergence by way of Constrained
Generating Procedures which recursively build-up emergent structure.
Let’s see how these insights can help us with the dilemmas involved with emergence.
-Structioning, Level Entanglement, Self-referentiality, and Paradox
In their algorithmic constructions, Fontana and Buss [26] take the property of selfmaintenance among sets of objects as a sign for emergent organization. They depict this
self-maintaining organization in terms of intricate interrelations among objects and levels of
objects. The circularity of this self-maintenance acts to form “boundaries” around the
organization. Similarly, Kauffman [49] places the emergence of life at the first
manifestation of autocatalytic sets out of a chemical "soup" of monomers -- two monomer
molecules join to form two dimer molecules, the latter acting as catalysts in the very
reaction that joins the two monomers, thus an autocatalytic loop. Self-maintenance and
autocatalysis are self-referential in the sense that they are recursive processes whose
outcome generates more of the components needed for the ongoing evolution. In other
words, the outcome is fed back into the operation like functional iteration in Feigenbaum’s
[24] sense.
Kauffman [49] also talks about the “boundary” creating effect of self-referentiality
when he cites the research of Farmer and Bagley on the way in which spatial compartments
enclosing an autocatalytic set enable the self-sustaining metabolic processes to increase the
amount of copies of each type of molecule in the system. When the amount has doubled, the
compartment can divide into two. Lemke [50] has developed a 3-level unit for
understanding emergence that is also couched in similar constructs as level mappings and
semiotic closure. We could keep listing similar examples, e.g., Rosen’s [21] (M/R)-systems,
Varela’s [56] self-referential logic of autopoiesis, and so on.
Furthermore, the claim of emergent phenomena having macro-determinative
influence on lower levels, the claim that Bedau’s distinction between weak and strong
emergence was meant to avoid, can be understood in terms of self-referentiality and level
entanglement. Wimsatt [51] talks of causal thickets associated with the difficulty of
unambiguously unambiguously localizing an entity on any one level and Humphries [13]
talks about arbitrariness as to where to draw a level distinction in emergent phenomena
since there is so much intercourse between levels. In emergence, upper levels influence
lower levels and lower levels influence upper levels, and there’s lateral interaction, and so
on. In the midst of this causal thicket, the feedback among components, macro- and micro-,
is so entrenched in what makes the system a complex system that macro-determination is
16
just another ingredient in the causal web. I think that we can trace the difficulties that the
neuroscientist Roger Sperry [52] ran into with his emergentist theory of the macrodetermination of mind on brain processes to his keeping mind and brain too apart
theoretically. Macro-determination only poses a conundrum when level distinctions are hard
and fast and non-arbitrary, but emergence is precisely that series of events which entangles
levels.
Indeed, “entanglement,” interaction”, “recursive,” “causal thickets” may all be too
tame when it comes to emergence. The intriguing thing here is that the very arising of the
new emergent level is characterized by just this transgression of the level distinction itself.
Emergence is involved with the production of a new emergent level of objects with the
radically novel property that they transgress their very assignment to this new distinct level.
The image I have of this characteristic of emergence is that of traveling on a bridge,
(1i ,2i ,...ni ), tk  }  (1i 1 ,2i 1 ,...ni1 ),t l  , but this bridge has the bizarre feature of
curling-up behind itself as the traveler proceeds to i+1. That is, once the higher level is
reached, the lower level has been subsumed into it so that lower level components are now
parts of this higher level coherence. The creative radical novelty of emergence is just this
level entanglement of the new emergent level.
Emergence then partakes of paradox, not just the paradox of the radically new
emerging in the first place, but the paradox that the new emergent level is characterized by
the transgression of level distinctions! Since I think that these are true paradoxes and not
products of linguistic inefficiency, emergence will require formulations that themselves
partake of paradox. The paradoxical nature of self-transcending constructions, that is, their
very ability to transcend themselves will require new “paraconsistent” logics like those
being developed by Nathaniel Hellerstein [35], Louis Kauffman [53], Ben Goertzel [54],
and N. da Costa [55]. As Hellerstein puts it, once the paradoxical logic is incorporated into a
basic building block, there is then the possibility of a productio ex absurdo which is really
an alternative way of expressing what we have been calling self-transcending constructions.
This is not identical to a pure bootstrapping phenomena like Goertzel's "magician" systems
which generate each other out of nothing. Instead, it is a self-transcending of previous
patterns by some kinds of  operators which operate on those patterns to bring about a
radically new pattern. Furthermore, paradoxical logics, because they are able to bring
together A and not A, establish a place for the negation that we saw was so critical in the
radical novelty of Cantor’s -Structioning. The black box of emergence would include
recursiveness plus some sort of negation to open up the space to the radically novel.
-Structionings and Emergent Order
One of the chief puzzles of emergence is where the order characterizing emergent
structures comes from. Various solutions of this putative puzzle include: symmetry and
ergodicity breaking [15]; spontaneous "order for free" [49]; and more arcane explanations
based on random graphs and so on. These explanations, of course, add important insights,
but if emergence is understood in terms of self-transcending constructions, the origin of the
order of emergence comes from the action of self-transcending constructional processes
building-upon and transforming preexisting order. One source of this pre-existing order that
-Structionings transform is usually overlooked: the order found in the “containers” of the
complex systems in question.
Self-transcending constructions creatively build on this already present order like
buildings taking on the shape of the ground and space allotted to them. The order found in
the emergent Benard Cells demonstrates this creative transformation of the order of the
container: the distance separating two neighboring currents is on the order of the vertical
17
height of the container [56] -- in fact, the number of convection rolls can be curtailed by
reducing the ratio of horizontal dimension to vertical height. Along the same lines, certain
instabilities in the thermal boundaries of liquid systems similar to the Benard system lead to
more complicated kinds of convection [Weiss, 57].
We can see the constructional nature of such container order in Ray's [58] Tierra
which includes the strategy of creating a container for the "organisms" that renders them
incapable of replicating outside the virtual computer. Moreover, in the "ancestor" in this
block of memory called the soup are three segments: the first counts its instructions to see
how long it is; the second reserves that much space in nearby memory by way of a
protective membrane; the third copies its code into the reserved space, creating a daughter
cell from the mother cell. And, as was mentioned above, Kauffman [49] cited research of
Farmer and Bagley on the way in which spatial compartments enclosing an autocatalytic set
enable the self-sustaining metabolic processes to increase the amount of copies of each type
of molecule in the system. When the amount has doubled, the compartment can divide into
two.
Finally, it needs to be emphasized that explanations of emergent order in Artificial
Life simulations need to includes the order contained tacitly in the actual “containers” of the
simulations provided by the hardware and software. For example, neural networks are wired
together like the cells of cellular automata are connected. This provides a container or arena
that the self-transcending constructions of the simulations transform into the order of the
emergent phenomena. Of course, there are many sources of order in bringing about the
radical novel order of emergent phenomena, but by conceiving emergence in terms of Structionings, the key is on how order is taken from anywhere it is presently available as
well as where it might show up along the way.
-Structionings and Irreducibility
One of the chief characteristics of emergent phenomena, from the earliest
formulations to now, is their supposed irreducibility to lower level components and process.
Indeed, emergence has been formulated as an alternative to micro-reductionism for studying
complex systems. In this light, Wimsatt [59] has proposed an aggregativity heuristic for
distinguishing emergents and aggregates. Simply put, whatever is invariant under
decomposition, rearrangement, inter-substitution, reaggregation, is aggregative, not
emergent. As an example, let’s consider an analogy with semantical decomposition.
According to Hintikka [45] claims made about decomposition in semantical analysis are
based on similar ideas on syntactical decomposability. Tarski was thus inspired to mirror the
semantic structure of a sentence by the way it is constructed syntactically from its basic
components. This included the notion that semantical constructions recursively go from the
simpler to the more complicated ones, enabling Tarski to recursively define truth in an
inside-out procedure for formal languages. In the face of claims that semantics is ultimately
decomposable to some basic semantic units just as syntax is supposed to be, Hintikka offers
a counter- example. Consider these two sentences:
"Jim can beat anyone."
"John doesn't believe that "Jim can beat anyone."
The second sentence changes the meaning of the italicized first sentence included
within it since now the meaning of "Jim can beat anyone" becomes "Jim can beat someone."
Therefore, the second sentence cannot be decomposed into the first since the decomposition
will shift the meaning of the second clause. If semantical units change their meaning upon
decomposition or recontextualization then they cannot be fundamental units to which
semantics can be reduced. In other words, semantical units are not aggregative, they are
18
emergent. Context is the higher level pattern that determines much of what the lower level
units mean, and after all semantics is about meaning not just syntax.
Similarly, emergenct phenomena can be interpreted as contextual in the sense of their
environments (e.g., the environment in which emergence aids the adaptability of an animal)
and are like semantics in their resistance to reduction. This, I believe, offers a good hint
about where to look for a theory of emergence: semantical, not syntactical constructs. For
this reason, Lemke’s [50] and other’s semiotic approach to emergent levels is offering
direction for future research.. Of course, this emphasis on semantics doesn't mean that
syntactical elements are not crucial to emergence, it simply means that it must be syntax
plus a whole lot more.
There is another key to emergent irreducibility in the construct of self-transcending
constructions. For example, if we consider Cantor’s constructed number to be {’s} and the
list of countable numbers from which it is generated {’s}, then {’s} cannot be reduced to
{’s} because they have been constructed to be radically different than {’s}! Someone
might say at this juncture that Cantor's number is nothing but numbers in a certain sequence
so it can be reduced to integers. But it is precisely this sequence which is lost if you say it is
nothing but numbers. It's like saying that everything is nothing but vibrating "m-branes", or,
that emergent patterns in cellular automata are nothing but the individual cells in different
patterns. You can say that, but, of course, such reductions get rid of the pattern which
defines the emergent phenomena. You can reduce as much as you like but you will thereby
be negating the structure you want to explain.
Finally, there is one more way to think about the irreducibility claim of emergent
phenomena relating to the paradoxical nature of the new level discussed above. Using our
analogy of the bridge curling up behind the traveler to the new level, we can say that
emergent phenomena are irreducible in the sense that there is nothing left to reduce them to
[13]. The former lower level components no longer exist as they did before but have a new
existence bound up and entangled with the new emergent level. Indeed, in some sense this
can be said about Cantor’s uncountable sets—they cannot be reduced to the countable sets
out of which they were generated because, at least as far as measure theory goes, the
countable sets have measure zero!
Conclusion
Research on mathematics education has shown that an overly tight hold on specific
constructs can hinder further insight. For example, Presmeg [60] found that students who
held equilateral triangles as their cognitive prototype of triangles had difficulty in
recognizing non-equilateral triangles as also triangles. In this paper, I have tried to present
some alternative ways of thinking about emergence that depart from the dominant constructs
now being used. As such, my remarks are just hints of a future theory of emergence.
A theory of emergence will of necessity be a complex one. Perhaps an appropriate
way of thinking about is the idea of paccaya or dependent co-origination of Buddhism. A
network of factors emerge given the co-emergence of the other factors. Given this factor,
then there is that factor. This doesn’t proceed in a lineal order, but in the sense that all
factors are needed for the emergence of each other. Emergent phenomena emerge for many
interrelated reasons. When all the factors are present, emergents emerge. Indeed, because of
the great diversity of nature, I don’t think there will be one formalization of emergence or
that one kind of self-transcending construction will explain them all.
As stated earlier, however, I do think that formalizations of emergence will be
forthcoming. But we have to keep in mind that the study of complex systems is changing the
very meaning of formalization. This shows itself in the role that simulations are playing to
19
gain insight into emergent phenomena. I think the role of simulations will be even more
radical in the future: rather than being simply tools used to investigate emergence, like the
use of telescopes in astronomy, simulations will enter in the very formulation of emergence
itself. What I mean here is that instead of classical theories which present principles,
constructs, equations, implications, explanations of experimental results, and so on, theories
of emergence will have all that plus simulations depending on the different contexts in
which different emergent phenomena are seen. These simulations will not just be
experiments that lead to theory, they will be inside the theory as an essential component.
This had to do in part with the non-analytic nature of the nonlinearities involved, but also
with the creative nature of self-transcending constructions. A theory of emergence will itself
need to be emergent in the sense of evolving as more is revealed about the emergence in
question. Thus, instead of Laws of Emergence structurally similar to the way the Laws of
Thermodynamcis were posed, the new theories of emergence will be, by their very nature,
evolving—they will include simulations in their very formulations. In such a selftranscending perspective, emergents themselves become a basic element in a theory of
emergence.
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