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The illusion of structure or insufficiency of approach? the un(3) of unruly problems

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Abstract

The ability to formalize problems in a quantitative manner is the key to predictive power. We characterize a lack of formality as unruliness, relate unruliness as a property of un(3) (undecidability, uncomputability, and unpredictability), and define a class of problems which even when well-posed remain highly informal in nature. Despite this lack of formalism, systems represented by these problems still exhibit significant structure. We call this class of problems hard-to-represent, and are characterized by the difficulties of quantification and symbolization, as well as the inherent un-physicality of a system. A significant part of this difficulty involves both finding the proper metaphor for such systems and a method for analyzing the system components. To counter these difficulties, we propose a new analytical paradigm called perceptual analysis, which brings an umbrella of diverse approaches to bear. These include neural-inspired modeling, visualization-based feature selection, and soft computation, which provide an alternate means to quantify features and discover structure in a manner that is less dependent on traditional mathematical presumptions.
The illusion of structure or insufficiency of approach? the un(3) of unruly problems
Bradly Alicea, Jesse Parent, and Ankit Gupta
Orthogonal Research and Education Laboratory
Abstract
The ability to formalize problems in a quantitative manner is the key to predictive power.
We characterize a lack of formality as unruliness, relate unruliness as a property of un(3)
(undecidability, uncomputability, and unpredictability), and define a class of problems which
even when well-posed remain highly informal in nature. Despite this lack of formalism, systems
represented by these problems still exhibit significant structure. We call this class of problems
hard-to-represent, and are characterized by the difficulties of quantification and symbolization,
as well as the inherent un-physicality of a system. A significant part of this difficulty involves
both finding the proper metaphor for such systems and a method for analyzing the system
components. To counter these difficulties, we propose a new analytical paradigm called
perceptual analysis, which brings an umbrella of diverse approaches to bear. These include
neural-inspired modeling, visualization-based feature selection, and soft computation, which
provide an alternate means to quantify features and discover structure in a manner that is less
dependent on traditional mathematical presumptions.
Introduction
A little bit of many things, but mostly the things that cannot be easily coded
This quote is taken from the description for Dr. Bradly Alicea's Github repository [1]. At
first look, this might be the perfect description of a generalist with a range of unruly interests.
This is of course a subjective take, but does bring up a conceptual point. Unruliness can be used
as a way to describe what we cannot formalize. Unruly problems are messy, but can also include
so-called wicked problems [2]. Such problems are hard to represent with mathematics and
sometimes even with words. However, they are not inherently qualitative or structureless! As
Wigner once proposed that mathematics is unreasonable in its effectiveness [3], so we might also
propose that poorly-represented problems have elusive answers. As a contrast, we propose that
unruliness is the main way we can identify instances of undecidability, uncomputability, and
unpredictability (which we refer to as the un(3) of unruly problems). Once identified, we can
analyze such problems using a range of alternative principles and analytical approaches.
In general, un(3) problems are hard to represent. Yet this is not always for the lack of
mathematical tools. Examples include systems with a large number of variables or high degrees
of nonlinearity. These types of systems stand in contrast to methods such as experimental design
or measurement of nearest-neighbor interactions. Inverse problems of highly-interacting systems
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such as swarm intelligence can also be classified in this way. While solutions can be proposed,
this essay focuses on the need for a new paradigm for solving such problems.
What makes a problem hard-to-represent?
The core of problem representation involves separating the subject of observation from
that which is being observed, computed, or predicted. This process, which is a form of
differentiation between subject and object, is similar to boundary demarcation in systems
analysis [4, 5]. In both cases, there is a fundamental difference between systems where
differentiation is easy and those in which it is difficult to impossible. This essay deals with the
latter case, and in particular what makes such systems so difficult to characterize. Specifically,
differentiation serves as an example of difficulties in quantifying, symbolizing, representing, and
metaphor-building such systems. At the core of incomplete differentiation is an inability to
transmit alternative forms of information to the observer [6]. These alternative forms of
information act as redundancies, and allow for the observer to more easily identify patterns and
structure in data generated by the system in question.
Difficult to quantify
While many problems can be easily quantified, many other problem domains cannot.
Even many quantifiable problems do not yield clear quantitative structure. While we study model
systems to yield broadly-applicable insights into similar systems, model systems also provide a
window into regularities that govern very general natural processes. In this essay, quantitative
structure will be limited to separability (gaps and difference), alignment (function and
curve-fitting), and correspondence (one set of data points correspond to an analogous set). While
these attributes are signatures of quantitative structure, they can also be identified in systems
with limited structure.
Quantitative structure also exists as a signature of dynamic organizational complexity.
We can examine this by considering the most extreme alternative to quantitative structure: grey
goo [7]. Grey goo (or the complete lack of structure) can be attributed to processes such as
runaway autophagy [8] and results in a smearing of categories and distinction. Kaufmann's
Noah's Vessel experiment suggests that if all biological cells on earth were ground up and poured
into a single vessel, we could test to see if the biocomplexity of cells and multicellular organisms
re-establishes itself. While Kaufmann suggests it might be possible due to the contents of our
vessel acting like a supercritical fluid [9]. This behavior is driven by interactions at the molecular
level, unbounded by cell membranes and other subsystem boundaries.
From this example, we might assume that grey goo results in a lack of structure. Yet
might we suggest that grey goo leads to a different kind of structure: qualitative structure.
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Qualitative structure is simply structure that cannot be easily identified with quantitative
methods. There are also unexplored links between qualitative structure, randomness, and chaos
so that random behavior leads to structure that is neither systematic nor replicable. Yet there is
even a more fundamental question: do quantitative methods rely upon our perceptual abilities
and cultural traditions to impose order upon a system and yield structure? If so,
qualitative-quantitative converter models based on models of human cognition could decode
much of this transient structure and turn it into quantitative and computable data.
As we rely on statistical models to identify structure, do we need a new field where
different types of metaphors yield unique structural observations, while ensemble models
produce consensus views of ontological structure? Clearly we cannot simply rely on ever more
sophisticated numeric or symbolic representations. Therefore, we propose a new approach called
perceptual analysis that utilizes symbolized metaphors to translate unidentifiable structure into
quantitative forms. Perceptual analysis relies on natural processes such as collective behaviors,
neuronal functional processes, and the fundamental properties of geometric structures. We might
also be able to couple perceptual analysis with verification methods borrowed from conventional
mathematics and even fields such as computer security or artificial intelligence safety. While we
will discuss the details of perceptual analysis later in this essay would form the basis for a new
way to compute, understand nature, and analyze human societies.
Difficult to symbolize
Aside from complex systems that are hard to quantify, there are also classes of complex
systems that are inherently difficult to symbolize. In the context of a complex system, symbols
include mathematical objects such as numbers, operators, and representations of selected features
and processes. Symbols provide a path to both theory and measurement. For example,
identifiable structure or discreteness can be characterized with a symbol, or symbolized. What
we might refer to as symbolization is the description of a complex system as a series of symbols.
The quest for symbolization is embodied in Newell and Simon’s Physical Symbol System
Hypothesis [10], but as application to the real-world has shown, a symbolization strategy is quite
unsuitable for many problem domains.
Nevertheless, symbolization is often a necessary step in the creation of models, theories,
programming abstractions, and data structures. Yet computational representation is also required
for model-building, and aside from having the same limitations as symbolization, is dependent
on available mathematical tools and cultural biases [11]. For example, one potential limiting
factor in symbolization and representation is the ability to reference congruent conceptual
material, or more generally, to use metaphors to casually describe the dynamics of a complex
system. These metaphors may be intentionally selected because they provide easy access to the
language of quantification. Julian Jaynes [12] considered theories to be metaphors between
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model and data. Therefore, we must also consider the metaphors we use to create theories and
interpret data. Simply extending existing models and metaphors to data is insufficient. Even
physical metaphors are insufficient to the task of making quantification easier. In the next
section, we will explicitly discuss problems for which there is no physical analogue.
Another stumbling block to symbolization are so-called "wicked" problems [2]. Wicked
problems incorporate many different system types, but have been characterized by large social
problems such as a planned economy, social change and persistence, or complex systems with
many parallel components [13]. Why are wicked problems so difficult to address? Let us
consider one example of a wicked problem: urban dynamics. In the attempts to solve this
problem, a variety of top-down approaches have been taken, including systems dynamics [14]
and machine learning [15]. Yet such approaches still only cast a broad net over the particular
sources of variation in the system. In addition, many of the variables and phenomena needed to
characterize the problem are not predictable. While it is tempting to suggest that these
phenomena simply exist in a chaotic regime, it is more likely that unpredictable components are
variables that produce unstructured but nevertheless important information in a system that
exhibits structure.
Un-physical systems and representational metaphors
We often use physical analogues to address complex biological and social problems.
Examples include the brain and neurological function as a "computer", "velocity" of spent
money, "jet engines" that propel organisms through water or air, or "gas" models of social
interactions. But how physical are these systems, and do analogues provide insight? Even if the
physical analogy does hold, to what extent is the physical analogue only modeling or speaking to
a fraction of the structure in our system relative to a holistic view? This was discussed earlier in
the difficult to quantify section. Un-physical problems pose a unique challenge in this respect:
the structure is not merely transient, but reliant on a global view in space and time. Therefore, a
subset of un-physical problems are those that cannot be abstracted easily into computational
form.
Returning to the issue of physical versus non-physical systems, we can compare
phenomena that are psychophysical versus phenomena that are metaphysical. Psychophysical
phenomena are interactions between the human perceptual system and regularities that result
from physical features of the stimulus. By contrast, metaphysical phenomena are those which
cannot be mapped to an objective frame of reference. An example of this could be a cultural
representation of an object’s physical attributes that do not map to physical laws [16]. Examples
of metaphysical representations can often be found in video games and religious myths. In both
physical and metaphysical representations, a physical analogue would require at least an
intermediate layer to translate the physical process to another type of process. This secondary
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process might be semi-physical (such as a set of pathways in the brain) or not at all physical
(philosophical narrative). In any case, the broad application of physical metaphors offer a
sometimes suboptimal series of trade-offs between quantification, symbolization, and
understandability.
It may also be that such metaphors rest on systems that are easy to quantify and
symbolize. Systems with discrete parts, consistent outputs, and deterministic behaviors fall into
such a category. Returning to our discussion on symbolization, metaphors that are easy to
symbolize but otherwise poorly descriptive of process complexity are actually preferable. Yet
this is done at significant cost to realism. In the realm of Diatom biomechanics (multicellular
motility), we have witnessed a history of diverse metaphors [17] being used to describe
organism-level movement and structural function. Potential mechanisms include jet engines,
rowing ships, bubble-driven movement, treadmilling, compressed air-driven movement, and
explosive propulsion. While not all of them are equally plausible, the diverse interpretations of a
singular biophysical phenomenon is evidence that even something that is straightforwardly
physical (diatom motility) is actually quite hard to symbolize.
Is a new paradigm for measurement and computation the answer?
In this essay, we have learned that a wide variety of systems have components that are
difficult to characterize at an epistemic level. Certain problems have a structure that are simply
not amenable to conventional mathematical approaches. Yanofsky [18] demonstrates this
through the example of solving Zeno’s paradoxes. While these paradoxes can be solved through
the introduction of novel ideas, these new approaches are not always compatible with
contemporary mathematical and empirical tools. Thus, we will now attempt to establish a set of
investigative approaches that transcend the limitations of quantification, symbolization, and
physical analogy. These involve the potential for new models, logical frameworks, and
replication of experimental methods.
When the Zeno’s paradox example is coupled with complexity in the form of problems
that are wicked, difficult to symbolize, and difficult to represent, it is no wonder that un(3)
phenomena rear their head. This leads us to propose a new paradigm (perceptual analysis) for
understanding empirical and conceptual limitations posed by this essay. Perceptual analysis is a
heterodox mix of neural-inspired modeling, visualization-based feature selection, and soft
computation. While this new computational and quantitative paradigm may be at times hard to
reconcile with more conventional mathematics and computer science, it may help us overcome
some of the limitations of tradition.
The first potential focus of perceptual analysis involves the development of continuous
(or hybrid continuous/discrete) methods that use the behavior of fluids, biological collectives, or
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soft materials as inspiration. While continuous approximation has become increasingly easier
with advances in computing technology, this requires an ability to capture the phenomenon of
interest using methods that make such systems quantifiable. To bridge this conceptual gap, we
can map such systems onto fluids, biological collectives [19], and soft materials [20]. While
these mappings are quasi-quantitative, our mathematical understanding of such systems is a bit
more tractable than the alternative.
A second focus of perceptual analysis involves the identification and representation of
intervals and interval-based data structures. In this way, we can convert difficult to quantify
systems into quantitative grist for interval and difference computing [21]. Interval and difference
computing requires us to define these features in the system of interest, often in ways that are
little better than guesswork. Techniques such as interval type-2 fuzzy sets [22] allow us to
discover natural intervals in an uncertain or poorly-defined context. Another promising technique
is to use the features of natural systems as a tool for computation. Interval computation can be
done in this way using neural models [23] and neural-inspired models such as dendritic
computing [24]. Neural-inspired models offer a key advantage to unlocking new analytical
vistas. While they are similar to neural networks in terms of function, neural-inspired models are
also not constrained to the typical neural network representation.
We can also engage in the construction of nested and overlapping shapes to facilitate
geometric generativity and ultimately isometric networks. Isometric networks derived in a
generative manner can be used to represent sets of relationships that possess non-transitive
properties, recursivity, and co-varying higher dimensionality. We will discuss isometric
networks first. Isometric networks consist of a series of interconnected isometric shapes.
Isometric geometries have been used in artistic representation, video game design, machine
learning [25], and topological physics [26]. Isometric networks are drawn in two dimensions, but
due to their orientation, project three dimensions of information. Due to this projection, the
second and third dimensions are entangled, leading to shared variance amongst both dimensions.
While isometric networks can represent transitive relationships, we can configure such
networks (and particularly their growth) in such a way so as to introduce non-transitive elements.
Nested and overlapping shapes drawn as a generative landscape might be a way to introduce
such properties. Much as injecting stochasticity into boolean models make them more lifelike, so
can injecting generativity into simple geometric systems gets us closer to what Douglas
Hofstadter refers to as a strange loop [27].
As traditional statistical methods allow us to discover correlations and correspondences,
we might also want to discover freer associations that are harder to define in terms of
curve-fitting or formal statistical tests. This can be done by considering exotic methods such as
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vague soft sets [28] and neural associative coding [29]. We can also return to using fuzzy logic
and interval measures [30], this time to characterize multidimensional space. Alternatively, we
might use agents that exhibit emotional valences in response to complex stimuli to discover
non-obvious associations [31]. The cumulative and collective behaviors of multiple agents, as
well as unexpected responses to complex stimuli, might aid in our quest for symbolization and
quantitative understanding. This allows us to build representations robust to the limitations
discussed in this essay, and can also be used to facilitate interval and difference computing.
Conclusions
While we have discussed many thought experiments, broad system classes, and potential
methods in this essay, there are more specific philosophically-oriented questions that might be
addressed. In the case of Noah’s Vessel, the assumption is made that there is a path from simple
chemical interactions to structure and order. While this is conventionally thought of in terms of
emergence, perhaps perceptual analysis can provide insights into this process of emergence from
complete randomness. In a similar fashion, we might also consider problems that are
conventionally thought to be beyond the scope of human comprehension. Human (and
potentially animal) consciousness is just but one example of this type of problem. Of course,
applying perceptual analysis in this way would also require innovations in empirical
investigation and simulation. Nevertheless, our approach could contribute to understanding better
the scope of universality and structure in such systems.
Returning to the title of this essay, we can pose a final question: are un(3) phenomena
due to a lack of structure or insufficient approaches to hidden structure? Based on our
assessment, the answer is clear. An enhanced ability to identify and characterize such systems
allows us to approximate a solution to un(3) problems, but does not solve them. Select problems
(such as wicked problems) might be unsolvable in the conventional sense. At best, we can offer
means by which to allow more differentiation and thereby more information to be gained through
expanding these techniques and approaches. To increase awareness of this, we have laid out the
reasons why such problems are hard-to-represent, open questions that arise from these
limitations, and a potential new paradigm for taming the unruliness of un(3) problem domains.
Further work involves refining the application of these analytical solutions to a new paradigm.
Acknowledgments
We would like to thank the Saturday Morning NeuroSim group (YouTube:
https://tinyurl.com/Sat-Morning-NeuroSim) for their comments and time to discuss the
development of this essay. Additional thanks go to Aidan Rocke for his discussion and
commentary on many problems related to what is presented here.
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