Illustration from the paper, “Mapping the structural core of human cerebral cortex” by Hagmann P, Cammoun L, Gigandet X, Meuli R, Honey CJ, Wedeen VJ, Sporns O (2008). PLoS Biology Vol. 6, No. 7, e159.
“The most incomprehensible thing about the universe is that it is comprehensible.”
— Albert Einstein
As Einstein noted, although it may be a curious fact, the world is not an impenetrable fog, nor are we prisoners in and of our own minds. Rather, we can tease out a reliable (if always partially incomplete) account of the structure of what is really going on, to a remarkable, even astonishing, degree. It is simply the case that nature does not give up its secrets easily, and the process of teasing out what is going on is immense. But fortunately for us, this process has culminated in recent years with a breathtaking kind of consilience — and a most useful one.
Perhaps nowhere do these developments have more practical effect than in the topic of mereology. Aristotle’s understanding of hylomorphism has been greatly deepened, and in some cases turned on its head, by over two millennia of discoveries into the universal atomic and molecular structure of the world. There is no real place for fundamental essence to enter the picture, nor any need for it to: the universe is perfectly comprehensible as an endless compositional structure. But this composition is not merely additive, but, in an important sense, transformative and inter-relational. We will discuss below the profound significance of this revelation.
Similarly, it is possible to make a perfectly useful description of the structure of the brain, and its relation to various problem-solving and representative functions of the animal (that is, ourselves). We can note interesting and useful structural relationships and isomorphic properties, without respect to any metaphysical or ontological assumptions.
This “new structuralism” (or it could be called “symmetric structuralism,” for reasons that will become apparent) is distinguished from the original most notably in its emphasis on process, and the complex symmetric transformations that occur within it. As Jean Piaget, a notable figure in this school of thought, put it in 1968, “there exists no structure without a construction, abstract or genetic.” Moreover, as we shall see, the new understanding can even find a structural home for value, meaning, agency, choice — those things that are thought of as exclusionary opposites of what had been assumed to be an inherently determinist structural perspective. But it is in the new structures revealed by modern science that things get very interesting indeed.
We begin with the neuro-structuralist’s view of the human brain. What is particularly useful about this picture of things is that the structure of the brain and its activities — let us assume, the equivalent of the mind, seen from beyond our own first-person perspective — is a structure in the world like any other, and it is not necessary to posit a transcendental realm of the mind, or of ideal forms, to explain what is going on in a plausible way. Such a realm may or may not exist; it is simply unnecessary to account for it from this perspective on things.
Furthermore, the relation between this structure of the brain and that of the rest of the world can now be seen to have an important relational structure in its own right. For one thing, there is a fundamental characteristic of partial symmetry. That is, the structures of the brain do have partial isomorphic correspondences with the structures of reality. These correspondences have their origin in the transformations of structures through symmetric progressions, beginning with the most direct kinds (say, instantaneous flashes of activity corresponding to instantaneous threats in the external world) and progressing to much more speculative threats (say, evasive patterns of instinctive behavior in response to a possible threat that is more remote, such as the agitated evasions of animal herds at the mere sight of a predator).
A similar thing can be seen to be going on in a more articulated form within language, its structure, and its structural processes. Indeed, the more we understand it, the more language seems to be a form of useful “software” running on our mental hardware, and, so to speak, written in its operating code. And language, too, displays this “progressive symmetry” — from vocalizations that originally stood for the patterns of events in nature, e.g. the glissandos of expectation and curiosity, the staccatos of action) to more sophisticated symmetric verbal elements and grammars, describing progressively more complex conditions within the world (including complex philosophical discussions like this one).
Language as “the architecture of possibility”
Thus, we arrive at a notion of language as a kind of model-making activity — mirroring and modeling some part of the world that is of interest to us. But these models are not static, but in fact define the realm of our action and possibility. Moreover, in doing so, they actually generate new possibility, as a real logical feature of the structure of nature within that landscape. And in that generation are the seeds of choice, and what is generally identified as free will.
This is an important but little-appreciated point. To bring the point home, let us consider a very simple example. X informs Y that Y can catch the bus on the other side of the street, but only when the hand on X’s watch passes between these two marks. Y then asks, “Can I catch another bus later?” “Yes,” X reports that Y can catch another one in another hour; or Y can catch another one down that street in two hours.
X and Y have both just built a dynamic model of Y’s bus-catching that is relevant and useful on that day. In constructing that model, they have defined new possibilities — and in so doing, in this exceedingly simple example, they have actually generated them.
Y has no possibility of catching a bus that Y doesn’t know is coming — or more precisely, but what is virtually the same thing, Y has infinite numbers of small random possibilities, including stumbling upon that bus at that hour; but that logical possibility is, like most others, vanishingly remote. Similarly, Y can discover a network of other possibilities that can be generated only through the dynamics of language. It is only through the power of language, the power of such mental symmetries creating systemic relationships, that X and Y have defined this particular — and, for them, novel — architecture of possibility.
It is in this generative power that we have also identified an architecture of choice, and of agency. For the linguistic power to generate possibility has also generated the possibility of selection. We have arrived at the old philosophic value of free choice, set loose from its anchor in a deterministic structure.
But one might object, are not the structures of the language, like the structures of the world, determining the outcomes? Why is this no less a matter of “reading a script written in advance”? It is at this point that the new sciences of complexity offer us two very useful and related insights.
One is the notion that the complex interconnectivity of a system creates ambiguity and multiple potential — or, in more familiar terminology, freedom. An example may be helpful, in the familiar children’s game, Rock, Paper, Scissors. These are three interactive elements within this system. Which is superior? Rock is superior to scissors, which is superior to paper, which is superior to rock. Any two couplings are unambiguous; but the system overall is ambiguous, because of these circular (but in fact purely structural) linkages.
This kind of ambiguity created by multiple inter-linkage is precisely what distinguishes complex systems from simple ones.
But one might object, if we know the initial state of the system, why can’t we determine the next step unambiguously? If we know the current object is a rock, for example, then we also know unambiguously that the next object will be inferior if scissors, and superior if paper.
But this is the second point from the sciences of complexity: in a non-trivial complex system, it is impossible to know precisely the initial set of conditions. This is the familiar principle of sensitivity to (unknowable) initial conditions. Although it seems possible, looking in rear-view fashion, to determine a course of events based upon initial conditions, it is (probably) impossible to represent initial conditions with absolute precision. And in a deterministic analysis of behavior of any but the most trivially complex systems, absolute precision is required. But in fact, absolute precision can never be achieved.
The proof of this can be found, once again, in Kurt Gödel’s 1931 paper, and a series of follow-up papers. Not only was Principia Mathematica necessarily incomplete — that is, incapable of representing even itself with absolute precision — but any such system must be equally incomplete. We cannot distill down to an “absolute blueprint” of the initial conditions.
In essence Gödel relied upon a kind of “Epimenides’ Paradox” — showing that any such system is capable of saying, in effect, “I am lying to you now.” But this is a logically untenable state of affairs, of course: if the system is lying, then it is telling the truth; if it is telling the truth, then it is lying! The problem is that in essence we are failing to account for the regressive nature of abstractions. Every abstraction is a secondary structure, only partially symmetrical with the first. It is not possible to get an abstraction that is completely congruent with its subject. This is true even if the subject is itself. This is the problem with the post-structuralists’ retreat into language, as 114 discussed earlier.
Moreover, it would not be of much use even if it were. A typically wonderful Lewis Carroll story demonstrates this clearly on an intuitive level. In the children’s tale Sylvie and Bruno Concluded, our protagonists meet a man from another country, who tells them about his country’s efforts to greatly improve the accuracy of their maps. As the maps get more and more complete, they become as large as the regions they represent. Finally they make the ultimate map, as large as the country itself! But do they use it much?
“It has never been spread out, yet,” said Mein Herr: “the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well.”
This delightfully absurd little fable reminds us that in order to be useful, maps — abstractions, models, linguistic structures, knowledge — must be in some sense different than, and indeed, simpler than, the regions they represent. Gödel’s incompleteness is not a defect of knowledge, but an essential and highly useful characteristic of it — but it is we who are already doing the selection and the omission. As Whitehead noted, “an abstraction is nothing other than an omission of part of the truth.”
Moreover, there is another major fly in the ointment of initial conditions. It is nothing other than ourselves. As the physicist Werner Heisenberg noted, first in a “hard” principle of quantum physics and later as a more general epistemological principle, we can never fully write ourselves, the observers, out of the observations. Our impenetrable subjectivity, uncertainty and ambiguity are unavoidably and, a priori part of the initial conditions at any point.
It is here that the essence of our own agency lies, in our own irreducible, uncomputable, undecidable participation. Any valid scientific description of what is going on must ultimately take this situation into account, as an a priori condition. All facts are derivative of this truth: we are embedded participants.
This is not bad news; indeed, it is the best possible news. It gives us ever more choice: meaningful choice. It means that we can make useful abstractions, and abstractions of abstractions, and grand and powerful theories — and that they can be extremely useful to us, in defining, and creating, new architectures of real possibility — real, because our choices are real, and our ability to cause them to occur is real.
This also means that we must be aware of the limitations of abstractions, and their fundamental derivative nature. Furthermore, we must be aware of the dangers of their seductive accuracy (too easily confused with perfection), and the perpetual tendency to become lost in the abstractions, and fail to understand their perpetual (if often undetectable) incongruences, limitations, and derivations. We may become lost in what Wittgenstein called “the bewitchment of intelligence by means of language.” As Whitehead noted in his remarkable book Modes of Thought, we must ensure “a right adjustment of the process of abstraction.”
A (neo) structuralist reinterpretation of idealism and hylomorphism
What can we conclude now about the primary forms, going back to Plato and Pythagoras? What of proportion and harmony, and their possible echoes in the moral order of things?
It might appear that they have been destroyed for good, or at best, left as simplified constructions of language, or of the mind. But if we remember that language tends to have symmetrical correspondences with the structure of nature, we may not be so surprised to learn that oddly similar kinds of structures have returned, in a very different set of circumstances.
The new emphasis on pattern-like structures has made us aware of certain recurrent classes of structural outlines that get created as a result of certain kinds of patterning processes. A long thin structure (such as a rod) that is moving otherwise freely about a fixed length fixed at the end (relative to other structures) delimits a kind of shell structure that takes on a characteristic pattern: a dome or sphere or hemisphere. A structure moving freely along the length of a tensioned cord and around it as an axis delimits a cylinder. And so on.
We need not posit a transcendent realm of such structures, other than to say that they are patterns that are consistently created by the interactive movements of other patterns — and made comprehensible by the symmetrical patterns of our own language and thinking. They are no more “independently real” than the word “sphere” is real, except as a generative possibility modeled within language. But so are vast numbers of other structures. These are simply structures that we find of particular interest, because they have a particularly simple and comprehensible genesis.
Moreover, the power of modern biology and mathematics is in its ability to move along a progression of complexity to much more esoteric structures — which, nonetheless, can also be comprehended through such a process of linguistic symmetry-mapping.
A particularly fascinating (and representative) example is what is known as an attractor. If one graphs complex phenomena according to their variable constituents (e.g. position, density, or other characteristics) it is often observed to be the case that certain patterns form around particular regions of the graph, for reasons that are not always obvious. These “attractor basins” are in fact similar to the limits of a radius that describes a sphere, but they are often much more complex and esoteric versions of the same kind of phenomenon. Indeed, some of these are so complex and odd that they have been dubbed “strange attractors” — quite odd graphical shapes that nonetheless take on strongly defined patterns (swirls, toruses, etc).
It turns out that such attractors occur all over — and they appear to be very important characteristics of the way things work, notably in biological processes. They are in fact the structures of the characteristic patterns that form within nature as the result of complex adaptive processes.
The biologist Brian Goodwin proposed that such “structural attractors” played a key role in the formation of biological structures within evolution. A dolphin’s dorsal fin, for example, took on a characteristic but very complex shape as the result of highly complex interactions of turbulence processes in water, laminar flows and so on. The dolphin was solving the problem (or more accurately, the dolphin’s evolutionary process was solving the problem) with a characteristic geometry, that was defined as the limit of the solution, just as the radius of a string defines the limit of a sphere. The shark, evolving from an entirely different animal class some 300 million years earlier, produced the nearly identical structure — for no other reason than that the complexity of the problem defined a similar structural attractor.
Going back to Plato, was the shape of a dorsal fin a timeless form, existing in some unseen realm? No, certainly not: it was a pattern that formed for comprehensible reasons, but one that formed repeatedly and consistently. We could name that pattern, and find use in the naming (i.e. repeated instances of application, or characteristic symmetries). The pattern may not be decomposable — and yet, it was comprehensible, through this new lens of nature.
Such patterns also have their counterpart in what physicists refer to as symmetry-breaking. A perfectly symmetrical universe would have no pattern at all: indeed, it would be a very dull place! But introduce a tiny break in that symmetry — a kind of grit within the oyster, so to speak — and fascinating things begin to happen. (This can be seen in a simple mathematical analogue, in the computer program called “Life” — a series of cells following rules “do nothing” until one tiny change is made in one cell — and instantly, a vast pattern is created across the screen.)
In a sense, this is a return to Plato’s idealism, and to Aristotle’s hylomorphism — but with a very notable twist. We are not appealing to any fundamental set of forms that are at the decomposable root of things, but rather, we are identifying a set of recurrent patterns that can be classed according to their tendency to coalesce into relative simplicity, or into what we may term “order.” A sphere is indeed simpler than a dorsal fin — but they are both the same kind of structure within reality. They are both generated by the interactions of ultimately comprehensible structures. (Though in the case of the dorsal fin, that takes a great deal of time and effort.)
There is even an implied resolution of the age-old mystery of the “place” of mathematics. Is it, as Pythagoras suggested, a fundamental level of reality underlying all we see? This is no longer necessary: we can see the world instead as an inter-related field of isomorphic structures transforming in time. Thus the mathematical laws and categories of order are not residing in an unseen realm, but instead, are generated simpler structures that arise from the interactions of other more complex structures. They are, in effect, limit domains of generative possibility. They are symmetrical features (exhibiting partial symmetry-breaking) within the world of structure and process. They are, in a word, patterns.
But what is the ultimate reality of such a “pattern,” then, as a distinct entity with generative possibility? And what is the ultimate reality of “possibility” itself? As far as we are concerned, ultimately these are “only” abstractions within our own brains (or in our computers, or on our papers.) But these are no less “real” structures in the world. It is our participatory interest in this isomorphic relationship — and our active deletion of parts of the reality in the isomorphism, through our powers of language and thought — that creates its generative power.
It is in the categorical confusion between what is “real” as a structure at the level at which our biological interest (and a priori participation) is usefully concerned, and the secondary or tertiary (or beyond) levels of abstractive self-reference, that the trouble begins. The trouble is in the seductive appearance of fundamental completeness, which does not exist.
So we can finally dispense with an external, transcendent realm of Forms, and shift our understanding to that of patterns. But in a powerful echo of Aristotle, Plato, and Pythagoras, we can classify these forms, or patterns, usefully, and see them as primary orders within certain contexts (again, being careful that we understand these abstractions for what they are, and taking care not to make categorical or fundamentalist errors).
Certainly these forms are primary with respect to the composition of a dolphin’s body, say, or a shark’s. And they are primary with respect to their repetition over time, and their ontogeny within an organism. Genetically speaking, they appear to be primary with respect to certain genome or proteome sequences within embryogenesis, which express themselves in the familiar form. They appear to originate as primary patterns within these structures.
Very interesting work suggests that these genetic sequences may in fact be clusters of pattern-like genetic sequences, functioning to bud and fold and shape dorsal fin structures, not unlike the sequences of origami. For example, Newman and Bhat (2008) have proposed a pattern-based model for the genesis of the Cambrian Explosion, which seems to offer a plausible (and very intriguing) model for the hylomorphic evolution of multi-cellular form.
If we are designers making houses, we might identify a corollary. Let us describe, for example, the shape of a column. Why does it exist as it does? We can ascertain structural tensions on its base and top that call for thickening at these points. We can describe efficient shapes that call for tapering of the shafts, or stiffening using rib-like flutes. We can describe the psychological needs of human beings who are in the presence of these columns, who may find a deep and pleasing biological resonance in their tree-like structure. This description of a column can amount to a structural attractor within architecture — and indeed, we can see columns with precisely this set of characteristics in cultures around the world.
We can now turn to larger structures — let us say, a porch structure — and consider similar questions. How does it connect the building to the public realm? What relation do people have with the street when they are within it, and simultaneously with the building? Are they able to interact with other people on the street, while having a sense of safe refuge, combined with an appealing sense of prospect? Does this add to the civic quality of the neighborhood?
And do the columns of such a porch also take on the shapes described above, which add to the psychological and structural coherence of the porch? Have we defined a structural type — a structural attractor of “porch” which has certain recurrent problem-solving value?
It becomes apparent that such recurrent patterns may have great value — if seen as such — and as elements that have components (columns, say) and that in turn combine into still larger elements (buildings, say). We need not take a rigidly hierarchical view of how such forms are composed, or pose a transcendent or fundamental class, but we can still see that there is a nested relationship between these recurrent and re-usable patterns — at least partially so. It presents us with a highly useful opportunity to make adaptive transformations of whole structures.
This “symmetric structuralist” view of nature does not diminish the capacity for meaning and value within a world of structure. On the contrary, as we will discuss in more detail below, it enriches the meaning and value of structure itself. Structure is thus not some dead shell: it is the domain where the phenomenon of life as we know it arises — and along with it, the related phenomena of quality, meaning and value.
And it appears to do so through the patterning of the interconnected wholes of structure, rather than through a simply conceived atomic assembly. The latter is a derivative abstraction. The former is, quite literally, where life happens — indeed, where we ourselves, living beings, already find ourselves immersed. Meaning is structural, and structure is meaningful. And there is much to say about how all this works, and how we may apply it to our own activities.