The ancient and cutting-edge topic of mereology

Flowers blooming in the author’s London garden. In spite of dramatic changes over the blooming process, the structure always displays a remarkably integrated relation of parts to wholes.

“First, the taking in of scattered particulars under one Idea, so that everyone understands what is being talked about… Second, the separation of the Idea into parts, by dividing it at the joints, as nature directs, not breaking any limb in half as a bad carver might.”
— Plato, Phaedrus

I previously asserted that both Jacobs and Alexander have taken up again the ancient philosophical topic of mereology — the relation of parts to wholes. This topic is a central theme in both of their works: how a city serves as an inter-connected fabric, how a design problem forms a network of related elements, or how people work together in polycentric groups to manage the growth of a complex whole. In this section I will consider mereology more directly, as a topic that is overdue for a re-assessment.

In spite of modernity’s radical technological and social advances—or perhaps precisely because of them — ancient questions of philosophy remain relevant, even urgent, and we would do well to revisit them. This is particularly true when we are dealing with problems of planning and design, working in groups and organizations of varying complexity. We need to be clear about our premises, our aims, and the efforts we are making individually and collectively to achieve them.

What is reliable knowledge? How do actions bring into being new structures? How can we work with others, and together as a civilization, to produce, and to sustain, a higher quality of life?

For Jacobs and Alexander, these are surprisingly dynamic questions. The old topics have certainly been informed, and transformed, by a centuries-long series of mind-boggling scientific discoveries into the structures of nature (as Jacobs described so lucidly in the last chapter of The Death and Life of Great American Cities). These discoveries have also served to reveal logically inherent weaknesses within modern technology itself and its progressive capacity for breakdown and malfunction.

Most disheartening, we have witnessed a series of unintended consequences and self-destructive outcomes, stemming directly from our own apparent best intentions and rational efforts. Instead of the hoped-for modernity of rational fulfillment, we are left with a “post-modernity” of geopolitical dysfunction, ecological devastation, and the grim specter of climate change. In this environment, the modern design professions seem to alternate between cynical despair and a manic faith in magical thinking.

And yet, as Jacobs and Alexander argued, within the logic of these scientific discoveries — particularly the more recent discoveries of organized complexity — lie the seeds of regeneration. As we begin to tease apart the secrets of living systems, we can begin to draw useful lessons indeed for the reform of our own failing technologies.

There are on offer, moreover, new ways of decomposing and analyzing the structure of our problems, the better to resolve them. As Jacobs put it, the way we think about, say, “the kind of problem a city is,” must correspond to the structure of the problem sufficiently for us to get somewhere in its solution. If it doesn’t — if we have made categorical errors in our own thinking — we will be doomed to a series of intractable and very possibly catastrophic failures.

Therefore, in order to get to the bottom of things, we need to ask fundamentally philosophical questions, following the principle that “as we think, we live.” For planning and design are profoundly philosophical activities, at once employing teleology (aims), epistemology (the nature of knowledge), ontology (the nature of reality), and logic (the structure of truth). This is a philosophy that, as we will discuss, has been informed, and profoundly transformed, by the structural revelations of modern science — and vice versa.

Birth in a Cave

Alfred North Whitehead famously observed that the European philosophical tradition can be characterized as “a series of footnotes to Plato.” Plato’s idealism — his understanding of a static class of “Forms” underlying the varied appearances of form we experience — has left its deep mark on the Western intellect, and nowhere more so than in the modern design professions.

We must recognize, of course, that Plato was not the real beginning of the story, and he owed a profound debt to Pythagoras’ earlier mathematical idealism. Similarly, Plato’s legacy was transformed by Aristotle and others to follow. (As Bertrand Russell put it, Aristotle could be thought of as “Plato with common sense.”) Nonetheless, Plato undoubtedly marks a seminal moment in the history of Western, indeed perhaps human, thought. And one metaphor stands out as most seminal.

It is, of course, the famous Allegory of the Cave, recounted by Plato in The Republic. We are, Socrates says, like men chained to the floor of a cave, watching only the flickering, ever-changing reflections on the wall before us, cast by unseen puppeteers behind, projected from the light of a great fire. We do not see the real, unchanging objects held by the puppeteers, nor the fire — even less, the bright sun outside the cave. Thus our knowledge of the nature of the world and its objects is derived only from a series of transitory reflections — mere instances — of the real, permanent truth, which we can only uncover slowly, through a kind of “ascent” from the depths of this cave. That ascent begins when we are freed from our chains, and are able to behold the objects of the puppeteers.

Those objects are, as students of philosophy know, Plato’s eide or Forms: the unseen but “really real” categorical structures that create the shadowy, impermanent perceptions of everyday experience. These are, we later learn, invariant and timeless mathematical and geometric structures, carrying universal aesthetic and moral implications. (It is here that Pythagoras’ earlier mathematical idealism is most keenly felt.) There is a unity to be discovered between the Good, the True, and the Beautiful.

Thus the allegory encapsulated Plato’s explanation of the central problem of epistemology: the variability of the world around us, the fragmentation and apparent disorder of it, and the incompleteness and even the corruption of our knowledge of it. What we encounter in the world is only a series of reflections of a deeper but no less knowable reality — but one that is precisely ordered and fundamental.

But as later philosophers noted, Plato’s idealism removed the epistemological problem to another realm, but did not get rid of it. After all, it is not clear how we will be any more able to establish the reality of the eide and ascertain their invariability and completeness. Even if we verify their existence, might they not turn out to be just as shadowy in their own way? (As I will discuss shortly, this was very much the sort of picture that modern science would later offer.)

Plato’s student Aristotle certainly cast the situation in a very different light — though his account still owed much to his great teacher. In his hylomorphism — his theory of matter, hylo-, taking form, morphe — he was still supremely concerned to explain the fact that many objects have the same universal form — many trees all have the form of a tree, and so on. But for Aristotle, the form was not located in a transcendent or unseen realm, but inhered within the object itself, as an instance of a potential universal becoming actual — and it is this, he said, that comprises its substance.

Moreover, this was a process that could repeat at different scales: the substance at one level could become the matter at another. Thus the matter of, say, a house — its bricks — could take on the triangular form of a pediment, or the cubic form of its walls. In turn, these forms could become the constituent matter for still other forms: a house, a city, and so on.

In this rudimentary structuralist account — which, like Plato’s, would become profoundly influential — Aristotle began to describe a logic of mereology: the way parts relate to wholes. This is a topic of central importance in modernity — all the more so, as the technology of modernity greatly magnifies the number of constituents, and the complexity of their relationships.

As noted, both Plato’s and Aristotle’s accounts owe much to the earlier Pythagoras and his mathematical “music of the spheres” — his faith that precise ratios underlay the orderly structures of the cosmos, and indeed, the order of all that is good and beautiful. Thus the mathematical properties of the forms — their proportions, their adherence to unchanging mathematical laws, and so on — were considered fundamental. In fact the five regular solids discovered by Pythagoras and his followers are now called “Platonic solids.”

In Plato’s case, however, such geometric forms were extrinsic, part of a transcendent reality, while for Aristotle they were intrinsic, the universal structural fundamentals of substance. This shift to the structures of daily experience is widely regarded as a watershed event in the history of philosophy, and earns Aristotle the title of “the father of empiricism.” But importantly for our discussion, it was an empiricism anchored in a no less Platonic kind of “fundamentalism of form.” In both cases, the categories of form (whether seen as ideals or universals) were fundamental, aggregate constituents.

This is a view of form that, in both cases, sees a particular form as reducible to simpler forms, and ultimately, to a primary and eternal set of mathematical forms and ratios. It follows that the act of creating form is the reverse: a kind of hierarchical process of applying primary structures according to correct rules of proportion, scale, fitness and so on — which, if executed correctly, delivered the Good, the True and the Beautiful, as part of a single ordered process.

Idealism and Hylomorphism in Early Modern Theory

This essentially reductionist view of structure held powerful sway over the coming tradition of Western science, and provided the mental framework to unlock a long series of astonishing secrets of nature — to say nothing of the prodigious industrial advances that ensued. But it slowly became clear that this reductionism was also, as we will discuss presently, not the entire story, and some of the minor bits left out are much more critical than we realized — a revelation that we are grappling with still today.

But in the meantime, this structural reductionism also held powerful sway over designers and architects for generations to come — and over their most influential theorists. For example, it can be seen clearly in the writings of Vitruvius in the Classical era, and his doctrine that architecture is the imitation of nature, and of its fundamental patterns — culminating in his conception of the“Vitruvian Man,” later drawn by Leonardo da Vinci: the human figure, decomposed into the square and the circle. It can also be seen clearly in Alberti’s early Renaissance classic De Re Aedificatoria, which also echoed Plato, Aristotle and Vitruvius, and described a fundamental ordering of circles, squares, and harmonic relations between them.

Indeed, the same reductionism can be seen readily from the writings of the most influential early Modernists — and in fact they took this reductionism to a remarkable extreme. Perhaps this reflected an over-confidence in the late nineteenth century, that the gifts of technological modernity, expressing the logical rigor and progressive spirit of the Enlightenment, had finally delivered on the ultimate promise of Hellenic rationality. The payoff was at hand.

Here is Le Corbusier’s critique, presented in his book Toward a New Architecture (1924), of Gothic architecture, presenting a remarkably Aristotelian argument that, in essence, Gothicism gets its hylomorphism all wrong:

Gothic architecture is not, fundamentally, based on spheres, cones and cylinders. Only the nave is an expression of a simple form, but of a complex geometry of the second order (intersecting arches). It is for that reason that a cathedral is not very beautiful and that we search in it for compensations of a subjective kind outside plastic art. A cathedral interests us as the ingenious solution of a difficult problem, but a problem of which the postulates have been badly stated because they do not proceed from the great primary forms.

The geometrical fundamentalism of the phrase “great primary forms” is striking. But if “a cathedral is not very beautiful,” then what is beautiful, and why? For Le Corbusier, it is those structures that arise directly from the Platonic fundamentals of mathematical calculation — like American grain elevators:

Thus we have the American grain elevators and factories, the magnificent FIRST FRUITS of the new age. THE AMERICAN ENGINEERS OVERWHELM WITH THEIR CALCULATIONS OUR EXPIRING ARCHITECTURE.

This picture is in contrast to the usual opposition of modernism to classicism. Certainly the former emphasis on ornament and decoration was stripped away, replaced by a new minimalist celebration of the logic of the machine. But the understanding of architecture as a composition of primary forms, bound by the timeless Pythagorean laws of mathematics, is consistent throughout. Indeed, modernism’s central narrative of the “end of history” — its faith in an arrival beyond the messiness of happenstance, delivered by logic and reason into a new age of mechanical perfection, can now be seen for the Hellenic idealism that it is. At last, we would achieve the rational ordering of our buildings, our machines, our very culture, in a glorious flowering of rational modernity.

Incompleteness, Uncertainty, and Complexity

And yet, at precisely the moment that architects were celebrating the triumph of reason in the new modernity, scientists and philosophers were busy making new discoveries that had a devastating effect upon it. These discoveries came inexorably from the further teasing out of the complex structure of things, in fields such as biology, physics, information theory and — most disheartening for logical idealists — mathematics and logic itself.

In 1912, Whitehead and his Cambridge colleague Bertrand Russell, in their classic treatise Principia Mathematica, attempted to lay out a reductionist scheme for all of logic and mathematics. Their result is widely considered to be one of the most important works of mathematical logic since Aristotle. But Kurt Gödel, in an earth-shaking 1931 paper, managed the neat trick of using the machinery of Whitehead’s and Russell’s logical scheme to disprove itself — more specifically, to prove that in spite of their intended aim, and whatever else its merits, their work is and must be logically “incomplete.” More than that, he managed to prove that any such system must be, at least to some degree, “incomplete.”

The implications, developed in further papers and in philosophical work since, were staggering. There can be no perfect blueprint of nature, no reductive model to fully explain the logical structure of things. This would seem to be a powerful blow to Platonic idealism, and no less, to Aristotelian hylomorphism.

At the same time, physicists were observing very strange behavior at the subatomic level, with powerful implications for the rest of nature’s structure. Far from assuring us, in Platonic fashion, of an unseen solid realm, experimental results seemed to point the way to ever more impermanence, endless variability, and even randomness. With each new discovery, the universe looked more and more complicated, more and more a cloud-like shower of energy particles (or were they waves?) that only temporarily coalesce into “things” or forms, and then fly off to coalesce into other things, following complex pattern-like rules. (This seemed to echo an earlier and decidedly non-Platonic theme in Hellenic thought, the dynamicism of Heraclitus.)

The lessons of biology were no less astounding, and humbling for fundamental reductionists. We ourselves turn out to be vast-trillion-member societies of semi-independent, cooperating micro-organisms, few of whom live longer than about seven years. (In fact every one of those cells contains two separate symbiotic organisms with separate genetic histories.) And while we do seem to be generated from simpler processes, the whole scheme is not at all the neat picture of an earlier age, but rather, one of coded transformations and interactions of stupendous complexity. The cells, rather than forming “top-down” according to neat divisions and assignments, seem to know how to “self-organize” into over-arching patterns that tend to resemble the primary forms of mathematics, but do not conform at all precisely. (For example, we have a sphere-like, but not precisely spherical, structure in our eyes, and a cylinder-like, but not precisely cylindrical, structure in our fingers, and so on).

Whitehead himself helped to articulate important outlines of a new “structuralist” (or “process”) philosophy that was rooted more in the workings of processes over time — generators of endless varieties of form whose products bore the marks of this diachronic and evolutionary process. In his later work he outlined a “process philosophy” that could explain the cloud-like particulate forms of nature that were more consistent with the powerful new picture emerging from the sciences — described in books such as Process and Reality and Science and the Modern World. And he sought to explain the relation of our own ideas to these realities of process, and the dangers of confusion that lay therein, in books like Modes of Thought and Adventures of Ideas.

The postmodernist debacle

Gradually, of course, these ideas seeped into everyday life — and, ultimately, our understanding of culture, the arts, and architecture. We came to see that we live in a less than rational world, where accidental political events could result in nuclear annihilation, or unintended economic forces could result in a global depression. Worse, even our own beautiful technological structures could malfunction in horrendous but unforeseen ways, and deliver not utopia, but pollution, ecological destruction, climate change, and economic decay.

A series of disastrous failures of modernist architectural and urbanist projects — insightfully critiqued by a generation of scholars like Jane Jacobs — reached a crisis point in the 1960s and 1970s. The achievements of rationality and logic had culminated in the modern technology of city-making: but it had not delivered utopia, but a persistent fragmentation, disorder and decay. There was a flaw in the neatly ordered system — and it was not merely in its surface application, but increasingly, could be seen to reside at the logical core of it. Gödel’s critique seemed to apply to the very logic of modernism: it was fatally, catastrophically, incomplete.

The ensuing counter-reaction to modernism was dubbed, of course, “postmodernism:” marked by an abandonment of the fundamentalist quest for ideals of form, and a return to the diachronic qualities of language and narrative. Architecture would now be a kind of story that a civilization tells about itself: highly symbolic, self-referential, ironic. The failure of Hellenic rationality would itself become part of the story.

The postmodernist critique of modernism was articulated particularly clearly by the architect Rem Koolhaas, in his essay, “Whatever Happened to Urbanism” (in S, M, L, XL, 1995):

Modernism’s alchemistic promise — to transform quantity into quality through abstraction and repetition — has been a failure, a hoax: magic that didn’t work. Its ideas, aesthetics, strategies are finished. Together, all attempts to make a new beginning have only discredited the idea of a new beginning. A collective shame in the wake of this fiasco has left a massive crater in our understanding of modernity and modernization. . . Since then, we have been engaged in two parallel operations: documenting our overwhelming awe for the existing city, developing philosophies, projects, prototypes for a preserved and reconstituted city and, at the same time, laughing the professional field of urbanism out of existence, dismantling it in our contempt for those who planned (and made huge mistakes in planning) airports, New Towns, satellite cities, highways, high-rise buildings, infrastructures, and all the other fallout from modernization.

This postmodernism in art and architecture was mirrored by a post-structuralism in philosophy, built upon an empirical and even logical suspicion of the presumed powers of knowledge and reason. It, too, had its roots deep in the story of Western philosophy. It owed a particular debt to Immanuel Kant’s earlier philosophy of Transcendental Idealism — itself a confrontation with the evident limits of knowledge.

Kant’s Transcendental Idealism can be thought of as a kind of inversion of Plato’s idealism. We do experience something roughly like Platonic forms, in a sense — in the experienced structures of words and ideas — but they are to be treated as a priori categories of mind, and not, in any sense deeper realities of the world — which, even if they existed, could never be understood except through the very same categories of mind. Thus, if we wanted to be logical, we must begin there in the realm of mind — and we must inevitably find ourselves trapped there. Ontology is forever bounded by epistemology.

The post-structuralists went a step farther, taking as their departure point the linguistic structuralism of anthropology. But whereas structuralists like Saussure and Levi-Strauss saw language as a malleable tool for the anchoring of an external meaning within a culturally variable structure, post-structuralists like Derrida and Foucault dispatched with external meaning altogether, and focused instead on the relative ways in which cultures and their sub-groups construct narratives of “meaning” (or reliable knowledge, or truth) as a wholly synthetic structural process. Echoing the tension between Plato and Aristotle, they dispensed with any external concept of meaning, and (if they considered it at all) saw it as something that inhered within the structure, and the structural differences, of language itself.

The post-structuralists also saw the ways that different groups use this construction process to gain political control of one another, by imposing the structural narratives by which cultural meanings are defined. Thus the task of a philosopher is now to analyze this construction, and deconstruct it into its constituents so as to expose its inner workings. We may not be able to determine the relative merits of one set of meanings over another — indeed, that would be “privileging” one group of constructors over another — but we can certainly reveal the hegemony of one group and its constructions (say, colonial powers) over the constructions of others (say, indigenous peoples).

In architecture and the arts, this philosophy expressed itself in acts of deconstruction of the narrative texts of power: exposing the constructions of power elites, so as to fuel an art of liberation. Thus we may decry the concentration of corporate power — perhaps finding it even among our own clientele, if we are architects — but we can still “deconstruct” this reality, and reveal it within our art. This kind of art may not be transcendent in any metaphysical sense — nor does it any longer claim to be. Rather, it is meant to derive an emotional power from its cathartic eloquence. If we cannot fix the world, we can at least understand its hidden structure, and celebrate its revelation.

Post-structuralism was thus a logical evolutionary step, and a useful cautionary narrative on the powers of reason. Like a good Zen koan, it served to break the spell of ideas over those who applied them too rigidly — in particular, the demonstrably specious claims of some philosophical absolutists (including some neo-Platonists) to have “got it right.” But beyond describing these limits, it did not go very far in explaining to us what is really going on with language, and its structural relation to the emerging scientific picture of nature — and in particular, the evident phenomenon of biological intelligence.

Indeed, there are fundamental logical flaws with the post-structuralist account that leave it, at best, highly compromised in its ability to provide a useful explanation of what is going on in the structure of things—at least beyond the narrow confines of cultural creativity. In its logically purest form, its doctrine against external meaning and truth is ultimately self-contradictory, since, as Gödel showed, we are no more able to escape the problem of incompleteness within language than outside it. The illusion of escape comes from a categorical confusion, forgetting that language is just as “real” a structure in the world as any other — with all the same epistemological perils. (We will come back to this logically confusing point in more detail in the next chapter.)

This categorical confusion leaves the post-structuralists with the inability to form a coherent theory of even what they themselves are doing, and how it applies to any “thing” — even their own construction — beyond a kind of elaborate word-game. But the process of science requires that we posit a structural model of “what is going on” — and then test that model for conformance, revision, and refinement. Thus we are required to have a meta-model of fidelity, or what some call “the ring of truth” — which is not the same thing as absolute congruence with truth. Thus, in its essentially all-or-nothing form, the epistemological doctrine of post-structuralism is at best severely limited in its empirical usefulness, and at worst, fundamentally irreconcilable with a coherent scientific project. (A point of harsh criticism during the so-called “science wars” debates.)

One may deny the validity of such a scientific project, of course — or, like many post-structuralists, argue that it is merely one more constructed narrative. But that seems charmingly oblivious to the practical urgency of the situation at hand. We are rather like passengers in a car that we ourselves have made, and we are finding it unusually hard to steer. Should we now propose to stop steering altogether, because we have found error, partiality, and bias in the car’s manual? This seems at best an over-reaction, and at worst, a kind of pervasive logical confusion about the useful, if imperfect, role of language. (We will discuss this “useful imperfection” in more detail below.)

Indeed, a more coherent picture of the situation is now being conveyed by fields like neuroscience and information theory, which have brought new insights into the workings of language and the brain, and their place in the structure of nature. As I will explore in the next chapter, these and other developments point toward a revived structuralist account of what is going on — bringing with it a series of profound new implications.