Participants: Emmanuelle Chiappone-Piriou, Jose Sanchez, Casey Rehm, Jordi Vivaldi, David Rozas, Giorgio Lando, Daniel Koehler with questions from the audience including Mario Carpo and Philippe Morel.

Daniel Koehler: The talks of the symposium were diverse and rich but also abstract, and intentionally external to architecture. At such a point it can be asked if, how, and what role Mereologies can play in architecture? For the discussion we are joined by additional architects with unique angles on composition and part-thinking in their work. Casey Rehm, a computational designer, Jose Sanchez, who is working actively with digital models of participation and Emmanuelle Chiappone-Piriou, an ecological thinker, experienced in the history of architecture. 

José Sanchez: My first reaction to the presentations is controversial. I think it presents well much of the work that is happening in architecture at the moment showing an interest in Mereology and discrete architecture. However, looking at the issue of parts is fundamentally a project where the idea of composition and the idea of structure is relevant as well. Patterns organised by parts can potentially deal with different forms of value. So, in a way, I find a surprising rejection in some of the ideas. 
Mereology seems to be giving us a framework for many different positions to coexist, and I think that we did an excellent presentation of a much clearer advocacy for a form of relations that we might desire that has to do with pre-production, more like an agnostic framework that allows to give us a vocabulary. Are we interested in having advocacy, in having that intentionality, or are we more interested in what the ontology should be or the framework that we are going to work in?

Daniel: I have learnt something from Giorgio’s book that when we define Mereology, it comes in different notions and ranges. On the one hand, you can see it as a distinct theory, as a specific project that has its own agenda. But also, and more crucial in the first place: you can take Mereology as a larger framework to talk about the relations of parts to wholes – simply compositions. OK, but you might ask: why don’t we use the term composition directly? Because, composition has a specific connotation in architecture and refers to the Ecole des Beaux Arts, classical means of relating objects. It was rejected by the Bauhaus, which promoted a different form of composition with modern means. We could continue this through the history of architecture. In architecture, composition is a specific style but not a history. How could we compare those different modes of architectural composition? Can we think of something parallel to morphology or typology which would allow us to compare a plurality of relations between parts and wholes without defending a certain style? When the formal readings of parts turn into their own project, it might be quite valuable that one can figure a figuration without predefining its value by imposing a structure. That might be Mereology as a project. But first of all, the question is how can we intentionally speak about parts? That would be Mereology as a methodology.

Giorgio Lando: I agree with Daniel that it is very important to distinguish various ways in which the word “Mereology” can be legitimately meant. In particular, the word “Mereology” stands in some cases for a specific theory of parthood and composition, and this theory may be such that structure has a role in it, or such that structure has no role in it. A historically important kind of mereological theory, Classical Mereology, is of the latter kind: it is deliberately blind to structure in providing existence and identity conditions for complex entities. In other cases, however, the word “Mereology” stands for an entire field of research, within which competing theories disagree about the role which structure should – or should not – play. If Mereology is seen as a field of research, then it is misleading to say that structure plays no role in it. This equivocation may explain some of José’s perplexities. 
However, some other perplexities are likely to persist even once we disambiguate the word “Mereology”, and we focus on Classical Mereology. Classical Mereology indeed includes some highly counterintuitive principles, and the usual reaction of the layman to these principles is to dismiss them rather quickly. For example, it might seem prima facie incredible that the order of the parts of something does not matter for the identity conditions of complex entities. However, this quick dismissal is usually determined by an equivocation: what is actually incredible is that the order of the parts of a building, or of a village, or of a car does not matter for its nature, for what that building, that village or that car is. However, this is not what Classical Mereology claims. What Classical Mereology claims is weaker and more reasonable: it says that the order of the parts does not matter for the identity conditions of complex entities, such as buildings, villages and cars. 
According to Classical Mereology, it never happens that there are two distinct entities which only differ because of their structure. Classical Mereology is not committed to the frankly incredible claim that structure has no impact on the nature of complex entities, but only to the more reasonable claim that complex entities are never distinct only in virtue of their structure. 
Moreover, this claim of Classical Mereology is restricted to single concrete entities. This might make the confrontation between Classical Mereology and other disciplines, such as architecture, troublesome, inasmuch as these disciplines are more interested in abstract types than in concrete tokens, more interested in repeatable entities than in their single, concrete instantiations. As far as I understand, when architects speak about the parts of a building or of a city, in most cases they are not speaking about a single piece of material and the way in which it is composed, but about a type of building and the fact that there are different types of buildings which result from the combination of the same types of architectural elements, differently combined. 
Once you move from this level of types and abstract entities to the level of concrete entities, the claim of Classical Mereology that structure has no role in the identity conditions of complex entities is much less incredible: consider a single, concrete building (not a type of a building) in a certain moment in time. In that moment, its parts are structured only in one way: the parts of a single, concrete building cannot be structured in two different ways at the same time.
Architects might legitimately retort that architecture is about repeatable types of buildings, about projects which can be applied several times. Given this approach, Classical Mereology is probably not the best tool for modelling repeatable types, and it is indeed desirable to look at different theories, which are not deliberately blind to structure. Mathematics is full of tools which can be employed to this purpose, including set theory and various kinds of algebras. Architects may legitimately wonder why philosophers focus on Classical Mereology instead, which is a serious candidate for the role of sound and exhaustive theory of parthood and composition for single concrete entities, but not for abstract types. The reason is probably a sort of deep-seated philosophical skepticism towards abstract entities, and the idea that fundamental reality consists of concrete entities, while abstract entities are less fundamental, or even a mere construct of the human mind.list or minimalistic inclinations
However, it is not the case that all the philosophers working on Mereology endorse the claims of Classical Mereology. In particular, in the literature of the last ten years, many prominent philosophers (such as Karen Bennett, Aaron Cotnoir and Katherine Hawley) have by contrast argued that Classical Mereology is completely misguided, and that we should also pay attention to structure within the realm of concrete entities. In my book I have defended the claim that, by contrast, Classical Mereology is a perfectly adequate theory of parthood and composition for concrete entities, but many other mereologists disagree with me. More in general, there is virtually no claim about parthood and philosophy about which every philosopher agrees! 

Mario Carpo: Giorgio, you have said that at some point Mereology merges with set theory. What exactly is here the overlay or intersection between Mereology and set theory? In reverse, where is Mereology separating itself from set theory, and where are the core differences?

Emmanuelle Chiappone-Piriou: Is there any way that relates Mereology to category theory?

Giorgio: For what concerns the relation between set theory and Classical Mereology (which, as we have seen, is a specific theory, which is mainly designed to characterise the realm of concrete entities and the way in which they are part one of another), the deepest difference consists in the transitivity of the relation: the relation of parthood in Classical Mereology is transitive, while the relation of elementhood in set theory is not transitive. Thus, if a first entity is part of a second entity and the second entity is part of a third entity, then – according to Classical Mereology – the first entity is part of the third entity. By contrast, it can happen that something is an element of a set, which in turn is an element of a second set, while that something is not an element of the second set. Sets are stratified: you have typically sets of sets of sets. In Classical Mereology, as a consequence of the transitivity of parthood, there are no stratified complex entities. 
While there are many interesting ties between set theory and Mereology, I am unaware of any connection between Mereology and category theory.

Mario: Can you give us maybe an example, like three inclusions in set theory and three inclusions in Mereology?

Giorgio: Consider the set of Italians. I am a member of this set. The set of Italians is also a member of the set of European people. However, I am not a member of this latter set, inasmuch as I am not a European people (I am not a people at all!). We thereby obtain a failure of transitivity of elementhood among sets. Nothing similar is admitted by Classical Mereology: I am part of the fusion of Italians, the fusion of Italians is part of the fusion of Europeans, and I am part of the fusion of Europeans as well.

Mario: So, in set theory, these don’t happen?

Giorgio: It does not happen in the sense that it does not always happen. There are indeed cases in which the same elements appear at different levels of the set-theoretical hierarchy, but this does not happen in general, and is not warranted by any principle of set theory. There are actually many varieties of set Theory, but in no variety of set Theory is elementhood transitive.

Philippe Morel: My feeling is that Mereology is a matter of “technicalities” about a relationship that exists in set theory. If you look at the inclusion as the property you are also looking for in Mereology, I don’t really get what Mereology brings on top of the purely mathematical “canonical” set theory. It gives me the feeling that Mereology is foremost a way (or a “trick”) for philosophers to take control of a theory that escapes them because it is a fully mathematical theory… So, this is why I have a bit of a problem with this notion because again, technically speaking, I still can’t make a clear distinction between the philosophical property and the mathematical property. It is like a layer of metaphysics that is brought on top of the mathematical theory and of course I can’t consider this as a great addition. My second issue is more of a general remark. Why don’t you speak about relational databases like SQL databases? At some point, to my understanding, it is a very practical implementation of what describes Mereology, because it is all about belonging, etc.
Though, I find the mereological approach interesting, especially if it prevents a reintroduction of composition, as I see a danger of bringing back this concept of composition in architectural discourse.

Giorgio: You are right: set-theoretical inclusion (i.e., the relation of being a subset) has precisely the same formal feature of mereological parthood. However, set-theoretical inclusion is not the fundamental relation of set theory: it is definable in terms of set-theoretical elementhood, while set-theoretical elementhood is not definable in terms of set-theoretical inclusion. Thus, the fundamental relation of set theory is elementhood and is not transitive, while the fundamental relation of Classical Mereology is parthood, which is transitive. 
There have been several attempts (for example in Parts of Classes, a book by David Lewis) to exploit the formal analogy between mereological parthood and set-theoretical inclusion in order to reduce set theory to Classical Mereology. The biggest obstacle for this project are set-theoretical singletons, i.e. sets with a single element. The relation between these single elements and their singletons is not easily reducible to Mereology: it is a kind of brute stratification (a form of structure), which has no place in Classical Mereology.
I agree with Philippe’s remark that Classical Mereology is nowadays a mathematically uninteresting theory, in spite of the fact that it has been originally elaborated by great mathematicians such as Stanisław Leśniewski and Alfred Tarski: it is simply a complete algebra without a zero object. The reason why philosophers discuss Classical Mereology does not depend on its alleged mathematical originality: some philosophers (including me) think that this very simple and unoriginal mathematical theory is the sound and complete theory of parthood and composition, at least in the realm of concrete entities. Thus, the reason to be interested in Classical Mereology is not its mathematical originality, but its plausible correspondence with the way in which parthood and composition really work.
As far as datasets are concerned, I think that it is prima facie preferable to construe them as sets rather than as mereological wholes. Indeed, the distinction between inclusion and elementhood is pivotal for datasets. This distinction characterises set theory, while there is no analogous distinction in Classical Mereology.

Daniel: I would like to extend on Giorgio’s point that Mereology offers mathematically an algebra without a zero object. Mereology starts with individuals without defining a set in the first place. In Mereology, you can’t have an empty set, a null set, a zero object. You can’t have a building without building parts. You need parts for thinking a building. This will become more dominant in future because with higher computing capabilities we are able to compute more and more without the need of abstract models. Take as an example the Internet of Things: a building environment where every building part has sensors and is connected. That means that very literally building parts can talk with each other. Such a building environment also participates, and will offer its own economy. Here, value begins with a building part as an active participant in the market. Already in daily BIM practice it is impossible to think of a building without its parts. So, we should also stop thinking of buildings as predefined sets.
To my understanding, a database is constructed on a very specific ontological worldview. Today’s databases take Composition-as-Identity. This principle says that everything is included in the distribution of data points. Nothing above the distribution of atoms exists, not any compound meaning. Whereas, compounds are fundamental to architecture. Just think of a typology; you can’t reduce a façade to windows. What does a courtyard actually exist of? This of course does not relate to math but to philosophy. It is controversial, otherwise it would not be philosophical. Every building is controversy, or call it multiplicitous, because architecture is pre-logical in a sense. We can’t reduce architecture to math. It is also the point where the discussions on beauty depart in architecture. With ease you can describe a building in the first instance through the distribution of its cells. You can describe a housing project just through the part-relation of a shared wall between two flats only. But how do you describe the mountain which Moshe Safdie designed by stitching together the shared walls of flats in such a way that their roofs turn into terraces? Architecture starts where it exceeds simplicity. Yes, we can design buildings with the use of databases with ease. We are able today to compute buildings without structures. But where are their compound meanings? It will be fundamental to find a way to compute what is common, what is collective between the parts. Therefore, I think we should be suspicious of databases or any kind of structural models which were thought without any compound meaning, so to say, without architecture in the first place.

Jose: I’ll re-bring some of the points that Jordi made to the conversation. Jordi, you brought up Graham Harman’s concept of a radical present. I find it kind of controversial that it seems to eradicate a form of speculation, a form of potential, a form of endless abstractness. If we’re moving from the classic Mereology towards a more abstract sense, I think that a lot of architecture production that we discuss especially with discrete projects – that has to do with parts – has to do with potential encounters of entities in that list and is not purely defined by the actual instantiation of the actual encounter of entities. So, we evaluate and design, also thinking that encounters might never happen. So, under the umbrella of that radical present, I wonder what do you see in them?

Jordi Vivaldi Piera: I would say that the term “potential” is misleading. Its meaning generally refers to its capacity to produce other realities, but at the same time it undermines the possibility of novelty because it assumes that an object already contains what it will become. In this sense, I emphasise radical presence in order to understand which object’s “actualities” permit the production of novelty, rather than understanding which are the hypothetical novelties that it contains and therefore at the same time undermines. In this sense, I interpret potentiality as a particular type of actuality.

Casey: I was interested in Daniel’s point; it reminds me of a recent article by Luciana Parisi called “Reprogramming Decisionism”, where she’s talking about machine learning, neural networks and that these technologies in essence assemble. With this, fact is accumulated, which says that something is probably something else. I’m interested in this relative to Mereology and also the statement that a human deals with abstraction but a machine deals with simple facts. How does the mereological project deal with probability? Is that something probably something rather than not? How does the part, certainly something like, you know, the models that you have shown us rely on clear logic? As I nearly understood there is a kind of model that you’re describing, but how does Mereology deal with improbability? I think it is also something that is going to face the design profession in relationship to the kinds of machines which deal with things. 

Giorgio: As far as probability is concerned, I do not envisage any specific, direct problem stemming from the interaction of probability and Mereology. A mereological claim can have a certain degree of probability, and the probability at stake can be either objective/statistical or subjective. In neither case are there specific problems: mereological claims are, from this viewpoint, on a par with other claims. 
While probability is not directly troublesome, there are some potential problems in the vicinity: Classical Mereology does not countenance the hypothesis that an entity is part of another, but only at a certain degree. Consider a cloud in the sky: the water molecules in the centre of the cloud are definitely parts of the cloud, and the molecules far away from the cloud definitely are not parts of the cloud. However, there seems to be a grey zone of molecules, which are neither definitely within the cloud nor definitely out of it. 
These scenarios can be treated in various ways, and the approach depends on the adoption of a certain theory of vagueness. According to the so-called epistemic theory of vagueness (set forth for example by Timothy Williamson), the fact that we are unable to identify the boundaries of a cloud depends on our epistemic limitations (we are unable to identify the boundaries of the cloud, but this does not show that the cloud has in itself no definite boundaries). According to the semantic theory of vagueness (in the version adopted for example by David Lewis), there are actually myriads of clouds and each cloud has precise boundaries; however, our discourses about the cloud are semantically underdetermined, inasmuch as we have not decided which among the myriads of clouds in the sky we are speaking about. Both the epistemic theory of vagueness and the semantic theory of vagueness are perfectly compatible with Classical Mereology, because they locate vagueness in our language or in our epistemic practices and not in reality: in reality, given two entities, either the former definitely is part of the latter, or the former definitely is not part of the latter.  
However, recently also the so-called ontological theory of vagueness (Michael Tye is one of the most ardent advocates of this approach to vagueness) has gained some traction. According to the ontological theory of vagueness, vagueness is in reality, and this happens also in the mereological case of the cloud: the molecules at the periphery of the cloud are neither definitely parts nor definitely non-parts of the cloud. The adoption of the ontological theory of vagueness indeed requires a revision of Classical Mereology. According to Classical Mereology, for example, two complex entities are identical if and only if they have the same proper parts (the proper parts of something are those parts of it which are not identical to it): but this principle is not applicable to entities which have no definite domain of proper parts. According to the ontological theory of vagueness, this is what happens in the case of the clouds and in similar cases. To sum up: probability and various theories of vagueness (such as the epistemic theory and the semantic theory) do not require any departure from Classical Mereology; only the ontological theory of vagueness requires a departure. 

Emmanuelle: It appears we are navigating and combining different sets of discourses that may or may not be consistent with one another, nor with Mereology as it appears here to be merge into a compositional paradigm: we are simultaneously addressing materiality and formal systems, social coherences and principles of governance, all at once.
I believe that, as in the 1950s and 1960s, architecture faces the risk of talking itself into an impasse, by resorting to certain languages and positions that may induce, and reproduce, a reification of social patterns. 
In this context I often think of a remark from Michel Ragon, the French architecture critic who wrote about and promoted experimental architecture in the 1960s. Looking back at those projects, twenty years later, he asked himself how a “life-like” macro-structure could be designed in advance, and if it could be designed at all, considering life is “rightly made of chance and unpredictability”. This remains a valid and important question, which is updated by our resort to instruments that allow us to think of, and manipulate, the world in terms of particles and parts. Quantum physics teaches us that there is irreducible uncertainty in our physical existence, an inherent contingency, and that there is a fundamental limit of precision with which you can actually measure a particle, hence a limit to the precision with which you can grasp the world. How is it that this uncertainty can be taken into account when dealing with matter or with information; and, when dealing with parts, how can we do so without first defining them? How can we account for interactions and relationality? How is it that we can account for change, for performance and transformation, all at once?
This brings me to a second point that stems from this a priori impossibility to capture the image of life without “to some extent captur[ing] life itself” (Ragon). I understand that Mereology makes a claim for exhaustibility and generality. But what if we take this claim into the architectural project? Do we think that we can actually design a system, a structure or a whole whose formal principles allow for it to be exhaustive? Following Gödel, I understand that you either have exhaustibility or consistency, but not both. 

Mario: Can I go back to the branch of theoretical philosophy to cover things? We more or less know why we in the design profession became interested in particles, and the relation between particles, in recent years. It seems he (Daniel) came across the term Mereology. He hijacked it and imported it into the architectural discourse. Like we always do. We take a more refined tool which comes from another discipline, and then we appropriate it and give it another meaning which means nothing to you (Giorgio). This we have been doing for a long time. This part of the story we know. The part of the story that we don’t know, that you can tell us in two lines is, does this happen with Mereology? Can you give us an outline of the history of analysis of Mereology in contemporary philosophical discourse? Because when I was a student nobody mentioned Mereology, and now everyone does? When did that happen? Where does this come from? And from a distance, from a critical point of view, why is it that you right now are talking about Mereology while many years ago nobody talked about it?

Giorgio: The word ‘Mereology’ is rather new and was made relatively popular by Stanisław Leśniewski at the beginning of the 20th century (according to Leśniewski, Mereology was more properly a branch of logic). However, philosophers (and in particular metaphysicians) have always used the notion of part and set forth theories about it. Plato’s theory of parthood has been recently analysed and defended by Verity Harte, while Aristotle’s theory of parthood is considered by several neo-Aristotelian metaphysicians a viable option in the contemporary mereological debate.

Mario: But, in math, there are fractions, proportions, modularity. These are all today discussed as mereological questions.

Giorgio: An important difference between many past theories of parthood (in particular in Ancient and Medieval philosophy) and contemporary Mereology concerns the expected domain of application: Plato, Aristotle, Abelard and Ockham were for example mainly interested in the parthood relation which connects a property with an individual instantiating those properties, or two properties one with another. These instances of parthood were important within metaphysics itself, for example when a theory of ideas or universals was elaborated. By contrast, contemporary Mereology is more focused on the concrete, spatio-temporal parts of concrete entities.
However, no matter what the original domain of application of the parthood relation was, the theories of parthood became progressively more abstract and formal: in some works of Leibniz (17th century), for example, it is possible to find a formally complex and highly abstract theory of parthood, whose principles are expected to hold irrespective of the domain of application. This is also the case of the theory of parthood developed by Bernard Bolzano in the 19th century. Thus, in spite of the fact that the word ‘Mereology’ became popular only in the 20th century, contemporary Mereology has solid roots in the history of philosophy. 
Nonetheless, it is true that – for example – forty years ago Mereology was much less popular than nowadays. This may have depended on the alternating fortunes of metaphysics (the wider branch of philosophy to which Mereology belongs) in analytic philosophy. Forty years ago analytic philosophers, in continuity with logical positivism, often despised metaphysics as an obsolete leftover from the past. This has changed dramatically in the later decades, thanks to the influence of thinkers such as David Lewis and Saul Kripke, and metaphysics is now back at the centre stage of contemporary analytic philosophy. The renewed popularity of Mereology is an aspect of the renewed popularity of metaphysics in general. This also depends on the fact that contemporary metaphysicians often attach great importance to the concepts of existence and identity. Classical Mereology has the ambition to provide existence and identity conditions for every complex entity. This makes Classical Mereology highly interesting for contemporary metaphysicians. 

Philippe: Let’s make a comparison with the discipline of architecture. In architecture, this last trend could be compared to what happened with Christopher Alexander, or before with Mies and then Peter Eisenman. The challenge for me is that I don’t consider Mereology an uninteresting philosophy in architecture, I just see it as a highly modernist theory.
My question is the following. According to you (Giorgio), in the field of philosophy, do you consider Mereology as a modernist philosophical trend or something that has nothing to do with philosophical modernism? Because in architecture, my feeling is that it directly corresponds to a highly modernist attitude, and the fact is that this modernist attitude is highly reductionist. It is defining what is the most elemental aspect of things, so it’s pure reductionism, and it’s still based on some concept of – maybe not order, but at least some attempt at bringing order into things (though sometimes “unpredictable order”).
For me, that is super modernist and my feeling is that we are living in a world built on this reductionist modernity. Right after this reduction – and we already had it in some form a hundred years ago –, let’s say after 1950 we were already going into the opposite direction: an explosion of models… That one is now based on statistical methods, on big data, as related by Mario in his book. So again, I’m not saying Mereology can’t be an important or at least a useful platform for debate, I am just wondering about the inherent nostalgia of going backward in the ordering of reality – in History. Maybe we can – and should – just accept absolute chaos and trillions of trillions of terabytes of data as a fact, without trying to put some order into that. So, my question finally on a purely philosophical level is: do you consider Mereology as modernist, or maybe as a new modern or late modern philosophical theory, or as something which has nothing to do with that?

Giorgio: There is indeed a modernist component in Mereology: the deliberate blindness to structure, which characterises Classical Mereology, is motivated by a form of “taste for desert landscapes”, which in turn might be seen as the outcome of a modernist appetite for order. However, it should also be considered that Classical Mereology includes either as an axiom or as a theorem (according to the way in which Classical Mereology is axiomatised) the principle of Unrestricted Composition, according to which – given some entities, no matter how sparse and gerrymandered they are – they compose something. Due to Unrestricted Composition, Classical Mereology is committed to the existence of all sorts of awkward entities, such as the fusion of my left arm, Barack Obama’s nose and the Great Pyramid of Giza!
On the other hand, a rather “modernist” thesis, which is often associated with Classical Mereology, is the thesis of Composition as Identity. According to the thesis of Composition as Identity, any whole is strictly speaking identical to its parts and is – so to say – no addition to being, with respect to them. This mereological thesis is expected to warrant a form of ontological economy, and can be seen, as a consequence, as the outcome of an appetite for order. 
However, Composition as Identity is not derivable from Classical Mereology, and is a highly problematic thesis in itself. A whole (for example, a chair) and its parts (the four legs, the back and the seat) are mutually discernible, inasmuch as – for example – the chair is one entity, while the four legs, the back and the seat are six entities. If they are discernible (i.e., if they have different properties), then it is not easy to make sense of the claim (entailed by Composition as Identity) that they are identical.

Casey: I think you have covered everything I want to say. Based on this I don’t think there is anything suggestively reductive about composition. I think that it is a ridiculous idea that unrestricted composition suggests that this property could be part of something.
My colleague Daniel is doing the mereological project, but it is certainly nothing reductive. I think it’s more that there is a very explicitness and straightforwardness about the roles and function of the thing, i.e. the function isn’t the exclusive part of the composition, especially according to the kind of lectures we saw today.

Mario: I have a suspicion. I see one main point of this symposium is that in the theory of parts of  today’s computation the parts we are dealing with are new in the history of architecture theory because they don’t need rules of application. These parts are different from Alberti’s or Eisenman’s because for the first time ever in the history of humankind or the history of design we can deal with parts without any rules or orders in them whether it is proportions, fractions, modules, geometrical symmetry, proportional symmetry, etc.
In the history of design, all these tricks and tools were needed to make sense out of parthood. We had to invent structures, like reductionism or data compression, to put some order into the chaos generated by the random accumulation of parts–to make order out of chaos; to manage parts in a “rational”, ie intelligible way: a way that made sense for the limited data-management skills of our own mind.  And now for the first time ever in many practical instances we are getting particles just as they are. We can put them flat on the table and each one of them stands, and that is all that we need. This the nobility of the parts that you’re dealing with. This is the novelty: parts without anthropocentric reduction and human-made intelligibility. 

Casey: Do you say that there are no rules for these parts or is it just that the rules are inherited in the parts and not applied to the total? I’m suspicious of saying that (the former) in dealing with parts. And again, we still have rules because we have generated something that is mereological. There are still rules but the rules are in the parts rather than trying to be imposed on them. And so actually, it is just where the rules are located in the design process. 

Mario: There must be rules of some sort somewhere, but the main difference, and again, I follow my suspicion, we no longer need rules to manage the accumulation of parts beyond the limit of computational (ie machinic) retrieval.  We don’t need to structure them in symmetrical parthood or any other strategy for part retrieval. We always needed some superposition over the structure to reduce the complexity of what was so big that we couldn’t deal with it. Now when dealing with something so big, we can just let the machines deal with it.  The generation process must have some rules somewhere, but my suspicion is these are no longer needed for any practical human purpose. Now we are capable of managing any messy random heap of disconnected parts–because if fact we don’t have to deal with that mess any more: we have machine to do it in our stead.

Emmanuelle: One simple question would be: what kind of parts are we dealing with? Are they not themselves wholes composed of other parts, entering into larger or different wholes? Are we talking solely about human-made parts, which designers can generate, craft and master, or we are considering opening up these wholes to other domains; thus, to what degree and within which limit are they potentially extendable?
You’ll excuse me for coming back to my previous point, regarding the notion of uncertainty and how it can be taken into account, and let’s hypothesise the wholes we consider are governmental ensembles. The researcher in philosophy of law Antoinette Rouvroy identifies how uncertainty and unpredictability are systematically considered as risk. She analyses how the cybernetic and algorithmic order that underlie our contemporary forms of governance attempt to systematically and preemptively tackle risk in order to eradicate it. On the other side, there is a reverse relationship to risk that, against risk management, consists in exploiting it and profiting from it, as you can see in high frequency trading. Risk here appears to be the motor of speculation, it plays with the asymmetric distribution of information within a system.
But if you consider chance, and hence uncertainty and unpredictability, as being not epistemic – as in both aforementioned cases – but objective, and furthermore, if you consider it to be at the source of all life in the biosphere – as Biology Nobel Prize Jacques Monod showed – how can it be taken into account and integrated in the elaboration of hybrid parts and wholes? Embracing this objectivity could allow us to conceptualise a commonality based on an open, decentralised notion of whole that is not subjected to social constructivism.

Giorgio: I owe an answer to Emmanuelle about unpredictability. Unpredictability can be either an epistemic phenomenon (it happens when some human subjects are de facto unable to foresee how things will go, and their inability to do so might be due to their contingent cognitive limitations), or a metaphysical phenomenon (there is metaphysical unpredictability when something is objectively indeterminate, independently of any fact concerning human subjects). If unpredictability is seen as an epistemic phenomenon, then it does not require any modification of Mereology: the fact that some human subjects are unable to determine whether x is part of y has no impact on the circumstance whether objectively x is part of y. 
The philosophical consequences of quantum indeterminacy are hard to interpret: according to some interpretations, it is indeed a kind of objective, metaphysical indeterminacy. However, as far as I can see, quantum indeterminacy does not concern mereological relations. Thus, it seems to me that neither epistemic nor metaphysical unpredictability have any specific bearing on Mereology.

Daniel: Unpredicted and indeterminant like a good building, it seems to me that Emmanuelle and Giorgio overcame the boundaries of the round table. I would like to use the moment to thank you all for your insights, contributions, and round up the discussion with an open ending.

Part relationships play an important role in architecture. Whether an aspect of a Classical order, a harmonious joining of building components, a representation of space, a partition of spaces, or as a body that separates us and identifies us as individuals. From the very outset, every form of architecture begins with an idea of how parts come together to become a whole and an understanding of how this whole relates to other parts. Architecture first composes a space as a part of a partitioning process well before defining a purpose, and before using any geometry.

The sheer performance of today’s computational power makes it possible to form a world without a whole, without any third party or a third object. Ubiquitous computing fosters peer-to-peer or better part-to-part exchange. It is not surprising then that today’s sharing represents an unfamiliar kind of partiality. From distributive manufacturing to the Internet of Things, new concepts of sharing promise systematic shifts, from mass-customisation to mass-individualisation: the computational enabled participations are foundational. It is no longer the performance or mode of an algorithm that drives change but its participatory capacities. From counting links, to likes, to seats, to rooms: tools for sharing have become omnipresent in our everyday lives. Thus, that which is common is no longer negotiated but computed. New codes – not laws or ideologies – are transforming our cities at a rapid pace, but what kind of parthood is being described? How does one describe something only through its parts today? To what extent do the automated processes of sharing differ from the partitioning of physical space? How can we add, intervene and design such parts through architecture?

The relationship between parts and their whole is called Mereology. In this issue of Prospectives, mereology’s theories and the specifics of part-relations are explored. The differences between parts and the whole, the sharing of machines and their aesthetics, the differences between distributive and collective, their ethical commitments, and the possibilities of building mereologies are discussed in the included articles and interviews.

Just as mereology describes objects from their parts, this issue is partial. It is not a holistic proposal, but a collection of positions. Between philosophy, computation, ecology and architecture, the texts are reminders that mereologies have always been part of architecture. Mereology is broadly a domain that deals with compositional possibilities, relationships between parts. Such an umbrella – analogue to morphology, typology, or topology – is still missing in architecture. Design strategies that depart part-to-part or peer-to-peer are uncommon in architecture, also because there is (almost) no literature that explores these topics for architectural design. This issue hopes to make the extra-disciplinary knowledge of mereology accessible to architects and designers, but also wishes to identify links between distributive approaches in computation, cultural thought and built space.

The contributions gathered here were informed by research and discussions in the Bartlett Prospectives (B-Pro) at The Bartlett School of Architecture, UCL London from 2016 to 2019, culminating in an Open Seminar on mereologies which took place on 24 April 2019 as part of the Prospectives Lecture Series. The contributions are intended as a vehicle to inject foundational topics such as mereology into architectural design discourse.

The Contributions

This compilation starts with Giorgio Lando’s text “Mereology and Structure”. Lando introduces what mereology is for philosophers, and why philosophers discuss mereological theses, as well as disagree one with another about them. His text focuses in particular on the role of structure in mereology outlining that from a formal point of view part relations are freed from structure. He argues that independence from structure might be the identifying link between mereology and architecture. The second article “From Partitioning to Partaking” is a plea for re-thinking the city. Daniel Koehler’s essay points to the differences between virtual and real parts. Koehler observes a new spatial practice of virtual representations that render previous models of urban governance obsolete. He argues that the hyper-dimensional spaces of a big data-driven economy demand a shift from a partitioning practice of governance to more distributed forms of urban design. In “Matter versus Parts: The Immaterialist Basis of Architectural Part-Thinking” Jordi Vivaldi Piera highlights the revival of matter in parallel to the discrete turn in contemporary discourses on experimental architecture. The essay gravitates around the notion of part-thinking in association with the notion of form. Fluctuating between continuous and discrete, the text sets out requirements for radical part-thinking in architecture. As a computational sociologist, David Rozas illustrates the potential of decentralised technologies for democratic processes at the scale of neighborhood communities. After an introduction to models of distributed computation, “Affordances of Decentralised Technologies for Commons-based Governance of Shared Technical Infrastructure” draws analogies to Elinor Ostrom’s principles of commons governance and how those can be computationally translated, turning community governance into fully decentralised autonomous organisations.

Departing from the Corbusian notion of a ‘machine for living’, Sheghaf Abo Saleh defines a machine for thinking. In “When Architecture Thinks! Architectural Compositions as a Mode of Thinking in the Digital Age” Abo Saleh states that the tectonics of a machine that thinks is brutal and rough. As a computational dialogue, she shows how roughness can enable posthumanism which, in her case, turns “tempered” parts into a well-tempered environment. Ziming He’s entry point for “The Ultimate Parts” is the notion of form as the relations between parts and wholes. He’s essay sorts architectural history through a mereological analysis, proposing a new model of part-to-part without wholes. Shivang Bansal’s “Towards a Sympoietic Architecture: Codividual Sympoiesis as an Architectural Model” investigates the potential of sympoiesis. By extending Donna Haraway‘s argument of “tentacular thinking” into architecture, the text shifts focus from object-oriented thinking to parts. Bansal argues for the limits of autopoiesis as a system and conceptualises spatial expressions of sympoiesis as a necessity for an adaptive and networked existence through “continued complex interactions” among parts.

Merging aspects of ‘collective’ and ‘individuality,’ in “Codividual Architecture within Decentralised Autonomous System” Hao Chen Huang proposes a new spatial characteristic that she coins as the “codividual”. Through an architectural analysis of individual and shared building precedents, Huang identifies aspects of buildings that merge shared and private features into physical form. Anthony Alviraz’s paper “Computation Within Codividual Architecture” investigates the history and outlook of computational models into architecture. From discrete to distributed computation, Alviraz speculates on the implications of physical computation where physics interactions overcome the limits of automata thinking. In “Synthesizing Hyperumwelten”, Anna Galika transposes the eco-philosophical concept of an HyperObject into a “Hyperumwelt”. While the Hyperobject is a closed whole that cannot be altered, a Hyperumwelt is an open whole that uses objects as its parts. The multiple of a Hyperumwelt offers a shift from one object’s design towards the impact of multiple objects within an environment.

Challenging the notion of discreteness and parts, Peter Eisenman asks in the interview “Big Data and the End of Architecture Being Distant from Power” for a definition of the cultural role of the mereological project. Pointing to close readings of postmodern architecture that were accelerated by the digital project, Eisenman highlights that the demand for a close reading is distanced from the mainstream of power. The discussion asks: ultimately, what can an architecture of mereology critique? The works of Herman Hertzberger are an immense resource on part-thinking. In the interview “Friendly Architecture: In the Footsteps of Structuralism”, Herman Hertzberger explains his principle of accommodation. When building parts turn into accommodating devices, buildings turn into open systems for staging ambiguity.**

The issue concludes with a transcript from the round table discussion at the Mereologies Open Seminar at The Bartlett School of Architecture on 24 April 2019.

Acknowledgments

The contributions evolved within the framework of Bartlett Prospectives (B-Pro) at The Bartlett School of Architecture, UCL. I want to thank Frédéric Migayrou for his vision, commitment and long years of building up a research program, not only by architecture but through computation. I would like to thank Roberto Bottazzi for the years of co-organising the Prospectives Lecture Series, where plenty of the discussions that form the backbone of this issue took place. Thanks to Mario Carpo for raising the right question at the right time for so many people within the program, thanks to Andrew Porter for enabling so many events, to Gilles Retsin, for without the discrete there are no parts, Mollie Claypool for the editing and development of Prospectives journal, and Vera Buehlmann, Luciana Parisi, Alisa Andrasek, Keller Easterling, Matthew Fuller, John Frazer, Philippe Morel, Ludger Hovestadt, Emmanuelle Chiappone-Piriou, Jose Sanchez, Casey Rehm, Tyson Hosmer, and Jordi Vivaldi Piera for discussions and insights. 

I want to thank Rasa Navasaityte, my partner in Research Cluster 17 at B-Pro, for driving the design research. Thank you for the research contributed by the researchers and tutors: Christoph Zimmel, Ziming He, Anqi Su, Sheghaf Abo Saleh, and to all participants, specifically to highlight: Genmao Li, Zixuan Wang, Chen Chen, Qiming Li, Anna Galika, Silu Meng, Ruohan Xu, Junyi Bai, Qiuru Pu, Anthony Alviraz, Shivang Bansal, Hao-Chen Huang, Dongxin Mei, Peiwen Zhan, Mengshi Fu, Ren Wang, Leyla El Sayed Hussein, Zhaoyue Zhang, Yao Chen, and Guangyan Zhu.

The issue includes articles that evolved from thesis reports conducted in the following clusters: Ziming He from Research Cluster 3 tutored by Tyson Hosmer, David Reeves, Octavian Gheorghiu, and Jordi Vivaldi in architecture theory. Sheghaf Abo Saleh, Anthony Alvidrez, Shivang Bansal, Anna Galika, Hao Chen Huang from Research Cluster 17 tutored by Daniel Koehler and Rasa Navasaityte. If not indicated directly, the featured images, graphics of this issue are by Daniel Koehler, 2020.

Parts, chunks, stacks and aggregates are the bits of computational architecture today. Why do mereologies – or buildings designed from part-to-whole – matter? All too classical, the roughness of parts seems nostalgic for a project of the digital that aims for dissolving building parts towards a virtual whole. Yet if parts shrink down to computable particles and matter, and there exists a hyper-resolution of a close to an infinite number of building parts, architecture would dissolve its boundaries and the capacity to frame social encounters. Within fluidity, and without the capacity to separate, architecture would not be an instrument of control. Ultimately, freed from matter, the virtual would transcend from the real and form finally would be dead. Therein is the prospect of a fluid, virtual whole.

The Claustrophobia of a City that Transcends its Architecture

In the acceleration from Data to Big Data, cities have become more and more virtual. Massive databases have liquefied urban form. Virtual communication today plays freely across the material boundaries of our cities. In its most rudimentary form virtuality is within the digital transactions of numbers, interests and rents. Until a few years ago,  financial investments in architectural form were equatable according to size and audience, e.g. as owner-occupied flats, as privately rented houses or as lease holding.[1] Today capital flows freely scatter across the city at the scale of the single luxury apartment. Beyond a certain threshold in computational access, data becomes big. By computing aggregated phone signal patterns or geotagged posts, virtual cities can emerge from the traces of individuals. These hyperlocal patterns are more representative of a city than its physical twin. Until recently, architecture staged the urban through shared physical forms: the sidewalk, lane or boulevard. Adjacent to cars, walkable for pedestrians or together as citizens, each form of being urban included an ideology of a commons, and grounded with that particular parts of encountering.

In contrast, a hyper-local urban transcends lanes and sidewalks. Detached from the architecture of the city, with no belonging left, urban speculation has withdrawn into the private sphere. Today, urban value is estimated by counting private belongings only, with claustrophobic consequences. An apartment that is speculatively invested displaces residents. The housing shortage in the big cities today is not so much a problem of lack of housing, but instead of vacant space, accessible not to residents but to interests they hold in the hyper-urban.[2] The profit from rent and use of space itself is marginal compared to the profit an embodied urban speculation adds to the property. The possibility of mapping every single home as data not only adds interest, like a pension to a home but literally turns a home into a pension.[3] However this is not for its residents but for those with access to resources. Currently, computing Big Data expands and optimises stakeholders’ portfolios by identifying undervalued building assets.[4] However, the notion of ‘undervalued’ is not an accurate representation of assets.

Hyper-localities increase real estate’s value in terms of how their inhabitants thrive in a neighbourhood through their encounters with one another and their surrounding architecture. The residents themselves then unknowingly produce extra value. The undervaluing of an asset is the product of its residents, and like housework, is unpaid labour. In terms of the exchange of capital, additional revenue from a property is usually paid out as a return to the shareholders who invested in its value. Putting big data-driven real estate into that equation would then mean that they would have to pay revenues to their residents. If properties create surplus value from the data generated by their residents, then property without its residents has less worth and is indeed over-, but not under-, valued.

The city uses vehicles for creating public revenue by governing the width of a street’s section or the height of a building. Architecture’s role was to provide a stage for that revenue to be created. For example the Seagram Building (van der Rohe, Johnson, 1958) created a “public” plaza by setting back its envelope in exchange for a little extra height. By limiting form, architecture could create space for not only one voice, but many voices. Today, however, the city’s new parameters hidden in the fluidity of digital traces cannot be governed by the boundaries of architecture anymore. Outlined already 40 years ago, when the personal computer became available, Gilles Deleuze forecasted that “Man is not anymore man enclosed”.[5] At that time, and written as a “Postscript on the Societies of Control”, the fluid modulation of space prospected a desirable proposition. By liquefying enclosures, the framework of the disciplinary societies of Foucault’s writings would disappear. In modern industrial societies, Deleuze writes, enclosures were moulds for casting distinct environments, and in these vessels, individuals became masses of the mass society.[6] For example, inside a factory, individuals were cast as workers, inside schools as students. Man without a cast and without an enclosure seemed to be freed from class and struggle. The freedom of an individual was interlinked with their transcendence from physical enclosures.

During the last forty years, the relation between a single individual and the interior framed architecture rightly aimed to dissolve the institutional forms of enclosures that represented social exclusion at their exterior. Yet, in this ambition alternative forms for the plural condition of what it means to be part of a city were not developed. Reading Deleuze further, a state without enclosures also does not put an end to history. The enclosures of control dissolve only to be replaced. Capitalism would shift to another mode of production. When industrial exchange bought raw materials and sold finished products, now it would buy the finished products and profit from the assemblies of those parts. The enclosure is then exchanged with codes that mark access to information. Individuals would not be moulded into masses but considered as individuals: accessed as data, divided into proper parts for markets, “counted by a computer that tracks each person’s position enabling universal modulation.”[7] Forty years in, Deleuze’s postscript has become the screenplay for today’s reality.

Hyper-parts: Spatial Practices of representations

A house is no longer just a neutral space, an enclosing interior where value is created, realised and shared. A home is the product of social labour; it is itself the object of production and, consequently, the creation of surplus value. By shifting from enclosure to asset, the big data-driven economy has also replaced the project behind modernism: humanism. Architecture today is post-human. As Rosi Braidotti writes, “what constitutes capital value  today is the informational power of living matter itself”.[8] The human being as a whole is displaced from the centre of architecture. Only parts of it, such as its “immanent capacities to form surplus-value”, are parts of a larger aggregation of architecture. Beyond the human, the Hyper-city transcends the humane. A virtual city is freed from its institutions and constituent forms of governance. Economists such as Thomas Piketty describe in painstaking detail how data-driven financial flows undermine common processes of governance, whether urban, regional, or national, in both speed and scale. Their analysis shows that property transactions shelled in virtual value-creation-bonds are opaque to taxation. Transcending regulatory forms of governance, one can observe the increase of inequalities on a global scale. Comparable to the extreme wealth accumulation at the end of the nineteenth century, Piketty identifies similar neo-proprietarian conditions today, seeing the economy shifting into a new state he coins as “hypercapitalism”.[9] From Timothy Morton’s “hyper-objects” to hypercapitalism,  hyper replaces the Kantian notion of transcendence. It expresses not the absorption of objects into humanism, but its withdrawal. In contrast to transcendence, which subordinates things to man’s will, the hyper accentuates the despair of the partial worlds of parts – in the case of Morton in a given object and in the case of Piketty in a constructed ecology.

When a fully automated architecture emerged, objects oriented towards themselves, and non-human programs began to refuse the organs of the human body. Just as the proportions of a data center are no longer walkable, the human eye can no longer look out of a plus-energy window, because it tempers the house, but not its user. These moments are hyper-parts: when objects no longer transcend into the virtual but despair in physical space. More and more, with increasing computational performance, following the acronym O2O (from online to offline),[10] virtual value machines articulate physical space. Hyper-parts place spatial requirements. A prominent example is Katerra, the unicorn start-up promising to take over building construction using full automation. In its first year of running factories, Katerra advertises that it will build 125,000 mid-rise units in the United States alone. If this occurred, Katerra would take around 30% of the mid-rise construction market in the company’s local area. Yet its building platform consists of only twelve apartment types. Katerra may see the physical homogeneity as an enormous advantage as it increases the sustainability of its projects. This choice facilitates financial speculation, as the repetition of similar flats reduces the number of factors in the valuing of apartments and allows quicker monetary exchange, freed from many variables. Sustainability refers not to any materiality but to the predictability of its investments. Variability is still desired, but oriented towards finance and not to inhabitants. Beyond the financialisation of a home, digital value machines create their own realities purely through the practice of virtual operations.

Here one encounters a new type of spatial production: the spatial practice of representations. At the beginning of what was referred to as “late capitalism”, the sociologist and philosopher Henri Lefebvre proposed three spatialities which described modes of exchange through capitalism.[11] The first mode, a spatial practice referred to a premodern condition, which by the use of analogies interlinked objects without any forms of representation—the second, representations of space linked directly to production, the organic schemes of modernism. The third representational spaces express the conscious trade with representations, the politics of postmodernism, and their interest in virtual ideas above the pure value of production. Though not limited to three only, Lefebvre’s intention was to describe capitalism as “an indefinite multitude of spaces, each one piled upon, or perhaps contained within, the next”.[12] Lefebvre differentiated the stages in terms of their spatial abstraction. Incrementally, virtual practices transcended from real-to-real to virtual-to-real to virtual-to-virtual. But today, decoupled from the real, a virtual economy computes physically within spatial practices of representations. Closing the loop, the real-virtual-real, or new hyper-parts, do not subordinate the physical into a virtual representation, instead, the virtual representation itself acts in physical space.

This reverses the intention of modernism orientated towards an organic architecture by representing the organic relationships of nature in geometric thought. The organicism of today’s hypercomputation projects geometric axioms at an organic resolution. What was once a representation and a geometry distant from human activity, now controls the preservation of financial predictability.

The Inequalities Between the Parts of the Virtual and the Parts of the Real

Beyond the human body, this new spatial practice of virtual parts today transcends the digital project that was limited to a sensorial interaction of space. This earlier understanding of the digital project reduced human activity to organic reflexes only, thus depriving architecture of the possibility of higher forms of reflection, thought and criticism. Often argued through links to phenomenology and Gestalt theory, the simplification of architectural form to sensual perception has little to do with phenomenology itself. Edmund Husserl, arguably the first phenomenologist, begins his work with considering the perception of objects, not as an end, but to examine the modes of human thinking. Examining the logical investigations, Husserl shows that thought can build a relation to an object only after having classified it, and therefore, partitioned it. By observing an object before considering its meaning, one classifies an object, which means identifying it as a whole. Closer observations recursively partition objects into more unaffected parts, which again can be classified as different wholes.[13] Husserl places parts before both thought and meaning.

Derived from aesthetic observations, Husserl’s mereology was the basisof his ethics, and was therefore concluded in societal conceptions. In his later work, Husserl’s analysis is an early critique of the modern sciences.[14] For Husserl, in their efforts to grasp the world objectively, the sciences have lost their role in enquiring into the meaning of life. In a double tragedy, the sciences also alienated human beings from the world. Husserl thus urged the sciences to recall that they ground their origins in the human condition, as for Husserl humanism was ultimately trapped in distancing itself further from reality.

One hundred years later, Husserl’s projections resonate in “speculative realism”. Coined By Levi Bryant as “strange mereology”,[15] objects, their belongings, and inclusions are increasingly strange to us. The term “strange” stages the surprise that one is only left with speculative access. However, ten years in, speculation is not distant anymore. That which transcends does not only lurk in the physical realm. Hyper-parts figurate ordinary scales today, namely housing, and by this transcend the human(e) occupation.

Virtual and physical space are compositionally comparable. They both consist of the same number of parts, yet they do not. If physical elements belong to a whole, then they are also part of that to which their whole belongs. In less abstract terms, if a room is part of an apartment, the room is also part of the building to which the apartment belongs. Materially bound part relationships are always transitive, hierarchically nested within each other. In virtual space and the mathematical models with which computers are structured today, elements can be included within several independent entities. A room can be part of an apartment, but it can also be part of a rental contract for an embassy. A room is then also part of a house in the country in which the house is located. But as part of an embassy, the room is at the same time part of a geographically different country on an entirely different continent than the building that houses the embassy. Thus, for example, Julian Assange, rather than boarding a plane, only needed to enter a door on a street in London to land in Ecuador. Just with a little set theory, in the virtual space of law, one can override the theory of relativity with ease.

Parts are not equal. Physical parts belong to their physical wholes, whereas virtual parts can be included in physical parts but don’t necessarily belong to their wholes.  Far more parts can be included in a virtual whole than parts that can belong to a real whole. When the philosopher Timothy Morton says “the whole is always less than the sum of its parts”,[16] he reflects the cultural awareness that reality breaks due to asymmetries between the virtual and the real. A science that sets out to imitate the world is constructing its own. The distance which Husserl spoke of is not a relative distance between a strange object and its observer, but a mereological distance, when two wholes distance each other because they consist of different parts. In its effort to reconstruct the world in ever higher resolution, modernism, and in its extension the digital project, has overlooked the issue that the relationship between the virtual and the real is not a dialogue. In a play of dialectics between thought and built environment, modernism understood design as a dialogue. In extending modern thought, the digital project has sought to fulfill the promise of performance, that a safe future could be calculated and pre-simulated in a parallel, parametric space. Parametricism, and more generally what is understood as digital architecture, stands not only for algorithms, bits, and rams but for the far more fundamental belief that in a virtual space, one can rebuild reality. However, with each resolution that science seeks to mimic the world, the more parts it adds to it.

The Poiesis of a Virtual Whole

The asymmetry between physical and virtual parts is rooted in Western classicism. In early classical sciences, Aristotle divided thinking into the trinity of practical action, observational theory and designing poiesis. Since the division in Aristotle’s Nicomachean Ethics, design is a part of thought and not part of objects. Design is thus a knowledge, literally something that must first be thought. Extending this contradiction to the real object, design is not even concerned with practice, with the actions of making or using, but with the metalogic of these actions, the in-between between the actions themselves, or the art of dividing an object into a chain of steps with which it can be created. In this definition, design does not mean to anticipate activities through the properties of an object (function), nor to observe its properties (materiality), but through the art of partitioning, structuring and organising an object in such a way that it can be manufactured, reproduced and traded.

To illustrate poiesis, Aristotle made use of architecture.[17] No other discipline exposes the poetic gap so greatly between theory, activity and making. Architecture first deals with the coordination of the construction of buildings. As the architecture historian Mario Carpo outlines in detail, revived interest in classicism and the humanistic discourse on architecture began in the Renaissance with Alberti’s treatise: a manual that defines built space, and ideas about it solely through word. Once thought and coded into words, the alphabet enabled the architect to physically distance from the building site and the built object.[18] Architecture as a discipline then does not start with buildings, but with the first instructions written by architects used to delegate the building.

A building is then anticipated by a virtual whole that enables one to subordinate its parts. This is what we usually refer to as architecture: a set of ideas that preempt the buildings they comprehend. The role of the architect is to imagine a virtual whole drawn as a diagram, sketch, structure, model or any kind of representation that connotates the axes of symmetries and transformations necessary to derive a sufficient number of parts from it. Architectural skill is then valued by the coherence between the virtual and the real, the whole and its parts, the intention and the executed building. Today’s discourse on architecture is the surplus of an idea. You might call it the autopoiesis of architecture – or merely a virtual reality. Discourse on architecture is a commentary on the real.

Partitioning Architectures

From the very outset, architecture distanced itself from the building, yet also aimed to represent reality. Virtual codes were never autonomous from instruments of production. The alphabet and the technology of the printing press allowed Alberti to describe a whole ensemble distinct from a real building. Coded in writing, printing allowed for the theoretically infinite copies of an original design. Over time, the matrices of letters became the moulds of the modern production lines. However, as Mario Carpo points out, the principle remained the same.[19] Any medium that incorporates and duplicates an original idea is more architecture than the built environment itself. Belonging to a mould, innovation in architecture research could be valued in two ways. Quantitatively, in its capacity to partition a building in increasing resolution. Qualitatively, in its capacity to represent a variety of contents with the same form. By this, architecture faced the dilemma that one would have to design a reproducible standard that could partition as many different forms as possible to build non-standard figurations.[20]

The dilemma of the non-standard standard moulds is found in Sebastiano Serlio’s transcription of Alberti’s codes into drawings. In the first book of his treatise, Serlio introduces a descriptive geometry to reproduce any contour and shape of a given object through a sequence of rectangles.[21] For Serlio, the skill of the architect is to simplify the given world of shapes further until rectangles become squares. The reduction finally enables the representation of physical reality in architectural space using an additive assembly of either empty or full cubes. By building a parallel space of cubes, architecture can be partitioned into a reproducible code. In Serlio’s case, architecture could be coded through a set of proportional ratios. However, from that moment on, stairs do not consist only of steps, and have to be built with invisible squares and cubes too.

Today, Serlio’s architectural cubes are rendered obsolete by 3D printed sand. By shrinking parts to the size of a particle of dust, any imaginable shape can be approximated by adding one kind of part only. 3D printing offers a non-standard standard, and with this, five hundred years of architectural development comes to an end.

Replicating: A Spatial Practice of Representations

3D printing dissolved existing partitioning parts to particles and dust. A 3D-printer can not only print any shape but can also print at any place, at any time. The development of 3D printing was mainly driven by DIY hobbyists in the Open Source area. One of the pioneering projects here is the RepRap project, initiated by Adrian Bowyer.[22] RepRap is short for replicating rapid prototyping machine. The idea behind it is that if you can print any kind of objects, you can also print the parts of the machine itself. This breaks with the production methods of the Modern Age. Since the Renaissance, designers have crafted originals and used these to build a mould from those so that they can print as many copies as possible. This also explains the economic valuation of the original and why authorship is so vehemently protected in legal terms. Since Alberti’s renunciation of drawings for a more accurate production of his original idea through textual encoding, the value of an architectural work consisted primarily in the coherence of a representation with a building: a play of virtual and real. Consequently, an original representation that cast a building was more valued than its physical presentation. Architecture design was oriented to reduce the amount of information needed to cast. This top-down compositional thinking of original and copy becomes obsolete with the idea of replication.

Since the invention of the printing press, the framework of how things are produced has not changed significantly. However, with a book press, you can press a book, but with a book, you can’t press a book. Yet with a 3D printer, you can print a printer. A 3D printer does not print copies of an original, not even in endless variations, but replicates objects. The produced objects are not duplicates because they are not imprints that would be of lower quality. Printed objects are replicas, objects with the same, similar, or even additional characteristics as their replicator.

A 3D printer is a groundbreaking digital object because it manifests the foundational principle of the digital – replication – on the scale of architecture. The autonomy of the digital is based not only on the difference between 0 and 1 but on the differences in their sequencing. In mathematics in the 1930s, the modernist project of a formal mimicry of reality collapsed through Godel’s proof of the necessary incompleteness of all formal systems. Mathematicians then understood that perhaps far more precious knowledge could be gained if we could only learn to distance ourselves from its production. The circle of scientists around John von Neumann, who developed the basis of today’s computation, departed from one of the smallest capabilities in biology: to reproduce. Bits, as a concatenation of simple building blocks and the integrated possibility of replication, made it possible, just by sequencing links, to build first logical operations, and connecting those programs to today’s artificial networks.[23] Artificial intelligence is artificial but it is also alive intelligence.

To this day, computerialisation, not computation is at work in architecture. By pursuing the modern project of reconstructing the world as completely as possible, the digital project computerised a projective cast[24] in high resolution. Yet this was done without transferring the fundamental principles of interlinking and replication to the dimensions of the built space.

From Partitioning to Partaking

The printing press depends on a mould to duplicate objects. The original mould was far more expensive to manufacture than its copies, so the casting of objects had to bundle available resources. This required high investments in order to start production, leading to an increasing centralisation of resources in order to scale the mass-fabrication of standard objects for production on an assembly line. Contrarily, digital objects do not need a mould. Self-replication provided by 3D printing means that resources do not have to be centralised. In this, digital production shifts to distributed manufacturing.[25]

Independent from any mould, digital objects as programs reproduce themselves seamlessly at zero marginal costs.[26] As computation progresses, a copy will then have less and less value. Books, music and films fill fewer and fewer shelves because it no longer has value to own a copy when they are ubiquitously available online. And the internet does not copy; it links. Although not fully yet integrated into its current TCP-IP protocol,[27] the basic premise of hyperlinking is that linked data adds value.[28] Links refer to new content, further readings, etc. With a close to infinite possibility to self-reproduce, the number of objects that can be delegated and repeated becomes meaningless. What then counts is hyper-, is the difference in kind between data, programs and, eventually, building parts. In his identification of the formal foundations of computation, the mathematician Nelson Goodman pointed out that beyond a specific performance of computation, difference, and thus value, can only be generated when a new part is added to the fusion of parts.[29] What is essential for machine intelligence is the dimensionality of its models, e.g., the number of its parts. Big data refers less to the amount of data, but more to the number of dimensions of data.[30]

With increasing computation, architecture shifted from an aesthetic of smoothness that celebrated the mastership of an infinite number of building parts to roughness. Roughness demands to be thought (brute). The architecture historian Mario Carpo is right to frame this as nostalgic, as “digital brutalism”.[31] Similar to brutalism that wanted to stimulate thought, digital roughness aims to extend spatial computability, the capability to extend thinking, and the architecture of a computational hyper-dimensionality. Automated intelligent machines can accomplish singular goals but are alien to common reasoning. Limited around a ratio of a reality, a dimension, a filter, or a perspective, machines obtain partial realities only. Taking them whole excludes those who are not yet included and that which can’t be divided: it is the absolute of being human(e).

A whole economy evolved from the partial particularity of automated assets ahead of the architectural discipline. It would be a mistake to understand the ‘sharing’ of the sharing economy as having something “in common”. On the contrary, computational “sharing” does not partition a common use, but enables access to multiple, complementary value systems in parallel.

Cities now behave more and more like computers. Buildings are increasingly automated. They use fewer materials and can be built in a shorter time, at lower costs. More buildings are being built than ever before, but fewer people can afford to live in them. The current housing crisis has unveiled that buildings no longer necessarily need to house humans or objects. Smart homes can optimise material, airflow, temperature or profit, but they are blind to the trivial.

It is a mistake to compute buildings as though they are repositories or enclosures, no matter how fine-grain their resolution is. The value of a building is no longer derived only from the amount of rent for a slot of space, but from its capacities to partake with. By this, the core function of a building changes from inhabitation to participation. Buildings do not anymore frame and contain: they bind, blend, bond, brace, catch, chain, chunk, clamp, clasp, cleave, clench, clinch, clutch, cohere, combine, compose, connect, embrace, fasten, federate, fix, flap, fuse, glue, grip, gum, handle, hold, hook, hug, integrate, interlace, interlock, intermingle, interweave, involve, jam, join, keep, kink, lap, lock, mat, merge, mesh, mingle, overlay, palm, perplex, shingle, stick, stitch, tangle, tie, unit, weld, wield, and wring.

In daily practice, BIM models do not highlight resolution but linkages, integration and collaboration. With further computation, distributed manufacturing, automated design, smart contracts and distributed ledgers, building parts will literally compute the Internet of Things and eventually our built environment, peer-to-peer, or better, part-to-part – via the distributive relationships between their parts. For the Internet of Things, what else should be its hubs besides buildings? Part-to-part habitats can shape values through an ecology of linkages, through a forest of participatory capacities. So, what if we can participate in the capacities of a house? What if we no longer have to place every brick, if we no longer have to delegate structures, but rather let parts follow their paths and take their own decisions, and let them participate amongst us together in architecture?

In this short paper I illustrate what is mereology for philosophers, and which reasons lead philosophers to discuss mereological theses and disagree one with another about them. I will focus in particular on the role of structure in mereology and propose a rather simple account of what structure is from the viewpoint of mereology. As we are going to see, many philosophical controversies in mereology concern the issue of whether mereology should account for structure or not, and which role (if any) structure plays in mereology. I will also present some examples of philosophical controversies about mereological principles, and of the reasons which might be brought for choosing one side or the other in these controversies.

An additional purpose of the paper is to suggest that structure is a broad topic of common concern between architecture and philosophical mereology. In their discussions about structure, mereologists do often resort to examples involving buildings and villages. These examples – I am going to use some of them in what follows – are extremely simple, to the point of naïveté. The frequent usage of these examples might depend on the rough intuition that there are indeed some connections between mereology and architecture, and the concept of structure might be the link between them.

What Is Mereology About?

Let us begin by asking what is mereology. Mereology is the theory of two (mutually related) relations: parthood and composition. Parthood is a one-one relation between a part and a whole. These are some instances of parthood: 

my left hand is part of my body; 

Portugal is part of Europe; 

an atom of oxygen is part of a molecule of water; 

a handle is part of a door; 

the word ‘salad’ is part of the sentence ‘I eat a salad’.

The other relation which, together with parthood, is the subject matter of mereology is composition, a many-one relation. Composition indeed connects many entities (the components) to a single entity (the composed entity). Consider a rather simplistic house with a base, four outer walls, a roof and nothing else. The base, the four walls and the roof compose the house. Another example is the following: the Netherlands, Belgium and Luxembourg compose Benelux.

Composition is definable in terms of parthood. The composed entity is expected to include all the components as parts and not to include anything extraneous to the components. Also the latter feature of the composed entity can be expressed in terms of parthood, namely by requiring that every part of the composed entity has at least a part in common with at least one of the components. 
In order to express the resulting definition of composition in terms of parthood we can first define the relation of overlap, which holds between two entities if and only if they have at least a part in common (P is the relation of parthood; ⚬ is the defined relation of overlap):

Overlap: x ० y ≡_def ∃z (z P x ⋀ z P y)  

Now, composition can be defined in terms of parthood and overlap (which has in turn been defined in terms of parthood above) as follows (C is the relation of composition; xx is a plural variable for the components; ≺ is the relation of being one of):

Composition: xx C y ≡_def ∀z (z ≺ xx → z P y) ⋀ ∀z (z P y → ∃w (w ≺ xx ⋀  z ० w))

According to this definition, some entities xx compose one entity y if and only if every entity z that is one of xx is also part of y and every entity z that is part of y is such that there is an entity w that is one of xx and overlaps z.

What do mereologists say about parthood and composition? They attribute some features to these relations. The attributed features are generally expected to be formal or topic-neutral, in the sense that the features are not expected to depend on which kinds of entities are part of one another or compose one another. For example, the features of parthood and composition are expected by mereologists to be independent of whether we are considering the parts of a human body, the parts of a chemical molecule, or the parts of a town. Mereology is also formal in a more general sense, inasmuch as mereology never attempts to identify the parts of something or to establish what composes what: it is not the expected duty of mereology to establish what are the parts of a car, or of a building, or what composes a sentence. 

Mereological principles consist in attributions of some formal features to parthood and composition. Let us consider some examples of mereological principles and of formal features which these principles attribute to parthood and composition. In considering these examples, it is important to keep in mind that every mereological principle (any attribution of formal features to parthood and composition) is controversial, including the following examples. It can also be controversial whether the principles at stake are really formal: given a candidate principle, it might be objected that it holds when parthood is instantiated by, say, buildings, while it fails when parthood is instantiated by animal organisms. 

A mereological principle about parthood defended by many philosophers is for example that parthood is a transitive relation (transitivity is the formal feature which this principle attributes to parthood): if a is part of b and b is part of c, then a is part of c. Here are three more or less controversial instances of transitivity (involving various kinds of entities, in coherence with the expected topic-neutrality of mereological principles): 

if a handle is part of a door and the door is part of a building, then the handle is part of the building;

if my left hand is part of my left arm and my left arm is part of my body, then my left hand is part of my body;

if an MP is part of the Italian Parliament and the Italian Parliament is part of the Inter-Parliamentary Association, then the MP is part of the Inter-Parliamentary Association. 

Principles about parthood also concern the problem of whether the chains of parthood terminate or not. Do the chains of parthood terminate downwards? Does everything have a part that is different from itself no matter how deep you go down? Or is there a bottom layer of mereological simples (i.e. entities without further parts)? In the other direction, do the chains of parthood terminate upwards? Is everything part of something different from itself? Or is there a top layer, the mereological universe (i.e. an entity such that everything is part of it, and it is part of nothing different from itself)? Mereological principles can dictate an answer to these questions. 

These principles can be applied to matters of potential architectural or urbanistic concern. As regards downward termination, a problem of potential architectural concern is whether there is a bottom layer of parts of a building (say: the layer of bricks, or the layer of the smallest pieces which are visible to the human eye). As regards upward termination, a problem of potential urbanistic concern is whether there is a top layer of entities, over which entities stop being interesting for urbanists and are at best interesting for other kinds of scholars, such as geographists.

Other principles concern composition (the other relation, which – together with parthood – is the subject matter of mereology). The principles about composition mainly provide existence conditions and identity conditions for composed entities. Existence and identity are in general two pivotal concepts in metaphysics, i.e. in the wider branch of philosophy to which mereology belongs. As far as composition is concerned, mereology asks whether, given some entities, there exists something which they compose (in answering this question, mereologists provide existence conditions for composed entities), and whether there may exist two or more entities composed by the same entities (in answering this latter question, mereologists provide identity conditions for composed entities). 

Thus, given some components, how many things do they compose? Consider Barack Obama’s nose, my left shoe and the Great Pyramid of Giza. Do they compose anything? Is there a spatially scattered object (a bit in Washington, a bit here, a bit in Egypt) they compose? 

Consider instead a building entirely made of bricks. Those bricks compose that building. Do the bricks compose only the building? Or do they compose also a different, less structured entity, which we might dub ‘heap of bricks’? Suppose that the building collapses: after the collapse – one might say – the building stops existing, while the heap of bricks continues existing. How can this happen, if they are identical, i.e. if the bricks compose only one entity? A single entity cannot both continue existing and stop existing.

The following are two examples of two rather popular (albeit controversial) mereological principles about composition (their usual label is indicated in the parenthesis):

given some entities – no matter how sparse and heterogeneous they are – there is at least an entity composed by them (Unrestricted Composition); 

given some entities, there is at most an entity composed by them (Uniqueness of Composition). 

Unrestricted Composition provides existence conditions for composed entities, while Uniqueness of Composition provides identity conditions for composed entities.

The Role of Structure in Mereology

In the above example of the brick building and of the heap of bricks, we have seen how the temptation arises to distinguish wholes according to their being structured or non-structured, or according to their being structured in different ways. In order to appreciate the role of structure in mereology, it is very important not to misidentify the subject matter of the philosophical controversies about this kind of example. 

Indeed, no mereologist doubts that, in order to compose a building, the bricks have to be in some mutual relations and that, more in general, not any heap of bricks is a building. There is no interesting philosophical controversy on the fact that the parts of many composed entities are arranged or structured in a certain way. 

The problem mereology is concerned with is different: does structure have any impact on the existence and identity of composed entities? Do some things need to be structured in a certain way (e.g. in a building-like way, or in a car-like way, or in a person-like way, or in a mountain-like way) in order to compose something? There is obviously a difference between the parts of a car on the one hand, and Barack Obama’s nose, my left shoe and the Great Pyramid of Giza on the other. The difference is that the parts of a car are in certain mutual relations and have certain designated functional roles. The relations among the parts of a car and their functional roles are constrained by the nature of cars in general, and by the project of that specific car model. The mutual relations among parts and the roles of the parts are roughly what structure consists in. 

To repeat: nobody doubts that the parts of a car have a kind of structure which, by contrast, disparate, sparse entities lack. Mereology is not about this. The mereological problem at stake is whether, on the basis of this difference between the parts of a car on the one hand, and Barack Obama’s nose, my left shoe and the Great Pyramid of Giza on the other, we should conclude that only the parts of a car compose something, while Barack Obama’s nose, my left shoe and the Great Pyramid of Giza do not compose anything. If this were the case, then, in contrast with the above principle of Unrestricted Composition, it would not be true that, given some entities – no matter how disparate and sparse they are – there exists something they compose: there would be nothing which Barack Obama’s nose, my left shoe and the Great Pyramid of Giza compose. 

Unrestricted Composition is a mereological principle according to which structures have no bearing on the existence conditions for composed entities. According to Unrestricted Composition, composed entities exist irrespective of whether and how their components are structured. 

Another mereological problem concerning structure is: does structure have any bearing on the identity conditions of composed entities? According to Uniqueness of Composition, given some entities there is at most one entity composed by them. Thus, there cannot be two different entities composed by the same entities. This entails that two composed entities cannot have exactly the same parts. If the composed entities are different, then they have a different part, and this different part is their difference maker. The fact that these parts are in different mutual relations and have different roles is not an admissible difference maker for composed entities. If Uniqueness of Composition is true, then the identity conditions for composed entities countenance only their parts, and not their structure. Thus, both Unrestricted Composition (for what concerns the existence conditions for composed entities) and Uniqueness of Composition (for what concerns the identity conditions for composed entities) exhibit a kind of deliberate blindness to structure.

Principles of Structure Obliteration

Before considering how it is possible to argue in favour or against this blindness to structure, it is useful to be a bit more precise on how structure is construed in this context. A theory of parthood and composition can countenance or obliterate various aspects of structure. This depends on whether, in a certain theory of parthood, certain principles of structure obliteration hold or not. We will consider four principles of structure obliteration.[1] These principles are interesting because they are of help in distinguishing various aspects of structure. These aspects of structure are obliterated in the identity conditions for composed entities, if the respective obliteration principle holds. They are by contrast countenanced in the identity conditions for composed entities, if the respective principle fails. 

The first principle is Absorption and claims that the repetition of parts is not a difference maker for composed entities. If Absorption is true, then the repetition of parts has no impact on the identity conditions for composed entities. Absorption can be formalised as follows (Σ is an operation of composition, whose inputs are the components and whose output is the composed entity; the formula expresses the fact that the multiple occurrences of the inputs – expressed by the multiple occurrences of the variables x and y in the left part of the formula – make no difference for the identity conditions of the output): 

Absorption: ∑ (…, x, x, …, y, y, …) = ∑ (…, x, …, y, …)

The second and the third principles are Collapse and Levelling and jointly claim that the stratification and the groupings of parts at different levels are not difference makers for composed entities (i.e. they have no impact on the identity conditions for composed entities). 

Collapse: ∑ (x) = x

Levelling: ∑ (…, ∑(x, y, z, …), …, ∑ (u, v, w, …), …) = ∑ (…, x, y, z, …, …, u, v, w, …, …)

The fourth principle is Permutation and claims that the order of the parts is not a difference maker for composed entities (i.e. it has no impact on the identity conditions for composed entities).

Permutation: ∑ (…, x, …, y, …, z,…) = ∑ (…, y, …, z, …, x, …)

This approach manages to differentiate various theories of parthood, according to their degree of blindness with respect to structure. What is usually called (for historical reasons) Classical Mereology abides by all the above four principles of structure obliteration and is, as a result, deliberately blind with respect to the repetition, to the stratification, to the groupings at different levels and to the order of the components in a composed entity. Only the parts matter for the identity conditions of composed entities, according to Classical Mereology. By contrast, the ways in which the parts are arranged/structured (e.g. repeated, stratified, grouped or ordered) do not matter. 

It is noteworthy that the four principles of structure obliteration are mutually independent. If you adopt a Non-Classical Mereology, you are not thereby forced to reject all the four principles as a single package. You can reject one or more of them, while keeping the others. In so doing, you sometimes end up adopting a theory which is no less well-established than Classical Mereology. For example, suppose that you think that the stratification and groupings at various levels of parts matter for the identity conditions of composed entities, while their order and repetition do not matter. What you obtain is Set Theory, a well-established theory, with a pivotal role in the foundations of mathematics.

Controversies in Mereology (and How to Argue About Them)

How should we argue about the formal features of parthood and composition? What reasons can be brought in favour of or against the mereological principles which attribute formal features to parthood and composition? And what reasons can – in particular – be brought in favour of or against the principles we have introduced above, such as Transitivity of Parthood, Unrestricted Composition, Uniqueness of Composition and the four principles of structure obliteration? 

Mereologists mainly proceed either by analysing and assessing alleged counterexamples to the mereological principles, or by analysing and assessing a priori arguments in support of or against them. As far as counterexamples are concerned, let us focus on Uniqueness of Composition and on its radical blindness to structure (i.e. – as we have seen in § 3 – blindness to repetition, stratification, groupings at different levels and order of components in a composed entity). One might be tempted to dismiss Uniqueness of Composition rather quickly, on the basis of the fact that some prima facie unavoidable counterexamples might seem fatal to it. 

Consider the components of the sentence ‘Gina loves Mario’: the words ‘Gina’, ‘loves’ and ‘Mario’. The same components can also form the sentence ‘Mario loves Gina’. ‘Gina loves Mario’ and ‘Mario loves Gina’ might seem two composed entities with the same components, in contrast with Uniqueness of Composition. The order of words (an aspect of structure, which belongs to the subject matter of syntax in linguistics) in sentences seems to matter for the identity conditions of sentences, in contrast with Uniqueness of Composition and with Permutation. 

Consider also a very small village, composed by a square, two streets and four buildings. It seems prima facie plain that Uniqueness of Composition fails for those components: the disposition of the streets with respect to the square, and the location of the buildings with respect to those of the square and of the streets (in general: the way in which the components of the village are arranged) would seem to matter for the identity of the village. 

However, the evaluation of these alleged counterexamples is not as easy as it seems, and Uniqueness of Composition should not be dismissed so quickly. Why? Because whether two composed entities (two sentences, two small villages) can be different while having the same parts and whether they can – as a consequence – be different only in virtue of their structure depends on a controversial identification of the entities at stake. 

In the case of the sentence, it depends on whether the entities at stake are linguistic tokens or types. Consider only single concrete tokens or inscriptions of ‘Mario’, ‘loves’ and ‘Gina’: sequences of sounds, stains of ink or groups of pixels on a screen. These concrete tokens are always in a single, specific order. Some of these inscriptions are ordered in a way such that ‘Mario’ is the first inscription (counting from the left) and ’Gina’ is the last inscription (this is the case of the word inscriptions in ’Mario loves Gina’). Others of these inscriptions are ordered in a way such that ’Gina’ is the first inscription (counting from the left) and ’Mario’ is the last inscription (this is the case of the word inscriptions in ’Gina loves Mario’). It never happens that the same inscriptions are arranged in two ways and thereby compose two different wholes. The initial impression that it is clear that two sentences can be different simply due to the arrangement of their parts (even if they have exactly the same parts) depends on seeing words not as tokens but as abstract word types, which occur in many different sentence types. 

Consider also the case of the small village. Given a specific small village, at a specific time, the square, the two streets and the four buildings are arranged in a single way (they have a single structure). There are not two small villages composed by that square, those streets and those buildings at that time. 

These considerations about the sentence and the small village can, as a matter of fact, be generalised. Whenever we are tempted to dismiss those principles of Classical Mereology which express its deliberate blindness to structure, it turns out that the temptation depends on a controversial characterisation of the involved entities.[2]

Please note that the philosophers objecting to Uniqueness of Composition might refine their counterexamples, and the defenders of Uniqueness of Composition might refine their analysis in order to deal with these counterexamples: the purpose of the above analysis is not to resolve the philosophical disputes about the role of structure in mereology in favour of blindness to structure, but to exemplify the way in which philosophers argue one with another about mereology and structure. The exemplifications are also meant to suggest that these controversies are unlikely to be easily solvable by adducing counterexamples: the analysis of these counterexamples is often arduous and depends on controversial assumptions. 

Finally, I would like to discuss an example of the second main way of arguing about mereological principles, the one which involves general, a priori arguments in support of or against mereological principles. Let us focus in this case on Unrestricted Composition. Suppose that you deny Unrestricted Composition. This denial will be based on the intuition that there is a patent difference between – say – the parts of a car on the one hand, and Barack Obama’s nose, my left shoe and the Great Pyramid of Giza on the other. What does this difference consist in? In order to obtain an alternative to Unrestricted Composition, this difference should correspond to a general condition which a plurality of entities should satisfy in order to have a fusion. 

We might try to extract this general condition from the examples, e.g. by observing that the parts of a car are spatially close one to another, while Barack Obama’s nose, my left shoe and the Great Pyramid of Giza are not; or by observing that there are causal links among the parts of a car (e.g. a movement in the steering causes a movement in its anterior wheels), while Barack Obama’s nose, my left shoe and the Great Pyramid of Giza are in no direct causal connection. On this basis, one might propose the following criteria for the restriction of composition: only mutually close entities compose something; only entities which move together (or act together) compose something. However, these criteria are unavoidably vague. There is no such thing as being definitely close in space or as being definitely causally connected. Every two parts of the universe are at some distance and have some kind of more or less remote causal connection. 

How is it possible to fix a threshold, i.e. a minimal degree of proximity or of causal connectedness? The threshold should be such that: entities above that threshold compose something; entities below that threshold do not compose anything. For example, we should determine once and for all at which maximum distance some buildings should be in order to contribute to compose a certain town, instead of belonging to two different towns. Any such threshold would be arbitrary, and would risk making arbitrary our existence claims about composed entities (such as towns) as well. 

A famous argument in support of Unrestricted Composition is based on the thesis that existence claims cannot be either vague or arbitrary. This means that no compelling motivation for restricting composition can be satisfied, and that, as a consequence, composition cannot be restricted. Thus, composition would be unrestricted. 

The general idea behind this famous argument for Unrestricted Composition (of which I have presented a simplified version)[3] is that existence conditions for composed entities should not be constrained by structural considerations about the mutual unity of parts. Why? Because it is arbitrary to delimit the domain of what is unitary. Every plurality of entity can be seen as unitary (or as non-unitary) according to certain criteria and/or from a certain standpoint. This is a general motivation why classical mereology is deliberately blind to structure: because the attribution of roles to structure risks introducing arbitrary and hardly justifiable thresholds. Thus, also, when mereologists proceed by analysing and assessing a priori arguments – as much as when they proceed by analysing and assessing counterexamples, as we have seen above – the problem whether structure has any role in mereology is pivotal, and has no easy solution.

Mereology is a formal concept which enters architecture as an additional formal category. Form is a rather ambiguous concept in architecture. So in this essay, first an investigation is conducted by contrasting two closely related concepts: shape and content.

Hans Trusack criticises the problem of shape for its shallow formalism and historical-theoretical indifference as a defensive strategy that evades the disciplines and difficulties of past and future.[1] The distinction between the terms “form” and “shape”, following Tursack’s argument, is a “matter of generative process”. Both terms point to the production of visual expression. Yet while shape refers to the appearance of an object, form reflects the logic of transformation and operation within historical and theoretical contexts such as political and religious ideology, economics and technological background. Tursack criticised the strategy of shape in architecture, stating its lack of reference, it being “plainly, and painfully, evident”,[2] and incapable of moving forward. Whereas form is difficult, disciplinary and requires historical and theoretical study, and yet promises the future. 

Form has the advantage of being able to deal with complex relations due to its deep and continuously evolving intervention with content. The term form derives from the Latin word forma, is understood as the combination of two Greek words: eidos, the conceptual form, and morphe, the physical form. The complexity of form can be attributed to these differentiated meanings, yet complexity is compatible with agencies and relations. This can emerge further by conducting a brief historical review.

Ancient Greek architecture pursues the ideality in mathematics and proportion. The efforts made by architects in designing the Parthenon provides evidence of this feature. These operations tried to approximate the physical shape of architecture to the “ideal” form. Form reflects the pursuit of ideality and perfection in this period. 

For Gothic architecture, there were more concerns about structure, and matter was pushed to its maximum capability to build as tall as possible for religious appeal. Consequently, structures were designed to be rigid and lightweight, and solid walls were replaced by glass windows, while flying buttresses supported the main structure to grow even taller. Consequently, astonishing space and fascinating transparency emerged.

Modernism claims that “form follows function”,[3] rejecting traditional architecture styles. The reality of matter and the logic of technology eschewed decorations, proportions, or any subjective distortion of matter. The emphasis on the term “function” illustrates an ideology of treating architecture as a machine. Each part is nothing more than a component that has a certain feature inside this machine, and redundant decorations and details are removed to deliver this idea clearly. Without distractions, space becomes evident.

In the shift to postmodernism, the uniformity and the lack of variety of modernist architectures were reacted against, and a great variety of approaches emerged to overcome the shortcomings of modernism. Parametricism, for instance, has been promoted by the thriving of digital technologies. Designers are capable of more complex formal production, and architectural elements have become variables that can be interdependently manipulated. In this formalism, rigidity, isolation, and separation are opposed, while softness, malleability, differentiation and continuity are praised.

From the examples above, form is the embodiment of the relations between architecture and its motive in specific historical scenarios, while for shape, only the results are accounted for – relations are ignored, and architecture is treated as isolated physical entities, incapable of producing new relations. Different methodologies of dealing with architectural form also imply the variation of ideology in compiling form with content.

Mereology – An Approach of Architectural Form

In recent philosophical texts, a third notion of form is brought forward. Contrary to a dialectic of form and content, here investigations deal with the resonance of parts: the description of objects by their ontological entanglement only. The writings of the philosopher Tristan Garcia are a strong example for such mereological considerations. In his treatise Form and Object: A Treatise on Things (2014), Garcia investigates the ontology of objects with two initial questions, “… what is everything compose of? … what do all things compose?”[4] The first question interrogates the internal, the elementary component of everything. The second interrogates the external, the totality of everything. For Garcia, the form of a thing is “the absence of the thing, its opposite, its very condition,”[5] form has two senses, the “beginning”, and the “end”, which never ends. Form begins when a thing ends, it begins with different forms; in the end, since it has “endless end”, form ultimately merges into one, which is “the world”. Garcia defines an object as “a thing limited by other things and conditioned by one or several things.”[6] The form of an object depends on what comprehends or limits this object. Every object is “embedded in a membership relation with one or several things”,[7] they can be divided by defining limits, which is also a thing distinguishing one thing from another. Garcia’s argument adapts the concept of mereology. Form has two extremes, one toward the fundamental element of matter, and the other toward the world, comprehending everything. All things can always be divided into an infinite number of parts, and they can always be parts of another thing. Identifying parts or wholes within a section we can operate on can establish a limit. The relevance between form and mereology opens a new opportunity to inspect architectural form from a different point of view.

One of the first discussions about parts and wholes in modern philosophy was posed by Edmund Husserl, in Logical Investigation (1st ed. 1900-1901, 2nd ed, 1913),[8] but the term “mereology” has not been put forward until Stanisław Leśniewski used it in 1927 from the Greek work méros (parts).[9] Mereology is considered as an alternative to set theory. A crucial distinction lies between mereology and set theory in that set theory concerns the relations between a class and its elements, while mereology describes the relations between entities. The mathematical axioms of mereology will be used as the fundamental theory of developing the method of analysing architectural form.

Following Roberto Casati and Achim Varzi, the four fundamental mathematical formularisations of mereology are: “Relations are reflexive, antisymmetric and transitive. (…) First, everything is part of itself. Second, two different objects cannot be part of each other. Third, each part of a part of a whole is also part of that whole. Fourth, an object can be a part of another object, if both exist.”[10] 

Mereology can be a promising approach also for the reading of architectural form, as it emphasises relationships without reducing buildings to their appearance or function. However, such philosophical descriptions consider wholes and parts as mostly abstract figures. Therefore, a supplement could be developed to properly categorise the mereological relations in the field of architecture. Having the relations between form and mereology addressed, methodologies can be developed to access the analysis of architectural form. Mereology as a specific methodology for architecture is quite new. One of the first introductions can be found in Daniel Koehler’s book The Mereological City: A Reading of the Works of Ludwig Hilberseimer (2016). Here, Koehler departs from the modern city, exemplified through the work of Ludwig Hilberseimer to illustrate mereological relations in the modernist city. From the room to the house to the city to the region, Hilberseimer canonically drew the city as a hierarchical, nested stack of cellular spaces.[11] However, through the close reading of its mereological relations it becomes clear that political, economic or social conditions are entangled in a circular composition between the parts of the city. Recalling Garcia’s discourse, and resonating with Leon Battista Alberti’s thesis, Koehler shows that the cells in Hilberseimer’s modernist city are interlocked. A house becomes the whole for rooms; a city becomes the whole for houses. By considering the city and its individual buildings equally, “the whole is a part for the part as a whole.”[12]

Architectural Relations Between Parts and Wholes

Parts are not only grouped, packed and nested through different scales, but also in different relations. Specific relationships have been developed in different architectural epochs and styles. Mathematically, four general classes of relations can be drawn: whole-to-whole, part-to-part, whole-to-parts and parts-to-whole, while more specific subclasses can be discovered from each. 

According to the mathematical definition, between wholes there exist complex relations, the whole could exist on any mereological level, and the complexity of relations between multiple levels are also accounted for. Whole-to-whole relations can become complex when considering multi-layer interaction, and more relations can be identified: juxtapose, overlap, contain, undercrossing, transitivity, partition, trans-boundary, intact juxtapose, compromised juxtapose.

A first glance of New York, gives the impression that it is quite heterogeneous, but underneath there is a city grid underlying the heterogeneity, and while the relations displayed in the grid are rather simple, all wholes juxtapose with one another. In comparison, in Siena, an Italian city, the urban space is quite complex, where boundaries of all wholes negotiate with others, the gaps in between are carefully treated, the nesting relations are extremely rich, and multiple relations from the diagram above can be found.

The whole-to-parts relation studies what the whole does to its part, namely in terms of top-down rules. The mathematical definition does not involve specific situations that a whole-part condition holds. Distinctions within individual contexts make a significant difference in clarifying an explicit relation. The situations for the whole can generally be classified into following types: fuse, fit and combine.

One of Zaha Hadid’s projects, Heydar Aliyev Centre, indicates the fusing relation. Architecture is represented as a smooth, fluid volume. The distinction between elements disappears, and this dominating power even extends to the external landscape. In order to maintain a continuous whole, parts are fabricated into a particular shape, having their unique unchangeable locations. The continuous whole excessively overwhelms the parts, yet not all parts are reshaped to fuse into the whole, and because the parts are small enough in relationship to the whole, the control from the whole is weakened, and parts are fit into the whole.

The third type is combining. An example for this relation is Palladio’s project Villa Rotonda. In this case, parts are obvious. The whole is a composition of the parts’ identities. However, the whole also holds a strong framework, in a rigorous geometric rule that decides positions and characters of parts. The arrangement of parts is the embodiment of this framework. 

The parts-to-whole relation studies what the parts do to the whole, or the power of bottom-up relationships. The different situations of parts are also key parameters in validating a given relation. The classification of situations for parts are as follows: frame, intrinsic frame, extrinsic frame, bounded alliance, unbounded alliance.

Emil Kaufmann thoroughly investigated the innovative works by Claude Nicholas Ledoux in Three Revolutionary Architects: Boullee, Ledoux and Lequeu (1952).[13] According to Kaufmann’s study, Ledoux’s works developed new compositional relations of elements from the Baroque. The characteristics of parts in Baroque architecture are rich, but tend to regulate the identities of all the elementary parts and fuse them together to serve the harmony of the whole, presenting the intrinsic framing. Ledoux’s work is an extrinsic framing, where the parts are relatively independent, with each element maintaining its own properties, and while consisting of the whole, they can be replaced with other identical components.

One of my projects in discrete aggregation of elements presents an unbounded alliance relation. The aggregation as a whole shows a form that is discretised (Figure 12), and does not pass any top-down instructions to its parts.

Part-to-Part Without Whole – The Ultimate Parts

For part-to-part relations, local interactions are emphasised, and interactions occur at multiple levels of compositions, where the part-to-part relations in some cases are similar to that between wholes. It has following classifications: juxtapose, interrelate, contain, partition, overlap, trans-juxtapose, over-juxtapose, over-partition, over-overlap.

Architects have been working on the possibility of removing the whole by studying the part-to-part relations. Several approaches have been developed, mainly through computation. Neil Leach considers the city as a “swarm intelligence”,[14] bringing forward the potential of developing urban form with computational method. Leach encourages swarm intelligence for the interactions between agents (parts), which “offers behavioral translations of topology and geometry”,[15] while fractals, L-systems or cellular automata are all constrained by some limitation. However, although swarm intelligence is based on the interaction of individual agents, it is always treated as a whole; all cells of CA are fixed in the background grid, which is also a whole. For fractals and L-systems, they can be subdivided into infinite parts, a transcendent whole where all parts grown from still exist. In the mereological sense, none of these cases can escape the shadow of the whole – strictly speaking, they are part-to-whole relations. To discuss the part-to-part relation in more depth, more investigation is needed to clarify the concept of part.

In The Democracy of Objects (2011), Levi Bryant claims that objects constitute a larger object by establishing relations with others, but this doesn’t alter the existence of objects, as he says, “all objects equally exist, but not all objects exist equally.” In Bryant’s discourse, this independence suggests the dissolution of the whole. Bryant proposes a concept of “regimes of attraction”, that includes the “endo-relation” and the “exo-relation”. The endo-relation indicates that the proper being of an object consists of its powers or what an object can do”, not the “qualities” emerging within an exo-relation. An object possesses “volcanic powers”, the stabilisation of the regime of attraction actualises it into a specific state.[16] The concept of the whole reduces objects to this state, which displays only a section of their proper beings. The concept of regimes of attraction is against this reduction.

The regime of attraction can be linked to the notion of “assemblage” from Manuel DeLanda, however, there is a distinction between the two. Assemblage holds only the relation of exteriority, whereas regime of attraction maintains both relations of interiority and exteriority. In Assemblage Theory (2016), DeLanda reassembled the concept “assemblage”, which was originated from the French agencement. Created by Gilles Deleuze and Félix Guattari, this original term refers to the following meanings: the “action of matching or fitting together a set of components” – the process, and the “result of such an action” – the product. 

DeLanda emphasised two aspects, heterogeneity and relations. As he indicated, the “contrast between filiations and alliances”[17] can be described in other words as intrinsic and extrinsic relations. 

The nature of these relations has different influences on the components. The intrinsic relation tends to define the identities of all the parts and fix them into exact location, while the extrinsic relation connects the parts in exteriority – without interfering with their identities. DeLanda summarised four characteristics of assemblage: 1) individuality, an assemblage is an individual entity, despite different scale or different number of components; 2) heterogeneity, components of an assemblage are always heterogeneous; 3) composable, assemblages can be composed into another assemblage; 4) bilateral-interactivity, an assemblage emerges from parts interactions, it also passes influences on parts.[18]

DeLanda then moved on to the two parameters of assemblage. The first parameter is directed toward the whole, the “degree of territorialisation and deterritorialisation”, meaning how much the whole “homogenises” its component parts. The second parameter is directed toward the parts, the “degree of coding and decoding”, meaning how much the identities of parts are fixed by the rules of the whole. The concept of assemblage provides us a new lens of investigating these mereological relations. With this model, the heterogeneities and particularity of parts are fully respected. The wholes become immanent, individual entities, existing “alongside the parts in the same ontological plane”,[19] while parts in a whole are included in the whole but not belonging to it, and according to Bryant’s discourse, the absence of belonging dispelled the existence of the whole.[20]

From the study of regime of attraction and assemblage, this essay proposes a new concept – “the ultimate parts” – in which a proper “part-to-part without whole” is embedded. A part (P) horizontally interacts with its neighbouring parts (Pn), with parts of neighbouring parts (Pnp), as well as interacting downwardly with parts that compose it (Pp) and upwardly with wholes it is constituting which are also parts (Pw). This concept significantly increases the initiatives of parts and decreases the limitations and reductions of them. It doesn’t deny the utilities of the whole, but considers the whole as another independent entity, another part. It’s neither top-down, nor bottom-up, but projects all relations from a hierarchical structure to a comprehensive flattened structure. The ultimate parts concept provides a new perspective for observing relations between objects from a higher dimension.

One application of this concept is TARSS (Tensegrity Adaptive Robotic Structure System), my research project in MArch Architectural Design in B-Pro at The Bartlett School of Architecture in 2017–2018. This project utilises the features of tensegrity structures of rigidity, flexibility and lightweight. The difference is that rather than fixing parts into a static posture and eliminating their movements, the project contrarily tries to increase the freedom of parts as much as possible. The tensile elements have the ability to adjust their lengths collaboratively to change the general shape of the aggregation. Reinforcement learning is employed to empower the parts with objective awareness. The training sessions were set up toward multiple objectives that are related to architectural concerns, including pathfinding, transformation, balance-keeping, self-assembling and structural load distributing. This approach brings obvious benefits, as architecture design in this sense is not only about an eventual result, but about the dynamic process of constantly responding to the environmental, spatial or functional requirements. The premise is to treat parts as ultimate parts whilst retaining their objectivity and being able to actively interact at all mereological levels without limitations.

The concept of ultimate parts brings forward a new relation of “part-to-part without whole”. This new relation belongs to a higher dimension. The details and essence of objects are simultaneously displayed, without being obscured by the compositional structure. Analogised with spatial dimensions, a 3-dimensional cube simultaneously shows all its faces and interior in 4-dimensional space. The significance is that it opens vast new perspectives and operational methodologies in the architectural design realm. Especially with the advancement in robotics and artificial intelligence, this type of new relationship enables greater opportunities by regarding machines as characters with immense potential to work with us, instead of for us. The role of designers would be very much like “breeders of virtual forms”,[21] who do not rule the form, but guide it towards the demands. This moves away from anthropocentric design by overcoming part-to-whole with part-to-part.