Computational Machines in a Coexistence with Concrete Universals and Data Streams

We discuss that how the majority of traditional modeling approaches are following the idealism point of view in scientific modeling, which follow the set theoretical notions of models based on abstract universals. We show that while successful in many classical modeling domains, there are fundamental limits to the application of set theoretical models in dealing with complex systems with many potential aspects or properties depending on the perspectives. As an alternative to abstract universals, we propose a conceptual modeling framework based on concrete universals that can be interpreted as a category theoretical approach to modeling. We call this modeling framework pre-specific modeling. We further, discuss how a certain group of mathematical and computational methods, along with ever-growing data streams are able to operationalize the concept of prespecific modeling. 1. How to Approach the Notion of Scientific Modeling? Modeling paradigms, as a necessary element of any scientific investigation, act like pairs of glasses, which impact the way in which we encode (conceive of) the real world. Therefore, any kind of intervention in real world phenomena is affected by the chosen modeling paradigm and the real phenomena under investigation. In the domain of urbanism and urban design, cities as complex and open environments with dynamic and multidimensional aspects are challenging cases for modeling scholars, as there are many distinct urban phenomena. Figure 1 shows a list of different functional aspects of urban phenomena in an indexical manner. Figure 1. Different functional aspects of urban phenomena In addition to the diversity of urban problems, there is a huge variety of competing paradigms for analyzing cities: the city as an ecological phenomena that is optimally adjusted to an environment (economic, political, cultural) assumed to be “natural” for it; the city as a thermodynamic system that needs to be balanced and which can be controlled; the city as a grammatical text with its own “syntactical laws”; the city as a biological organism following fractal growth patterns. Further, historical perspective provide additional city models such as the City of Faith, the City as Machine, or the Organic City, and especially, since the advent of computers from the second half of the twentieth century, city as information. Although comparing to classical science and its engineering disciplines such as Physics, Chemistry and Mechanics, urban design, planning and modeling is a rather young discipline, when one does a quick search of the keywords central to this field, one is quickly confused by the number of approaches and the variety of practical problems within the reaches of the discipline. For example, A. G. Wilson’s five volume text on urban modeling is over 2,600 pages long. A broad range of casebased canonization has thus emerged, and applied techniques are developed for specific urban functions such as urban land use, urban transportation, urban economy, urban social patterns, and so on. As a result, the lack of a more abstract categorization of applied techniques makes comparison between them very hard. Beginning in mid-twentieth century, General Systems Theory emerged as one of the main theories for working toward unification of different disciplinary modeling practices. In principle, the underlying idea of systems theory is the promotion of a unified view to modernist-reductionist science, which was diversified around a variety of application and functional domains. Although, interdisciplinary collaborations such as making analogies within disciplines, (e.g., hydraulic theories to describe biological systems) was not new, systems theory’s formalization, as an orthogonal view to classically diversified scientific and practical problems, reached to a point in which, according to George Klir, systemic tasks such as modeling, optimization, and simulation have emerged as distinct scientific disciplines. However, taking systems theory as a body of knowledge (rather than a specific and singular theory), one could expect a gradual divergence of its methods, starting from its unified principles. The advent of computational methods by Alan Turing in 1940s and later the democratization of computational methods in 1980s created a new diversified landscape of system modeling approaches. As a result, after fifty years we encounter a competitive ecosystem of different modeling species with different capacities and trade-offs. Figure 2 shows a list of different modeling methodologies in an indexical manner. Figure 2. Competitive ecosystem of different modeling methodologies. Therefore, the first motivation of this present essay is to find a unifying (abstract) perspective for assessment of different (inter-disciplinary) modeling approaches, while keeping the diversities. We think toward this aim we need to investigate the mathematical and philosophical grounds of scientific modeling. 2. Formal Definitions and Categories of Scientific Modeling Because there is such a wide variety of modeling approaches in different scientific domains, formalizing and theorizing the practice of scientific modeling is an active research area in philosophy of science. For example, according to Roman Frigg and Stephen Hartmann, there exist the following types of models: Probing models, phenomenological models, computational models, developmental models, explanatory models, impoverished models, testing models, idealized models, theoretical models, scale models, heuristic models, caricature models, didactic models, fantasy models, toy models, imaginary models, mathematical models, substitute models, iconic models, formal models, analogue models and instrumental models are but some of the notions that are used to categorize models. Nevertheless, these categories are not still abstract enough, but rather labels for different (not necessarily exclusive) modeling approaches. To better understand of models, one can look at the interpretation of their roles and functions, and to distinguish the presets on which the different points of view are based. One of the main issues by which models have been extensively discussed is the relation between models and the way of representation of real phenomena under study (the target system). According to Frigg and Hartmann, from a representational point of view there are “models of phenomena” and “models of data” and within these categories there are subcategories such as “scale models,” “idealized models,” “analogical models,” like the hydraulic model of an economic system, which are further divided to material analogy, where there is a direct similarity between the properties (or relations between properties) of two phenomena, and formal analogy, where two systems are based similarly on a formalization such as having the same mathematical equations that describe both systems. (Hesse 1963). Further, one can refer to phenomenological models, which are focused on the behavior of the particular phenomena under investigation rather than on underlying causes and mechanisms. Further, there are models of theories, like the dynamic model of the pendulum, which is based on Newtonian laws of motion. Models can also be divided into ontological classes like physical objects, fictional objects, set-theoretic structures, descriptions, and equations. However, these categories of models and modeling approaches overlap and they are rather descriptive and neutral classifications than critical. They do not give us a measure or a gauge to compare different modeling approaches in terms of their capacities and their limits in dealing with different levels of complexity in real world problems. In this essay I am looking for a way to condition modeling approaches in different levels of complexity to examine their theoretical capacities. Among the above-mentioned categories, the crucial, but somewhat commonly accepted shared property of the majority of traditional scientific modeling approaches is that they are all based on some sorts of idealization. In what follows, I explain different aspects of idealization in scientific modeling and, following the issues of idealizations, directs us to the problem of universals, which is an old philosophical issue. 3. Idealization in Scientific Modeling In the context of philosophy of science, idealization in modeling has been discussed extensively. In principle, idealization is considered to be equal to an intended (over) simplification in the representation of the target system. Although there are different ways of explaining or defining the notion of idealization, Michael Wiseberg discusses three kinds of idealization that we refer to in this work: minimalist idealization, Galilean idealization, and multiple-model idealization. Minimalist idealization is the practice of building models of real world phenomena by focusing only on main causal factors. Therefore, as is inferred from its name, minimalist models usually end to very simple elements that are informative enough for further decision-making. For example, the aim in domain of networks analytics is to explain complex behaviors that happen in the real phenomena by means of network properties such as centrality measures, integration, closeness, between-ness, etc. As an example, in urban theory, in the city science approach or urban scaling laws the final goal is to find a few main informative factors in cities such as city size or population in order to explain other aspects of cities such as energy consumption in a linear equation. Even though it seems obvious that cities are complex phenomena with many observable aspects and many exceptions, minimalist models attract attention exactly because they identify and state very general rules. Figure 3. Network analytics: Structure oriented modeling (minimalist idealization), Central Place Theory (left) and Space Syntax (right). City theories that seek to create archetypical city models are in a way minimalist idealized models. For example, Lynch’s City of Faith, City of Machine, or City as Organism or Cedric Price’s egg analogies of the city (city as boiled egg, city as fried egg, or city as scrambled egg) are characterized by few urban elements that are informative enough to explain each model and to discriminate that city model from the other models. David Grahame Shane shows how three above-mentioned models could be identified by linear combinations of three recombinant elements, called Enclave, Armature, and Heterotopia. The second category of Galilean idealization as the most pragmatic type of idealizations happens when the modeler intentionally simplifies the conditions of a complicated situation toward more computational tractability and simplicity. For example, it is common in economic models to assume that agents are rational maximizers, or in transportation models to assume that commuters take the shortest path, or to assume there is no friction in motion models of the particles. The basic idea of Galilean idealization is that by understanding the modeling environment gradually, it is possible to de-idealize or to build more comprehensive models on top of previous ones. Therefore, the majority of engineering approximation methods such as systems of differential equations or computational fluid dynamics or biological reaction networks are among this category of idealized models. Further, figure 4 shows how the idealization process in a complex phenomena (here, the agent based modeling of land-use transportation dynamics of a city) leads to a parametric and feature based representation of the real phenomena. This layering and parameterization gives the modeler the option to adjust the resolution (levels of details) of the model based on the needs and the purposes of the modeling process and the constraints and limitations, including the availability of data or prior knowledge or time and scale resolutions. Figure 4. Parametricism: Idealization of the interactions between different agencies through layering and parameterization of the real phenomena. The third category of idealization, multiple-model idealization, results to those models that consist of several (not necessarily compatible) models or several models with different assumptions and different properties. This type of idealization is in fact a combination of two other idealizations and it can be very useful when understanding the final output (the behavior) of the model is more important than knowing the underlying mechanisms of the target phenomena. For example, in weather forecasting, ensemble models, which (Gneiting and Raftery 2005) include several predictors with different parameters or even different structures, are used to predict weather conditions. Further, from a systemic and functional point of view there are many models in which idealization is happening in (one) main aspects of real phenomena. To just name a few: static or dynamic models, structure-oriented idealization (in network models), process-oriented idealization (such as system dynamics, system of differential equations), rule-based idealization (such as cellular automata or fractals), and decentralized interactions (such as agent based), all are placed in the above mentioned categories of idealizations. Figure 5. System dynamics: process-oriented idealization. However, considering the size and the variety of parameters and aspects in the target phenomena, idealized models create a dichotomy, where on one extreme the models are all general, simple and tractable, and on the other, models become complicated, specific and high-resolution. In fact, multiple model idealization becomes necessary whenever the selected parameters and aspects of the target system in each individual model (out of Galilean idealization for example) are not sufficient, but also add more aspects to an individual model, either making it more complicated or resulting in model inconsistency. This issue seems to be a never-ending debate in many scientific fields including biology, ecology, economics, and cognitive and social science, where one group believe in the explanatory power of models and the other group believes in model accuracy and the level of details comparing to the real phenomena. Although idealized models have been applied successfully in many classical modeling problems, but this type of debate cannot be fruitful in dealing with complex systems as long as there is no abstraction from the current paradigm of scientific modeling (i.e. idealization). Analogically, an onion-like model of numbers explains what I mean by the abstraction in the concept of modeling. For example, with natural numbers (or more generally, integers) one can never grasp the richness of proportions and fractions in rational numbers (e.g., 2.6, which is neither 2 or 3 from a natural number perspective), while the introduction to the concept of rational numbers as the ratio of two integer numbers (e.g., 26/10) solved this problem. Therefore, by choosing 1 as the denominator, one can show that all the integers are rational numbers; while with rational numbers we have new capacities in addition to integers. Similarly, if we take an idealized model as an arbitrary representation of real phenomena by adding several of them together (which is the case in multiple model idealization), we still cannot grasp the whole complexity. Therefore, our hypothesis is that an abstraction to the concept of modeling is needed in order to conceptually encapsulate all the potential arbitrary views in an implicit way. However, I do not claim that one can introduce a new concept as such, but in fact in this work I am trying to identify and discover new aspects of a potential body of thinking in scientific modeling. In order to highlight this conceptual abstraction from the current idealization paradigm, first we need to explain the notion of universals, including abstract and concrete universals, followed by our interpretations of these concepts in relation to the notion of scientific modeling. In the next section, after presenting the connections between the notions of idealization and abstract universals, I will formally describe the concepts of abstract universals and concrete universals, which can be interpreted as set theoretical and category theoretical definitions of these two notions. Further, I will show how the concept of concrete universals from category theory can open up a new level of modeling paradigm. 4. Universals and Modeling In the majority of texts written about idealization in the domain of scientific modeling, the notion of idealization is equal to simplification and the elimination of empirical details and deviations from a general theory that is the base for the final model. At the same time, the word “ideal” literally comes along with “those perfections that cannot be fully realized.” For example, circle-ness as a property is an ideal that cannot be fully realized, and any empirical circular shape has, to a degree, the circle-ness property. Figure 6. Enso (circle of Zen): Toward the ideal circle. Therefore, the idealization process in scientific modeling can be explained as a form of purification of empirical observations toward a set of given (assumed) ideal properties. In statistical data analysis, it is always assumed that collected empirical data follows a normal distribution function. Thus, one can convert the empirical data to a normal distribution function and utilizes from the machinery of this ideal mathematical representation (i.e. the normal distribution function). Applications of idealizations in many mathematical approaches such as linear algebra are enormous. For example, a Fourier transformation (Figure 7) can be seen as a form of idealization by which any observed time-varying data can be reconstructed (approximately) by a set of time-varying vectors (a set of pure sinusoidal waves with different frequencies and phases). From this perspective, any waveform phenomenon is a linear combination of a set of ideal prototypes. Figure 7. Fourier decomposition: Any observed form is a linear combination of some ideal cyclic form. However, these ideal forms (a wave with a certain frequency in the case of the Fourier analysis) as the set of aspects (properties) of real phenomena are abstract. This means that there is no concrete (empirical) instance that fully matches one or several of these a priori, ideal properties. From this point of view, idealized models are models that are based on the notion of abstract universals. The notions of “universals” and “property” are old topics in philosophy that can be approached differently, namely through realism, idealism, or nominalism. However, in this work I focus on the distinctions between concrete and abstract universals in relation to the paradigms of scientific modeling. According to David Ellerman, “In Plato's Theory of Ideas or Forms (ειδη), a property F has an entity associated with it, the universal uF, which uniquely represents the property. Therefore, an object X has the property F i.e. F(X), if and only if it participates in the universal uF to a degree (μ).” For example, “whiteness” is a universal and the set of white objects that participate in “whiteness” property (i.e., with different degrees of whiteness) are represented by this property. Further, “Given a relation μ, an entity uF is said to be a universal for the property F (with respect to μ) if it satisfies the following universality condition: For any x, x μ uF if and only if F(x).” This condition is called universality, and it means that the universal is the essence of that property. In addition to universality, a universal should be unique. “Hence there should be an equivalence relation (≈) so that universals satisfy a uniqueness condition: If uF and uF' are universals for the same F, then uF ≈ uF'.” Therefore, any entity that satisfies the conditions of universality and uniqueness for a certain property is a universal for that property. Now, if a universal is self-participating, it is called a concrete universal; if it does not have self-participatory properties, it is an abstract universal. For example, whiteness is an abstract universal as there is no empirical (concrete instance) to be “whiteness.” In language models, being a “verb” is a property that can be assigned to many words, but “verb” itself is an external definition and it is not self-participating in the sets of concrete verbs. The same argument goes for the above example of the Fourier analysis and ideal forms. On the other hand, defining a property as being part of set A and set B has a concrete universal, which is the intersection of two sets A and B (A∩B). It means that any object from set A and B (including all the potential subsets) that has this property (being part of A and B) participates in the intersection set A∩B, and since A∩B is participating in itself, then it is a concrete universal. Further, Ellerman shows that how modern set theory is the language of abstract universals and how category theory can be developed as the mathematical machinery of concrete universals. Finally, he summarizes that, Category theory as the theory of concrete universals has a different flavor from set theory, the theory of abstract universals. Given the collection of all the elements with a property, set theory can postulate a more abstract entity, the set of those elements, to be the universal. But category theory cannot postulate its universals because those universals are concrete. Category theory must find its universals, if at all, among the entities with the property. In the past few decades there have been many theoretical works to further the new field of category theory in terms of this fundamental difference between set theory and category theory. For example, currently the main categorical approaches in mathematics are topos theory and sheaf theory, which are generalizations of topology and geometry to an algebraic level. It seems that applications of these general frameworks in different domains should be one of the main future research areas in the field of modeling. On the other hand, Ellerman concludes that, Topos theory is important in its own right as a generalization of set theory, but it does not exclusively capture category theory’s foundational relevance. Concrete universals do not “generalize” abstract universals, so as the theory of concrete universals, category theory does not try to generalize set theory, the theory of abstract universals. Category theory presents the theory of the other type of universals, the self-participating or concrete universals. Now that we have defined the concepts of abstract and concrete universals, we need to formalize two different approaches of modeling, which are based on these notions of the universal. As stated earlier, idealized models are models that are based on the notion of abstract universals and consequently idealized models can be interpreted as set theoretical models. In the next section, by focusing on the idea of representation in idealized models, I show their theoretical consequences and their limits in dealing with complex systems, with the definition of the abstract universal being crucial. Next, I show another conceptual representational framework that is matched with the concept of concrete universals. Further, I will introduce an alternative line of modeling to