This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

I argue that the contrast between models and theories is important for public policy issues. I focus especially on the way a mathematical model explains just one aspect of the data.

Prior to statistical mechanics and especially before the advent of the new quantum mechanics, it was traditionally held, following in the main the Kantian philosophy, that the task of science is to attain a unique quantitative representation of reality. It was thought—and with justifiable zeal—that a scientific discipline is exact to the extent to which a mathematical pattern yielding quantitative relations can be applied to its subject matter. This view was based on the implicit assumptions that functional relatedness (...) conducive to quantitative evaluation is bound to reveal the most generic, and hence, the most essential traits of events. (shrink)

Mathematical models provide explanations of limited power of specific aspects of phenomena. One way of articulating their limits here, without denying their essential powers, is in terms of contrastive explanation.

Mathematics from the electromagnetic field quantization procedure and the soliton models of photons are used to construct a new 3D model of photons. Besides the interaction potential between the charged particle and the photons, which contains the annihilation and creation operators of photons, the new function for a description of free propagating photons is derived. This function presents the vector potential of the field, the function is a product of the harmonic oscillator eigenfunction with the well-defined coordinate of the (...) oscillator and the Gaussian function of the polar radius in the transverse direction. In the article, the difference between the quantum mechanics of particles and photons is discussed. (shrink)

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That (...) series rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology. (shrink)

This article contributes to the revision of the procedure of robustness analysis of mathematical models in epistemic democracy using the systematic review method. It identifies the drawbacks of robustness analysis in epistemic democracy in terms of sample universality and inference from samples with the same results. To exemplify the effectiveness of systematic review, this article conducted a pilot review of diversity trumps ability theorem models, which are mathematical models of deliberation often cited by epistemic democrats. (...) A review of nine models extracted from 352 papers exemplifies the effectiveness of robustness analysis supplemented by systematic review in epistemic democracy. (shrink)

Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a (...) central role, namely homotopy theory. I argue that mathematicians introduce genuine models and I offer a rough classification of these models. (shrink)

Current understanding of the development of quantity representations is based primarily on performance in the number‐line task. We posit that the data from number‐line tasks reflect the observer's underlying representation of quantity, together with the cognitive strategies and skills required to equate line length and quantity. Here, we specify a unified theory linking the underlying psychological representation of quantity and the associated strategies in four variations of the number‐line task: the production and estimation variations of the bounded and unbounded number‐line (...) tasks. Comparison of performance in the bounded and unbounded number‐line tasks provides a unique and direct way to assess the role of strategy in number‐line completion. Each task produces a distinct pattern of data, yet each pattern is hypothesized to arise, at least in part, from the same underlying psychological representation of quantity. Our model predicts that the estimated biases from each task should be equivalent if the different completion strategies are modeled appropriately and no other influences are at play. We test this equivalence hypothesis in two experiments. The data reveal all variations of the number‐line task produce equivalent biases except for one: the estimation variation of the bounded number‐line task. We discuss the important implications of these findings. (shrink)

Projections of future climate change cannot rely on a single model. It has become common to rely on multiple simulations generated by Multi-Model Ensembles (MMEs), especially to quantify the uncertainty about what would constitute an adequate model structure. But, as Parker points out (2018), one of the remaining philosophically interesting questions is: “How can ensemble studies be designed so that they probe uncertainty in desired ways?” This paper offers two interpretations of what General Circulation Models (GCMs) are and how (...) MMEs made of GCMs should be designed. In the first interpretation, models are combinations of modules and parameterisations; an MME is obtained by “plugging and playing” with interchangeable modules and parameterisations. In the second interpretation, models are aggregations of expert judgements that result from a history of epistemic decisions made by scientists about the choice of representations; an MME is a sampling of expert judgements from modelling teams. We argue that, while the two interpretations involve distinct domains from philosophy of science and social epistemology, they both could be used in a complementary manner in order to explore ways of designing better MMEs. (shrink)

This paper sets out to show how mathematical modelling can serve as a way of ampliating knowledge. To this end, I discuss the mathematical modelling of time in theoretical physics. In particular I examine the construction of the formal treatment of time in classical physics, based on Barrow’s analogy between time and the real number line, and the modelling of time resulting from the Wheeler-DeWitt equation. I will show how mathematics shapes physical concepts, like time, acting as a (...) heuristic means—a discovery tool—, which enables us to construct hypotheses on certain problems that would be hard, and in some cases impossible, to understand otherwise. (shrink)

In this essay I argue against I. Bernard Cohen's influential account of Newton's methodology in the Principia: the 'Newtonian Style'. The crux of Cohen's account is the successive adaptation of 'mental constructs' through comparisons with nature. In Cohen's view there is a direct dynamic between the mental constructs and physical systems. I argue that his account is essentially hypothetical-deductive, which is at odds with Newton's rejection of the hypothetical-deductive method. An adequate account of Newton's methodology needs to show how Newton's (...) method proceeds differently from the hypothetical-deductive method. In the constructive part I argue for my own account, which is model based: it focuses on how Newton constructed his models in Book I of the Principia. I will show that Newton understood Book I as an exercise in determining the mathematical consequences of certain force functions. The growing complexity of Newton's models is a result of exploring increasingly complex force functions (intra-theoretical dynamics) rather than a successive comparison with nature (extra-theoretical dynamics). Nature did not enter the scene here. This intra-theoretical dynamics is related to the 'autonomy of the models'. (shrink)

To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality, personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment of (...) mathematics to informal reality is not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed. (shrink)

Taking scientific practice as its starting point, this book charts the complex territory of models used in science. It examines what scientific models are and what their function is. Reliance on models is pervasive in science, and scientists often need to construct models in order to explain or predict anything of interest at all. The diversity of kinds of models one finds in science – ranging from toy models and scale (...) class='Hi'>models to theoretical and mathematical models – has attracted attention not only from scientists, but also from philosophers, sociologists, and historians of science. This has given rise to a wide variety of case studies that look at the different uses to which models have been put in specific scientific contexts. By exploring current debates on the use and building of models via cutting-edge examples drawn from physics and biology, the book provides broad insight into the methodology of modelling in the natural sciences. It pairs specific arguments with introductory material relating to the ontology and the function of models, and provides some historical context to the debates as well as a sketch of general positions in the philosophy of scientific models in the process. (shrink)

This paper describes model building and manipulation in the solution of problems in mechanics. An automatic problem solver, MECHO, solving problems in several areas of mechanics, employs (1) a knowledge base representing the semantic content of the particular problem area, (2) a means-ends search strategy similar to GPS to produce sets of simultaneous equations and (3) a “focusing” technique, based on the data within the knowledge base, to guide the GSP-like search through possible equation instantiations. Sets of predicate logic statements (...) are employed to describe this model building activity. These clauses are used to give information content to the knowledge base and provide both basis and guidance for the goal driven search.It is hypothesized that humon subjects solving mechanics problems employ similar model building techniques. Protocols of several subjects are presented and comparisons are drawn with the traces of the automated problem solver. Adjustments are made to the program to provide a better fit of traces to protocols. Some implications are presented for using a rule based model for describing human problem solving performance in solving mechanics problems. (shrink)

The ArgumentWhen economists report on research using mathematical models, they use a literary form similar to the experimental report in the laboratory sciences. This form consists of a narrative of a series of events, with a clear temporal segregation of the agency of the author and the agency of the objects of study. Existing explanations of this literary form treat it as a rhetorical device that either conceals the agency of the author in constructing and interpreting the findings, (...) or simply appropriates the appearance of accepted scientific method. This article — based on analysis of a research program in economics, a single article that issued from that program, and in-depth interviews with the authors — proposes an alternate interpretation. Drawing on the praxeological “laboratory studies” tradition in science studies, we treat work with mathematical models as involving the interaction of economists with objects that act independently of the analyst's will. The clear separation of the economist's and the models agency, as depicted in the published report, is not the result of a rhetorical rewriting of actual events, but is a practical accomplishment. Every step in the analytical work that preceded the paper is devoted to developing a procedure in which the economists' agency will be completely accountable in terms of accepted practices, and the performance of the model will be distinct and compelling. (shrink)

We argue that dynamical and mathematical models in systems and cognitive neuro- science explain (rather than redescribe) a phenomenon only if there is a plausible mapping between elements in the model and elements in the mechanism for the phe- nomenon. We demonstrate how this model-to-mechanism-mapping constraint, when satisfied, endows a model with explanatory force with respect to the phenomenon to be explained. Several paradigmatic models including the Haken-Kelso-Bunz model of bimanual coordination and the difference-of-Gaussians model of (...) visual receptive fields are explored. (shrink)

The philosophy of linguistics is a rich philosophical domain which encompasses various disciplines. One of the aims of this thesis is to unite theoretical linguistics, the philosophy of language, the philosophy of science and the ontology of language. Each part of the research presented here targets separate but related goals with the unified aim of bringing greater clarity to the foundations of linguistics from a philosophical perspective. Part I is devoted to the methodology of linguistics in terms of scientific (...) modelling. I argue against both the Conceptualist and Platonist interpretations of linguistic theory by means of three grades of mathematical involvement for linguistic grammars. Part II explores the specific models of syntactic and semantics by an analogy with the harder sciences. In Part III, I develop a novel account of linguistic ontology and in the process comment on the type-token distinction, the role and connection with mathematics and the nature of linguistic objects. In this research, I offer a structural realist interpretation of linguistic methodology with a nuanced structuralist picture for its ontology. This proposal is informed by historical and current work in theoretical linguistics as well as philosophical views on ontology, scientific modelling and mathematics. (shrink)

Statistical learning refers to the ability to identify structure in the input based on its statistical properties. For many linguistic structures, the relevant statistical features are distributional: They are related to the frequency and variability of exemplars in the input. These distributional regularities have been suggested to play a role in many different aspects of language learning, including phonetic categories, using phonemic distinctions in word learning, and discovering non-adjacent relations. On the surface, these different aspects share few commonalities. Despite this, (...) we demonstrate that the same computational framework can account for learning in all of these tasks. These results support two conclusions. The first is that much, and perhaps all, of distributional statistical learning can be explained by the same underlying set of processes. The second is that some aspects of language can be learned due to domain-general characteristics of memory. (shrink)

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunctions being the primary lens. If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets. By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. Semiadjunctions (...) (essentially a formulation of universal mapping properties using hets) can then be combined in a new way to define the notion of a brain functor that provides an abstract model of the intentionality of perception and action (as opposed to the passive reception of sense-data or the reflex generation of behavior). (shrink)

Are there genuine mathematical explanations of physical phenomena, and if so, how can mathematical theories, which are typically thought to concern abstract mathematical objects, explain contingent empirical matters? The answer, I argue, is in seeing an important range of mathematical explanations as structural explanations, where structural explanations explain a phenomenon by showing it to have been an inevitable consequence of the structural features instantiated in the physical system under consideration. Such explanations are best cast as deductive (...) arguments which, by virtue of their form, establish that, given the mathematical structure instantiated in the physical system under consideration, the explanandum had to occur. Against the claims of platonists such as Alan Baker and Mark Colyvan, I argue that formulating mathematical explanations as structural explanations in this way shows that we can accept that mathematics can play an indispensable explanatory role in empirical science without committing to the existence of any abstract mathematical objects. (shrink)

This volume represents a magnum opus by Wolfgang Weidlich, summarizing his long work in the area of sociodynamics. It lays out the origins and development of his ideas on this topic, presents a variety of applications drawn from his previous work, and offers some new insights and suggestions. For those acquainted with Professor Weidlich’s work it is a satisfying summing up. For those unacquainted with it, the book provides a good overview and discussion of what is involved in it, both (...) its weaknesses and its strengths. It has a definite predecessor, Weidlich’s 1983 book with his frequent coauthor, Günter Haag, Concepts and Models of a Quantitative Sociology, but goes well beyond the arguments and models presented in that volume. (shrink)

Although the use of mathematical models is ubiquitous in modern science, the involvement of mathematical modeling in the sciences is rarely seen as cases of interdisciplinary research. Often, mathematics is “applied” in the sciences, but mathematics also features in open-ended, truly interdisciplinary collaborations. The present paper addresses the role of mathematical models in the open-ended process of conceptualizing new phenomena. It does so by suggesting a notion of cultures of mathematization, stressing the potential role (...) of the mathematical model as a boundary object around which negotiations of different desiderata can take place. This framework is then illustrated by a case study of the early efforts to produce a mathematical model for quasi-crystals in the first two decades after Dan Shechtman’s discovery of this new phenomenon in 1984. (shrink)

Extract from Hofstadter's revew in Bulletin of American Mathematical Society : http://www.ams.org/journals/bull/1980-02-02/S0273-0979-1980-14752-7/S0273-0979-1980-14752-7.pdf -/- "Aaron Sloman is a man who is convinced that most philosophers and many other students of mind are in dire need of being convinced that there has been a revolution in that field happening right under their noses, and that they had better quickly inform themselves. The revolution is called "Artificial Intelligence" (Al)-and Sloman attempts to impart to others the "enlighten- ment" which he clearly regrets not (...) having experienced earlier himself. Being somewhat of a convert, Sloman is a zealous campaigner for his point of view. Now a Reader in Cognitive Science at Sussex, he began his academic career in more orthodox philosophy and, by exposure to linguistics and AI, came to feel that all approaches to mind which ignore AI are missing the boat. I agree with him, and I am glad that he has written this provocative book. The tone of Sloman's book can be gotten across by this quotation (p. 5): "I am prepared to go so far as to say that within a few years, if there remain any philosophers who are not familiar with some of the main developments in artificial intelligence, it will be fair to accuse them of professional incom- petence, and that to teach courses in philosophy of mind, epistemology, aesthetics, philosophy of science, philosophy of language, ethics, metaphysics, and other main areas of philosophy, without discussing the relevant aspects of artificial intelligence will be as irresponsible as giving a degree course in physics which includes no quantum theory." -/- (The author now regrets the extreme polemical tone of the book.). (shrink)

A case study of quantum mechanics is investigated in the framework of the philosophical opposition “mathematical model – reality”. All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be “complete” (versus Einstein’s opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Gödel’s opinion) can be interpreted (...) furthermore as the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely “transparent” and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr’s postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them “conservation of energy conservation” in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of “quantum gravity” as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two “parts” or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are “flat” and only “properties”: they need only a single separable complex Hilbert space to be defined. (shrink)

Wilson [Dialectica 63:525–554, 2009], Moore [Int Stud Philos Sci 26:359–380, 2012], and Massin [Br J Philos Sci 68:805–846, 2017] identify an overdetermination problem arising from the principle of composition in Newtonian physics. I argue that the principle of composition is a red herring: what’s really at issue are contrasting metaphysical views about how to interpret the science. One of these views—that real forces are to be tied to physical interactions like pushes and pulls—is a superior guide to real forces (...) than the alternative, which demands that real forces are tied to “realized” accelerations. Not only is the former view employed in the actual construction of Newtonian models, the latter is both unmotivated and inconsistent with the foundations and testing of the science. (shrink)

Metaphors and models involve correspondences between events in separate domains. They differ in the form and precision of how the correspondences are expressed. Examples include correspondences between phylogenic and ontogenic selection, and wave and particle metaphors of the mathematics of quantum physics. An implication is that the target article's metaphors of resistance to change may have heuristic advantages over those of momentum.

Although every emerging infectious disease occurs in a unique context, the behaviour of previous pandemics offers an insight into the medium- and long-term outcomes of the current threat. Where an informative historical analogue exists, epidemiologists and policymakers should consider how the insights of the past can inform current forecasts and responses.

In this paper I consider some data of Covid-19 infection in Italy from the 20th of February to the 29th of June 2020. Data are analyzed using some fits based on mathematical models. This analysis may be proposed to students of the last class of the Liceo Scientifico in order to debate a real problem with mathematical tools.

Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting (...) two cases: Cramér’s random model of the prime numbers and the function field model of the integers. These cases show that mathematicians, like empirical scientists, rely on unrealistic models to gain understanding of complex phenomena. They also have important implications for some much-discussed theses about scientific understanding. First, modeling practices in mathematics confirm that one can gain understanding without obtaining an explanation. Second, these cases undermine the popular thesis that unrealistic models confer understanding by imparting counterfactual knowledge. (shrink)

Does the materiality of a three-dimensional model have an effect on how this model operates in an exploratory way, how it prompts discovery of new mathematical results? Material mathematical models were produced and used during the second half of the nineteenth century, visualizing mathematical objects, such as curves and surfaces—and these were produced from a variety of materials: paper, cardboard, plaster, strings, wood. However, the question, whether their materiality influenced the status of these models—considered as (...) exploratory, technical, or representational—was hardly touched upon. This article aims to approach this question by investigating two case studies: Beltrami’s paper models vs. Dyck’s plaster ones of the hyperbolic plane; and Chisini’s string models of braids vs. Artin’s and Moishezon’s algebraization of these braids. These two case studies indicate that materiality might have a decisive role in how the model was taken into account mathematically: either as an exploratory or rather as a technical or pedagogical object. (shrink)

An introduction to the model-theoretic approach in the philosophy of science is given and it is argued that this program is further enhanced by the introduction of partial structures. It is then shown that this leads to a natural and intuitive account of both "iconic" and mathematical models and of the role of the former in science itself.

Scientists have used models for hundreds of years as a means of describing phenomena and as a basis for further analogy. In Scientific Models in Philosophy of Science, Daniela Bailer-Jones assembles an original and comprehensive philosophical analysis of how models have been used and interpreted in both historical and contemporary contexts. Bailer-Jones delineates the many forms models can take (ranging from equations to animals; from physical objects to theoretical constructs), and how they are put to (...) use. She examines early mechanical models employed by nineteenth-century physicists such as Kelvin and Maxwell, describes their roots in the mathematical principles of Newton and others, and compares them to contemporary mechanistic approaches. Bailer-Jones then views the use of analogy in the late nineteenth century as a means of understanding models and to link different branches of science. She reveals how analogies can also be models themselves, or can help to create them. The first half of the twentieth century saw little mention of models in the literature of logical empiricism. Focusing primarily on theory, logical empiricists believed that models were of temporary importance, flawed, and awaiting correction. The later contesting of logical empiricism, particularly the hypothetico-deductive account of theories, by philosophers such as Mary Hesse, sparked a renewed interest in the importance of models during the 1950s that continues to this day. Bailer-Jones analyzes subsequent propositions of: models as metaphors; Kuhn's concept of a paradigm; the Semantic View of theories; and the case study approaches of Cartwright and Morrison, among others. She then engages current debates on topics such as phenomena versus data, the distinctions between models and theories, the concepts of representation and realism, and the discerning of falsities in models. (shrink)

Biologists, climate scientists, and economists all rely on models to move their work forward. In this book, I explore the use of models in these and other fields to introduce readers to the various philosophical issues that arise in scientific modeling. I show that paying attention to models plays a crucial role in appraising scientific work. -/- After surveying a wide range of models from a number of different scientific disciplines, I demonstrate how focusing on (...) class='Hi'>models sheds light on many perennial issues in philosophy of science and in philosophy in general. For example, reviewing the range of views on how models represent their targets introduces readers to the key issues in debates on representation, not only in science but in the arts as well. Also, standard epistemological questions are cast in new and interesting ways when we confront the question, "What makes for a good (or bad) model?". (shrink)

A new diagnostic modeling system for automatically synthesizing a deep‐structure model of a student's misconceptions or bugs in his basic mathematical skills provides a mechanism for explaining why a student is making a mistake as opposed to simply identifying the mistake. This report is divided into four sections: The first provides examples of the problems that must be handled by a diagnostic model. It then introduces procedural networks as a general framework for representing the knowledge underlying a skill. The (...) challenge in designing this representation is to find one that facilitates the discovery of misconceptions or bugs existing in a particular student's encoding of this knowledge. The second section discusses some of the pedagogical issues that have emerged from the use of diagnostic models within an instructional system. This discussion is framed in the context of a computer‐based tutoring/gaming system developed to teach students and student teachers how to diagnose bugs strategically as well as how to provide a better understanding of the underlying structure of arithmetic skills. The third section describes our uses of an executable network as a tool for automatically diagnosing student behavior, for automatically generating “diagnostic” tests, and for judging the diagnostic quality of a given exam. Included in this section is a discussion of the success of this system in diagnosing 1300 school students from a data base of 20.000 test items. The last section discusses future research directions. (shrink)