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)

The present contribution offers an overview of the main problems concerning the representation of the planets in the pre-Islamic Iranian world, the origin of their denominations, their astral roles and the reasons behind their demonization in the Zoroastrian and Manichaean frameworks. This is a preliminary attempt to resume the planetary iconography and iconology in western and eastern Iranian sources, involving also external visual data, such as those coming from Dunhuang and the Chinese world. The article offers an intellectual journey into (...) a net of mutual cultural and spiritual relations, focusing on the image of the heaven (and of its celestial beings), thereby proposing a new synthesis and highlighting a number of intercultural contaminations. (shrink)

The present contribution offers an overview of the main problems concerning the representation of the planets in the pre-Islamic Iranian world, the origin of their denominations, their astral roles and the reasons behind their demonization in the Zoroastrian and Manichaean frameworks. This is a preliminary attempt to resume the planetary iconography and iconology in western and eastern Iranian sources, involving also external visual data, such as those coming from Dunhuang and the Chinese world. The article offers an intellectual journey into (...) a net of mutual cultural and spiritual relations, focusing on the image of the heaven (and of its celestial beings), thereby proposing a new synthesis and highlighting a number of intercultural contaminations. (shrink)

Microbial model systems have a long history of fruitful use in fields that include evolution and ecology. In order to develop further insight into modelling practice, we examine how the competitive exclusion and coexistence of competing species have been modelled mathematically and materially over the course of a long research history. In particular, we investigate how microbial models of these dynamics interact with mathematical or computational models of the same phenomena. Our cases illuminate the ways in which microbial systems (...) and equations work as models, and what happens when they generate inconsistent findings about shared targets. We reveal an iterative strategy of comparative modelling in different media, and suggest reasons why microbial models have a special degree of epistemic tractability in multimodel inquiry. (shrink)

Templates travel because they offer a tractable format that can be used for model-building in a variety of domains. It is often because of this quality that a particular template is chosen. But one cannot assume that there are always templates ready to model a new phenomenon, and moreover, templates have also been designed at some point. A critical aspect of this designing process is the choice of the mathematical objects with which one hopes to capture this phenomenon. This (...) means that one has to assess which of the properties of different kinds of mathematical objects and which combinations of them are most appropriate. The selection of these objects is based on whether their properties enable the model to perform its function. This selection of mathematical objects is similar to the selection of materials in mechanical design. (shrink)

Three-dimensional material models of molecules were used throughout the 19th century, either functioning as a mere representation or opening new epistemic horizons. In this paper, two case studies are examined: the 1875 models of van ‘t Hoff and the 1890 models of Sachse. What is unique in these two case studies is that both models were not only folded, but were also conceptualized mathematically. When viewed in light of the chemical research of that period not only were both of these (...) aspects, considered in their singularity, exceptional, but also taken together may be thought of as a subversion of the way molecules were chemically investigated in the 19th century. Concentrating on this unique shared characteristic in the models of van ‘t Hoff and the models of Sachse, this paper deals with the shifts and displacements between their operational methods and existence: between their technical and epistemological aspects and the fact that they were folded, which was forgotten or simply ignored in the subsequent development of chemistry. (shrink)

Many scientific models are representations. Building on Goodman and Elgin’s notion of representation-as we analyse what this claim involves by providing a general definition of what makes something a scientific model, and formulating a novel account of how they represent. We call the result the DEKI account of representation, which offers a complex kind of representation involving an interplay of, denotation, exemplification, keying up of properties, and imputation. Throughout we focus on material models, and we illustrate our claims with the (...) Phillips-Newlyn machine. In the conclusion we suggest that, mutatis mutandis, the DEKI account can be carried over to other kinds of models, notably fictional and mathematical models. (shrink)

Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part (...) II supplements the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. (shrink)

This unique compendium presents a comprehensive and self-contained theory of material development under imperfect information and its applications. The book describes new approaches to synthesis and selection of materials with desirable characteristics. Such approaches provide the ability of systematic and computationally effective analysis in order to predict composition, structure and related properties of new materials. The volume will be a useful advanced textbook for graduate students. It is also suitable for academicians and practitioners who wish to have fundamental (...) models in new material synthesis and selection. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and (...) for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning. (shrink)

Templates travel because they offer a tractable format that can be used for model-building in a variety of domains. It is often because of this quality that a particular template is chosen. But one cannot assume that there are always templates ready to model a new phenomenon, and moreover, templates have also been designed at some point. A critical aspect of this designing process is the choice of the mathematical objects with which one hopes to capture this phenomenon. This (...) means that one has to assess which of the properties of different kinds of mathematical objects and which combinations of them are most appropriate. The selection of these objects is based on whether their properties enable the model to perform its function. This selection of mathematical objects is similar to the selection of materials in mechanical design. (shrink)

In this paper, I distinguish scientific models in three kinds on the basis of their ontological status—material models, mathematical models and fictional models, and develop and defend an account of fictional models as fictional objects—i.e. abstract objects that stand for possible concrete objects.

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

This article develops a framework for analysing how digital software and models become mediums for creative imagination in architectural design. To understand the hermeneutics of these relationships, we develop key concepts from Material Engagement Theory (MET) and Postphenomenology (PP). To push these frameworks into the realm of digital design, we develop the concept of Digital Materiality. Digital Materiality describes the way successive layers of mathematics, code, and software come to mediate enactive perception, and the possibilities of creative material engagement actualised (...) in work with software. Just as molecular materials come to transform action with material objects, so digital materiality comes to enable and transform creative practices with computers. Digital architectural design form a new space for ongoing enactive discovery and creativity through manipulation of digital models and their underlying software environments. By shifting relationships within their digital models, architects can direct their attention, intention, and imagination towards widely different aspects of the model. Here, creative imagination becomes a fundamentally situated activity where mind emerges through dynamic interaction between a variety of embodied, material, and cultural domains. (shrink)

This paper is devoted to some generalizations of explanatory potential of connectionist approaches to theoretical problems of the philosophy of mind. Are considered both strong, and weaknesses of neural network models. Connectionism has close methodological ties with modern neurosciences and neurophilosophy. And this fact strengthens its positions, in terms of empirical naturalistic approaches. However, at the same time this direction inherits weaknesses of computational approach, and in this case all system of anticomputational critical arguments becomes applicable to the connectionst models (...) of mind. The last developments in the field of deep learning gave rich empirical material for cognitive sciences. Multilayered networks, mathematical models of associative dynamics of learning, self-organizing neuronets and all that allow to explain the principles of human conceptual organizing and after this to emulate these processes in computer systems. At all engineering achievements of this technology there is a traditional criticism from representatives of cognitive psychology who cannot accept a thesis about learning ability of a neuronet on the basis of redistribution of scales. Process of learning of natural intelligence, according to cognitive models, happens due to attraction of knowledge broadcast in a symbolical form (mental representations, concepts) at the expense of the systems of output knowledge expressed in the propositional contents. Some philosophical aspects of «neural metaphor» in modern cognitive sciences create the problem field which demands comprehensive understanding, the first step towards which is taken in this work. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann hypothesis, have had to be considered in terms of the evidence for and against them. In recent decades, massive increases in computer power have permitted the gathering of huge amounts of numerical evidence, both for conjectures in pure mathematics and (...) for the behavior of complex applied mathematical models and statistical algorithms. Mathematics has therefore become (among other things) an experimental science (though that has not diminished the importance of proof in the traditional style). We examine how the evaluation of evidence for conjectures works in mathematical practice. We explain the (objective) Bayesian view of probability, which gives a theoretical framework for unifying evidence evaluation in science and law as well as in mathematics. Numerical evidence in mathematics is related to the problem of induction; the occurrence of straightforward inductive reasoning in the purely logical material of pure mathematics casts light on the nature of induction as well as of mathematical reasoning. (shrink)

This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology is (...) kept to a minimum, no background in formal set-theory is required, and the book contains proofs of all the required set theoretical results. The reader is taken on a journey starting with König's Lemma, and progressing via order relations, Zorn's Lemma, Boolean algebras, and propositional logic, to completeness and compactness of first-order logic. As applications of the work on first-order logic, two final chapters provide introductions to model theory and nonstandard analysis. (shrink)

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)

In modern mathematics the value of applied research increases, for this reason, modern mathematics is initially focused on resolving the situation actually arose in this respect on a par with other disciplines. Using a new tool - computer systems, applied mathematics appealed to the new object: not to nature, not to society or the practical activity of man. In fact, the subject of modern applied mathematics is a problem situation for the actor-person, and the study is aimed at solving the (...) material and practical problems. Developing the laws of its internal logic, mathematical science today covers various practices and makes them its own requirements, subjugating and organizing in their own way a subject of study. This math activity is influenced by such external factors as the condition of the actor-person; capacity of computers together with service professionals; characteristics of the object, and the external circumstances act as constituting the essence of the case and not as minor issues. The modern applied mathematics is becoming more and more similar to the engineering sciences, and the importance of mathematical modeling problem is rising. In the studies the author bases on the broader context of modern science, including works of philosophy and methodology, as well as mathematicians and specialists in the field of natural sciences. (shrink)

This monograph offers a critical introduction to current theories of how scientific models represent their target systems. Representation is important because it allows scientists to study a model to discover features of reality. The authors provide a map of the conceptual landscape surrounding the issue of scientific representation, arguing that it consists of multiple intertwined problems. They provide an encyclopaedic overview of existing attempts to answer these questions, and they assess their strengths and weaknesses. The book also presents a comprehensive (...) statement of their alternative proposal, the DEKI account of representation, which they have developed over the last few years. They show how the account works in the case of material as well as non-material models; how it accommodates the use of mathematics in scientific modelling; and how it sheds light on the relation between representation in science and art. The issue of representation has generated a sizeable literature, which has been growing fast in particular over the last decade. This makes it hard for novices to get a handle on the topic because so far there is no book-length introduction that would guide them through the discussion. Likewise, researchers may require a comprehensive review that they can refer to for critical evaluations. This book meets the needs of both groups. (shrink)

Any computer can create a model of reality. The hypothesis that quantum computer can generate such a model designated as quantum, which coincides with the modeled reality, is discussed. Its reasons are the theorems about the absence of “hidden variables” in quantum mechanics. The quantum modeling requires the axiom of choice. The following conclusions are deduced from the hypothesis. A quantum model unlike a classical model can coincide with reality. Reality can be interpreted as a quantum computer. The physical processes (...) represent computations of the quantum computer. Quantum information is the real fundament of the world. The conception of quantum computer unifies physics and mathematics and thus the material and the ideal world. Quantum computer is a non-Turing machine in principle. Any quantum computing can be interpreted as an infinite classical computational process of a Turing machine. Quantum computer introduces the notion of “actually infinite computational process”. The discussed hypothesis is consistent with all quantum mechanics. The conclusions address a form of neo-Pythagoreanism: Unifying the mathematical and physical, quantum computer is situated in an intermediate domain of their mutual transformation. (shrink)

The word “model” is highly ambiguous, and there is no uniform terminology used by either scientists or philosophers. Here, a model is considered to be a representation of some object, behavior, or system that one wants to understand. This article presents the most common type of models found in science as well as the different relations—traditionally called “analogies”—between models and between a given model and its subject. Although once considered merely heuristic devices, they are now seen as indispensable to modern (...) science. There are many different types of models used across the scientific disciplines, although there is no uniform terminology to classify them. The most familiar are physical models such as scale replicas of bridges or airplanes. These, like all models, are used because of their “analogies” to the subjects of the models. A scale model airplane has a structural similarity or “material analogy” to the full scale version. This correspondence allows engineers to infer dynamic properties of the airplane based on wind tunnel experiments on the replica. Physical models also include abstract representations which often include idealizations such as frictionless planes and point masses. Another, but completely different type of model, is constituted by sets of equations. These mathematical models were not always deemed legitimate models by philosophers. Model-to-subject and model-to-model relations are described using several different types of analogies: positive, negative, neutral, material, and formal. (shrink)

This textbook is a second edition of the successful, Mathematical Logic: On Numbers, Sets, Structures, and Symmetry. It retains the original two parts found in the first edition, while presenting new material in the form of an added third part to the textbook. The textbook offers a slow introduction to mathematical logic, and several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions. Part I, Logic Sets, and Numbers, shows how mathematical logic (...) is used to develop the number structures of classical mathematics. All necessary concepts are introduced exactly as they would be in a course in mathematical logic; but are accompanied by more extensive introductory remarks and examples to motivate formal developments. The second part, Relations, Structures, Geometry, introduces several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions, and shows how they are used to study and classify mathematical structures. The added Part III to the book is closer to what one finds in standard introductory mathematical textbooks. Definitions, theorems, and proofs that are introduced are still preceded by remarks that motivate the material, but the exposition is more formal, and includes more advanced topics. The focus is on the notion of countable categoricity, which analyzed in detail using examples from the first two parts of the book. This textbook is suitable for graduate students in mathematical logic and set theory and will also be of interest to mathematicians who know the technical aspects of the subject, but are not familiar with its history and philosophical background. (shrink)

This paper examines two parallel discussions of scientific modeling which have invoked experimentation in addressing the role of models in scientific inquiry. One side discusses the experimental character of models, whereas the other focuses on their exploratory uses. Although both relate modeling to experimentation, they do so differently. The former has considered the similarities and differences between models and experiments, addressing, in particular, the epistemic value of materiality. By contrast, the focus on exploratory modeling has highlighted the various kinds of (...) exploratory functions of models in the early stages of inquiry. These two perspectives on modeling are discussed through a case study in the field of synthetic biology. The research practice in question explores biological control by making use of an ensemble of different epistemic means: mathematical models and simulations, synthetic genetic circuits and intracellular measuring devices, and finally electronic circuits. We argue that the study of exploratory modeling should trace the ways different epistemic means, in different materialities, are being combined over time. Finally, the epistemic status of such novel investigative objects as synthetic genetic circuits is evaluated, with the conclusion that they can function as both experiments and models. (shrink)

Elementary set theory accustoms the students to mathematical abstraction, includes the standard constructions of relations, functions, and orderings, and leads to a discussion of the various orders of infinity. The material on logic covers not only the standard statement logic and first-order predicate logic but includes an introduction to formal systems, axiomatization, and model theory. The section on algebra is presented with an emphasis on lattices as well as Boolean and Heyting algebras. Background for recent research in natural language (...) semantics includes sections on lambda-abstraction and generalized quantifiers. Chapters on automata theory and formal languages contain a discussion of languages between context-free and context-sensitive and form the background for much current work in syntactic theory and computational linguistics. The many exercises not only reinforce basic skills but offer an entry to linguistic applications of mathematical concepts. For upper-level undergraduate students and graduate students in theoretical linguistics, computer-science students with interests in computational linguistics, logic programming and artificial intelligence, mathematicians and logicians with interests in linguistics and the semantics of natural language. (shrink)

Traditional logic as a part of philosophy is one of the oldest scientific disciplines. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, Russell and others to create a logistic foundation for mathematics. It steadily developed during the 20th century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy. While there are already several well-known textbooks on mathematical logic, this book is unique in that (...) it is much more concise than most others, and the material is treated in a streamlined fashion which allows the professor to cover many important topics in a one semester course. Although the book is intended for use as a graduate text, the first three chapters could be understood by undergraduates interested in mathematical logic. These initial chapters cover just the material for an introductory course on mathematical logic combined with the necessary material from set theory. This material is of a descriptive nature, providing a view towards decision problems, automated theorem proving, non-standard models and other subjects. The remaining chapters contain material on logic programming for computer scientists, model theory, recursion theory, Godel’s Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. The author has provided exercises for each chapter, as well as hints to selected exercises. About the German edition: …The book can be useful to the student and lecturer who prepares a mathematical logic course at the university. What a pity that the book is not written in a universal scientific language which mankind has not yet created. - A.Nabebin, Zentralblatt. (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 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)

Investigation into the sequence structure of the genetic code by means of an informatic approach is a real success story. The features of human language are also the object of investigation within the realm of formal language theories. They focus on the common rules of a universal grammar that lies behind all languages and determine generation of syntactic structures. This universal grammar is a depiction of material reality, i.e., the hidden logical order of things and its relations determined by natural (...) laws. Therefore mathematics is viewed not only as an appropriate tool to investigate human language and genetic code structures through computer sciencebased formal language theory but is itself a depiction of material reality. This confusion between language as a scientific tool to describe observations/experiences within cognitive constructed models and formal language as a direct depiction of material reality occurs not only in current approaches but was the central focus of the philosophy of science debate in the twentieth century, with rather unexpected results. This article recalls these results and their implications for more recent mathematical approaches that also attempt to explain the evolution of human language. (shrink)

The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several (...) weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. (shrink)

This article questions the effects on urban research dynamics of the Big Data and AI turn in urban management. Increasing access to large datasets collected in real time could make certain mathematical models developed in research fields related to the management of urban systems obsolete. These ongoing evolutions are the subject of numerous works whose main angle of reflection is the future of cities rather than the transformations at work in the academic field. Our article proposes grasp the scientific (...) dynamics in areas of research related to two urban systems: transportation and water. The article demonstrates the importance of grasping these dynamics if we want to be able to apprehend what the urban management of tomorrow's cities will be like. To analyse these research areas’ dynamics, we use two complementary materials: bibliometric data and interviews. The interviews conducted in 2018 with academics and higher education officials in Paris and Edinburgh suggest avenues for hybridization between traditional modelling approaches and research in machine learning, artificial intelligence and Big Data. The bibliometric analysis highlight the trends at work: it shows that traffic flow as well as transportation studies are focussing more and more on AI and Big Data and that traffic flow studies are arousing a growing interest among computer scientists, while, so far, this interest is less pronounced in the water research area, and more especially regarding water quality. The differences observed between research on transportation and that on water confirm the multifaceted nature of the developments at work and encourage us to reject overly hasty and simplistic generalisations about the transformations underway. (shrink)

Demonstrating the different roles that logic plays in the disciplines of computer science, mathematics, and philosophy, this concise undergraduate textbook covers select topics from three different areas of logic: proof theory, computability theory, and nonclassical logic. The book balances accessibility, breadth, and rigor, and is designed so that its materials will fit into a single semester. Its distinctive presentation of traditional logic material will enhance readers' capabilities and mathematical maturity. The proof theory portion presents classical propositional logic and (...) first-order logic using a computer-oriented (resolution) formal system. Linear resolution and its connection to the programming language Prolog are also treated. The computability component offers a machine model and mathematical model for computation, proves the equivalence of the two approaches, and includes famous decision problems unsolvable by an algorithm. The section on nonclassical logic discusses the shortcomings of classical logic in its treatment of implication and an alternate approach that improves upon it: Anderson and Belnap's relevance logic. Applications are included in each section. The material on a four-valued semantics for relevance logic is presented in textbook form for the first time. Aimed at upper-level undergraduates of moderate analytical background, Three Views of Logic will be useful in a variety of classroom settings. Gives an exceptionally broad view of logic. Treats traditional logic in a modern format. Presents relevance logic with applications. Provides an ideal text for a variety of one-semester upper-level undergraduate courses. (shrink)

Needless to say, Charles Parsons’s long awaited book1 is a must-read for anyone with an interest in the philosophy of mathematics. But as Parsons himself says, this has been a very long time in the writing. Its chapters extensively “draw on”, “incorporate material from”, “overlap considerably with”, or “are expanded versions of” papers published over the last twenty-five or so years. What we are reading is thus a multi-layered text with different passages added at different times. And this makes for (...) a rather bumpy read. There is another route Parsons could have taken: he could have reprinted the relevant papers with postscripts, and then top-and-tailed the collection with a preface and added concluding reflections. It must sound very ungrateful, but I rather suspect that that might have worked better. Much of the book is about arithmetic. But Parsons has woven into the discussion claims about mathematics more generally and about set theory in particular. We might well have a basic worry about this structure: for do defensible claims about the ontology and epistemology of arithmetic have to be generalizable to apply to more infinitary mathematics? For example, suppose you are attracted to a Hellman-like modal structuralist account of arithmetic: then should you think it a problem if you suspect that such an account can’t readily be extended to cope e.g. with set theory (since you boggle at the idea of possible worlds free of abstracta but with enough structure to somehow model ZFC)? Parsons himself seems to waver over such questions. So for present purposes, I’ll focus just on arithmetic, and in what follows I’ll revisit two of the most familiar Parsonian themes, his views on structuralism as an account of the ontology of arithmetic, and his exploration of the role of intuition in grounding arithmetical knowledge. (shrink)

Recently, Bechtel and Abrahamsen have argued that mathematical models study the dynamics of mechanisms by recomposing the components and their operations into an appropriately organized system. We will study this claim through the practice of combinational modeling in circadian clock research. In combinational modeling, experiments on model organisms and mathematical/computational models are combined with a new type of model—a synthetic model. We argue that the strategy of recomposition is more complicated than what Bechtel and Abrahamsen indicate. Moreover, synthetic (...) modeling as a kind of material recomposition strategy also points beyond the mechanistic paradigm. (shrink)

Continuing his series of books on the mathematics of gambling, the author shows how a simple-rule game such as roulette is suited to a complex mathematical model whose applications generate improved betting systems that take into account a player's personal playing criteria. The book is both practical and theoretical, but is mainly devoted to the application of theory. About two-thirds of the content is lists of categories and sub-categories of improved betting systems, along with all the parameters that might (...) stand as the main objective criteria in a personal strategy - odds, profits and losses. The work contains new and original material not published before. The mathematical chapter describes complex bets, the profit function, the equivalence between bets and all their properties. All theoretical results are accompanied by suggestive concrete examples and can be followed by anyone with a minimal mathematical background because they involve only basic algebraic skills and set theory basics. The reader may also choose to skip the math and go directly to the sections containing applications, where he or she can pick desired numerical results from tables. The book offers no new so-called winning strategies, although it discusses them from a mathematical point of view. It does, however, offer improved betting systems and helps to organize a player's choices in roulette betting, according to mathematical facts and personal strategies. It is a must-have roulette handbook to be studied before placing your bets on the turn of either a European or American roulette wheel. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Advances in Peircean Mathematics: The Colombian School ed. by Fernando ZalameaGianluca CaterinaFernando Zalamea (Ed.) Advances in Peircean Mathematics: The Colombian School Berlin, Boston: De Gruyter, 2022. 212 pp. (incl. index).The volume Advances in Peircean Mathematics is an important, very much needed contribution towards a deeper understanding of the impact of Peirce's work especially in the fields of mathematics, logic, and semiotic. It fills a gap in the current (...) literature by recasting some of [End Page 373] Peirce's profound mathematical insights into a formal framework. This is a finely edited editorial project, which summarizes the work of more than two decades done by a group of scholars—the Colombian School—lead by Fernando Zalamea, a pioneer of Peirce studies, and pragmatism in general, in Latin America.The structure of the volume unfolds in three dense chapters, each corresponding to a specific mathematical context connected to various aspects of Peirce's work. Broadly speaking, in the first chapter, Gustavo Arengas proposes the theory of higher category theory as a natural framework to represent the main traits of Peirce's semiotic; in the second chapter, Francisco Vargas develops a thorough set-theoretical model of Peirce's continuum; in the third chapter, Arnold Oostra, in a truly Peircean spirit, formalizes the notion of Intuitionistic Existential Graphs, emphasizing the idea that intuitionistic logic is the most natural context to represent and understand Peirce's diagrammatic logic. The fourth and last chapter, penned by Zalamea, provides the reader with a solid summary of the main results presented in the volume along with a robust road map to navigate through the technical details of the text.Mathematics, as implied by the title of the volume, is at the core of this project, and the authors do not shy away from the complexity, details, and depth of it in order to shed light on some of the most fundamental and groundbreaking nature of Peirce's ideas. That level of rigor and formal soundness is certainly one of the strengths of the volume, although readers who may not be acquainted with at least some basic notions of category theory or with the mathematics of Peirce's work, may find some of the text a bit terse—this is not a casual reading and it does require a certain level of commitment in order to appreciate the depth of the provided insights. On the other hand, all the background material is clearly laid out, making the volume a self-contained work which can be thoroughly enjoyed by a general audience.The real upshot of recasting some of Peirce's intuitions within the fabric of formal mathematics is not only that of providing the instruments to clarify those ideas, but also that of highlighting the role that Peirce's work has been playing in the development of modern mathematics in many relevant aspects, from the very genesis of category theory—which today is widely regarded as a necessary and unifying language in several fields of the mathematical and physical sciences—to the development of non-standard analysis, just to mention a few of them.An elegant instance of such perspective is offered by Gustavo Arengas in the first chapter, where it is argued that ∞-categories can be used to model the natural order of hierarchies implicit in Peirce's conception of the basic triadic relation between sign, object, and interpreter. What, at first glance, may seem just a simple analogy, unfolds onto a methodical and enlightening analysis of Peirce's semiotic, as the author builds [End Page 374] the connections with the relevant categorical constructions in a precise manner. Most of the rest of the chapter focuses on Peirce's notions of the pragmatic and pragmaticist maxim, and the use of basic structures such as monads, limits, presheaves and sheaves as the natural environment to better convey their meaning, both from a mathematical and a philosophical perspective. The reader will find it interesting to realize that, by delving deeper into the understanding of the relational nature of category theory, a great chunk of Peirce's work can be seen as fundamentally intertwined with the apparatus and the language of modern mathematics... (shrink)

New mechanical philosophy posits that explanations in the life sciences involve the decomposition of a system into its entities and their respective activities and organization that are responsible for the explanandum phenomenon. This mechanistic account of explanation has proven problematic in its application to mathematical models, leading the mechanists to suggest different ways of aligning abstract models with the mechanist program. Initially, the discussion centered on whether the Hodgkin-Huxley model is explanatory. Network models provided another complication, as they apply (...) to a wide number of materially diverse systems. In this article, we examine the various attempts to integrate abstract models within the mechanist program, also presenting a further challenge: the Heimburg-Jackson model, which was introduced as an alternative to the Hodgkin-Huxley model. We argue that although the notion of abstraction as the omission of irrelevant mechanistic details appears to give a mechanistic solution for accommodating abstract models, this notion does not suit models whose epistemic strategy is not decompositional. As a result, the mechanist has to choose whether to dilute the mechanistic approach nearly beyond recognition or to claim that many, if not most, abstract theoretical models do not deliver mechanistic explanations, or qualify as explanatory at all. (shrink)

The assessment of models in an experiment depends on their material nature and their function in the experiment. Models that are used to make the phenomenon under investigation visible - sensors - are assessed by calibration. However, calibration strategies assume material intervention. The experiment discssed in this paper is an experiment in economics to measure the influence of technology shocks on business cycles. It uses immaterial, mathematical instruments. It appears that calibration did not work for these kinds of models, (...) it did not provide reliable evidence for the facts of the business cycle. (shrink)

Accounts of the relation between theories and models in biology concentrate on mathematical models. In this paper I consider the dual role of models as representations of natural systems and as a material basis for theorizing. In order to explicate the dual role, I develop the concept of a remnant model, a material entity made from parts of the natural system(s) under study. I present a case study of an important but neglected naturalist, Joseph Grinnell, to illustrate the extent (...) to which mundane practices in a museum setting constitute theorizing. I speculate that historical and sociological analyses of institutions can play a specific role in the philosophical analysis of model-building strategies. (shrink)

In the range of his intellectual interests and the profundity of his mathematical thought Hermann Weyl towered above his contemporaries, many of whom viewed him with awe. This volume, the most ambitious study to date of Weyl's singular contributions to mathematics, physics, and philosophy, looks at the man and his work from a variety of perspectives, though its gaze remains fairly steadily fixed on Weyl the geometer and space‐time theorist. Structurally, the book falls into two parts, described in the (...) general introduction by the editor: Part 1 contains four essays on particular aspects of Weyl's work, highlighting ideas he developed in various editions of his classic Raum‐Zeit‐Materie. Part 2 presents a lengthy study by Robert Coleman and Herbert Korté covering nearly the whole gamut of Weyl's mathematical research, an impressive feat. Both in the introduction as well as in footnotes to the articles Erhard Scholz's editorial voice chimes in discreetly, helping tie all five studies together.Coleman and Korté begin chronologically with Weyl's early work in analysis and the modern theory of Riemann surfaces before turning to differential geometry, unified field theory, and the space problem, a topic they use as a springboard for a discussion of their own recent work on the foundations of space‐time. They then take up Weyl's shift to group representation theory and its applications to quantum mechanics, ending with his much earlier research on the structure of the continuum. All of these topics are well handled, but the authors' own agendas coupled with their penchant for overlooking chronology in order to package Weyl's work into neat little bundles leave one feeling rather stranded and far removed from the sources of Weyl's inspiration. Moreover, the narrative style makes this part of the volume read like a technical appendix, albeit a most informative one. Readers who tackle Scholz's far more contextualized essay will be amply rewarded by comparing his views with the opinions set forth by Coleman and Korté in Part 2.Scholz gives a masterful account of Weyl's intellectual journeys from 1917 to 1925 in a study that serves as a fulcrum for the entire volume. Drawing on a number of recently published studies, including his own, on the interplay between mathematics and physics inspired by Einstein's theory of general relativity, Scholz describes how Weyl responded to this challenge by developing a truly infinitesimal space‐time geometry that generalized classical Riemannian geometry. Although unconvinced by Einstein's critique of his unified field theory, Weyl shifted his focus from this realm to the classical space problem, analyzed earlier with more primitive techniques by Hermann Helmholtz and Sophus Lie. In this connection, it should be mentioned that Thomas Hawkins has given a probing analysis of Weyl's related work on the representation of Lie groups in his tour‐de‐force work, Emergence of the Theory of Lie Groups . Scholz argues that Weyl's struggle to tame his modernized version of the space problem stemmed from a deep‐seated belief in his geometrical ideas, which in turn were nourished by philosophical musings. By demonstrating the closely related conceptual links that motivated Weyl's research in infinitesimal geometry, space‐time physics, and the foundations of mathematics, Scholz nicely illuminates the underlying fabric of epistemological concerns that occupied Weyl's attention during this fertile period.The three remaining essays in Part 1 focus on other aspects of Weyl's work in mathematical physics and cosmology. Skuli Sigurdsson's “Journeys in Spacetime” offers a broad interpretation of Weyl's career, one that emphasizes Weyl's sensitivity to cultural tensions as reflected in his philosophical roots, which combined phenomenology with facets of German idealism. Shaken by the annihilation of cultural values in Nazi Germany, Weyl became deeply aware of the gulf that separated his earlier life in Göttingen and Zurich from the one he took up at Princeton's Institute for Advanced Study in 1933. He tried to adapt, but felt out of place in an Anglo‐American scientific culture openly hostile toward metaphysics and speculative philosophy. Sigurdsson stresses these tensions, contrasting the introspective, creative individual against the backdrop of the collective in the age of the machine, but without spelling out which collective were most important for him. Wolfgang Pauli thought he knew and, like Einstein before him, he had no compunction about bluntly telling Weyl he was a mathematician, not a physicist.Pauli's opinions notwithstanding, Weyl did far more than just dabble around the mathematical edges of the new physics. If Coleman and Korté perhaps press their case for his visionary accomplishments too far, Norbert Straumann's essay “Ursprünge der Eichtheorien” suggests why Weyl's reputation among physicists has risen steadily ever since the advent of Yang‐Mills theory in the 1950s. In the course of describing Weyl's adaptation of his gauge transformation formalism to Dirac's electron theory, Straumann sheds considerable light on Pauli's role as self‐appointed watchman guarding the disciplinary boundary that separated theoretical physics from physical mathematics . He further suggests that disciplinary jealousy was a major reason why Pauli dismissed Weyl's two‐component formalism for spinors out of hand.In the realm of cosmology, on the other hand, Weyl's work has long since passed into the dustbins of history, as Hubert Goenner remarks in recounting a fascinating chapter in the infancy of space‐time physics. While doing so, Goenner shows how initially Weyl almost slavishly adopted what Einstein called Mach's principle, which asserts that the metric structure of space‐time is solely determined by the distribution of matter in the universe. This notion was quickly challenged by Willem De Sitter, who showed that Einstein's matter‐free field equations admitted a global solution with non‐zero constant curvature. Both Einstein and Weyl tried to argue that invisible masses must be present just over the “spatial horizon” of De Sitter's world in order to account for its curvature. Goenner meticulously analyzes the physical and mathematical issues at stake in this debate, stressing how Weyl gradually moved away from a strong physical interpretation to one in which mathematics models rather than physics models simply reveal natural phenomena. He argues further that Weyl's cosmological principle arose as the final expression of his search for a deeper physical meaning.Given the quality of these essays, it is regrettable that this book contains so little about Weyl's professional career, a weakness the editor could have redressed at least partially in his general introduction. This omission is all the more unfortunate given the dearth of readily accessible information about Weyl's life available elsewhere. For however mundane his outward existence may have been, the reader cannot be expected to appreciate the interplay between the world Weyl knew and his creative responses to it without fairly detailed knowledge of his biography. Shorn from these contexts, it becomes difficult to form a flesh‐and‐blood image of Weyl beyond the cliché‐ridden stereotype that sees him as a “heroic thinker in the grand German tradition.” While none of the authors falls into this trap, the collective impression they leave suggests a most enigmatic figure. Either Weyl the man tends to get lost in the shadows of his collected scientific output or he appears as a mystic loner, an outcast who abhorred the machine age in which he lived. Closer attention to the people in his life would no doubt produce a very different picture of the man and his interests. This major lacuna notwithstanding, the present volume will surely remain an indispensable resource for any future investigations of Weyl's staggering intellectual achievements. (shrink)

New mechanical philosophy posits that explanations in the life sciences involve the decomposition of a system into its entities and their respective activities and organization that are responsible for the explanandum phenomenon. This mechanistic account of explanation has proven problematic in its application to mathematical models, leading the mechanists to suggest different ways of aligning abstract models with the mechanist program. Initially, the discussion centered on whether the Hodgkin-Huxley model is explanatory. Network models provided another complication, as they apply (...) to a wide number of materially diverse systems. In this article, we examine the various attempts to integrate abstract models within the mechanist program, also presenting a further challenge: the Heimburg-Jackson model, which was introduced as an alternative to the Hodgkin-Huxley model. We argue that although the notion of abstraction as the omission of irrelevant mechanistic details appears to give a mechanistic solution for accommodating abstract models, this notion does not suit models whose epistemic strategy is not decompositional. As a result, the mechanist has to choose whether to dilute the mechanistic approach nearly beyond recognition or to claim that many, if not most, abstract theoretical models do not deliver mechanistic explanations, or qualify as explanatory at all. (shrink)

This chapter contains sections titled: Introduction Discovering a Defensible Kind of Philosophical Writing Imitators vs. Constitution‐Painters The Necessary and Sufficient Criterion of Philosophical Writing.

Algorithmic software for mathematical modeling of structural stability is considered, which is reduced to solving a partial generalized eigenvalues problem of sparse matrices, with automatic parallelization of calculations on modern parallel computers with graphics processors. Peculiarities of realization of parallel algorithms for different structures of sparse matrices are presented. The times of solving the problem of stability of composite materialsusing a three-dimensional model of "finite size fibers" on computers of different architectures are given. In mathematical modeling of physical (...) and technical processes in many cases there is a need to solve problems of algebraic problem of eigenvalues with sparse matrices of large volumes. In particular, such problems arise in the analysis of the strength of structures in civil and industrial construction, aircraft construction, electric welding, etc. The solving to these problems is to determine the eigenvalues and eigenvectors of sparse matrices of different structure. The efficiency of solving these problems largely depends on the effectiveness of mathematical modeling of the problem as a whole. Continuous growth of task parameters, calculation of more complete models of objects and processes on computers require an increase in computer productivity. High-performance computing requirements are far ahead of traditional parallel computing, even with multicore processors. High-performance computing requirements are far ahead of traditional parallel computing, even with multicore processors. Today, this problem is solved by using powerful supercomputers of hybrid architecture, such as computers with multicore processors and graphics processors, which combine MIMD and SIMD architectures. But the potential of high-performance computers can be used to the fullest only with algorithmic software that takes into account both the properties of the task and the features of the hybrid architecture. Complicating the architecture of modern high-performance supercomputers of hybrid architecture, which are actively used for mathematical modeling means a significant complication of efficient use of these resources in creating parallel algorithms and programs. here are problems with the creation of algorithmic software with automatic execution of stages of work, which are associated with the efficient use of computing resources, ways to store and process sparse matrices, analysis of the reliability of computer results. This makes it possible to significantly increase the efficiency of mathematical modeling of practical problems on modern high-performance computers, as well as free users from the problems of parallelization of complex problems. he developed algorithmic software automatically implements all stages of parallel computing and processing of sparse matrices on a hybrid computer. It was used at the Institute of Mechanics named after S.P. Tymoshenko NAS of Ukraine in modeling the strength problems of composite material. A significant improvement in the time characteristics of mathematical modeling was obtained. Problems of mathematical modeling of the properties of composite materials has an important role in designing the processes of deformation and destruction of products in various subject areas. Algorithmic software for mathematical modeling of structural stability is considered, which is reduced to solving a partial generalized problem of eigen values of sparse matrices of different structure of large orders, with automatic parallelization of calculations on modern parallel computers with graphics processors. The main methodological principles and features of implementation of parallel algorithms for different structures of sparse matrices are presented, which ensure effective implementation of multilevel parallelism of a hybrid system and reduce data exchange time during the computational process. As an example of these approaches, a hybrid algorithm of the iteration method in subspace for tape and block-diagonal matrices with a frame for computers of hybrid architecture is given. Peculiarities of data decomposition for matrices of profile structure at realization of parallel algorithms are considered. The proposed approach provides automatic determination of the required topology of the hybrid computer and the optimal amount of resources for the organization of an efficient computational process. The results of testing the developed algorithmic software for problems from the collection of the University of Florida, as well as the times of solving the problem of stability of composite materials using a three-dimensional model of "finite size fibers" on computers of different architectures. The results show a significant improvement in the time characteristics of solving problems. (shrink)

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis (...) of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction. (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.

Novikov is one of Russia's leading logicians and the appearance of this fine textbook is a good indicator of increasing American interest in Soviet logic. The book contains some new material, including a new independence proof of the rule of complete induction from the remaining axioms of first-order arithmetic. The first third of this work consists in chapters on propositional algebra and the propositional calculus. The first-order predicate calculus comes next under discussion: here a number of important classical results—Gödel's incompleteness (...) theorem, the Compactness theorem, Skolem-Löwenheim theorem—are proved rigorously. The author throughout the book leans fairly heavily on model-theoretic techniques, hence knowledge of some algebra, especially elementary field theory, will be useful. The last two chapters are concerned with first order arithmetic and the elements of proof theory. The only shortcoming is the lack of a bibliography and related scholarly apparatus; the student using this text may have difficulty locating material in other publications relevant to that in the book without outside assistance. The translation is very smooth and clear. Altogether, a first-class job.—P. J. M. (shrink)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. (...) Several attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)