Philosophy of Mathematics

This volume represents a contribution to the philosophy of economics with a distinctive point of view. The contributors to the volume have not by and large taken up the general and abstract issues that have usually occupied philosophers in their discussions of the philosophy of economics in the past. Instead, they have selected particular areas of economics and have probed these areas for the philosophical and methodological issues that they raise: for example, the mean­ing of causal ascriptions in econometrics, the reliability of large economic models, the status of the assumptions of rationality that go into the public goods argu­ment, the explanatory import of equilibrium analysis, and the role of experiment in economics. The primary essays are written by philosophers, who were invited to concentrate on philosophical issues that arise at the level of the everyday theoretical practice of working economists. Commentary essays have been pro­vided by working economists, who were asked to respond to the philosophical arguments from the standpoint of their own disciplines. The volume thus repre­sents something of an “experiment” in the philosophy of science, striving as it does to explore methodological issues across two research communities. The fruits of the experiment are available for the reader’s own assessment; but it is the editor’s judgment that these exchanges between philosopher and economist have been fruitful indeed.

In recent years there has been an increased focus on the need for teachers to develop learning communities where all students have opportunities to engage in interactive discourse. However, there are few studies that focus on student perceptions and accounts of mathematical reasoning in classrooms with interactive mathematical talk as a focus of reform. A framework of teacher actions to develop classroom and mathematical practices was developed from classroom observations. Photo-elicitation interviews were used to investigate student perception and accounts of mathematical explanations and reasoning. The professional development programme, shifts in the teacher actions, and subsequent shifts in student perception and their recall of their own and peers’ mathematical reasoning over a school year are highlighted. Developing interactive dialogue in the classroom took considerable time and attention. Facilitating change to the way students both participated and understood their obligations required constant, ongoing attention to both the classroom and mathematical practices.

Colyvan, M., S. Linquist, W. Grey, P. Griffiths, J. Odenbaugh, and H. P. Possingham. 2009. Philosophical issues in ecology: recent trends and future directions. Ecology and Society 14(2): 22. https://doi.org/10.5751/ES-03020-140222

My aim in this thesis is to reveal the true nature of mathematical reasoning and to show the necessity of pictorial representations in mathematics. In order to support my thesis I argue that using only the formal language, without fully integrating visual reasoning in mathematical communication, mathematics operates insufficiently and it can never reach its full potential. First of all, I provide a short historical approach to philosophy of mathematics as well as a summary of current state of visualization in mathematics. Then, I bring forth a complete picture of philosophy of mathematics and mathematical objects, which consistently allows visual reasoning, intuitions and constructions in mathematical communication. This interpretation makes it easy to see the pictorial representations as a natural part of mathematical reasoning. Moreover, I lay out the medium for visual reasoning in mathematics: Space. After providing several definitions of space I conclude that Kant’s characterization is the finest one as a base for mathematical reasoning, in particular for visual reasoning. Furthermore, I bring out the method of visual reasoning: Synthetic a priori. In order to show the link between visualization and synthetic a priori I provide hitherto definitions of visualization and argue that these lead up to Kantian characterization of constructions and synthetic a priori. Then, I provide several interpretations to show that Kant’s characterization of mathematics has elements valuable for modern mathematics that modern logic cannot capture. Finally, I offer three main arguments leading to the main thesis “pictorial repre- sentations/ diagrams are necessary parts of mathematical reasoning”: (1) There are formal systems with diagrams. Hence, not all uses of diagrams are synthetic. (2) There are necessary non-formal uses of diagrams in mathematical proofs. (3) There is a strong link between visualization and Kantian philosophy of mathematics. As a result, I claim that the real capacity of mathematics is not in turning it into a machine language but integrating and appreciating the aesthetic part of it as well. I also indicate the necessity of having a theory of diagrammatics, which should be set up as a branch of mathematics.

This book guides readers by gradual steps through the central concepts and debates in the philosophy of science. Using concrete examples from the history of science, Kent W. Staley shows how seemingly abstract philosophical issues are relevant to important aspects of scientific practice. Structured in two parts, the book first tackles the central concepts of the philosophy of science, such as the problem of induction, falsificationism, and underdetermination, and important figures and movements, such as the logical empiricists, Thomas Kuhn, and Paul Feyerabend. The second part turns to contemporary debates in the philosophy of science, such as scientific realism, explanation, the role of values in science, the different views of scientific inference, and probability. This broad yet detailed overview will give readers a strong grounding whilst also providing opportunities for further exploration. It will be of particular interest to students of philosophy, the philosophy of science, and science.

This book guides readers by gradual steps through the central concepts and debates in the philosophy of science. Using concrete examples from the history of science, Kent W. Staley shows how seemingly abstract philosophical issues are relevant to important aspects of scientific practice. Structured in two parts, the book first tackles the central concepts of the philosophy of science, such as the problem of induction, falsificationism, and underdetermination, and important figures and movements, such as the logical empiricists, Thomas Kuhn, and Paul Feyerabend. The second part turns to contemporary debates in the philosophy of science, such as scientific realism, explanation, the role of values in science, the different views of scientific inference, and probability. This broad yet detailed overview will give readers a strong grounding whilst also providing opportunities for further exploration. It will be of particular interest to students of philosophy, the philosophy of science, and science.

Notes on Contributors

The numerical solution of the Navier-Stokes (NS) equations relies on complex algorithms such as operator splitting and multi-physics field coupling. However, traditional mathematical tools (e.g., set theory, category theory) struggle to rigorously describe the algorithm selection logic and execution process: set theory cannot characterize the dynamic operation-result relationships, while category theory, despite its abstract and unified perspective, lacks computational feasibility. This leads to algorithm design often depending on experience, resulting in an "black-box" execution process. To address this issue, this paper proposes a mathematical framework based on the Generalized Mapping Theory (GMT). Through a fourelement structure of "object set-operation set-result set-generation relationship", it transforms the static parameters (physical quantities, grids), dynamic operations (convection/viscous discretization, pressure reconstruction), and branch results (multiple algorithm selections) in the numerical solution of NS equations into computable tensor mappings. This framework not only provides a rigorous mathematical language for algorithms like category theory but also, through the explicit nature of operations and traceable path characteristics, makes sub-steps of operator splitting and probability weights of multi-model fusion transparent and controllable mapping units. Taking automotive external flow field simulation and thrombus formation blood flow simulation as cases, it is verified that the framework can accurately describe the execution logic of different NS algorithms and support the quantitative tracking of flow field results and computational performance. Experiments show that compared with traditional description methods, this framework enables the realization of mathematical consistency in algorithm selection and interpretability in the execution process, providing a new tool for the standardization and transparency of NS equation numerical algorithms.

Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Godel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material.

\n The notion of surreal number was introduced by J.H. Conway in the mid 1970\\'s: the surreal numbers constitute a linearly ordered (proper) class No containing the class of all ordinal numbers (On) that, working within the background set theory NBG, can be defined by a recursion on the class On. Since then, have appeared many constructions of this class and was isolated a full axiomatization of this notion that been subject of interest due to large number of interesting properties they have, including model-theoretic ones. Such constructions suggests strong connections between the class No of surreal numbers and the classes of all sets and all ordinal numbers. In an attempt to codify the universe of sets directly within the surreal number class, we have founded some clues that suggest that this class is not suitable for this purpose. The present work is an attempt to obtain an \\"algebraic (set) theory for surreal numbers\\" along the lines of the Algebraic Set Theory - a categorial set theory introduced in the 1990\\'s: to establish abstract and general links between the class of all surreal numbers and a universe of \\"surreal sets\\" similar to the relations between the class of all ordinals (On) and the class of all sets (V), that also respects and expands the links between the linearly ordered class of all ordinals and of all surreal numbers. We have introduced the notion of (partial) surreal algebra (SUR-algebra) and we explore some of its category theoretic properties, including (relatively) free SUR-algebras (SA, ST). We have established links, in both directions, between SUR-algebras and ZF-algebras (the keystone of Algebraic Set Theory). We develop the first steps of a certain kind of set theory based (or ranked) on surreal numbers, that expands the relation between V and On.\n

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important and well-known case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer some evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate and successful foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences.

The Interface Between Statistics and the Philosophy of Science

It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

An argument that there are perceptual mechanisms that retrieve information in cognitively and conceptually unmediated ways and that this sheds light on various philosophical issues. In Cognition and Perception, Athanassios Raftopoulos discusses the cognitive penetrability of perception and claims that there is a part of visual processes (which he calls “perception”) that results in representational states with nonconceptual content; that is, a part that retrieves information from visual scenes in conceptually unmediated, “bottom-up,” theory-neutral ways. Raftopoulos applies this insight to problems in philosophy of science, philosophy of mind, and epistemology, and examines how we access the external world through our perception as well as what we can know of that world. To show that there is a theory-neutral part of existence, Raftopoulos turns to cognitive science and argues that there is substantial scientific evidence. He then claims that perception induces representational states with nonconceptual content and examines the nature of the nonconceptual content. The nonconceptual information retrieved, he argues, does not allow the identification or recognition of an object but only its individuation as a discrete persistent object with certain spatiotemporal properties and other features. Object individuation, however, suffices to determine the referents of perceptual demonstratives. Raftopoulos defends his account in the context of current discussions on the issue of the theory-ladenness of perception (namely the Fodor-Churchland debate), and then discusses the repercussions of his thesis for problems in the philosophy of science. Finally, Raftopoulos claims that there is a minimal form of realism that is defensible. This minimal realism holds that objects, their spatiotemporal properties, and such features as shape, orientation, and motion are real, mind-independent properties in the world. Bradford Books imprint

The text- and argument-focused histories of philosophy that we have are mainly interested in teasing out the details of the positions taken on philosophical issues by individual philosophers. But this is a long way from having a historical explanation of the larger-scale trajectory of philosophical development. An empirical history of philosophy, however, examines the institutionalized places and venues for philosophical work that provide a rich, shared structure for the promotion of particular sorts of work. Mid-twentieth-century philosophers of science such as Herbert Feigl and Philipp Frank knew what they were doing—they were creating the space to promote certain sorts of work in venues such as the Institute for the Unity of Science and the Minnesota Center for Philosophy of Science and, ultimately, in publication venues such as Daedalus and the Minnesota Studies in Philosophy of Science. Here the engine of progress is the institution-building activity of committed philosophers making a space within th...