On the Completeness Interpretation of Representation Theorems

I work in areas of formal epistemology, philosophy of mathematics, decision theory, and am increasingly interested in issues of social epistemology and collective action, both as they relate to my earlier areas and in other ways. I've done work on various paradoxes of the infinite in probability and decision theory, on the foundations of Bayesianism, on the social epistemology of mathematics, and written one weird paper using metaphysics to derive conclusions about physics. Links of Interest: My research website including links and descriptions to most of my papers. My appearance (in 2015) on Julia Galef's "Rationally Speaking" podcast, discussing Newcomb's Paradox, its connection to other issues in decision theory and free will, and what I call a "tragedy of rationality". A discussion (from 2011) with Jonathan Weisberg about the role of accuracy in constraining beliefs and probabilities, and their connection, on Philosophy TV. The idea of this discussion eventually became my Dr. Truthlove paper in Nous (paper available from Philosophers' Annual - 10 Best Papers of 2015) My paper "Decision Theory without Representation Theorems", at the open access journal Philosophers' Imprint. My old blog, Antimeta, which I ran for several years in graduate school, discussing issues in philosophy of mathematics, probability, and occasionally metaphysics. My posts from the period 2005-2009 on Brian Weatherson's blog, Thoughts, Arguments, and Rants.

Stephen KleeneKleene, Stephen (1909–1994) once praised the Hungarian mathematician and logician Rózsa Péter (1905–1977) as “the leading contributor to the special theory of recursive functionsKleene, Stephen” (Kleene, 1952). Her works broke new ground and helped to establish recursion theory as a mathematical discipline. Today, KleeneKleene, Stephen is much better known in the philosophy of mathematics than Péter. In this chapter, I provide an overview of Rózsa Péter’s work and describe her active role in communicating these results to a broader audience in her book Playing with Infinity (1944/1961). First, I will briefly summarize the content and key statements of Péter’s book, including an overview of its reception. Second, I will focus on a case study: Péter’s popular sketch of Gödel’sGödel, Kurt proof. Third, I will contextualize this case study against the historical background of Hilbert’sHilbert, David program and the decision problem. This includes Péter’s research on recursion theory. Last, but not least, I will discuss Church’sChurch, Alonzo thesis and Péter’s interpretation of it. I conclude with some reflections on the relevance of Péter’s work from a philosophical, conceptual and practical perspective.

Preface 1. Introduction: a role for history Part I. Human and Artificial Mathematicians: 2. Communicating with automated theorem provers 3. Automated conjecture formation 4. The role of analogy in mathematics Part II. Plausibility, Uncertainty and Probability: 5. Bayesianism in mathematics 6. Uncertainty in mathematics and science Part III. The Growth of Mathematics: 7. Lakatos's philosophy of mathematics 8. Beyond the methodology of mathematical research programmes 9. The importance of mathematical conceptualisation Part IV. The Interpretation of Mathematics: 10. Higher dimensional algebra Appendix Bibliography Index.

History and Philosophy of Modern Mathematics was first published in 1988. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions.The fourteen essays in this volume build on the pioneering effort of Garrett Birkhoff, professor of mathematics at Harvard University, who in 1974 organized a conference of mathematicians and historians of modern mathematics to examine how the two disciplines approach the history of mathematics. In History and Philosophy of Modern Mathematics, William Aspray and Philip Kitcher bring together distinguished scholars from mathematics, history, and philosophy to assess the current state of the field. Their essays, which grow out of a 1985 conference at the University of Minnesota, develop the basic premise that mathematical thought needs to be studied from an interdisciplinary perspective.The opening essays study issues arising within logic and the foundations of mathematics, a traditional area of interest to historians and philosophers. The second section examines issues in the history of mathematics within the framework of established historical periods and questions. Next come case studies that illustrate the power of an interdisciplinary approach to the study of mathematics. The collection closes with a look at mathematics from a sociohistorical perspective, including the way institutions affect what constitutes mathematical knowledge.

The integration of children's literature in mathematics teacihng and learning has important pedagogical benefits, but many factors are effective in integrating children's literature into mathematics education. These factors include teachers' perspectives, knowledge, attitudes, beliefs and eagerness to implement it. It is thought that if teacher training programs cover courses on the use of children's literature books or other products, it would positively affect their attitudes towards such a practice. The aim of this study is to examine the opinions of preservice primary school teachers about the integration of children's literature in mathematics teacihng and learning. The views of the participants were collected before and after the use of a course outline based on the integration of mathematics with children's literature. The study was designed in a qualitative research design, and the opinions of the preservice teachers were evaluated extensively through the case study. The participants identified by convenience sampling method are 98 pre-service teachers who were studying in the last year of the teacher training program at a public university. The data of the study were obtained through mathematics lesson plans and open-ended questionnaires developed in relation to children's literature. The items in the first form were concerned with the experiences of the preservice teachers about integration of children's literature in mathematics, their views on the selection process of children's books and their expectations and evaluations regarding the practice. In the second form aimed to share the experiences of the participants following the implementation and to get their opinions about the teaching process which reflected the relationship between children's literature and mathematics. The data were analysed by using the descriptive statistics. The findings indicate that the views of the participants' are grouped into following four dimensions: book selection criteria and process, pedagogical effect, integration process, possible barriers and limitations. It is found that the participants have difficulty in choosing children's books that provide children opportunities for learning mathematics. Following the implementation, they emphasized the pedagogical benefits of the practice such as increasing children's motivation and interests in learning mathematics. During the integration process they mostly made use of the context of the books. Some of the participants employed the books for the purposes of teaching mathematical concepts and skills. However, the books were mostly used to get attraction, make courses more fun and increase student motivation. The participants stated that they had difficulties selecting and relating the books with mathematics subjects, and finding enough time for implementation. In order to use children's books as an effective tool to support conceptual understanding in mathematics teaching process, preservice and inservice teachers should be provided with necessary training and experience.

The paper describes the introductory approach to some questions of differential calculus (the program of mathematics for the final year of the course of the secondary humanistic studies), focussing on the question on the graph of a function. While the point is to find the graph of the function using the fundamental notions of calculus such as equations, limits, and derivatives, the conceptual framework is enriched by notions from historiography (Carlo Ginzburg), literature (Agatha Christie), and philosophy of mathematics. The aim is to teach students to solve a problem of graphing a function as Hercule Poirot would solve a case of murder.

Notes on Contributors

This chapter describes a constraints-based philosophy of mathematical practice and shows that mathematics can be so many different things, even if we look at a particular branch of mathematics in a particular time and place. It introduces a philosophical approach to mathematics that can serve as an integrative framework for the insights of the various philosophies of mathematics and demonstrates what kind of plurality the philosophy of mathematics must embrace, if it is to be faithful to the phenomenon that it seeks to explicate. The chapter reflects on the function of mathematical statements, consensus in mathematics, and mathematical interpretation and semiosis. It also considers various constraints that apply to mathematical practice and how they are negotiated by different mathematical cultures. Finally, it examines more mainstream notions of reality and truth of mathematical entities and statements and suggests how a takeoff on Hilary Putnam's notion of relevance might relativize them.

In the mathematical literature, a scalar integral equation with a degenerate kernel is well described (see below (1)), where all the written functions are scalar quantities). The authors are not aware of publications where systems of integral equations of (1) type with kernels in the form of a product of matrices would be considered in detail. It is usually said that the technique for solving such systems is easily transferred from the scalar case to the vector one (for example, in the monograph A.L. Kalashnikov "Methods for the approximate solution of integral equations of the second kind" (Nizhny Novgorod: Nizhny Novgorod State University, 2017), a brief description of systems of equations with degenerate kernels is given, where the role of degenerate kernels is played by products of scalar rather than matrix functions). However, as the simplest examples show, the generalization of the ideas of the scalar case to the case of integral systems with kernels in the form of a sum of products of matrix functions is rather unclear, although in this case the idea of reducing an integral equation to an algebraic system is also used. At the same time, the process of obtaining the conditions for the solvability of an integral system in the form of orthogonality conditions, based on the conditions for the solvability of the corresponding algebraic system, as it seems to us, has not been previously described. Bearing in mind the wide applications of the theory of integral equations in applied problems, the authors considered it necessary to give a detailed scheme for solving integral systems with degenerate kernels in the multidimensional case and to implement this scheme in the Maple program. Note that only scalar integral equations are solved in Maple using the intsolve procedure. The authors did not find a similar procedure for solving systems of integral equations, so they developed their own procedure.

The mathematical literature contains a good description of a scalar integral equation with a degenerate kernel (1), in which all of the written functions are scalar quantities. We are not aware of publications in which systems of integral equations of type (1) with kernels in the form of a product of matrices would be considered in detail. It is usually said that the technique for solving such systems is easily transferred from a scalar case to a vector one. For example, in A.L. Kalashnikov’s text book “Methods for Approximate Solution of Integral Equations of the Second Kind,” systems of equations with degenerate kernels are briefly described, in which the products of scalar rather than matrix functions play the role of degenerate kernels. However, as the simplest examples show, the generalization of the ideas of the scalar case for the case of integral systems with kernels represented by the sum of products of matrix functions is rather unclear, although in this case the idea of reducing an integral equation to an algebraic system is also used. In our opinion, the process of obtaining the conditions for the solvability of an integral system in the form of orthogonality conditions, based on the conditions for the solvability of the corresponding algebraic system, has not been described previously. Bearing in mind the great capabilities of the theory of integral equations in applied problems, we considered it necessary to give a detailed scheme for solving integral systems with degenerate kernels in a multidimensional case and to implement this scheme in the Maple math software. It should be noted that only scalar integral equations are solved in Maple using the intsolve procedure. Since we could not find a similar procedure for solving systems of integral equations, the have developed our own procedure.

A probabilistic approach for safety risk analysis in metro construction

We introduce the notion of a convex measure of risk, an extension of the concept of a coherent risk measure defined in Artzner et al. (1999), and we prove a corresponding extension of the representation theorem in terms of probability measures on the underlying space of scenarios. As a case study, we consider convex measures of risk defined in terms of a robust notion of bounded shortfall risk. In the context of a financial market model, it turns out that the representation theorem is closely related to the superhedging duality under convex constraints.

About primitive recursive algorithms

Despite recent advances in High Performance Computing (HPC), numerical simulation of high frequency (e.g. 1 Hz or higher) seismic wave propagation at the global scale is still prohibitive. To overcome this difficulty, we propose a hybrid method to efficiently compute teleseismic waveforms with 3-D source-side structures. By coupling the Spectral Element Method (SEM) with the Direct Solution Method (DSM) based on the representation theorem, we are able to limit the costly SEM simulation to a small source-side region and avoid computation over the entire space of the Earth. Our hybrid method is benchmarked against 1-D DSM synthetics and 3-D SEM synthetics. We also discuss numerical difficulties in the implementation, including slow DSM convergence near source depth, discretization error, Green’s function interpolation and local 3-D wavefield approximations. As a case study, we apply our hybrid method to two subduction earthquakes and show its advantage in understanding 3-D source-side effects on teleseismic P-waves. Our hybrid method reduces computational cost by more than two orders of magnitude when only source-side 3-D complexities are of concern. Thus our hybrid method is useful for a series of problems in seismology, such as imaging 3-D structures of a subducting slab or a mid-ocean ridge and studying source parameters with 3-D source-side complexities using teleseismic waveforms.

There are a number of studies which consider the relation between military spending and economic growth using Granger causality techniques rather than a well‐defined economic model. Some have used samples of groups of countries, finding no consistent results. Others have focused on case studies of individual countries, which has the advantage of the researchers bringing to bear much more data than the cross country samples and a greater knowledge of the structure of the economy and the budget. This paper adds to the literature by providing an analysis of two countries, Greece and Turkey, which are particularly interesting case studies given their high military burdens, the poor relations between the two and the resulting arms race in the area. In addition to analysing the data using standard “pre‐cointegration” Granger causality techniques, this paper employs modern vector autoregressive (VAR) methodology that utilises cointegration via Granger's representation theorem. The standard Granger causality tests suggest a positive effect of changing military burden on growth for Greece, but this is not sustained when the cointegration between output and military burden is taken into account. The only evidence of significant Granger causality is a negative impact of military burden on growth in Turkey.