This textbook for postgraduate students learning mathematical methods in economics provides a comprehensive account of mathematics required to analyse and solve problems of choice encountered by economists. It looks at a wide variety of decision-making problems, both static and dynamic, in various contexts and provides mathematical foundations for the relevant economic theory.

A thought experiment involving an omniscient being and quantum mechanics is used to justify non-deductive methods in mathematics. The twin prime conjecture is used to illustrate what can be achieved.

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)

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, and (...) that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect. (shrink)

The Prague Philosopher Bernard Bolzano (1781-1848) has long been admired for his groundbreaking work in mathematics: his rigorous proofs of fundamental theorems in analysis, his construction of a continuous, nowhere-differentiable function, his investigations of the infinite, and his anticipations of Cantor's set theory. He made equally outstanding contributions in philosophy, most notably in logic and methodology. One of the greatest mathematician-philosophers since Leibniz, Bolzano is now widely recognised as a major figure of nineteenth-century philosophy. Praised by Husserl as “one of (...) the greatest logicians of all times,” he has also been recognised by Michael Dummett as one of the first modern analytic philosophers and by Alberto Coffa as the founder of the “semantic tradition.” This volume contains English translations of the essay “On the Mathematical Method,” a concise introduction to Bolzano’s logic and philosophy of mathematics, as well as substantial selections from his correspondence with Franz Exner, Professor of Philosophy at the Charles University in Prague in the 1830s and 40s. It will be of interest to students of Austrian philosophy, the development of analytic philosophy, the philosophy of language, and the history and philosophy of logic and mathematics. (shrink)

This book examines the interface between mathematics and applied sciences. The editor examines the present and future needs for the interaction between various science and engineering areas. This edited book brings together the cutting-edge research on mathematics, combining various fields of science and engineering. The book begins with an introduction to computing and modeling. Next, computation and modeling trends are covered, along with chapters on structural static and vibration problems, heat conduction and diffusion problems, and fluid dynamics problems. Soft computing (...) and machine intelligence based modeling are also discussed, along with bio medical problems. The book concludes with a chapter on modeling in mining, electrical, electronics, and chemical problems. This book is a reference for students and researchers in various fields of engineering such as computer, mechanical, civil, aerospace, electrical, and chemical, as well as and other sciences such as applied mathematics, physics, chemistry, and life sciences. (shrink)

Regions-based theories of space aim—among others—to define points in a geometrically appealing way. The most famous definition of this kind is probably due to Whitehead. However, to conclude that the objects defined are points indeed, one should show that they are points of a geometrical or a topological space constructed in a specific way. This paper intends to show how the development of mathematical tools allows showing that Whitehead’s method of extensive abstraction provides a construction of objects that (...) are fundamental building blocks of specific topological spaces. (shrink)

This text is an ideal introduction for students to the basic mathematics of operations research as well as a valuable source of references to early literature ...

The generative and governing "idea" of radical modernity is spawned by the technique of mathematical construction deployed and interpreted by the major early-modern thinkers and their legatees. ;Chapter I is a survey of this legacy as it appears in Vico, Kant, Fichte, Marx and Nietzsche and in the post-Nietzschean inheritance of contemporary philosophy, hyperbolic in the case of Derrida et al., elliptical, in the case of Carnap and Goodman. ;In Chapter II I try to show how the pre-modern (...) class='Hi'>mathematical tradition, represented by Euclid, aimed at keeping the enticements of technical facility in check by means of didactic phronesis and how the post-Kantian interpretation of "existence" in Euclid as constructibility betrays his usage and self-understanding. I suggest that his focus in the postulates and elsewhere is on the undistorted iterability of graphic evocations of the items already intelligible thanks to the definitions or to the pre-understanding shared by the teacher and student. ;In Chapter III, devoted to Descartes the principal claims of modern constructivism are brought to sight. After examining Descartes' fabulous autobiography and its emphasis on self-origination, I turn to the style, contents and under-pinnings of the Geometry in an effort to extract from that text what he once referred to as "the metaphysics of geometry." The latter yields the conditions of successful problem-solving, i.e., dimensional homogeneity and kinematic continuity. These conditions, in turn, find their justification in Descartes' theses in the Rules concerning order, measure and the uniformity of "mental" activity. In the final section I apply the lessons learned from the Geometry and the Rules to one critical issue in the later Meditations, the transition from essence to existence. Descartes' "solution" generates a sequence of perplexities with Hobbes, Leibniz, Kant and other radical moderns continue to wrestle. (shrink)

I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to (...) provide an account of the practice and the applicability of mathematics while avoiding the conflation of metaphysical considerations with mathematical ones. (shrink)

According to Kant, “mathematical knowledge is the knowledge gained by reason from the construction of concepts.” In this paper, I shall make a few suggestions as to how this characterization of the mathematical method is to be understood.

Mathematics and philosophy have historically enjoyed a mutually beneficial and productive relationship, as a brief review of the work of mathematician–philosophers such as Descartes, Leibniz, Bolzano, Dedekind, Frege, Brouwer, Hilbert, Gödel, and Weyl easily confirms. In the last century, it was especially mathematical logic and research in the foundations of mathematics which, to a significant extent, have been driven by philosophical motivations and carried out by technically minded philosophers. Mathematical logic continues to play an important role in contemporary (...) philosophy, and mathematically trained philosophers continue to contribute to the literature in logic. For instance, modal logics were first investigated by philosophers and now have important applications in computer science and mathematical linguistics. (shrink)

Wolff advocates the mathematical method, which consists in chains of syllogisms that proceed from axioms and definitions to theorems, for achieving scientific certainty in branches of philosophy like ontology and physics. By contrast, in ‘The Discipline of Pure Reason in its Dogmatic Use’ Kant significantly limits the efficacy of this method in philosophy. In this paper I investigate an under-examined result of the Discipline: Kant’s claim that his system of philosophy does not contain “dogmata”. By identifying “dogmata” (...) in Wolff’s system of physics, I argue that, for Kant, they are propositions that uncritically deploy ideas of reason. I conclude that the Discipline extends criticisms raised in the Transcendental Dialectic to any erroneous use of the mathematical method in philosophy. (shrink)

This article examines Hobbes’s conception of mathematical method, situating his methodological writings in the context of disputed mathematical issues of the seventeenth century. After a brief exposition of the Hobbesian philosophy of mathematics, it investigates Hobbes’s attempts to resolve three important mathematical controversies of the seventeenth century: the debates over the status of analytic geometry, disputes over the nature of ratios, and the problem of the “angle of contact” between a curve and tangent. In the course (...) of these investigations, Hobbes’s account of mathematics and its method is contrasted with the those of Descartes, Isaac Barrow, John Wallis, and Christopher Clavius. (shrink)

RésuméLe problème de la détermination de la Qibla est l'une des questions cruciales qui se posent à la culture scientifique de l'Islam médiéval; le résoudre correctement nécessite tant des théories mathématiques que des observations. Les mathématiques relèvent de deux chapitres: la trigonométrie plane et la trigonométrie sphérique. L'observation et les instruments d'observation sont indispensables à la détermination des coordonnées géographiques de La Mecque et du lieu donné; ces coordonnées sont en effet les données que l'on entre dans les formules donnant (...) la Qibla. Dans son Almageste, Abū al-Wafāʾ résout brillamment ce problème. Son travail porte tant sur les mathématiques que sur l'observation; il obtient des solutions modernes, adéquates et faciles. En trigonométrie plane, il donne de nouvelles définitions des fonctions trigonométriques, démontre les formules des sinus, donne une approximation de sin 1° grâce à laquelle il construit des tables de sinus et de tangentes d'une grande précision. En trigonométrie sphérique, il démontre quatre nouveaux théorèmes dont la règle des tangentes ; cette règle lui permet de trouver, comme nous le montrerons, des solutions simples au problème de la détermination de la Qibla. En ce qui concerne les observations, il décrit trois instruments utilisés par lui plusieurs années de suite à Bagdad. Cet article, nourri de nombreux textes arabes originaux traduits et commentés, donne une description détaillée, technique et analytique, des méthodes mathématiques d'Abū al-Wafāʾ pour la détermination de la Qibla. (shrink)

Two families of mathematical methods lie at the heart of investigating the hierarchical structure of genetic variation in Homo sapiens: /diversity partitioning/, which assesses genetic variation within and among pre-determined groups, and /clustering analysis/, which simultaneously produces clusters and assigns individuals to these “unsupervised” cluster classifications. While mathematically consistent, these two methodologies are understood by many to ground diametrically opposed claims about the reality of human races. Moreover, modeling results are sensitive to assumptions such as preexisting theoretical commitments to (...) certain linguistic, anthropological, and geographic human groups. Thus, models can be perniciously reified. That is, they can be conflated and confused with the world. This fact belies standard realist and antirealist interpretations of “race,” and supports a pluralist conventionalist interpretation. (shrink)

This paper deals with the problem of reconstruction of the intergenic interaction graph from the raw data of genetic co-expression coming with new technologies of bio-arrays (DMA-arrays, protein-arrays, etc.). These new imaging devices in general only give information about the asymptotical part (fixed configurations of co-expression or limit cycles of such configurations) of the dynamical evolution of the regulatory networks (genetic and/or proteic) underlying the functioning of living systems. Extracting the casual structure and interaction coefficients of a gene interaction network (...) from the observed configurations is a complex problem. But if all the fixed configurations are supposedly observed and if they are factorizable into two or more subsets of values, then the interaction graph possesses as many connected components as the number of factors and the solution is obtained in polynomial time. This new result allows us for example to partly solve the topology of the genetic regulatory network ruling the flowering in Arabidopsis thaliana. (shrink)

This book reconstructs key aspects of the early career of Descartes from 1618 to 1633; that is, up through the point of his composing his first system of natural philosophy, Le Monde, in 1629-33. It focuses upon the overlapping and intertwined development of Descartes’ projects in physico-mathematics, analytical mathematics, universal method, and, finally, systematic corpuscular-mechanical natural philosophy. The concern is not simply with the conceptual and technical aspects of these projects; but, with Descartes’ agendas within them and his construction (...) and presentation of his intellectual identity in relation to them. Descartes’ technical projects, agendas and senses of identity shifted over time, entangled and displayed great successes and deep failures, as he morphed from a mathematically competent, Jesuit trained graduate in neo-Scholastic Aristotelianism to aspiring prophet of a systematised corpuscular-mechanism, passing through stages of being a committed physico-mathematicus, advocate of a putative ‘universal mathematics’, and projector of a grand methodological dream. In all three dimensions—projects, agendas and identity concerns—the young Descartes struggled and contended, with himself and with real or virtual peers and competitors, hence the title ‘Descartes-Agonistes’. (shrink)

This book is designed to make accessible to nonspecialists the still evolving concepts of quantum mechanics and the terminology in which these are expressed. The opening chapters summarize elementary concepts of twentieth century quantum mechanics and describe the mathematical methods employed in the field, with clear explanation of, for example, Hilbert space, complex variables, complex vector spaces and Dirac notation, and the Heisenberg uncertainty principle. After detailed discussion of the Schrödinger equation, subsequent chapters focus on isotropic vectors, used to (...) construct spinors, and on conceptual problems associated with measurement, superposition, and decoherence in quantum systems. Here, due attention is paid to Bell's inequality and the possible existence of hidden variables. Finally, progression toward quantum computation is examined in detail: if quantum computers can be made practicable, enormous enhancements in computing power, artificial intelligence, and secure communication will result. This book will be of interest to a wide readership seeking to understand modern quantum mechanics and its potential applications. (shrink)

This book argues that Descartes’ physics was a milestone on the road to modern mathematical physics. After Newton introduced a completely different approach to mathematical description of motion, Descartes’ physics became obsolete and even difficult to comprehend. This text follows the language of Descartes and the means of which motion can be described. It argues that Descartes achieved almost everything that later Newton was able to do—to describe the motion of interacting bodies- by different (i.e. algebraic) means. This (...) volume completely refutes the received view according to which Descartes’ physics was merely a kind of discursive natural philosophy. To make this interpretation more plausible the book follows Descartes’ ideas from his early work in mathematics, through his invention of the analytic method towards his mature physics. It shows that Descartes followed a similar heuristic pattern. The volume appeals to students and researchers; it invites the reader equippedwith minimal understanding of college mathematics to follow Descartes on his intellectual journey through the Scientific Revolution. The reader will gain a deeper understanding of the role of mathematical language in the creation of modern physics and a glimpse into the fascinating world of Descartes’ scientific thought. Several of Descartes’ philosophical ideas can be traced back to his scientific interests and thus the book elucidates the motivation behind some of Descartes’ key positions in the area of epistemology and method. In the penultimate chapter the book presents four arguments in favor of seeing Descartes as a physicist on par with Galileo and Newton. (shrink)

Outlines a method of solving mathematical problems for teachers and students based upon the four steps of understanding the problem, devising a plan, carrying out the plan, and checking the results.

This article is mainly a critique of Philip Kitcher's book, The Nature of Mathematical Knowledge. Four weaknesses in Kitcher's objection to Kant arise out of Kitcher's failure to recognize the perspectival nature of Kant's position. A proper understanding of Kant's theory of mathematics requires awareness of the perspectival nuances implicit in Kant's theory of pure intuition.

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors (...) tried to naturalize mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

Book review of John Schuster: Descartes-agonistes: Physico-mathematics, method and corpuscular-mechanism 1618-33. (Studies in History and Philosophy of Science, Volume 27.) Dordrecht: Springer, 2013, xix + 631pp. Descartes-Agonistes is the magnum opus of John Schuster, formerly of the University of New South Wales, honorary fellow at the University of Sydney. Its roots go back to the dissertation he wrote 35 years ago under Thomas Kuhn at Princeton University. As Schuster correctly remarks, some regard his dissertation as an underground classic. I (...) count myself among them: Schuster’s work has been directional in my work on the history of early modern science. Schuster himself prefers to regard his dissertation as ‘a vestige’ of a history of science of the 1970s that he has since gone beyond. In the 1980s, he elaborated and published on some central insights in a series of seminal articles, but his work on Descartes really got into its stride again around the turn of the millennium (Schuster 1980, 1990; Gaukroger et al. 2000). This book is the acme of this work, and grand it is: clocking 600 pages (excluding appendices) it extends, deepens and reinforces his analysis of Descartes into a dense and .. (shrink)

This paper focuses on the distinction between methods which are mathematically "clever", and those which are simply crude, typically repetitive and computer intensive, approaches for "crunching" out answers to problems. Examples of the latter include simulated probability distributions and resampling methods in statistics, and iterative methods for solving equations or optimisation problems. Most of these methods require software support, but this is easily provided by a PC. The paper argues that the crunchier methods often have substantial advantages from the perspectives (...) of user-friendliness, reliability (in the sense that misuse is less likely), educational efficiency and realism. This means that they offer very considerable potential for simplifying the mathematical syllabus underlying many areas of applied mathematics such as management science and statistics: crunchier methods can provide the same, or greater, technical power, flexibility and insight, while requiring only a fraction of the mathematical conceptual background needed by their cleverer brethren. (shrink)

The method of mathematical induction: The method of mathematical induction -- Examples and exercises -- The proof of induction of some theorems of elemetary algebra -- Solutions.