Contemporary philosophy of mathematics offers us an embarrassment of riches. Among the major areas of work one could list developments of the classical foundational programs, analytic approaches to epistemology and ontology of mathematics, and developments at the intersection of history and philosophy of mathematics. But anyone familiar with contemporary philosophy of mathematics will be aware of the need for new approaches that pay closer attention to mathematical practice. This book is the first attempt to give a coherent and unified presentation of this new wave of work in philosophy of mathematics. The new approach is innovative at least in two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non-constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics which escape purely formal logical treatment--such as visualization, explanation, and understanding--can nonetheless be subjected to philosophical analysis.
The Philosophy of Mathematical Practice comprises an introduction by the editor and eight chapters written by some of the leading scholars in the field. Each chapter consists of a short introduction to the general topic of the chapter followed by a longer research article in the area. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: diagrammatic reasoning and representational systems; visualization; mathematical explanation; purity of methods; mathematical concepts; the philosophical relevance ofcategory theory; philosophical aspects of computer science in mathematics; the philosophical impact of recent developments in mathematical physics.

John Cleary here explores the role which the mathematical sciences play in Aristotle's philosophical thought, especially in his cosmology, metaphysics, and epistemology. He also thematizes the aporetic method by means of which he deals with philosophical questions about the foundations of mathematics. The first two chapters consider Plato's mathematical cosmology in the light of Aristotle's critical distinction between physics and mathematics. Subsequent chapters examine three basic aporiae about mathematical objects which Aristotle himself develops in his science of first philosophy. What emerges from this dialectical inquiry is a different conception of substance and of order in the universe, which gives priority to physics over mathematics as the cosmological science. Within this different world-view, we can better understand what we now call Aristotle's philosophy of mathematics.

The Equation of Knowledge: From Bayes' Rule to a Unified Philosophy of Science introduces readers to the Bayesian approach to science: teasing out the link between probability and knowledge. The author strives to make this book accessible to a very broad audience, suitable for professionals, students, and academics, as well as the enthusiastic amateur scientist/mathematician. This book also shows how Bayesianism sheds new light on nearly all areas of knowledge, from philosophy to mathematics, science and engineering, but also law, politics and everyday decision-making. Bayesian thinking is an important topic for research, which has seen dramatic progress in the recent years, and has a significant role to play in the understanding and development of AI and Machine Learning, among many other things. This book seeks to act as a tool for proselytising the benefits and limits of Bayesianism to a wider public. Features Presents the Bayesian approach as a unifying scientific method for a wide range of topics Suitable for a broad audience, including professionals, students, and academics Provides a more accessible, philosophical introduction to the subject that is offered elsewhere

This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperuniverse Project was supported by the John Templeton Foundation from January 2013 until September 2015

Table of Contents: D. Duffie: Martingales, Arbitrage, and Portfolio Choice * J. Frohlich: Mathematical Aspects of the Quantum Hall Effect * M. Giaquinta: Analytic and Geometric Aspects of Variational Problems for Vector Valued Mappings * U. Hamenstadt: Harmonic Measures for Leafwise Elliptic Operators Along Foliations * M. Kontsevich: Feynman Diagrams and Low-Dimensional Topology * S.B. Kuksin: KAM-Theory for Partial Differential Equations * M. Laczkovich: Paradoxical Decompositions: A Survey of Recent Results * J.-F. Le Gall: A Path-Valued Markov Process and its Connections with Partial Differential Equations * I. Madsen: The Cyclotomic Trace in Algebraic K-Theory * A.S. Merkurjev: Algebraic K-Theory and Galois Cohomology * J. Nekovar: Values of L-Functions and p-Adic Cohomology * Y.A. Neretin: Mantles, Trains and Representations of Infinite Dimensional Groups * M.A. Nowak: The Evolutionary Dynamics of HIV Infections * R. Piene: On the Enumeration of Algebraic Curves - from Circles to Instantons * A. Quarteroni: Mathematical Aspects of Domain Decomposition Methods * A. Schrijver: Paths in Graphs and Curves on Surfaces * B. Silverman: Function Estimation and Functional Data Analysis * V. Strassen: Algebra and Complexity * P. Tukia: Generalizations of Fuchsian and Kleinian Groups * C. Viterbo: Properties of Embedded Lagrange Manifolds * D. Voiculescu: Alternative Entropies in Operator Algebras * M. Wodzicki : Algebraic K-Theory and Functional Analysis * D. Zagier: Values of Zeta Functions and Their Applications

Table of contents: Plenary Lectures * V.I. Arnold: The Vassiliev Theory of Discriminants and Knots * L. Babai: Transparent Proofs and Limits to Approximation * C. De Concini: Poisson Algebraic Groups and Representations of Quantum Groups at Roots of 1 * S.K. Donaldson: Gauge Theory and Four-Manifold Topology * W. Muller: Spectral Theory and Geometry * D. Mumford: Pattern Theory: A Unifying Perspective * A.-S. Sznitman: Brownian Motion and Obstacles * M. Vergne: Geometric Quantization and Equivariant Cohomology * Parallel Lectures * Z. Adamowicz: The Power of Exponentiation in Arithmetic * A. Bjorner: Subspace Arrangements * B. Bojanov: Optimal Recovery of Functions and Integrals * J.-M. Bony: Existence globale et diffusion pour les modeles discrets * R.E. Borcherds: Sporadic Groups and String Theory * J. Bourgain: A Harmonic Analysis Approach to Problems in Nonlinear Partial Differatial Equations * F. Catanese: (Some) Old and New Results on Algebraic Surfaces * Ch. Deninger: Evidence for a Cohomological Approach to Analytic Number Theory * S. Dostoglou and D.A. Salamon: Cauchy-Riemann Operators, Self-Duality, and the Spectral Flow

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Godel's platonism, up to the the current debate on Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem.

Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas.
This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical.
The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.

The book emerges from several contemporary concerns in mathematics, language, and mathematics education. However, the book takes a different stance with respect to language by combining discussion of linguistics and mathematics using examples from each to illustrate the other. The picture that emerges is of a subject that is much more contingent, much more relative, much more subject to human experience than is usually accepted. Another way of expressing this, is that the thesis of the book takes the idea of mathematics as a human creation, and, using the evidence from language, comes to more radical conclusions than most writers allow.

This is a concise introductory textbook for a one semester course in the history and philosophy of mathematics. It is written for mathematics majors, philosophy students, history of science students and secondary school mathematics teachers. The only prerequisite is a solid command of pre-calculus mathematics. It is shorter than the standard textbooks in that area and thus more accessible to students who have trouble coping with vast amounts of reading. Furthermore, there are many detailed explanations of the important mathematical procedures actually used by famous mathematicians, giving more mathematically talented students a greater opportunity to learn the history and philosophy by way of problem solving. Several important philosophical topics are pursued throughout the text, giving the student an opportunity to come to a full and consistent knowledge of their development. These topics include infinity, the nature of motion, and Platonism. This book offers, in fewer pages, a deep penetration into the key mathematical and philosophical aspects of the history of mathematics.

* Examines the history and philosophy of the mathematical sciences in a cultural context, tracing their evolution from ancient times up to the twentieth century * 176 articles contributed by authors of 18 nationalities * Chronological table of main events in the development of mathematics * Fully integrated index of people, events and topics * Annotated bibliographies of both classic and contemporary sources * Unique coverage of Ancient and non-Western traditions of mathematics

The book presents in a mathematical clear way the fundamentals of algorithmic information theory and a few selected applications. This 2nd edition presents new and important results obtained in recent years: the characterization of computable enumerable random reals, the construction of an Omega Number for which ZFC cannot determine any digits, and the first successful attempt to compute the exact values of 64 bits of a specific Omega Number. Finally, the book contains a discussion of some interesting philosophical questions related to randomness and mathematical knowledge. "Professor Calude has produced a first-rate exposition of up-to-date work in information and randomness." D.S. Bridges, Canterbury University, co-author, with Errett Bishop, of Constructive Analysis "The second edition of this classic work is highly recommended to anyone interested in algorithmic information and randomness." G.J. Chaitin, IBM Research Division, New York, author of Conversations with a Mathematician "This book is a must for a comprehensive introduction to algorithmic information theory and for anyone interested in its applications in the natural sciences." K. Svozil, Technical University of Vienna, author of Randomness & Undecidability in Physics

A compact survey, at the elementary level, of some of the most important concepts of mathematics. Attention is paid to their technical features, historical development and broader philosophical significance. Each of the various branches of mathematics is discussed separately, but their interdependence is emphasised throughout. Certain topics - such as Greek mathematics, abstract algebra, set theory, geometry and the philosophy of mathematics - are discussed in detail. Appendices outline from scratch the proofs of two of the most celebrated limitative results of mathematics: the insolubility of the problem of doubling the cube and trisecting an arbitrary angle, and the GAdel incompleteness theorems. Additional appendices contain brief accounts of smooth infinitesimal analysis - a new approach to the use of infinitesimals in the calculus - and of the philosophical thought of the great 20th century mathematician Hermann Weyl. Readership: Students and teachers of mathematics, science and philosophy. The greater part of the book can be read and enjoyed by anyone possessing a good high school mathematics background.

This essential companion volume to Chaitin's highly successful "The Limits of Mathematics", also published by Springer, gives a brilliant historical survey of the work of this century on the foundations of mathematics, in which the author was a major participant. The Unknowable is a very readable and concrete introduction to Chaitin's ideas, and it includes a detailed explanation of the programming language used by Chaitin in both volumes. It will enable computer users to interact with the author's proofs and discover for themselves how they work. The software for The Unknowable can be downloaded from the author's Web site.

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras' Theorem or Fermat's Last Theorem? The metaphysical question of what numbers are and the epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics' unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true-no one doubts that 2 + 3 = 5-but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about.

This wordless collection of strips by renowned artist/designer Rian Hughes reveals the lighter side of our obsession with social rankings. When everyone has a number, everyone knows their place. Lower numbers are better, higher numbers are less important, and that's just the way it is. But what if that number could change? You might try to buck the system and assert your individuality... or you might end up with a big fat zero. Big questions are explored and unexpected answers found in the first solo comics collection from award-winning designer & illustrator Rian Hughes. His whimsical, witty, and insightful strips will make you both smile and consider. Where do you stand in the pecking order? Is your number up?

The aim of this book is to present and analyze philosophical conceptions concerning mathematics and logic as formulated by Polish logicians, mathematicians and philosophers in the 1920s and 1930s. It was a remarkable period in the history of Polish science, in particular in the history of Polish logic and mathematics. Therefore, it is justified to ask whether and to what extent the development of logic and mathematics was accompanied by a philosophical reflection. We try to answer those questions by analyzing both works of Polish logicians and mathematicians who have a philosophical temperament as well as their research practice. Works and philosophical views of the following Polish scientists will be analyzed: Waclaw Sierpinski, Zygmunt Janiszewski, Stefan Mazurkiewicz, Stefan Banach Hugo Steinhaus, Eustachy Zylinsk and Leon Chwistek, Jan Lukasiewicz, Zygmunt Zawirski, Stanislaw Lesniewski, Tadeusz Kotarbinski, Kazimierz Ajdukiewicz, Alfred Tarski, Andrzej Mostowski and Henryk Mehlberg, Jan Sleszynski, Stanislaw Zaremba and Witold Wilkosz. To indicate the background of scientists being active in the 1920s and 1930s we consider in Chapter 1 some predecessors, in particular: Jan Sniadecki, Jozef Maria Hoene-Wronski, Samuel Dickstein and Edward Stamm.

Many artists are unaware of the mathematics that bubble beneath their craft, while some consciously use it for inspiration. Our instincts might tell us that these two subjects are incompatible forces with nothing in common, but what if we’re wrong?

Marcus du Sautoy, acclaimed mathematician and Simonyi Professor for the Public Understanding of Science at the University of Oxford, looks to art, music, design and literature to uncover the key mathematical structures that underpin both human creativity and the natural world.

Blueprints takes us from the earliest stone circles to the modernist architecture of Le Corbusier, from Bach’s circular compositions to Radiohead’s disruptive soundscapes, and from Shakespeare’s hidden numerical clues to the Dada artists who embraced randomness. Instead of polar opposites we find a complementary relationship that spans a vast historical and geographic landscape.

Whether we are searching for meaning in an abstract painting or deciphering poetry, there are blueprints everywhere: prime numbers, symmetry, fractals and the weirder worlds of Hamiltonian cycles and hyperbolic geometry. Nature similarly exploits these structures to achieve the wonders of our universe.

In this innovative and delightfully bold exploration of creativity, Marcus explains how we make art, why a creative mindset is vital for discovering new mathematics and how a fundamental connection to the natural world intrinsically links these two subjects.

Quadratic equations, Pythagoras' theorem, imaginary numbers, and pi - you may remember studying these at school, but did anyone ever explain why? Never fear - bestselling science writer, and your new favourite maths teacher, Michael Brooks, is here to help. In The Maths That Made Us, Brooks reminds us of the wonders of numbers: how they enabled explorers to travel far across the seas and astronomers to map the heavens; how they won wars and halted the HIV epidemic; how they are responsible for the design of your home and almost everything in it, down to the smartphone in your pocket. His clear explanations of the maths that built our world, along with stories about where it came from and how it shaped human history, will engage and delight. From ancient Egyptian priests to the Apollo astronauts, and Babylonian tax collectors to juggling robots, join Brooks and his extraordinarily eccentric cast of characters in discovering how maths made us who we are today.

What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field.

Mathematics has stood as a bridge between the Humanities and the Sciences since the days of classical antiquity. For Plato, mathematics was evidence of Being in the midst of Becoming, garden variety evidence apparent even to small children and the unphilosophical, and therefore of the highest educational significance. In the great central similes of The Republic it is the touchstone ofintelligibility for discourse, and in the Timaeus it provides in an oddly literal sense the framework of nature, insuring the intelligibility ofthe material world. For Descartes, mathematical ideas had a clarity and distinctness akin to the idea of God, as the fifth of the Meditations makes especially clear. Cartesian mathematicals are constructions as well as objects envisioned by the soul; in the Principles, the work ofthe physicist who provides a quantified account ofthe machines of nature hovers between description and constitution. For Kant, mathematics reveals the possibility of universal and necessary knowledge that is neither the logical unpacking ofconcepts nor the record of perceptual experience. In the Critique ofPure Reason, mathematics is one of the transcendental instruments the human mind uses to apprehend nature, and by apprehending to construct it under the universal and necessary lawsofNewtonian mechanics.

This work offers a re-edition of twelve mathematical tablets from the site of Tell Harmal, in the borders of present-day Baghdad. In ancient times, Tell Harmal was Saduppum, a city representative of the region of the Diyala river and of the kingdom of Esnunna, to which it belonged for a time. These twelve tablets were originally published in separate articles in the beginning of the 1950s and mostly contain solved problem texts. Some of the problems deal with abstract matters such as triangles and rectangles with no reference to daily life, while others are stated in explicitly empirical contexts, such as the transportation of a load of bricks, the size of a vessel, the number of men needed to build a wall and the acquisition of oil and lard. This new edition of the texts is the first to group them, and takes into account all the recent developments of the research in the history of Mesopotamian mathematics. Its introductory chapters are directed to readers interested in an overview of the mathematical contents of these tablets and the language issues involved in their interpretation, while a chapter of synthesis discusses the ways history of mathematics has typically dealt with the mathematical evidence and inquires how and to what degree mathematical tablets can be made part of a picture of the larger social context. Furthermore, the volume contributes to a geography of the Old Babylonian mathematical practices, by evidencing that scribes at Saduppum made use of cultural material that was locally available. The edited texts are accompanied by translations, philological, and mathematical commentaries.

What was the basis for the adoption of mathematics as the primary mode of discourse for describing natural events by a large segment of the philosophical community in the seventeenth century? In answering this question, this book demonstrates that a significant group of philosophers shared the belief that there is no necessary correspondence between external reality and objects of human understanding, which they held to include the objects of mathematical and linguistic discourse. The result is a scholarly reliable, but accessible, account of the role of mathematics in the works of (amongst others) Galileo, Kepler, Descartes, Newton, Leibniz, and Berkeley. This impressive volume will benefit scholars interested in the history of philosophy, mathematical philosophy and the history of mathematics.