This book is meant as a part of the larger contemporary philosophical project of naturalizing logico-mathematical knowledge, and addresses the key question that motivates most of the work in this field: What is philosophically relevant about the nature of logico-mathematical knowledge in recent research in psychology and cognitive science? The question about this distinctive kind of knowledge is rooted in Plato’s dialogues, and virtually all major philosophers have expressed interest in it. The essays in this collection tackle this important philosophical query from the perspective of the modern sciences of cognition, namely cognitive psychology and neuroscience. Naturalizing Logico-Mathematical Knowledge contributes to consolidating a new, emerging direction in the philosophy of mathematics, which, while keeping the traditional concerns of this sub-discipline in sight, aims to engage with them in a scientifically-informed manner. A subsequent aim is to signal the philosophers’ willingness to enter into a fruitful dialogue with the community of cognitive scientists and psychologists by examining their methods and interpretive strategies.
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.
This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts.
Why is the number seven lucky--even holy--in almost every culture? Why do we speak of the four corners of the earth? Why do cats have nine lives (except in Iran, where they have seven)? From literature to folklore to private superstitions, numbers play a conspicuous role in our daily lives. But in this fascinating book, Annemarie Schimmel shows that numbers have been filled with mystery and meaning since the earliest times, and across every society. In The Mystery of Numbers Annemarie Schimmel conducts an illuminating tour of the mysteries attributed to numbers over the centuries. She begins with an informative and often surprising introduction to the origins of number systems: pre-Roman Europeans, for example, may have had one based on twenty, not ten (as suggested by the English word "score" and the French word for 80, quatrevingt --four times twenty), while the Mayans had a system more sophisticated than our own. Schimmel also reveals how our fascination with numbers has led to a rich cross-fertilization of mathematical knowledge: "Arabic" numerals, for instance, were picked up by Europe from the Arabs, who had earlier adopted them from Indian sources ("Algorithm" and "algebra" are corruptions of the Arabic author and title names of a mathematical text prized in medieval Europe). But the heart of the book is an engrossing guide to the symbolism of numbers. Number symbolism, she shows, has deep roots in Western culture, from the philosophy of the Pythagoreans and Platonists, to the religious mysticism of the Cabala and the Islamic Brethren of Purity, to Kepler's belief that the laws of planetary motion should be mathematically elegant, to the unlucky thirteen. After exploring the sources of number symbolism, Schimmel examines individual numbers ranging from one to ten thousand, discussing the meanings they have had for Judaic, Christian, and Islamic traditions, with examples from Indian, Chinese, and Native American cultures as well. Two, for instance, has widely been seen as a number of contradiction and polarity, a number of discord and antithesis. And six, according to ancient and neo-platonic thinking, is the most perfect number because it is both the sum and the product of its parts (1+2+3=6 and 1x2x3=6). Using examples ranging from the Bible to the Mayans to Shakespeare, she shows how numbers have been considered feminine and masculine, holy and evil, lucky and unlucky. A highly respected scholar of Islamic culture, Annemarie Schimmel draws on her vast knowledge to paint a rich, cross-cultural portrait of the many meanings of numbers. Engaging and accessible, her account uncovers the roots of a phenomenon we all feel every Friday the thirteenth.
The Asian Logic Conference (ALC) is a major international event in mathematical logic. It features the latest scientific developments in the fields of mathematical logic and its applications, logic in computer science, and philosophical logic. The ALC series also aims to promote mathematical logic in the Asia-Pacific region and to bring logicians together both from within Asia and elsewhere for an exchange of information and ideas. This combined proceedings volume represents works presented or arising from the 14th and 15th ALCs.
This examination of a series of philosophical issues arising from the applicability of mathematics to science consists of scientifically-informed philosophical analysis and argument. One distinctive feature of this project is that it proposes to look at issues in philosophy of mathematics within the larger context of philosophy of science.
An anthology of the year's finest writing on mathematics from around the world, featuring promising new voices as well as some of the foremost names in mathematics.
This book offers a plurality of perspectives on the historical origins of logicism and on contemporary developments of logicist insights in philosophy of mathematics. It uniquely provides up-to-date research and novel interpretations on a variety of intertwined themes and historical figures related to different versions of logicism. The essays, written by prominent scholars, are divided into three thematic sections. Part I focuses on major authors like Frege, Dedekind, and Russell, providing a historical and theoretical exploration of such figures in the philosophical and mathematical milieu in which logicist views were first expounded. Part II sheds new light on the interconnections between these founding figures and a number of influential other traditions, represented by authors like Hilbert, Husserl, and Peano, as well as on the reconsideration of logicism by Carnap and the logical empiricists. Finally, Part III assesses the legacy of such authors and of logicist themes for contemporary philosophy of mathematics, offering new perspectives on highly debated topics—neo-logicism and its extension to accounts of ordinal numbers and set-theory, the comparison between neo-Fregean and neo-Dedekindian varieties of logicism, and the relation between logicist foundational issues and empirical research on numerical cognition—which define the prospects of logicism in the years to come. This book offers a comprehensive account of the development of logicism and its contemporary relevance for the logico-philosophical foundations of mathematics. It will be of interest to graduate students and researchers working in philosophy of mathematics, philosophy of logic, and the history of analytic philosophy.
This book offers an alternative to current philosophy of mathematics: heuristic philosophy of mathematics. In accordance with the heuristic approach, the philosophy of mathematics must concern itself with the making of mathematics and in particular with mathematical discovery. In the past century, mainstream philosophy of mathematics has claimed that the philosophy of mathematics cannot concern itself with the making of mathematics but only with finished mathematics, namely mathematics as presented in published works. On this basis, mainstream philosophy of mathematics has maintained that mathematics is theorem proving by the axiomatic method. This view has turned out to be untenable because of Gödel’s incompleteness theorems, which have shown that the view that mathematics is theorem proving by the axiomatic method does not account for a large number of basic features of mathematics. By using the heuristic approach, this book argues that mathematics is not theorem proving by the axiomatic method, but is rather problem solving by the analytic method. The author argues that this view can account for the main items of the mathematical process, those being: mathematical objects, demonstrations, definitions, diagrams, notations, explanations, applicability, beauty, and the role of mathematical knowledge.
