Matematikkens filosofi

"This book addresses the material devices used to represent and manipulate numerical concepts. Fingers, tallies, tokens, and written notations, invented in both ancestral and contemporary societies, explain what numbers are, why they are the way they are, and how we get them"--

Discourse on the Method of Rightly Conducting One's Reason and of Seeking Truth in the Sciences (French: Discours de la Méthode Pour bien conduire sa raison, et chercher la vérité dans les sciences) is a philosophical and autobiographical treatise published by René Descartes in 1637. It is best known as the source of the famous quotation 'Je pense, donc je suis' (English: 'I think, therefore I am', or 'I am thinking, therefore I exist'), which occurs in Part IV of the work. The similar Latin statement, Cogito, ergo sum, is found in Meditations on First Philosophy (1641) and Principles of Philosophy (1644). Discourse on the Method is one of the most influential works in the history of modern philosophy, and important to the development of natural sciences. In this work, Descartes tackles the problem of skepticism, which had previously been studied by Sextus Empiricus, Al-Ghazali and Michel de Montaigne. Descartes modified it to account for a truth he found to be incontrovertible. Descartes started his line of reasoning by doubting everything, so as to assess the world from a fresh perspective, clear of any preconceived notions.The book was originally published in Leiden, in the Netherlands. Later, it was translated into Latin and published in 1656 in Amsterdam. The book was intended as an introduction to three works: Dioptrique, Météores and Géométrie. La Géométrie contains Descartes's initial concepts that later developed into the Cartesian coordinate system. The text was written and published in French rather than the language in which philosophical and scientific texts were most frequently written and published in at the time, namely, Latin. Most of Descartes' other works were written in Latin.Together with Meditations on First Philosophy, Principles of Philosophy and Rules for the Direction of the Mind, it forms the base of the epistemology known as Cartesianism. The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, the knowledge of which is competent to man, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another.

Evolutionary game theory originated in population biology from the realisation that frequency-dependent fitness introduced a strategic element into evolution. Since its development, evolutionary game theory has been adopted by many social scientists, and philosophers, to analyse interdependent decision problems played by boundedly rational individuals. Its study has led to theoretical innovations of great interest for the biological and social sciences. For example, theorists have developed a number of dynamical models which can be used to study how populations of interacting individuals change their behaviours over time. In this introduction, this Element covers the two main approaches to evolutionary game theory: the static analysis of evolutionary stability concepts, and the study of dynamical models, their convergence behaviour and rest points. This Element also explores the many fascinating, and complex, connections between the two approaches.

This Element shows that Plato keeps a clear distinction between mathematical and metaphysical realism and the knife he uses to slice the difference is method. The philosopher's dialectical method requires that we tether the truth of hypotheses to existing metaphysical objects. The mathematician's hypothetical method, by contrast, takes hypotheses as if they were first principles, so no metaphysical account of their truth is needed. Thus, we come to Plato's methodological as-if realism: in mathematics, we treat our hypotheses as if they were first principles, and, consequently, our objects as if they existed, and we do this for the purpose of solving problems. Taking the road suggested by Plato's Republic, this Element shows that methodological commitments to mathematical objects are made in light of mathematical practice; foundational considerations; and, mathematical applicability. This title is also available as Open Access on Cambridge Core.

This book is a re-assessment of the structure and reach of symmetry, by an interdisciplinary group of specialists from the arts, humanities, and sciences at Oxford University. This book aims to open up the scope of interdisciplinary work in the study of symmetry and is intended for scholars of any background.

An entertaining and enlightening history of irrational numbers, from ancient Greece to the twenty-first centuryThe ancient Greeks discovered them, but it wasn't until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. In The Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define-and why so many questions still surround them. Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.

An engrossing look at the history and importance of a centuries-old but still unanswered math problemFor centuries, mathematicians the world over have tried, and failed, to solve the zeta-3 problem. Math genius Leonhard Euler attempted it in the 1700s and came up short. The straightforward puzzle considers if there exists a simple symbolic formula for the following: 1+(1/2)^3+(1/3)^3+(1/4)^3+. . . . But why is this issue-the sum of the reciprocals of the positive integers cubed-so important? With In Pursuit of Zeta-3, popular math writer Paul Nahin investigates the history and significance of this mathematical conundrum.Drawing on detailed examples, historical anecdotes, and even occasionally poetry, Nahin sheds light on the richness of the nature of zeta-3. He shows its intimate connections to the Riemann hypothesis, another mathematical mystery that has stumped mathematicians for nearly two centuries. He looks at its links with Euler's achievements and explores the modern research area of Euler sums, where zeta-3 occurs frequently. An exact solution to the zeta-3 question wouldn't simply satisfy pure mathematical interest: it would have critical ramifications for applications in physics and engineering, such as quantum electrodynamics. Challenge problems with detailed solutions and MATLAB code are included at the end of each of the book's sections.Detailing the trials and tribulations of mathematicians who have approached one of the field's great unsolved riddles, In Pursuit of Zeta-3 will tantalize curious math enthusiasts everywhere.

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface.We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

The new series of Ideen&Argumente subscribes to the ideal of a pluralist and open culture of argument and debate and presents well-produced volumes on topics and questions which make substantive or methodologically important contributions to contemporary philosophy. The publications are designed to effect a productive synergy between the Anglo-Saxon and Continental European philosophical traditions. Ideen&Argumente provides a platform for outstanding systematically oriented original editions and German first editions from all areas of Theoretical and Practical Philosophy. A welcome is extended to programmatic monographs from whatever philosophical direction. The aim is to highlight anew the thematic and methodological richness of contemporary philosophy.