2014 Reprint of 1954 American Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. This two volume classic comprises two titles: "Patterns of Plausible Inference" and "Induction and Analogy in Mathematics." This is a guide to the practical art of plausible reasoning, particularly in mathematics, but also in every field of human activity. Using mathematics as the example par excellence, Polya shows how even the most rigorous deductive discipline is heavily dependent on techniques of guessing, (...) inductive reasoning, and reasoning by analogy. In solving a problem, the answer must be guessed at before a proof can be given, and guesses are usually made from a knowledge of facts, experience, and hunches. The truly creative mathematician must be a good guesser first and a good prover afterward; many important theorems have been guessed but no proved until much later. In the same way, solutions to problems can be guessed, and a god guesser is much more likely to find a correct solution. This work might have been called "How to Become a Good Guesser."-From the Dust Jacket. (shrink)

The correspondence between Richard von Mises and George Pólya of 1919/20 contains reflections on two well-known articles by von Mises on the foundations of probability in the Mathematische Zeitschrift of 1919, and one paper from the Physikalische Zeitschrift of 1918. The topics touched on in the correspondence are: the proof of the central limit theorem of probability theory, von Mises' notion of randomness, and a statistical criterion for integer-valuedness of physical data. The investigation will hint at both the fruitfulness and (...) the limits of several of von Mises' notions such as ``collective'', ``distribution'' and ``complex adjuncts'' (characteristic functions) for further developments in probability theory and in ``directional statistics''. By pointing to the selectiveness of Pólya's criticism, the historical analysis shows the differing expectations of the two men with respect to the further development of the theory of probability and its applications. The paper thus gives a glimpse of the provisional state of the theory around 1920, before others such as P. Lévy (1886–1971) and A. N. Kolmogorov (1903–1987) stepped in and created a new paradigm for probability theory. (shrink)

La siguiente ponencia-taller tiene por finalidad explicar cómo podría aplicarse el Método de George Polya para resolver problemas (matemáticos) en el contexto de la Lógica Matemática, especialmente en la lógica Matemática elemental (la lógica de primer orden con identidad). Es una propuesta pedagógica experimental (además de las ya existentes) que tal vez pueda ser útil para la enseñanza de la lógica matemática en ciencias o en humanidades. Dicha ponencia se presentó (vía web) con motivo de la celebración del Día (...) Mundial de Lógica 2022 en dos ocasiones: En la Universidad Central de Venezuela (Venezuela) y en la Universidad Mayor de San Andrés (Logic & Analytics Group - La Paz y La Sociedad Científica de Filosofía de la UMSA, Bolivia). Es un proyecto pedagógico en construcción. (shrink)

A brief correction to a review of my book Imre Lakatos and the Guises of Reason published in Philosophia Mathematica, regarding the role of George Polya's notion of heuristic in Lakatos' Proofs and Refutations.

Philosophers have paid little attention to mathematical explanations . I present a variety of examples of mathematical explanation and examine two cases in detail. I argue that mathematical explanations have important implications for the philosophy of mathematics and of science. ;The first case study compares many proofs of Pick's theorem, a simple geometrical result. Though a simple proof surfaces to establish the result, some of the proofs explain the result better than others. The second case study comes from George (...) class='Hi'>Polya's Mathematics and Plausible Reasoning. He gives a proof that, while entirely satisfactory in establishing its conclusion, is insufficiently explanatory. To provide a better explanation, he supplements the proof with additional exposition. ;These case studies illustrate at least two distinct explanatory virtues, and suggest there may be more. First, an explanatory improvement occurs when a sense of "arbitrariness" is reduced in the proofs. Proofs more explanatory in this way place greater restrictions on the steps that can be used to reach the conclusion. Second, explanatoriness is judged by directness of representation. More explanatory proofs allow one to ascribe geometric meaning to the terms of Pick's formula as they arise. ;I trace the lack of attention to mathematical explanations to an implicit assumption, justificationism, that only justificational aspects of mathematical reasoning are epistemically important. I propose an anti-justificationist epistemic position, the epistemic virtues view, which holds that justificational virtues, while important, are not the only ones of philosophical interest in mathematics. Indeed, explanatory benefits are rarely justificational. I show how the epistemic virtues view and the recognition of mathematical explanation can shed new light on philosophical debates. ;Mathematical explanations have consequences for philosophy of science as well. I show that mathematical explanations provide serious challenges to any theory, such as Bas van Fraassen's, that considers explanations to be fundamentally answers to why-questions. I urge a closer interaction between philosophy of mathematics and philosophy of science; both will be needed for a fuller understanding of mathematical explanation. (shrink)

Hans Reichenbach, Philosophic Foundations of Quantum Mechanics. University of California Press. Berkeley and Los Angeles, 1944.Georg Pólya, How to Solve It, a New Aspect of Mathematical Method; Princeton University Press, 1945.

Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Le;vi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the (...) existence of unconscious mental processes in mathematical invention and other forms of creativity. Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field , remains an important tool for exploring the increasingly complex problem of mental life. The roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. His discussion of this process comprises a wide range of topics, including the use of mental images or symbols, visualized or auditory words, "meaningless" words, logic, and intuition. Among the important documents collected is a letter from Albert Einstein analyzing his own mechanism of thought. (shrink)

In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called — in mathematics — the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is that there (...) exists a direct line leading from Cantor’s diagonal argument to constructions that underlies of the proofs of several important theorems of the mathematical logic (in particular, Church’s theorem concerning the undecidability of formal arithmetic, Gödel’s theorem concerning the incopleteness of formal arithmetic, Tarski’s theorem concerning truth, and Turing’s theorem concerning the Halting Problem), and that the line could be described as an analogical mapping. In other words, Cantor’s diagonal argument and the proofs of the limitative theorems are structurally the same. Hence they can be represented as instances (or special cases) of the same general scheme. (shrink)

This lively introductory text exposes the student in the humanities to the world of discrete mathematics. A problem-solving based approach grounded in the ideas of George Pólya are at the heart of this book. Students learn to handle and solve new problems on their own. A straightforward, clear writing style and well-crafted examples with diagrams invite the students to develop into precise and critical thinkers. Particular attention has been given to the material that some students find challenging, such as proofs. (...) This book illustrates how to spot invalid arguments, to enumerate possibilities, and to construct probabilities. It also presents case studies to students about the possible detrimental effects of ignoring these basic principles. The book is invaluable for a discrete and finite mathematics course at the freshman undergraduate level or for self-study since there are full solutions to the exercises in an appendix. (shrink)

As the work of thinkers such as Michel Foucault, François Jacob, Louis Althusser, and Pierre Bourdieu demonstrates, Georges Canguilhem has exerted tremendous influence on the philosophy of science and French philosophy more generally. In Knowledge of Life, a book that spans twenty years of his essays and lectures, Canguilhem offers a series of epistemological histories that seek to establish and clarify the stakes, ambiguities, and emergence of philosophical and biological concepts that defined the rise of modern biology. How do (...) transformations in biology and modern medicine shape conceptions of life? How do philosophical concepts feed into biological ideas and experimental practices, and how are they themselves transformed? How does knowledge "undo the experience of life so as to help man remake what life has made without him, in him or outside of him?" Knowledge of Life is Canguilhem's effort to explain how the movements of knowledge and life come to rest upon each other. Published at the dawn of the genetic revolution and still pertinent today, the book tackles the history of cell theory, the conceptual moves toward and away from mechanical understandings of the organism, the persistence of vitalism, and the nature of normality in science and its objects. (shrink)

C'est dans l'esprit des travaux d'Octave Hamelin que se situent ces Etudes de philosophie grecque: l'activite propre de l'historien de la philosophie s'y manifeste comme une reconstruction des pensees du dedans, a partir d'une minutieuse preparation historique. Ces essais donnent une idee assez precise de l'ampleur des travaux de Georges Rodier dans le domaine de la philosophie grecque: le volume s'ouvre en effet sur une etude approfondie de la figure emblematique qu'est Socrate, envisage ensuite de maniere chronologique certains aspects (...) des pensees d'Antisthene, de Platon, d'Aristote et du systeme stoicien, avant de s'achever sur une etude de la pensee de Plotin, notamment dans la perspective singuliere de son influence dans la genese du systeme leibnizien. C'est ainsi que des questions aussi determinantes que celles de la nature et de l'evolution de la dialectique platonicienne, des preuves de l'immortalite de l'ame dans le Phedon, de la composition de la Physique d'Aristote, de la notion de substance dans la Metaphysique se trouvent ici abordees et jointes a des exposes argumentes de la morale aristotelicienne ou stoicienne. (shrink)

His is a journey marked by the questioning of experience itself, until what is reached is sovereign laughter, non-knowledge, and a Presence in no way distinct from Absence, where "The mind moves in a strange world where anguish and ecstasy ...

The author of this major study compares the significantly different approaches to business ethics on both sides of the Atlantic and considers what they have to learn from each other. He has considerable experience of business ethics in both Europe and North America, having taught and researched the subject at the University of St Gallen in his native Switzerland before his appointment as Professor of International Business Ethics in the College of Business Administration, University of Notre Dame, Indiana 46556, USA. (...) Professor Enderle was also the founding Honorary Treasurer of the European Business Ethics Network and is an Associate Editor of this Review. (shrink)

Kant's masterpiece, Critique of Pure Reason, is universally recognized to be among the most difficult of all philosophical writing, and yet it is required reading in almost every course that covers modern philosophy. Most students find Critique of Pure Reason impenetrable without the help of secondary sources. While there are numerous advanced scholarly works on the topic, Dicker's is the first treatment explicitly designed for undergraduates to read alongside the primary text, rendering Kant's views accessible without oversimplifying them. His book (...) will be useful to both undergraduate and graduate students tackling this notoriously difficult yet highly influential thinker in courses in modern philosophy, epistemology, and Kant. (shrink)

"La sociología jurídica —disciplina de origen más reciente que la sociología de la que es una rama esencial— se halla aún en plena vía de formación. A pesar del interés siempre creciente que ha atraído durante los últimos decenios, a pesar de su indudable actualidad, cuyas razones trataremos de precisar, la sociología jurídica no posee aún límites claramente fijos; sus diferentes representativos no están de acuerdo sobre su objeto mismo ni sobre los problemas que se trata de resolver, no sobre (...) sus relaciones con las otras disciplinas que estudian el derecho. ¿A qué se debe este retardo en el desarrollo de la sociología jurídica? A la circunstancia de que esta nueva disciplina ha tenido que combatir, en dos frentes contrarios, por su existencia mis-ma. Ha encontrado adversarios terribles tanto en el campo de los juristas como en el de los sociológicos, los que, a veces, a pesar de proceder de dos horizontes opuestos, se han unido para negar la oportunidad de la sociología del Derecho". Georges Gurvitch. (shrink)

The essay aims to show how business ethics—understood as a three-level approach—can strengthen the social cohesion of a society, which is jeopardized today in many ways. In the first part, the purpose of business and the economy is explained as the creation of wealth defined as a combination of private and public wealth that includes natural, economic, human, and social capital. Special emphasis is placed on the implications of the creation of public wealth which requires institutions other than the market (...) and motivations other than self-regarding ones. In the second part, the question of what holds a society together is discussed through different approaches: enlightened self-interest, a new game-theoretical approach, and the concept of the common good advanced by Catholic Social Teaching, followed by my own proposal. The third part presents several perspectives for business ethics to strengthen social cohesion of a society by focusing on the purpose of business and the economy to create natural, economic, human, and social capital; by advancing public goods that stand the test of ethical scrutiny; and by securing human rights conceptualized as public goods. (shrink)

Source et limites de l'unité dans la mathématique, par Georges Bouligand. Bibliographie (p. 16)--La crise de l'unité dans la physique contemporaine, par Jean Abelé.--A la recherche de l'unité de la matière; point de vue du chimiste, par Jean Moretti.--Le problème de l'unité de la vie. Polygénisme ou monogénisme; le problème de l'unité de l'espèc humaine, par J. Carles.

This paper offers a new concept of the firm that aims at balancing the corporate economic, social, and environmental responsibilities and goes beyond the stakeholder approach. It intends to provide a conceptual and operationalizable basis to fairly assess corporate conduct from both inside and outside the companies. To a large extent these different responsibilities may overlap and reinforce each other. However, if they conflict, they should be clearly evaluated for their own sake and in terms of wealth creation. Only then (...) can a balanced approach be realized. Section 1 briefly discusses some general aspects of the relationship between concepts and measurement. In Section 2, a concept of the firm is developed that is based on the notion of responsibility and balances economic, social and environmental responsibilities. According to these concepts, different ways of measuring corporate planning and performance are examined in Section 3, followed up by a summary and conclusions. (shrink)