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)

Here the author of How to Solve It explains how to become a "good guesser." Marked by G. Polya's simple, energetic prose and use of clever examples from a wide range of human activities, this two-volume work explores techniques of guessing, inductive reasoning, and reasoning by analogy, and the role they play in the most rigorous of deductive disciplines.

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.

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)

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 (...) 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)

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)

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.

George Molnar came to see that the solution to a number of the problems of contemporary philosophy lay in the development of an alternative to Hume's metaphysics. This alternative would have real causal powers at its centre. Molnar set about developing a thorough account of powers that might persuade those who remained, perhaps unknowingly, in the grip of Humean assumptions. He succeeded in producing something both highly focused and at the same time wide-ranging. He showed both that the notion (...) of a power was central and that it could serve to dispel a number of long-standing philosophical problems.Molnar's account of powers is as realist as any that has so far appeared. He shows that dispositions are as real as any other properties. Specifically, they do not depend for their existence on their manifestations. Nevertheless, they are directed towards such manifestations. Molnar thus appropriates the notion of intentionality, from Brentano, and argues that it is the essential characteristic of powers. He offers a persuasive case for there being some basic and ungrounded powers, thus ruling out the reducibility of the dispositional to the non-dispositional. However, he does allow that there are non-power properties as well as power properties. In this respect, his final position is dualistic.This is contemporary metaphysics of the highest quality. It is a work that was almost complete when its author died. It has been edited for publication by another specialist in the subject, Stephen Mumford, who has also provided an introduction that will allow non-specialists to become acquainted with the issues. David Armstrong, one of the greatest living metaphysicians and personal friend of George Molnar, has provided a Foreword. (shrink)

George Herbert Mead (1863-1931) had a powerful influence on the development of American pragmatism in the twentieth century. He also had a strong impact on the social sciences. This classic book represents Mead's philosophy of experience, so central to his outlook. The present as unique experience is the focus of this deep analysis of the basic structure of temporality and consciousness. Mead emphasizes the novel character of both the present and the past. Though science is predicated on the assumption (...) that the present is predictable based on a thorough knowledge of the past, the experience of the present, says Mead, is an utterly unique moment comparable to no other, and when it is past the novel character of that unique experience is irrevocable. The emergence of novelty within the perceived rational order of reality is the crux of the problem that Mead explores. The present, in his words, is "the emergent event . . . something which is more than the processes that have led up to it and which by its change, continuance, or disappearance, adds to later passages a content they would not otherwise have possessed." The present as "the seat of reality" heavily conditions our retrospective view of the past as much as it helps to shape the future. The novelty of every present experience causes us to reconstruct our preceding experiences to make sense of the past, which is naturally assumed to be the main cause of what we presently experience. Our perspective on reality is thus relative to the conditioning of each new event and it changes continuously as the effects of the present shift our view of the past and future. This emphasis on the integrative, holistic nature of reality, in which everything past, present, and future is a condition of everything else, makes Mead's philosophy highly relevant to today's scientific picture of a quantum universe, where chance and probability play a role in the emergence of reality. Also of great interest is the way in which he extends his basic analysis of temporal-spatial reality to the emergence of mind and consciousness as a natural development of the evolutionary process. This stimulating and provocative work attests to John Dewey's praise of Mead as "the most original mind in philosophy in America" of his generation. (shrink)

In his response to my essay “Out of Control,” Neil Levy contests my claims that (1) we are often responsible for acts that we do not consciously choose to perform, and that (2) despite the absence of conscious choice, there remains a relevant sense in which these actions are within our control. In this reply to Levy, I concede that claim (2) is linguistically awkward but defend the thought that it expresses, and I clarify my defense of claim (1) by (...) distinguishing my position from attributionism. (shrink)

George Boolos was one of the most prominent and influential logician-philosophers of recent times. This collection, nearly all chosen by Boolos himself shortly before his death, includes thirty papers on set theory, second-order logic, and plural quantifiers; on Frege, Dedekind, Cantor, and Russell; and on miscellaneous topics in logic and proof theory, including three papers on various aspects of the Gödel theorems. Boolos is universally recognized as the leader in the renewed interest in studies of Frege's work on logic (...) and the philosophy of mathematics. John Burgess has provided introductions to each of the three parts of the volume, and also an afterword on Boolos's technical work in provability logic, which is beyond the scope of this volume. (shrink)

Leading thinkers from both traditions explore common philosophical topics.

Kateb asserts that the defense of universal human rights requires two indispensable components: morality and human dignity. For Kateb, morality and justice have sound theoretical underpinnings; human dignity, by virtue of its “existential” quality, lacks its own theoretical framework. This he proceeds to establish with a critique of the writings of canonical Western political philosophers and contemporary thinkers like Peter Singer and Thomas Nagel. The author argues that while morality compels just governments to prevent, reduce, or eliminate human suffering inasmuch (...) as it is possible, people possess and are entitled to dignity by mere virtue of their “status” as human beings. Homo sapiens, he maintains, have a “stature,” manifest in the species's “great achievements,” that exceeds that of other creatures, even in the secular cosmos. (shrink)

Fraudulent financial reporting, financial statements with errors so material as to require restatement, and biased reporting marred by defects such as managed earnings have plagued financial reporting in many countries in recent years. All of those failures are ethics failures that represent breaches of fiduciary duties by individuals who accepted responsibilities but did not fulfill them. The financial reporting system practiced in America is viewed by the parties involved in it as generally satisfactory. However, according to another view, the interests (...) of those primarily and secondarily responsible for those reports conflict with the interests of the intended beneficiaries, or users, of the corporate financial statements. A more realistic view of the actual operation of that reporting system shows that it is fundamentally flawed. Primary responsibility for failures rests with top management and financial management of the reporting corporation who are so strongly motivated to render favorable reports on their stewardship that they neglect their fiduciary responsibilities to investors. Secondary responsibility falls on independent auditors who are so heavily influenced by enterprise management that they, too, fail in carrying out their responsibilities to users of the audited financial statements. Ethics compromises are also found in the performances of academic accountants and members of accounting standards-setting bodies. The conflict between managements interest in reporting its performance in a favorable light and investors interest in decision-useful financial information will always exist and require regulation. However, changes in those regulations and in the basic governance arrangements involving shareholders, management, and auditors could reduce the opportunities and temptations for failures in carrying out fiduciary responsibilities. Most importantly, the close relationships between auditors and management must be loosened in favor of closer relationships between auditors and investors. (shrink)

This chapter reiterates the point that thinking about God as a counter-movement to technology is not the same as rejecting technology, but as contributing to the attempt to live humanly with technology. The idea of God explored in the book is of God as free and giving freedom, as the one who is pure possibility and yet a possible object of active remembrance within the movement of historical time. However, the question remains open whether this God will be identical to (...) the God of Christian faith. (shrink)

This chapter explores the philosophy of Heidegger, for whom the question of technology was central, and whose views typify a wide range of critical views. Heidegger sees technology as the ultimate outworking of Greek metaphysics, with Nietzsche as its ultimate ideologue. In technology, the world is subject to enframing by the goals of the technological project as the condition of its experienceability. This approach permeates the contemporary university, including the humanities. The poetry of Hölderlin, however, provides Heidegger with another perspective, (...) one in which, in dialogue with the Pre-Socratics, we are invited to thinking as a radical counter-movement to technology. (shrink)

Language must be led by what is extra-linguistic. This chapter explores ideas of vision that might inform thinking about God in language. Drawing from Bakhtin and Tillich, from an icon of Andrei Rublev, and from the film After Life, the idea of a reverse vision is developed; vision that flows back upon itself, as offering one idea of vision that might aid non-technological thinking. In light of P. Florensky’s reflections on truth, this is connected with the notion of liturgy as (...) a site of active remembrance and hope. (shrink)

This chapter surveys the history of modern radical theology from John Robinson through the theology of the Death of God, to deconstruction and radical orthodoxy. It argues that even when positioning itself as answering to contemporary concerns, theology has typically overlooked the technological nature of contemporary society. This undermines any claims theology might have to leadership in the contemporary thought.

This chapter explores the ideas of theologians who have attended to the question of theology. Classically, theology sought the subordination of technology to spiritual values. Despite some techno-optimists such as Teilhard de Chardin, the pessimism of Jacques Ellul has had most influence in recent periods, particularly on green, liberation, and feminist theologies. A narrativist approach to the question is examined but found wanting.

Since early modern times, art has paralleled religion in its response to technology as illustrated by Ruskin’s thoughts on the colour purple. Heidegger also turned to art, especially the poetry of Hölderlin, as an alternative to technology. Against the background of Benjamin’s essay on ‘The Work of Art in the Age of Technical Reproducibility’, the question is asked whether the thoroughly technicized art of film can become a focus for such creative counter-technological thinking. A positive answer is developed with reference (...) to Andrei Tarkovsky’s film Nostalgia. (shrink)

This chapter sketches possibilities for thinking about God as a counter-movement to technology, that nevertheless accepts the reality of the technological society. One still remains free to think about God, even if it is uncertain that one’s thinking answers to anything ‘real’. Such thinking is allowed by the power to think beyond any provisional world-picture that fails to do justice to the wholeness of one’s experience of the world. In mysticism, there is a perennial emphasis on allowing ourselves to be (...) led by the mind’s longing for God. Yet, rather than self-assertion, thinking about God would have to wait upon the self-revealing gift of its object. (shrink)

This expanded edition of John Stuart Mill's _Utilitarianism_ includes the text of his 1868 speech to the British House of Commons defending the use of capital punishment in cases of aggravated murder. The speech is significant both because its topic remains timely and because its arguments illustrate the applicability of the principle of utility to questions of large-scale social policy.

This chapter asks what blaming someone adds to believing that he has acted badly. It examines three of the most popular accounts of the additional element: roughly, those which construe it as a public expression of one’s disapproval, as a belief that the agent’s misdeeds have marred his moral record, and as a negative emotional reaction. Of these familiar accounts, each is shown to be inadequate.

This chapter examines the Humean thesis that agents can only be blamed for their bad acts insofar as those acts are manifestations of defects in their characters. Several versions of this thesis are distinguished and criticized. The criticisms include both the familiar charge that the Humean can’t explain how someone can deserve blame for an act whose badness is “out of character” and the less familiar charge that on the Humean account, the badness of the act itself drops out as (...) irrelevant. It is argued, however, that although Hume was wrong to say that every blameworthy act reflects a flaw in the agent’s character, it may be right to say that every blameworthy act is rooted in the agent’s character. (shrink)

In this essay I argue that Malone-France’s anti-realistic interpretation of the Hartshorne-Peirce theory of possibles can be challenged in a number of ways. While his interpretation does suggest that there are in fact two distinct accounts of possibility in Hartshorne’s philosophy, one that is vulnerable to an antirealistic interpretation and one that is not, Hartshorne does have a consistent and defensible doctrine of possibles. I argue that Whitehead’s contrasting “nonprotean” theory of possibles or “eternal objects” has its own set of (...) conundrums to face, including problems with the coherence of the notion of the completeness of eternal objects and problems with infinite regresses. Whether Whitehead’s or Hartshorne’s account of possibles is correct, I concur with Malone-France that the Molinist doctrine of divine knowledge of future contingents is flawed. (shrink)

This paper is a brief reply to Henry Simoni's ‘Divine passibility and the problem of radical particularity: does God feel your pain?’ in Religious Studies, 33 (1997). I treat his discussion of my paper entitled ‘Hartshorne and Creel on impassibility’, Process Studies, 21 (1992). I argue that Simoni's examples used to illustrate the purportedly contradictory nature of the experiences of a God who universally feels creaturely states fail. For Simoni tacitly employs an inadequate notion of the law of non-contradiction, and (...) thereby misses the relevant phenomenological fact that it is possible for human beings to have integrated mental states that contain spatially distinctive but conflicting hedonic properties. Thus, it is possible for God (at least under Hartshornean descriptions) to have such experiences. I also argue that I have not ‘exploited an isolated passage’ in Hartshorne to make his views seem more palatable. The point of the passage in question is in fact repeated by Hartshorne and is systematically connected with his doctrine of the ‘objective and subjective form of feeling’. (shrink)