This chapter analyzes two short texts in which Peirce sketches out an anti-skeptical argument inspired by Kant’s refutation of idealism. The chapter will first consider why Peirce found Kant’s argument interesting and promising, given that it is often regarded as problematic and unsuccessful. It will then briefly reconstruct Kant’s refutation, highlighting its most problematic passages. Moreover, since Peirce’s own version of the argument relies on Kant’s views regarding the temporal structure of consciousness, the chapter will explain how Peirce (...) tackles this issue in “The Law of Mind.” Finally, the chapter considers Peirce’s own anti-skeptical argument and examines whether and how it can be seen as appropriating Kant’s strategy. (shrink)

“Marx did not reject the idea of a human nature. He was right not to do so.” That is the conclusion of this passionate and polemical new work by Norman Geras. In it, he places the sixth of Marx’s Theses on Feuerbach under rigorous scrutiny. He argues that this ambiguous statement—widely cited as evidence that Marx broke with all conceptions of human nature in 1845—must be read in the context of Marx’s work as a whole. His later writings are informed (...) by an idea of a specifically human nature that fulfills both explanatory and normative functions. The belief that Marx’s historical materialism entailed a denial of the conception of human nature is, Geras writes, “an old fixation, which the Althusserian influence in this matter has fed upon … Because this fixation still exists and is misguided, it is still necessary to challenge it.” One hundred years after Marx’s death, this timely essay—combining the strengths of analytical philosophy and classical Marxism—rediscovers a central part of his heritage. (shrink)

Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refutation systems.

A general theory of refutation systems is given. Some applications (concerning maximality and minimality in lattices of logics) are also discussed.

The paper deals with an argument reported by Razi (d. 1210) that was used to attempt to refute the immateriality of human nature. This argument is based on an epistemic asymmetry between our self-knowledge and our knowledge of immaterial things. After some preliminary remarks, the paper analyzes the structure of the argument in four steps. From a methodological point of view, the argument is similar to a family of epistemological arguments (notably, the Cartesian argument from doubt) and is vulnerable to (...) the same objection that can be raised against that form of reasoning. The last section points out that the argument can be used indirectly to highlight the weakness in some arguments for the claim that there is something immaterial in human beings. (shrink)

In the ‘Refutation of Idealism’ chapter of the first Critique, Kant argues that the conditions required for having certain kinds of mental episodes are sufficient to guarantee that there are ‘objects in space’ outside us. A perennially influential way of reading this compressed argument is as a kind of causal inference: in order for us to make justified judgements about the order of our inner states, those states must be caused by the successive states of objects in space outside (...) us. Here I consider the best recent versions of this reading, and argue that each suffers from apparently fatal flaws. -/- . (shrink)

This paper refutes two important and influential views in one fell stroke. The first is G.E. Moore’s view that assertions of the form ‘Q but I don’t believe that Q’ are inherently “absurd.” The second is Gareth Evans’s view that justification to assert Q entails justification to assert that you believe Q. Both views run aground the possibility of being justified in accepting eliminativism about belief. A corollary is that a principle recently defended by John Williams is also false, namely, (...) that justification to believe Q entails justification to believe that you believe Q. (shrink)

Confucianism defined benevolence with “feelings” and “ love.” “Feelings” in Confucianism can be mainly divided into three categories: feelings in general, love for one’s relatives, and compassion. The seven kinds of feeling in which people respond to things can be summarized as “likes and dislikes.” The mind responds to things through feelings; based on the mind of benevolence and righteousness or feelings of compassion, the expression of feelings can conform to the principle of the mean and reach the integration of (...) self and others, and of self and external things. The “relations between the seven kinds of feelings and the Four Commencements,” however, was not developed into a theoretical idea in Confucianism. After Confucius, the relationship between the universality of natural sympathies and the gradation of love for relatives gradually became an important subject in Confucian ideas of benevolence and love. By “refuting Yang Zhu and Mozi,” Mencius systematically expounded on this issue. Love had two ends: self- love and natural sympathies, between which existed the love for relatives. These two ends were not the two extremes of Yang’s self-interest and Mozi’s universal love. Love for relatives not only implied a gradation, but also contained universality and transcendence that came from self- love. Love for relatives, natural sympathies and self- love had a kind of tension and connectivity between two dynamic ends. The Confucian idea of benevolence and love hence demonstrated differences and interconnectivity. An accurate understanding of such “feelings” and “ love ” is important for us to grasp Confucian thoughts on benevolence and its realization. (shrink)

Confucianism defined benevolence with “feelings” and “love.” “Feelings” in Confucianism can be mainly divided into three categories: feelings in general, love for one’s relatives, and compassion. The seven kinds of feeling in which people respond to things can be summarized as “likes and dislikes.” The mind responds to things through feelings; based on the mind of benevolence and righteousness or feelings of compassion, the expression of feelings can conform to the principle of the mean and reach the integration of self (...) and others, and of self and external things. The “relations between the seven kinds of feelings and the Four Commencements,” however, was not developed into a theoretical idea in Confucianism. After Confucius, the relationship between the universality of natural sympathies and the gradation of love for relatives gradually became an important subject in Confucian ideas of benevolence and love. By “refuting Yang Zhu and Mozi,” Mencius systematically expounded on this issue. Love had two ends: self-love and natural sympathies, between which existed the love for relatives. These two ends were not the two extremes of Yang’s self-interest and Mozi’s universal love. Love for relatives not only implied a gradation, but also contained universality and transcendence that came from self-love. Love for relatives, natural sympathies and self-love had a kind of tension and connectivity between two dynamic ends. The Confucian idea of benevolence and love hence demonstrated differences and interconnectivity. An accurate understanding of such “feelings” and “love” is important for us to grasp Confucian thoughts on benevolence and its realization. (shrink)

Causal refutations of external-world scepticism start from our ability to make justified judgements about the order of our own experiences, and end with the claim that there must be perceptible external objects, some of whose states can be causally correlated with that order. In a recent paper, I made a series of objections to this broadly Kantian anti-sceptical strategy. Georges Dicker has provided substantive replies on behalf of a version of the causal refutation of idealism. Here I offer a (...) few final remarks about issues at the heart of our disagreement. -/- . (shrink)

In this paper I refute an apparently obvious objection to Frankfurt-type counterexamples to the Principle of Alternate Possibilities according to which if in the counterfactual scenario the agent does not act, then the agent could have avoided acting in the actual scenario. And because what happens in the counterfactual scenario cannot count as the relevant agent's actions given the sort of external control that agent is under, then we can ground responsibility on that agent having been able to avoid acting. (...) I illustrate how this objection to Frankfurt's famous counterexample is motivated by Frankfurt's own 'guidance' view of agency. My argument consists in showing that even if we concede that the agent does not act in the counterfactual scenario, that does not show that the agent could have avoided acting in the actual scenario. This depends on the crucial distinction between 'not φ-ing' and 'avoiding φ-ing'. (shrink)

Luca Castagnoli, Ancient Self-Refutation. The Logic and History of the Self- Refutation Argument from Democritus to Augustine, Cambridge: Cambridge University Press 2010, pp. XX+394. Hardback, ISBN 9780521896313. In his book Ancient Self-Refutation L. Castagnoli rightly observes that selfrefutation is not falsification; it overturns the act of assertion but does not prove that the content of the act is false. He argues against the widely spread belief that Sextus Empiricus accepted the self-refutation of his own expressions. Castagnoli (...) also claims that Sextus was effective in answering to the self-refutation charge. The achievement of the book is the discovery that in passages where Sextus seems to embrace the selfrefutation of his expressions (PH 1.14-15), he does not use the term peritropé, technical for self-refutation, but the term perigraphé, which means self-bracketing. Selfbracketing is weakening one’s own thesis, but not overturning it. Castagnoli claims that Sextus embraces the self-bracketing of his expressions, but never accepts their selfrefutation. However, Castagnoli is not right in saying that self-refutation is a shameful mistake for everybody. The mature skeptic cannot even think that self-refutation is wrong, because it would be a dogmatic view. Sextus seems to accept self-refutation at the end of Against the Logicians, where he presents the argument against proof and the metaphor of the ladder (M 8.480-1). Regardless of Sextus’ declarations, we have reason to think that he does not avoid self-refutation in a pragmatic sense. Self-bracketing of his position is not a consistent dialectical strategy, as Castagnoli writes, but the end of a rational discussion. Sextus avoids absolute self-refutation (we cannot falsify what he suggests), but he is unable to avoid pragmatic self-refutation (there is no way to assert his position without contradiction). This is the case even if Sextus refuses to assert his position. (shrink)

Elgin offers an influential and far-reaching challenge to veritism. She takes scientific understanding to be non-factive and maintains that there are epistemically useful falsehoods that figure ineliminably in scientific understanding and whose falsehood is no epistemic defect. Veritism, she argues, cannot account for these facts. This paper argues that while Elgin rightly draws attention to several features of epistemic practices frequently neglected by veritists, veritists have numerous plausible ways of responding to her arguments. In particular, it is not clear that (...) false propositional commitments figure ineliminably in understanding in the manner supposed by Elgin. Moreover, even if scientific understanding were non-factive and false propositional commitments did figure ineliminably in understanding, the veritist can account for this in several ways without thereby abandoning veritism. (shrink)

Socrates’ refutations of Thrasymachus in Republic I are unsatisfactory on a number of levels which need to be carefully distinguished. At the same time several of his arguments are more powerful than they initially appear. Of particular interest are those which turn on the idea of a craft, which represents a shared norm of practical rationality here contested by Socrates and Thrasymachus.

A 'self-refutation argument' is any argument which aims at showing that a certain thesis is self-refuting. This study was the first book-length treatment of ancient self-refutation and provides a unified account of what is distinctive in the ancient approach to the self-refutation argument, on the basis of close philological, logical and historical analysis of a variety of sources. It examines the logic, force and prospects of this original style of argumentation within the context of ancient philosophical debates, (...) dispelling various misconceptions concerning its nature and purpose and elucidating some important differences which exist both within the ancient approach to self-refutation and between that approach, as a whole, and some modern counterparts of it. In providing a comprehensive account of ancient self-refutation, the book advances our understanding of influential and debated texts and arguments from philosophers like Democritus, Plato, Aristotle, Epicurus, the Stoics, the Academic sceptics, the Pyrrhonists and Augustine. (shrink)

RésuméJe me penche sur un aspect de la philosophie sociale de Popper, à savoir les principes d'évaluation des interprétations historiques. Ma thèse globale est que suivant la perspective poppèrienne, notre choix parmi des interprétations historiques doit user d'au moins deux des critères qu'applique Popper au choix parmi diverses théories scientifiques : une interprétation devrait logiquement se prêter à une réfutation et elle devrait être consistante. Afin de montrer la pertinence et la fécondité de cette approche, je me concentre sur l'interprétation (...) historique des causes de la montée du nazisme que propose Erich Fromm, en particulier dansEscape from Freedom.Je montrerai que l'application de notions psychanalytiques dans le cadre de son interprétation historique a une valeur explicative faible, voire nulle. Son échec à expliquer certaines questions sociales tient à plusieurs raisons. Premièrement, l'auteur ne fournit pas, quoi qu'il en dise, d'explication causale convaincante. Ensuite, les concepts et les multiples hypotheèsesad hocauxquels il a recours tendent è rendre son interprétation irréfutable. Enfin, Fromm s'appuie sur des assomptions théoriques contradictoires. Mon analyse est basée sur la notion fondamentale de Popper voulant que la connaissance scientifique empirique exige au strict minimum une ouverture des interprétations historiques à la critique intersubjective. C'est cette exigence qui permet de les critiquer — c'est-à-dire les accepter, les réfuter et les corriger — sur une base rationnelle. (shrink)

_Conjectures and Refutations_ is one of Karl Popper's most wide-ranging and popular works, notable not only for its acute insight into the way scientific knowledge grows, but also for applying those insights to politics and to history. It provides one of the clearest and most accessible statements of the fundamental idea that guided his work: not only our knowledge, but our aims and our standards, grow through an unending process of trial and error.

_Conjectures and Refutations_ is one of Karl Popper's most wide-ranging and popular works, notable not only for its acute insight into the way scientific knowledge grows, but also for applying those insights to politics and to history. It provides one of the clearest and most accessible statements of the fundamental idea that guided his work: not only our knowledge, but our aims and our standards, grow through an unending process of trial and error.

The goal of this paper is to generalize specific techniques connected with refutation rules involving certain normal forms. In particular, a method of axiomatizing both a logic L and its complement −L is introduced.

My purpose is to refute the intentionalist approach to perception. Drawing from mainstream literature, I identify a principle on which any version of intentional theory relies. My paper is a detailed attack on the truth of the principle. In the first section I will introduce terminology and will taxonomize various statements of the intentional view. In the second section I will briefly outline a sketch of the skeletal intentionalist theory that develops from the assumption of the principle alone. Then, in (...) the third section, I will advance my reasons against this theory. In the fourth section, I will set forth anintuitive and definitive counterexample to the adequacy of the principle of intentionalism to accounting for ordinary perception. Moving from this, in the fifth section, I will provide some reasons explaining why intentionalism is condemned at being unsuccessful. Finally, in the last section of the paper, I will give my conclusions. (shrink)

The way in which knowledge progresses, and especially our scientific knowledge, is by unjustified anticipations, by guesses, by tentative solutions to our problems, by conjectures. These conjectures are controlled by criticism: that is, by attempted refutations, which include severely critical tests. They may survive these tests; but they can never be positively justified: they can neither be established as certainly true nor even as 'probable'. Criticism of our conjectures is of decisive importance: by bringing out our mistakes it makes us (...) understand the difficulties of the problems which we try to solve. This is how we become better acquainted with our problem, and able to propose more mature solutions: the very refutation of a theory - that is, of a tentative solution to our problem - is always a step forward that takes us nearer the truth. And this is how we can learn from our mistakes. As we learn from our mistakes our knowledge grows, even though we may never know - that is, know for certain. Since our knowledge can grow, there can be no reason here for despair of reason. And since we can never know for certain, the can be no authority here for any claim to authority, for conceit over our knowledge, or for smugness. The essays and lectures of which this book is composed apply this thesis to many topics, ranging from problems of the philosophy and history of the physical and social sciences to historical and political problems. (shrink)

This study introduces refutation-aware Gentzen-style sequent calculi and Kripke-style semantics for propositional until-free linear-time temporal logic. The sequent calculi and semantics are constructed on the basis of the refutation-aware setting for Nelson’s paraconsistent logic. The cut-elimination and completeness theorems for the proposed sequent calculi and semantics are proven.

This paper claims that Socrates’ refutation of Gorgias in the eponymous dialogue is designed not to find out the truth about the nature of the art of rhetoric itself but to refute the master of rhetoric himself. I try to justify this claim by displaying some major contradictions between the conclusions reached at with Gorgias and those reached at with Polus. When these contradictions are taken into account, the discussion with Polus is to be seen as reflecting the genuine (...) Socratic position about rhetoric, whereas the discussion with Gorgias seems only to be devised as a dexterity in refutation. (shrink)

Cover -- Half Title Page -- Title Page -- Copyright Page -- Original Title Page -- Original Copyright Page -- Contents -- Preface -- 1 Introduction -- 2 Probability And Possibility For Choice -- 1 Introductory -- 2 A Theory About Personal Power -- 3 A Criticism Of Keynes -- 4 Some More Theories About Personal Power -- 5 An Analogy Between Two Kinds Of Possibility -- 3 Probability And Natural Powers -- 1 Introductory -- 2 The Relation Between Epistemic (...) And Natural Possibility -- 3 A Criticism Of The Doctrine That 'Probable' Is Ambiguous -- 4 A Comparison Of Possibility With Probability -- 4 Some Unobservable Properties -- 1 Introductory -- 2 Certainty, Necessity And Hume -- 3 Epistemology And Determinism -- 4 An Analysis Proposed And Defended -- 5 Power And Law -- 5 Some Puzzles About Potentiality -- 1 Introductory -- 2 The idea of a 'circumstance' -- 3 The Distinction Between Intrinsic And Extrinsic Properties -- 4 The Refutation Of Actualism -- 5 The Refutation Of The Theory That 'All Power Is Conditional' -- 6 The Powers Of People Andthe Powers Of Things -- 1 Introductory -- 2 The Distinction Between Natural And Personal Power -- 3 The Refutation Of Ryle's Dispositionalist Account Of Human Capacities -- 7 IFS And Cans -- 1 Introductory -- 2 The Refutation Of The Theory That All Personal Power is Conditional -- 3 A Refutation Of The Orthodox Account Of 'I Can', As Equivalent To A Conditional Statement -- 4 A Contradiction In Dispositionalism -- 5 Further Objections To The Orthodox View -- 6 The Verification Of 'I Can': More About Trying -- 8 Deliberation, Freedom And Meaning -- 1 Introductory -- 2 More About Possibility And The Context Of Deliberation -- And About The Relation Between Power And Will -- 3 On Not Being Able To Help It: A: Discussion Of The Relationship Between Power And Responsibility -- 4 On The Meaning Of Words Like 'Can'. (shrink)

Popper's account of refutation is the linchpin of his famous view that the method of science is the method of conjecture and refutation. This thesis critically examines his account of refutation, and in particular the practice he deprecates as avoiding a refutation. I try to explain how he comes to hold the views that he does about these matters; how he seeks to make them plausible; how he has influenced others to accept his mistakes, and how (...) some of the ideas or responses to Popper of such people are thus similarly mistaken. I draw some distinctions necessary to the provision of an adequate account of the so-called practice of avoiding a refutation, and try to rid the debate about this practice of at least one red herring. I analyse one case of 'avoiding' a refutation in detail to show how the rationality of scientific practice eludes both Popper and many of his commentators. Popper's skepticism about contingent knowledge prevents him from providing an acceptable account of contingent refutation, and so his method is really the method of conjecture and conjecture. He cannot do without the concepts of knowledge and refutation, however, if his account of science is to be plausible or persuasive, and so he equivocates between, amongst other things, refutation as disproof and refutation as the weaker notion of discorroboration. I criticise David Stove's account of this matter, in particular to show how he misses this point. An additional advantage Popper would secure from this equivocation is that if refutations were mere discorroborations they would be easier to achieve, and hence more common in science, than is the case. On Popper's weak notion of refutation, it would be possible to refute true theories since corroboration does not entail truth. There are two other related doctrines Popper holds about refutation which, if accepted, make some refutations seem easier to obtain than is the case. I call these doctrines 'Strong Popperian Falsificationism' (SPF) and 'Weak Popperian Falsificationism' (WPF). SPF is the false doctrine that if a prediction from some theory is refuted then that theory is refuted. Popper does not always endorse SPF. In particular, when confronted with a counterexample to it, he retreats to WPF, which is the false doctrine that if a prediction from some theory is refuted then that theory is prima facie refuted. WPF , or even SPF, can seem plausible if one has in mind predictions derived from theories in strong or conclusive tests of those theories, which I suggest Popper characteristically does. v Popper is disposed to describe any such case of predictive failure which does not lead to the refutation of the theory concerned as one in which that refutation has been avoided. To reinforce his portrayal of the refutation, or the attempted refutation, of major scientific theories as the rational core of scientific practice, Popper treats the so-called practice of avoiding a refutation as untypical of science, and much so-called avoidance he dismisses as unscientific or pseudo-scientific. I argue that his notion of avoiding a refutation is incoherent. Popper is further driven to believe that such avoidance is possible, however, because he conflates sentences with propositions and propositions with propositional beliefs. Also, he wishes to avoid being saddled with the relativisim that is a consequence of his weak account of refutation as discorroboration. Popper believes that ad hoc hypotheses are the most important of the unscientific means of avoiding a refutation. I argue that his account of such hypotheses is also incoherent, and that several hypotheses thought to be ad hoc in his sense are not. Such hypotheses appear to be so largely because of Popper's use of rhetoric and partly because these hypotheses are unacceptable for other reasons. I conclude that to know that a hypothesis is ad hoc in Popper's sense does not illuminate scientific practice. Popper has also attempted to explicate ad hocness in terms of some undesirable, or allegedly undesirable, properties of hypotheses or the explanations they would provide. The first such property is circularity, which is undesirable; the second such property is reduction in empirical content, which is not. In the former case I argue that non-circularity is clearly preferable to non-ad hocness as a criterion for a satisfactory explanation or explanans, as the case may be, and in the latter case that Popper is barking up the wrong tree. Some cases of so-called avoidance are obviously not unscientific. The discovery of Neptune from a prediction based on the reasonable belief that there were residual perturbations in the motion of Uranus is an important case in point, and one that is much discussed in the literature. The manifest failure of astronomers to account for Uranus's motion did not lead to the refutation of Newton's law of gravitiation, yet significant scientific progress obviously did result. Retreating to WPF, Popper claims that Newton's law was prima facie refuted. In general, astronomers have never shared this view, and they are correct in not doing so. I argue that the law of gravitation would have been prima facie refuted only if there had been good reason at the time to believe as false what is true, namely, that an unknown trans-Uranian planet was the cause of those Uranian residuals. Knowledge of the trans-Uranian region was then so slight that it was merely a convenient assumption, one which there was little reason to believe was false, that the known influences on Uranus's motion were the only such influences. I conclude that in believing vi or supposing that it was this assumption that was false, rather than the law of gravitation, Leverrier and Adams, the co-predictors of Neptune, were acting rationally and intelligently. Popper's commentators offer a variety of accounts of the alleged practice of avoiding a refutation, and of this case in particular. I analyse a sample of their accounts to show how common is the acceptance of some of Popper's basic mistakes, even amongst those who claim to reject his falsificationism, and to display the effects on their accounts of this acceptance of his mistakes. Many commentators recognize that anomalies are typically dealt with by changes in the boundary conditions or in other of the auxiliary propositions employed. Where many still go wrong, however, is in retaining the presupposition of WPF which encouraged Popper to hold the contradictory view about anomalies in the first place. Thus Imre Lakatos and others, for example, have developed a 'siege mentality' about major scientific theories; they see them as under continual threat of refutation from anomalies, and so come to believe that dogmatism is essential in science if such theories are to survive as they do. I examine various such doomed attempts to reconcile Popper with the history of science. It is a common failure in this literature to conflate or to fail to see the need to distinguish a belief from a supposition, and an epistemic reason from a pragmatic reason. I argue that only if one does draw these distinctions can one give an adequate account of how anomalies are rationally dealt with in science. The other important strand in Popper's thinking about 'avoidance' of refutation which has seriously misled some of his commentators is his unfounded belief in the dangers of ad hoc hypotheses. I examine the accounts that a sample of such commentators provide of the trans-Uranian planet hypotheses of Leverrier and Adams. These commentators imply or assert what Popper only hints at, namely, that there is something fishy about this hypothesis. I provide a further defence of the rationality of entertaining this hypothesis at the time. I conclude with a few remarks about Popper's dilemma in respect of scientific practice and his long standing emphasis on refutations. (shrink)

Both the pessimistic inductions over scientific theories and over scientists are built upon what I call proportional pessimism: as theories are discarded, the inductive rationale for concluding that the next theories will be discarded grows stronger. I argue that proportional pessimism clashes with the fact that present theories are more successful than past theories, and with the implications of the assumptions that there are finitely and infinitely many unconceived alternatives. Therefore, the two pessimistic inductions collapse along with proportional pessimism.

According to a doctrine that I call “Cartesianism”, knowledge – at least the sort of knowledge that inquirers possess – requires having a reason for belief that is reflectively accessible as such. I show that Cartesianism, in conjunction with some plausible and widely accepted principles, entails the negation of a popular version of Fallibilism. I then defend the resulting Cartesian Infallibilist position against popular objections. My conclusion is that if Cartesianism is true, then Descartes was right about this much: for (...) S to know that p, S must have reasons for believing that p which are such that S can know, by reflection alone, that she has those reasons, and that she could not possibly have those reasons if p is not true. Where Descartes went wrong was in thinking that our ordinary, fallible, non-theologically grounded sources of belief (e.g., perception, memory, testimony), cannot provide us with such reasons. (shrink)

This paper discusses the method when an argument is refuted by a parallel argument since the flaw of the parallel argument is clearly displayed. The method is explicated, examined and compared with two other general methods.

An environmental ethic holds that some entities in nature or in natural states of affairs are intrinsically valuable. I argue that proposals for an environmental ethic either fail to satisfy requirements which any ethical system must satisty to be an ethic or they fail to give us reason to suppose that the values they promote are intrinsic values. If my arguments are correct, then environmental ethics is not properly ethics at all.

The purpose of the paper is to study syntactic refutation systems as a way of characterizing normal modal propositional logics. In particular it is shown that there is a decidable modal logic without the finite model property that has a simple finite refutation system.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...) Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)