This article discusses the set of Harsanyi payoff vectors of a cooperative TU-game, also known as the Selectope. We reconsider some results on Harsanyi payoff vectors within a more general framework. First, an intuitive approach is used, showing that the set of Harsanyi payoff vectors is the core of an associated convex game. Next, the set of individual rational Harsanyi payoff vectors, the Harsanyi imputations in short, is considered. Existence conditions are provided, and if non-empty, we provide a description as (...) the core of a well-defined convex game, and show that it is an externally stable set. (shrink)

The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for ...

Gathered together here are the fundamental texts of the great classical period in modern logic. A complete translation of Gottlob Frege's Begriffsschrift--which opened a great epoch in the history of logic by fully presenting propositional calculus and quantification theory--begins the volume, which concludes with papers by Herbrand and by Gödel.

The complex story of mathematics in the Tractatus In this paper some thoughts are presented about the treatment of mathematics in the Tractatus Logico-Philosophicus of Ludwig Wittgenstein. After introducing a metaphor for the mathematical ‘building’, we look at the scattered ideas about mathematics in the Tractatus itself. Although the general consensus is that Wittgenstein rejects the entire ‘building’, there are recent insights that suggest that a more coherent view of ‘Tractarian’ mathematics can be presented, if we are willing to leave (...) behind a foundational form of thinking. What this means will be outlined in some detail. The concluding general assessment is that the final word on the status of mathematics in the Tractatus is still pending. (shrink)

It has been observed many times before that, as yet, there are no encompassing, integrated theories of mathematical practice available.To witness, as we currently do, a variety of schools in this field elaborating their philosophical frameworks, and trying to sort out their differences in the course of doing so, is also to be constantly reminded of the fact that a lot of epistemic aspects, extremely relevant to this task, remain dramatically underexamined. This volume wants to contribute to the stock of (...) studies filling this perceived lacuna. It contains papers by established, upcoming, as well as beginning scholars, covering general, metaphilosophical themes such as naturalism, semiotics, pragmaticism, or empiricism, next to more specific topics including the unity of mathematical theories, thruth-flow in mathematics, diagrammatic reasoning, erroneous argumentation, or numerical analysis. (shrink)

Kurt Gödel’s incompleteness theorems and the limits of knowledge In this paper a presentation is given of Kurt Gödel’s pathbreaking results on the incompleteness of formal arithmetic. Some biographical details are provided but the main focus is on the analysis of the theorems themselves. An intermediate level between informal and formal has been sought that allows the reader to get a sufficient taste of the technicalities involved and not lose sight of the philosophical importance of the results. Connections are established (...) with the work of Alan Turing and Hao Wang to show the present-day relevance of Gödel’s research and how it relates to the limitations of human knowledge, mathematical knowledge in particular. (shrink)

On savait que c’était l’entité la plus complexe de l’univers connu. Mais le feu d’artifice de découvertes récentes dépasse l’entendement et fait exploser tous les schémas. Votre cerveau est (beaucoup) plus fabuleux que vous le croyez. Il est : totalement élastique, même âgé, handicapé, voire amputé de plusieurs lobes, le système nerveux central peut se reconstituer et repartir à l’assaut des connaissances et de l’action sur le monde ; totalement social, un cerveau n’existe jamais seul, mais toujours en résonance avec (...) d’autres. Mieux : nous sommes neuronalement constitués pour entrer en empathie avec autrui et aller à son secours. Ce livre aborde ces questions passionnantes avec cinq spécialistes. (shrink)

We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one’s means/ends ratio. Our (...) story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like. (shrink)

Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presentation inviting the contextualization of pieces of mathematical reasoning within encompassing bodies of (...) explicit and implicit, formal and informal background knowledge. (shrink)

Four experiments examined the strategies that individuals develop in sentential reasoning. They led to the discovery of five different strategies. According to the theory proposed in the paper, each of the strategies depends on component tactics, which all normal adults possess, and which are based on mental models. Reasoners vary their use of tactics in ways that are not deterministic. This variation leads different individuals to assemble different strategies, which include the construction of incremental diagram corresponding to mental models, and (...) the pursuit of the consequences of a single model step by step. Moreover, the difficulty of a problem (i.e. the number of mental models required by the premises) predisposes reasoners towards certain strategies. Likewise, the sentential connectives in the premises also bias reasoners towards certain strategies, e.g., conditional premises tend to elicit reasoning step by step whereas disjunctive premises tend to elicit incremental diagrams. (shrink)

Someone asked ‘What time is it?’ when her watch reads 3:08 is likely to answer ‘It is 3:10.’ We argue that a fundamental factor that explains such rounding is a psychological disposition to give an answer that, while not necessarily strictly truthful or accurate, is an optimally relevant one (in the sense of relevance theory) i.e. an answer from which hearers can derive the consequences they care about with minimal effort. A rounded answer is easier to process and may carry (...) the same consequences as one that is accurate to the minute. Hence rounding is often a way of optimising relevance. Three simple experiments give support and greater precision to the view that relevance is more important than strict truthfulness in verbal communication. (shrink)

Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought (1879), by G. Frege.--Some metamathematical results on completeness and consistency; On formally undecidable propositions of Principia mathematica and related systems I; and On completeness and consistency (1930b, 1931, and 1931a), by K. Gödel.--Bibliography (p. [111]-116).

In a series of highly publicized articles in 1952, Jean-Paul Sartre engaged Albert Camus in a bitter public confrontation over the ideas Camus articulated in his renowned work,. This volume contains English translations of the five texts constituting this famous philosophical quarrel. It also features a biographical and critical introduction plus two essays by contemporary scholars reflecting on the cultural and philosophical significance of this confrontation.

The paper gives an impression of the multi-dimensionality of mathematics as a human activity. This 'phenomenological' exercise is performed within an analytic framework that is both an expansion and a refinement of the one proposed by Kitcher. Such a particular tool enables one to retain an integrated picture while nevertheless welcoming an ample diversity of perspectives on mathematical practices, that is, from different disciplines, with different scopes, and at different levels. Its functioning is clarified by fitting in illustrations based on (...) actual research. (shrink)

Philosophy of mathematics today has transformed into a very complex network of diverse ideas, viewpoints, and theories. Sometimes the emphasis is on the "classical" foundational work (often connected with the use of formal logical methods), sometimes on the sociological dimension of the mathematical research community and the "products" it produces, then again on the education of future mathematicians and the problem of how knowledge is or should be transmitted from one generation to the next. The editors of this book felt (...) the urge, first of all, to bring together the widest variety of authors from these different domains and, secondly, to show that this diversity does not exclude a sufficient number of common elements to be present. In the eyes of the editors, this book will be considered a success if it can convince its readers of the following: that it is warranted to dream of a realistic and full-fledged theory of mathematical practices, in the plural. If such a theory is possible, it would mean that a number of presently existing fierce oppositions between philosophers, sociologists, educators, and other parties involved, are in fact illusory. (shrink)

In this paper several arguments are discussed and evaluated concerning the possibility of discrete space and time.

Researchers currently working on relational reasoning typically argue that mental model theory (MMT) is a better account than the inference rule approach (IRA). They predict and observe that determinate (or one-model) problems are easier than indeterminate (or two-model) problems, whereas according to them, IRA should lead to the opposite prediction. However, the predictions attributed to IRA are based on a mistaken argument. The IRA is generally presented in such a way that inference rules only deal with determinate relations and not (...) with indeterminate ones. However, (a) there is no reason to presuppose that a rule-based procedure could not deal with indeterminate relations, and (b) applying a rule-based procedure to indeterminate relations should result in greater difficulty. Hence, none of the recent articles devoted to relational reasoning currently presents a conclusive case for discarding IRA by using the well-known determinate vs indeterminate problems comparison. (shrink)

A solution of the zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by hermann weyl, The so-Called tile argument. This note shows that, Given a set of reasonable assumptions for a discrete geometry, The weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The pythagorean theorem is shown to hold for arbitrary right triangles.

Suppose you attend a seminar where a mathematician presents a proof to some of his colleagues. Suppose further that what he is proving is an important mathematical statement Now the following happens: as the mathematician proceeds, his audience is amazed at first, then becomes angry and finally ends up disturbing the lecture (some walk out, some laugh, …). If in addition, you see that the proof he is presenting is formally speaking (nearly) correct, would you say you are witnessing an (...) extraordinary event in urgent need of explanation? Surely, your answer would be yes. But do events of this type actually occur? The matter of the fact is, yes, they do. This paper presents the details of such an event, suggests a possible explanation and examines its implications for our understanding of mathematical practice. (shrink)

It is a generally accepted idea that strict finitism is a rather marginal view within the community of philosophers of mathematics. If one therefore wants to defend such a position (as the present author does), then it is useful to search for as many different arguments as possible in support of strict finitism. Sometimes, as will be the case in this paper, the argument consists of, what one might call, a “rearrangement” of known materials. The novelty lies precisely in the (...) rearrangement, hence on the formal-axiomatic level most of the results presented here are not new. In fact, the basic results are inspired by and based on Mycielski (1981). (shrink)

Four experiments examined the strategies that individuals develop in sentential reasoning. They led to the discovery of five different strategies. According to the theory proposed in the paper, each of the strategies depends on component tactics, which all normal adults possess, and which are based on mental models. Reasoners vary their use of tactics in ways that have no deterministic account. This variation leads different individuals to assemble different strategies, which include the construction of incremental diagrams corresponding to mental models, (...) and the pursuit of the consequences of a single model step by step. Moreover, the difficulty of a problem (i.e., the number of mental models required by the premises) predisposes reasoners towards certain strategies. Likewise, the sentential connectives in the premises also bias reasoners towards certain strategies, e.g., conditional premises tend to elicit reasoning step by step whereas disjunctive premises tend to elicit incremental diagrams. (shrink)