Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent (in an informal sense) and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

Between 1906 and 1911, as a response to Betrand’s Russell’s review of La Science et l’Hypothèse, Henri Poincaré launched an attack on the movement to formalise the foundations of mathematics reducing it to logic. The main point is the following: the universality of logic is based on the idea that their truth is independent of any context including epistemic and cultural contexts. From the free context notion of truth and proof it follows that, given an axiomatic system, nothing new can (...) follow. One of the main strategies of Poincaré’s solution to this dilemma is based on the notions of understanding and of grasping the architecture of the propositions of mathematics. According to this view mathematic rigour does not reduce to “derive blindly” without gaps from axioms,mathematical rigour is, according to Poincaré, closely linked to the ability to grasp the architecture of mathematics and contribute to an extension of the meaning embedded in structures that constitute the architecture of mathematical propositions. The focus of my paper relates precisely to the notion of architecture and to the notion of understanding. According to my reconstruction, Poincaré’s suggestions could be seen as pointing out that understanding is linked to reason not only within a structure but reasoning about the structure. (shrink)

In its origins Dialogical logic constituted the logical foundations of an overall new movement called the Erlangen School or Erlangen Constructivism that should provide a new start to a general theory of language and of science. In relation to the theory of language, according to the Erlangen-School, language is not just a fact that we discover, but a human cultural accomplishment whose construction reason can and should control. The constructive development of a scientific language was called the Orthosprache-project. Unfortunately, the (...) Orthosprache-project was not further developed and somehow seemed to fade away. Perhaps, one could say that one of the reasons is that the semantic links between dialogical logic, that was restricted to logical constants, and the more general linguistic aspirations of the Orthosprache were not sufficiently developed and then the new theory of meaning of dialogical logic seemed to be cut off from the project of setting the basis for a general theory of meaning.In the present paper we would like to contribute to precise one possible way in which a general dialogical theory of meaning could be linked to dialogical logic. More precisely, the main aim of the article is to set the basis for the meaning of elementary sentences in the context of dialogical interaction. (shrink)

Two crucial concepts of the methodology and philosophy of mathematics are considered: proof and truth. We distinguish between informal proofs constructed by mathematicians in their research practice and formal proofs as defined in the foundations of mathematics (in metamathematics). Their role, features and interconnections are discussed. They are confronted with the concept of truth in mathematics. Relations between proofs and truth are analysed.

The infinite regress of Carroll’s ‘What the Tortoise said to Achilles’ is interpreted as a problem in the epistemology of mathematical proof. An approach to the problem that is both diagrammatic and non-logical is presented with respect to a specific inference of elementary geometry.

Resumen: Janet Folina ha propuesto una interpretación del convencionalismo de Poincaré contraria a la que ofrecen Michael Friedman y Robert DiSalle. Ambos afirman que la propuesta de Poincaré queda refutada por la relati-vidad general pues supone una noción restrictiva de los principios a priori. Folina sostiene que el convencionalismo de Poincaré no es contradictorio con la relatividad general porque permite una noción relativizada de los princi-pios a priori. Intento mostrar que la estrategia de Folina es ineficaz porque Poincaré no puede (...) explicar el papel de la variedad de Riemann en la rela-tividad general. Concluyo que la tentativa de reconstruir el desarrollo de la relatividad general a partir de la filosofía de Poincaré puede constituir un obs-táculo para comprender la radicalidad del cambio que la relatividad generó en las relaciones entre física y geometría.: Janet Folina has proposed an interpretation of Poincaré’s conven-tionalism against Michael Friedman and Robert DiSalle. They both state that Poincaré’s proposal is refuted by general relativity, since it implies a restrictive notion of a priori principles. Folina claims that Poincaré’s conventionalism is not in conflict with general relativity because from Poincaré’s proposal it is possible to extract a relativized notion of a priori principles. I intend to show that Folina’s strategy is ineffective due to Poincaré’s inability to explain the role of Riemann manifold in general relativity. I conclude that the attempt to reconstruct the development of general relativity from Poincaré’s philosophy may be an obstacle to understanding the radical nature of the change that relativity brought to the relationship between physics and geometry. (shrink)

often insisted existence in mathematics means logical consistency, and formal logic is the sole guarantor of rigor. The paper joins this to his view of intuition and his own mathematics. It looks at predicativity and the infinite, Poincaré's early endorsement of the axiom of choice, and Cantor's set theory versus Zermelo's axioms. Poincaré discussed constructivism sympathetically only once, a few months before his death, and conspicuously avoided committing himself. We end with Poincaré on Couturat, Russell, and Hilbert.

This paper is composed of two independent parts. The first is concerned with Russell’s early philosophy of mathematics and his quarrel with Poincaré about the nature of their opposition. I argue that the main divergence between the two philosophers was about the nature of definitions. In the second part, I briefly present Le!niewski’s Ontology and suggest that Le!niewski’s original treatment of definitions in the foundations of mathematics is the natural solution to the problem that divided Russell and Poincaré.

D’un colloque à un autre, puis à un autre... En mai 1994, les Archives Henri-Poincaré, qui n’avaient alors que deux ans d’existence, organisaient à Nancy un important colloque dédié à l’œuvre scientifique et philosophique du savant [Greffe, Heinzmann et al. 1996]. En janvier 2012, à l’occasion du centenaire du décès d’Henri Poincaré, le laboratoire inaugurait à Nancy une année marquée par d’innombrables manifestations scientifiques et grand public avec un colloque « Vers une biographie d’Henr...

The goal of this paper is to answer the following question: Does it make sense to speak of thought experiments not only in physics, but also in mathematics, to refer to an authentic type of activity? One may hesitate because mathematics as such is the exercise of reasoning par excellence, an activity where experience does not seem to play an important role. After reviewing some results of the research on thought experiments in the natural sciences, we turn our attention to (...) experiments and thought experiments in mathematics, especially in fundamental mathematics, and investigate what thought experiments can teach us there. If we accept the principle that mathematical practice can sometimes be best described by a pragmatist approach where the concept of experience in mathematics is considered from a new perspective, thought experiments can in some cases be a useful instrument different from both ‘mathematical experiments’ and ‘formal’ proofs: they are mathematical experiments using deviant methods. (shrink)

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a (...) necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain. (shrink)

In the past 10 years, contemporary philosophy of mathematics has seen the development of a trend that conceives mathematics as first and foremost a human activity and in particular as a kind of practice. However, only recently the need for a general framework to account for the target of the so-called philosophy of mathematical practice has emerged. The purpose of the present article is to make progress towards the definition of a more precise general framework for the philosophy of mathematical (...) practice by exploring two strategies. A first strategy is to turn to philosophy of mind and Edwin Hutchins' view of distributed cognition in order to better understand the cognitive issues at play when considering a community of mathematicians; a second strategy is to refer to philosophy of language and focus on Robert Brandom's inferentialism and mathematical conceptual content. A possible combination of these two views, called enhanced material inferentialism, is then put forward as a promising framework to account for the philosophy of mathematical practice. (shrink)

In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians who interacted with Luitzen Egbertus Jan Brouwer in order to compare their conceptions of intuition. We will see how to the same word “intuition” very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty of considering concepts (...) that habitually occur in our thinking separately. Furthermore, we will discover that these different meanings had a philosophical, very relevant counterside: they passed from a racial characterization of mathematics to a pluralistic view of it. (shrink)

The main goal of part 1 is to challenge the widely held view that Poincaré orders the sciences in a hierarchy of dependence, such that all others presuppose arithmetic. Commentators have suggested that the intuition that grounds the use of induction in arithmetic also underlies the conception of a continuum, that the consistency of geometrical axioms must be proved through arithmetical induction, and that arithmetical induction licenses the supposition that certain operations form a group. I criticize each of these readings. (...) More fully, I argue that the justification Poincaré offers for the use of the group notion in geometry appears to extend to set-theoretic notions that would suffice to put arithmetic on a logical foundation, thus undermining his own case for the necessity of intuition in arithmetic. In part 2, I offer an interpretation of intuition’s role on which it justifies the use of group-theoretic, but not set-theoretic, notions. (shrink)

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent (...) with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument. (shrink)

This book explores the interplay between logic and science, describing new trends, new issues and potential research developments.

In this work I study the main tenets of the logicist philosophy of mathematics. I deal, basically, with two problems: (1) To what extent can one dispense with intuition in mathematics? (2) What is the appropriate logic for the purposes of logicism? By means of my considerations I try to determine the pros and cons of logicism. My standpoint favors the logicist line of thought. -/- .

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

Inferential communities are communities using specific substantial argumentative schemes. The religious or scientific communities are examples. I discuss the status of the mathematical community as it appears through the position held by the French mathematician Henri Poincaré during his famous ar-guments with Russell, Hilbert, Peano and Cantor. The paper focuses on the status of complete induction and how logic and psychology shape the community of mathematicians and the teaching of mathematics.