Mathieu Marion

Why Play Logical Games

Game semantics has almost achieved the status of a paradigm in computer science but philosophers are slow to take notice. One reason for this might be the lack of a convincing philosophical account of logical games, what it means to play them, for the proponent to win, etc., pointedly raised by Wilfrid Hodges as the ‘Dawkins question’. In this paper, I critically examine two available answers: after a brief discussion of an argument by Tennant against Hintikka games, I focus on Lorenzens attempt at providing a direct foundation for his game rules in the life-world, showing some of the difficulties inherent to that project. I then propose an alternative based on the theory of assertions developed by Dummett and Brandom.

Radical Anti-Realism and Substructural Logics

We first provide the outline of an argument in favour of a radical form of anti-realism premised on the need to comply with two principles, implicitness and immanence, when trying to frame assertability-conditions. It follows from the first principle that one ought to avoid explicit bounding of the length of computations, as is the case for some strict finitists, and look for structural weakening instead. In order to comply with the principle of immanence, one ought to take into account the difference between being able to recognize a proof when presented with one and being able to produce one and thus avoid the idealization of our cognitive capacities that arise within Hilbert-style calculi. We then explore the possibility of weakening structural rules in order to comply with radical anti-realist strictures.

Sraffa and Wittgenstein: Physicalism and constructivism

Abstract After a brief review of facts and hypotheses concerning Piero Sraffas intellectual exchanges with the philosopher Ludwig Wittgenstein and their content, a brief presentation of some of the basic ideas of Productions of Commodities by Means of Commodities is given, on the basis of which I argue, first, that Sraffas ‘objectivism’ in economics is closely related to the ‘physicalism’ towards which Wittgenstein moved soon after his return to Cambridge and, secondly, that the mathematics of that book are in line with Wittgensteins constructivist stance, as it is already found in his Tractatus Logico-Philosophicus.

Radical anti-realism, Wittgenstein and the length of proofs

After sketching an argument for radical anti-realism that does not appeal to human limitations but polynomial-time computability in its definition of feasibility, I revisit an argument by Wittgenstein on the surveyability of proofs, and then examine the consequences of its application to the notion of canonical proof in contemporary proof-theoretical-semantics.

Wittgenstein and finitism

In this paper, elementary but hitherto overlooked connections are established between Wittgensteins remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgensteins rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolems primitive recursive arithmetic and Goodsteins equational calculus, Wittgenstein represented generality by the use of free variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgensteins criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgensteins remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum.

Game Semantics and the Manifestation Thesis

This paper begins with a discussion of the idea of a successor programme to Dummett’s original anti-realist challenge, which should take into account recent developments, e.g., accommodating logical pluralism; I suggest furthermore that one explores a possible contribution from game semantics in the tradition of Lorenzen’s dialogical logic. The purpose of the paper is to examine what could possibly happen to Dummett’s Manifestation Argument within this new programme. After an analysis of the argument, I extract from it a Manifestation Thesis and argue that it fits game semantics, once some basic ideas concerning sense and force in the theory of meaning are reformulated and the second person standpoint is fully taken into account.

Reasoning About Knowledge In Linear Logic: Modalities and Complexity

In this paper, we briefly argue, following ideas set forth by Jacques Dubucs, for a radical version of anti-realism and claim that it leads to the adoption of a ‘substructural’ logic, linear logic. We further argue that, in order to avoids problems such as that of ‘omniscience’, one should develop an epistemic linear logic, which would be weak enough so that the agents could still be described as omniscient, while this would not be problematic anymore. We then examine two possible ways to develop an epistemic linear logic, and eliminate one. We conclude on some remarks about complexity. The paper contains a coding in Coq of fragments of modal linear logic and a proof of the ‘wise men’ puzzle.

Wittgenstein and Brouwer

In this paper, I present a summary of the philosophical relationship betweenWittgenstein and Brouwer, taking as my point of departure Brouwers lecture onMarch 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not doneserious philosophical work for years, if one is to understand the impact of thatlecture on him, it is better to compare its content with the remarks on logics andmathematics in the Tractactus. I thus show that Wittgensteins position, in theTractactus, was already quite close to Brouwers and that the points of divergence are the basis to Wittgensteins later criticisms of intuitionism. Among the topics of comparison are the role of intuition in mathematics, rule following, choice sequences, the Law of Excluded Middle, and the primacy of arithmetic over logic.

Aristotle on Universal Quantification: A Study from the Point of View of Game Semantics

In this paper we provide an interpretation of Aristotles rule for the universal quantifier in Topics Θ 157a34–37 and 160b1–6 in terms of Paul Lorenzens dialogical logic. This is meant as a contribution to the rehabilitation of the role of dialectic within the Organon. After a review of earlier views of Aristotle on quantification, we argue that this rule is related to the dictum de omni in Prior Analytics A 24b28–29. This would be an indication of the dictum’s origin in the context of dialectical games. One consequence of our approach is a novel explanation of the doctrine of the existential import of the quantifiers in dialectical terms. After a brief survey of Lorenzens dialogical logic, we offer a set of rules for dialectical games based on previous work by Castelnérac and Marion, to which we add here the rule for the universal quantifier, as interpreted in terms of its counterpart in dialogical logic. We then give textual evidence of the use of that rule in Platos dialogues, thus showing that Aristotle only made explicit a rule already implicit in practice, while providing a new interpretation of ‘epagogic’ arguments. Finally, we show how a proper understanding of that rule involves further rules concerning counterexamples and delaying tactics, stressing again the parallels with dialogical logic.

Kronecker’s ‘Safe Haven of Real Mathematics’

The mathematical legacy of Kronecker is impressive. The list of mathematicians who took up his problems includes Adolph Hurwitz, David Hilbert, Kurt Hensel, Julius Konig, Ernst Steinitz, Erich Hecke, Helmut Hasse, Carl Ludwig Siegel, Hermann Weyl and, recently, Andre Weil, Robert Langlands, Harold Edwards. But, although nobody has ever doubted the brilliance of Kronecker’s results and insights, it is relatively safe to say with Edwards that, more than a century after the publication of his complete works, “there are important passages in Kronecker’s work that no one, ever has fully understood, other than Kronecker himself” (1987, p. 29). This situation is the result of a lack of interest caused by disdain for Kronecker’s well-known foundational stance and for his style of mathematics. Indeed, Kronecker’s mathematical practice features an insistence on providing algorithms which is alien to the abstract, axiomatic approach which has been so dominant in the past hundred years, in particular in Bourbaki’s treatises. From this point of view, algebra consists in the study of (highly) abstract notions — the structures-meres — such as groups, rings, field, and so forth. Dedekind presented and refined his theory of ideals in successive versions of his Supplement XI to Lejeune-Dirichlet’s Vorlesungen uber Zahlentheorie (1879). Adopted by Hilbert in his Zahlbericht and studied carefully by emmy Noether and her students, Dedekind’s ideals played a crucial role in the development of modern algebraic number theory.