Preface
Preface to the first volume, first issue of the Annals of Mathematics and Philosophy.
Marquis, J., & Patras, F. (2023). Preface. M×Φ — Annals of Mathematics and Philosophy, 1(1), 1–4.
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Volume I, number 1, 2023
This issue, by invitation, gives an overview of the major issues open to mathematical philosophy through the testimonies and commitments of personalities representative of the two communities, mathematical and philosophical.
Cite this issue
M×Φ — Annals of Mathematics and Philosophy, 1(1), 2023.
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Introduction
Préface
Preface to the first volume, first issue of the Annals of Mathematics and Philosophy.
Marquis, J., & Patras, F. (2023). Préface. M×Φ — Annals of Mathematics and Philosophy, 1(1), online version, 4 p.
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Articles
What is axiomatics?
The article investigates axiomatics as a complex mathematical practice whose inquiry, while taking its cue from the analysis of some specific mathematical theories, requires an interdisciplinary approach. Axiomatics, if analyzed in detail through a study of its foundational component, of the styles with which it is associated and of the rules that govern it, performs a plurality of functions. It serves heuristic, descriptive, genetic-historical, pedagogical and architectural aims. But it can also play the role of conceptual analysis, modular analysis and coordination tool, soliciting a quest for rigor. An example taken from Peano's investigation of axiomatic systems illustrates the kind of results that this interdisciplinary approach to mathematical practice might produce, showing what can be achieved by considering axiomatics as research on the foundations of mathematics, or as a mathematical style, or as a social institution.
Cantù, P. (2023). What is axiomatics?. M×Φ — Annals of Mathematics and Philosophy, 1(1), 1–24.
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Introducing Heuristic Philosophy of Mathematics
Mainstream philosophy of mathematics, namely the philosophy of mathematics that has prevailed for the past century, holds that mathematics is theorem proving by the axiomatic method. But this is incompatible with Gödel's incompleteness theorems, and cannot account for many features of mathematics. This article proposes an alternative approach, heuristic philosophy of mathematics, according to which mathematics is problem-solving by the analytic method. The article argues that this is compatible with Gödel's incompleteness theorems, and can account for the features of mathematics not accounted for by mainstream philosophy of mathematics, such as the nature of mathematical objects and mathematical definitions.
Cellucci, C. (2023). Introducing Heuristic Philosophy of Mathematics. M×Φ — Annals of Mathematics and Philosophy, 1(1), 25–46.
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Some Personal Remarks
Mathematics is neither the contemplation of an independently given realm nor a mere creation of the mind. It arises from passive and active experience, yet grows through the exploration of hypothetical states of affairs: structures partly constituted by assumptions, representations, symbols, and methods, whose consequences can nevertheless be discovered. This standpoint calls into question sharp separations between pure and applied mathematics, since mathematical concepts and techniques have developed in close interaction with natural knowledge, scientific modelling, and human practices. It also shifts the philosophical focus from the ontology of mathematical objects to the objectivity of mathematical discourse, understood as a historically situated but intersubjectively valid form of conceptual work. Particular attention is given to the role of semiotic systems, notation, diagrams, formulae, and practices in the growth of mathematical knowledge, and to the irreducible complementarity between formal manipulation and conceptual meaning. Foundational questions concerning logic, set theory, continuity, and the continuum problem are approached from the same perspective: not as demands for a single ultimate framework, but as reasons for adopting a pluralistic view of mathematical foundations and of the role of mathematics within science.
Ferreiros, J. (2023). Some Personal Remarks. M×Φ — Annals of Mathematics and Philosophy, 1(1), 47–55.
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What makes mathematicians believe unproved mathematical statements?
This paper considers the reasons mathematicians give for making probabilistic judgments about unproved mathematical statements, and discusses how one might interpret and justify such judgments more formally. Following Pólya, I argue that we update our probabilistic judgments in a broadly Bayesian way, while to explain what they mean in the first place, I argue that they are referring not so much to the truth of the statements as to the likely existence or otherwise of reasons for them. The link between the two is provided by a "no-miracle" principle, which says that a surprising mathematical statement will not be true unless it is true for a reason. This principle applies only to statements that are sufficiently natural, so the paper also sets out criteria for a statement to be more or less natural.
Gowers, T. (2023). What makes mathematicians believe unproved mathematical statements?. M×Φ — Annals of Mathematics and Philosophy, 1(1), 57–110.
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Ways of thinking in mathematics
All mathematicians, whether they are engaged in teaching, in researching or in both at the same time, must master specific ways of thinking, of reasoning. These are ways of thinking that can be divided into three categories: reasoning and problem-solving; organizing already acquired knowledge; and characterizing, constructing or defining new elements and concepts. These are the modes that are presented in this essay, with a very brief indication of the historical moment in which they are constructed. The obligatory mastery of these modes of thinking, by itself, does not lead to mathematical creation; the imaginative and creative power of the mathematician is involved in this. Finally, we point out the appearance of an ontological problematic from the inversion that took place in the 19th century: are mathematical entities discovered or constructed? It is a problematic that has become a theme throughout the twentieth century about mathematical doing.
Lorenzo, J. (2023). Ways of thinking in mathematics. M×Φ — Annals of Mathematics and Philosophy, 1(1), 111–126.
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Logic, Mathematics, and Philosophy
Logic occupies a special position at the intersection of mathematics and philosophy because mathematics has repeatedly provided both the paradigm and the occasion for transformations in logic. The historical movement considered here runs from Greek diagrammatic practice and Aristotelian term logic, through Descartes’ symbolic algebra and Kant’s quantificational logic, to nineteenth-century deduction from defined concepts and Frege’s Begriffsschrift. The guiding problem is how mathematical knowledge can be at once strictly deductive and genuinely ampliative. Frege’s distinction between Sinn and Bedeutung, and between objectivity and relation to objects, makes possible a renewed answer: mathematical reasoning need not concern mathematical objects, but can reveal necessary relations among mathematical concepts. A mathematical language can display the contents of such concepts in signs that function both as a lingua and as a calculus ratiocinator. Some proofs extend knowledge because definitions do not merely supply premises; in fruitful cases they license materially valid, necessary inferences not licensed by logic alone. Defined concepts, theorems, and chains of reasoning can thereby be understood as intelligible unities: real wholes of real parts. Frege’s logic thus offers not only a new account of mathematical knowledge, but also a possible transformation of mathematical practice, teaching, and discovery.
Macbeth, D. (2023). Logic, Mathematics, and Philosophy. M×Φ — Annals of Mathematics and Philosophy, 1(1), 127–139.
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A Philosophical Task in our times
Starting from Ian Hacking’s diagnosis of a task for philosophy in our times, I address a neglected problem in the philosophy of mathematics: the stability of reference across historically and culturally situated mathematical practices. Mathematical knowledge appears strongly cumulative, yet its objects, methods, and descriptions vary locally, sometimes even among mathematicians who take themselves to be working in the same domain. Examples drawn from algebraic geometry, Greek geometry, infinitesimal analysis, symbolic algebra, group theory, and set theory show that the difficulty cannot be reduced to controversy, incommensurable paradigms, or changing epistemic values. Nor can the semantic externalism developed for the empirical sciences be transferred directly, since mathematical objects are not available through causal interaction with exemplars. Against both a purely descriptive account of reference and a vague appeal to practice, I propose to analyze the modes by which mathematical reference is stabilized. Leibniz’s theory of blind or symbolic knowledge offers a guiding thread: stabilization may depend on conceptual analysis, on genetic characterizations of objects through operations and constructions, or on the material anchoring of reasoning in symbolic and diagrammatic systems. These plural modes make room for an internal realism in mathematics, able to account at once for historical variability, trans-theoretical identity, and the cumulative character of mathematical knowledge.
Rabouin, D. (2023). A Philosophical Task in our times. M×Φ — Annals of Mathematics and Philosophy, 1(1), 141–160.
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Une tâche philosophique pour notre temps
À partir du diagnostic de Ian Hacking d’une « tâche philosophique de notre temps », j’examine un problème encore trop peu pris en compte dans la philosophie des mathématiques : la stabilité de la référence à travers la diversité historique et culturelle des pratiques mathématiques. Les mathématiques se présentent comme un savoir fortement cumulatif, mais leurs objets, leurs méthodes et les descriptions qui en sont données varient localement, parfois entre des mathématiciens appartenant à un même domaine. Des exemples empruntés à la géométrie algébrique, à la géométrie grecque, à l’analyse infinitésimale, à l’algèbre symbolique, à la théorie des groupes et à la théorie des ensembles montrent que la difficulté ne se laisse réduire ni à des controverses, ni à des paradigmes incommensurables, ni à une simple histoire des valeurs épistémiques. L’externalisme sémantique élaboré pour les sciences empiriques ne peut pas davantage être transposé tel quel, puisque les objets mathématiques ne sont pas accessibles par interaction causale avec des exemplaires. Contre une théorie purement descriptive de la référence comme contre une invocation trop indéterminée de la « pratique », je propose d’étudier les modes de stabilisation référentielle propres aux mathématiques. La théorie leibnizienne de la connaissance aveugle ou symbolique fournit ici un fil conducteur : la référence peut être stabilisée par l’analyse conceptuelle, par la caractérisation génétique des objets au moyen d’opérations et de constructions, ou par l’ancrage matériel du raisonnement dans les systèmes symboliques et diagrammatiques. Ce pluralisme ouvre la voie à un réalisme interne capable de rendre compte à la fois de la variabilité historique, de l’identité transthéorique et du caractère cumulatif du savoir mathématique.
Rabouin, D. (2023). Une tâche philosophique pour notre temps. M×Φ — Annals of Mathematics and Philosophy, 1(1), online version, 22 p.
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Towards Mathematical Criticism
We introduce and defend Mathematical Criticism as a field of inquiry that should stand to Mathematics as Literary, Artistic, and Musical Criticism stand to literature, art, and music. Against the frequent reduction of Philosophy of Mathematics to linguistic, logical, and set-theoretical questions, mathematical thought is approached through the creative works, techniques, diagrams, styles, and conceptual forces that form its fabric. The mathematical critic must first enter, know, feel, and live the works themselves, and only then describe, calibrate, evaluate, and explain them. The critical task is threefold: to describe the construction of mathematical thought, to assess the entanglement of such constructions with ideal realms of possibilia, and to contrast those entanglements with the real in a Peircean, inductive sense. The contrast with Analytic Philosophy of Mathematics is explored through Galois and Riemann, whose ambiguity theory, uniformization, algebraic structures, and complex variables disclose dimensions inaccessible to any external reduction of mathematical substance. We also recover neglected pioneers — Shaw, Lautman, and Javier de Lorenzo — and propose RTSK models, combining sheaves, topoi, Kripke models, and Riemann-surface ramifications, to preserve the local-global, archetypal, historical, and cultural stratifications of mathematical creativity.
Zalamea, F. (2023). Towards Mathematical Criticism. M×Φ — Annals of Mathematics and Philosophy, 1(1), 151–167.
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Interviews
Regards and Insights on the Universe, an Interview with Sir Roger Penrose
This paper is the transcript of an interview with Sir Roger Penrose, which took place on November 15, 2021. The main aim of the paper is to explore with him the various aspects of his work.
Szczeciniarz, J., & Kouneiher, J. (2023). Regards and Insights on the Universe, an Interview with Sir Roger Penrose. M×Φ — Annals of Mathematics and Philosophy, 1(1), 179–227.
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Letters and Communications
Mathematics and cognition
A short communication on the cognitive dimensions of mathematical thought, by one of the most distinguished mathematicians of the twentieth century.
Manin, Y. (2023). Mathematics and cognition. M×Φ — Annals of Mathematics and Philosophy, 1(1), 229–231.
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