IHPST (UMR 8590)

We identify four different minimal versions of the indispensability argument, falling under four difference varieties: an epistemic argument for semantic realism, an epistemic argument for platonism, and a non-epistemic version of both. We argue that most current formulations of the argument can be reconstructed by building upon the suggested minimal versions. Part of our discussion relies on a clarification of the notion of (in)dispensability as relational in character. We then present some substantive consequences of our inquiry for the philosophical significance of the indispensability argument, the most relevant of which being that both naturalism and confirmational holism can be dispensed with, contrary to what is held by many. 1 This paper builds on the analysis of indispensability arguments presented in Panza, Sereni [2013], chs. 6-7; some of its conclusions, together with the four versions of the minimal indispensability argument discussed in §3, have thus been anticipated in that work. 2 argument to be a form of platonism, or rather a conclusion in favour of mathematical realism 2. For our present purpose, it suffices that the two theses are acknowledged as distinct and both plausible: by platonism we mean, the thesis that there exist objects of a certain sort, namely such that our current mathematical theories can be taken to be about them, in short that there exist mathematical objects 3 ; by mathematical realism we mean a particular form of semantic realism, i.e. the thesis that the statements encompassed by our current mathematical theories, or better its theorems or consequences, are true (without specific commitment to what makes them true) 4. Puntnam's own views apart, his quotation above is commonly seen as a paradigmatic example of an argument for platonism. Prima facie, the Putnam's version of the argument appeals only two notions, indispensability and quantification. However, many believe that beside these notions, IA relies on some additional theses of Quinean provenance: confirmational holism and naturalism. The most debated formulation of IA that is faithful to this conception has been advanced by Mark Colyvan's 5 : i) We ought to have ontological commitment to all and only those entities that are indispensable to our best scientific theories; ii) Mathematical entities are indispensable to our best scientific theories;

We reconstruct essential features of Lagrange's theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange's notions of function and algebraic quantity, and concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem-especially the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method-the generality and internal organization of the discipline. 8 If Lagrange's program is so understood, it is not surprising that, after 1797, he often relied on infinitesimal devices, for example in the second edition of the Mécanique analytique (Lagrange, 1811-1815). Concerning the internal organization of the subject, the Mécanique analytique is indeed quite close to the frame of the Théorie, particularly its mechanical part (see Panza 1991-1992). And for Lagrange this was certainly more relevant, in establishing the place of mechanics and its mathematical methods within the mathematical corpus as a whole, than the local (and non-substantive) use of infinitesimals. 9 We shall use the term 'proof' and cognates for referring to arguments used, in the context of Lagrange's theory, to establish that something holds,

The indispensability argument (ia) comes in many different versions that all reduce to a general valid schema. Providing a sound ia amounts to providing a full interpretation of the schema according to which all its premises are true. Hence, arguing whether ia is sound results in wondering whether the schema admits such an interpretation. We discuss in full details all the parameters on which the specification of the general schema may depend. In doing this, we consider how different versions of ia can be obtained, also through different specifications of the notion of indispensability. We then distinguish between schematic and genuine ia, andargue that no genuine (non-vacuously and non-circularly) sound ia is available or easily forthcoming. We then submit that this holds also in the particularly relevant case in which indispensability is conceived as explanatory indispensability. * Many thanks to the audience of the Indispensability and Explanation workshop held at IHPST in Paris in November 19-20, 2012, where this paper was first presented. We are particularly grateful to Henri Galinon for his useful rejoinder at the conference and for subsequent discussion. The authors would like to thank the projects which financially supported their research for this article. Marco Panza wishes to thank the ANR-DFG Project Mathematical objectivity by representation. Andrea Sereni wishes to thank the Italian National Project PRIN 2010 Realism and Objectivity. 1 3 Something akin to a form of indispensability argument for abstract entities in semantics can be traced back to [Church, 1951]. Thanks to an anonymous reviewer for pointing us to Church's classic paper in this connection. As noticed by [Psillos, 1999], pp.10-11, the indispensability of the use of theoretical terms in the formulation of "efficacious" systems of laws is claimed in [Carnap, 1939], p.64. 4 Thanks to Maria Paola Sforza Fogliani for bringing this to our attention.

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how$AC$generalizes Frege’s views while$FC$comes closer to his original conceptions. Different authors diverge on the interpretation of$FC$and on whether it applies to definitions of both natural and real numbers. Our aim is to trace the origins of$FC$and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish$AC$from$FC$(§2). We discuss six rationales which may motivate the adoption of...

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ( ) or ‘Frege Constraint’ ( ), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how generalizes Frege’s views while comes closer to his original conceptions. Different authors diverge on the interpretation of and on whether it applies to definitions of both natural and real numbers. Our aim is to trace the origins of and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish from (§2). We discuss six rationales which may motivate the adoption of different instances of ...

— François Vi`ete considered most of his mathematical treatises to be part of a body of texts which he entitled Opus restitutæ Mathematicæ Analyseos Seu Algebrâ novâ. Despite this title and the fact that the term “algebra” has been often used to designate what is ...

Up to the 1740s, problems of equilibrium and motion of material systems were generally solved by an appeal to Newtonian methods for the analysis of forces. Even though, from the very beginning of the century—thanks mainly to Varignon (on which cf. [Blay 1992]), Jean Bernoulli, ...

Na‘īm ibn Mūsā’s lived in Baghdad in the second half of the 9th century. He was probably not a major mathematician. Still his Collection of geometrical propositions – recently edited and translated in French by Roshdi Rashed and Christian Houzel – reflects quite well the mathematical practice that was common in Thābit ibn Qurra’s school. A relevant characteristic of Na‘īm’s treatise is its large use of a form of inferences that can be said ‘algebraic’ in a sense that will be explained. They occur both in proofs of theorems and in solutions of problems. In the latter case, they enter different sorts of problematic analyses that are mainly used to reduce the geometrical problems they are concerned with to al-Khwārizmī’s equations.

Dans son intervention au colloque Koyré (Paris, 1986), Ernest Coumet a suggéré que le terme « révolution scientifique » ne désigne pas chez Koyré un événement historique, mais un idéaltype, au sens de Max Weber. L'auteur discute d'abord cette thèse de Coumet et expose les arguments que ce dernier apporte pour la soutenir. Dans la deuxième partie de l'article, il critique l'usage de la notion de révolution en histoire des sciences, en s'opposant en particulier à la possibilité de distinguer dans les productions des savants une « pensée scientifique » qui serait influencée par la « pensée philosophique » et dont les bouleversements marqueraient l'avènement d'une révolution.