Introduction: Logicism Today

Abstract

All the articles of this issue come from the international conference ‘Logicism Today’ organized in Besse-en-Chandesse (the very place where the first Bourbaki Congress was held) in June 2011.1 The wish of the organizers was to explore the internal diversity of the logicist movement, and to delineate its coherence and conceptual core: is there something, beyond the allegiance to the same forefathers, Frege, Russell and Carnap, and beyond confronting common adversaries, which would constitute a shared conceptual framework? Or, on the contrary, should one give up trying to find conceptual unity in what is really an archipelago of different programs? This was a risky undertaking since it looked very much like a family meeting, where one's dirty linen can be washed in private. Happily, this did not happen, and we warmly thank Patricia Blanchette, Valentin Sorin Costreie, Bob Hale, Gregory Landini, Øystein Linnebo, Steven Methven, Marco Panza, Robert Trueman, Sean Walsh, Mark Wilson, and all the students attending the talks for having always preferred constructive criticisms to lazy dismissings. What emerged finally was the multiform implementation of logicism according to three main parameters: the founding father taken as a reference (Frege, Russell, Ramsey, …), the connection established between logicism and philosophy of mathematics, the connection established between logicism and philosophy of language. The papers collected in this issue give a fair picture of the diversity of all talks. The first two, Hale's and Wilson's, pertain to properties, extensions, and sets. The last three get back to Russell, Landini's to clarify the lesson usually learned from Russell's paradox, Gandon's and Halimi's to focus on Russell's substitutional theory. Despite its multifarious aspects, logicism always corresponds to some balance sought between speculative constraints (concerning the use of unrestricted variables, the rejection of ontological types, or the status of properties) and mathematical goals (the abstractionist reconstruction of mathematical objects, the logical analysis of mathematical theories, the conceptual grasp of mathematical-physical processes). In ‘Properties and the interpretation of second-order logic’, Bob Hale offers a fresh look at the neo-Fregean abstractionist program in philosophy of mathematics through a discussion of the second-order comprehension scheme and an analysis of the very nature of the properties that are quantified over in that scheme. Hale's paper defends a deflationary approach where a property exists if and only if there stands for it a predicate that can be given appropriately determinate satisfaction conditions. Accordingly, it puts forward a nonstandard semantics for second-order logic, where the range of monadic second-order variables is the collection of all the subsets of the individual domain that are definable in the metalanguage. By doing so, he departs from Frege on two scores: first because he endorses an intensional view of properties and defines identity of two properties through a modal clause; secondly, because he does not take second-order quantification to be unrestricted. Above all, the main thread of the paper is a metaphysical analysis of properties whose consequences for logic and mathematics are only derivative. Hale concludes his analysis with two points: the first is that the usual metalogical features of compactness and completeness still hold in his semantical framework; the second is that categoricity of second-order axiomatization of important mathematical theories fails but can be given up without great loss. Hale finally focuses on the issue of impredicativity of second-order comprehension and argues both that his conception refers to a determinate range of possible values and that it is not inflationary. In ‘Enlarging one's stall or How did all of these sets get in here?’, Mark Wilson claims that two very different reasons led mathematicians to introduce sets in their theories. Set theory was first needed to extend the use of our concepts, ‘to shore up the so-called “free creativity of the mathematician” ’. Foundational programs based on abstraction principles still follow this old path. According to Wilson, however, another fundamental, yet often neglected, rationale has to be recognized behind the introduction of sets, namely ‘the desirability of understanding our computational relationships to the sorts of physical behaviors captured in differential equations (and allied forms of dynamical recipe)’. Wilson takes and deepens several examples of functions introduced by differential equations: the recourse to sets has to be understood in the context of the ‘algorithmic adjustments’ needed to extract reliable information from the equation and to compute as accurately as possible the solutions of the function at stake. Now, the language of set theory has proved to be the most natural to capture ‘our computational place in nature’, i.e., to describe the ‘mildly transcendental circumstances’ under which our ever-improving computations surround some target curve, asymptotically home in on natural processes or, on the contrary, fail to converge to the actual solution. Wilson then underlines that physical structures cannot generally be described as structures for a formal language, because the most important physical features of reality cannot be reduced to a list of primitive predicates. Set theory enters the scene precisely to capture previously unconceptualized properties, i.e., to handle physically relevant quantities to which no available linguistic predicate corresponds. The rest of Wilson's paper turns toward the other motivation for set theory, but illustrates the role of sets as supplementary entities by concrete detailed examples that point once again to geometry and mathematical physics. Wilson's conclusion is that hostility toward sets as abstract objects stems only from the stress put on one aspect of sets at the expense of the other, and from a philosophical blindness to the computational involvement of sets. Which is all grist for the partisan of mathematical naturalism. In ‘Zermelo and Russell's paradox’, Gregory Landini sets out to defuse Zermelo's priority claim, according to which Zermelo would have discovered, before Russell, what has become to be known as Russell's paradox. Through a careful examination of Zermelo's unpublished 1899 proof, Landini argues that Zermelo's result does not amount to Russell's paradox and does not show that there is no universal set: whereas Zermelo presupposed the full truth of the axiom of separation in order to prove that there is no universal set, Russell considered on the contrary the existence of a universal set as granted and diagnosed a ‘very subtle fallacy’ in the proof of Cantor's powerset theorem, due to the falsity of one instance of the axiom of separation. Indeed, as a consequence of a purely logical theorem proved by Russell in 1905 in ‘On some difficulties in the theory of transfinite ordinals and order types’, the introduction of the diagonal set corresponding to the universal set has to be precluded. Of course, Cantor's powerset theorem is true in modern set theory (with the axiom of separation available), but admission of set theory just begs the question. It remains that Zermelo's argument may be construed as a refutation, at least, of the extensional conception of sets (according to which any set is the extension of some propositional function, and any extension, a set). Yet the full refutation of the extensional conception, and the genuine Russell's paradox as well, only come from the Appendix B of the Principles of Mathematics: in short, Russell is the sole genuine author of Russell's paradox. In ‘Variable, structure and restricted generality’ and ‘Structured variables’, Sébastien Gandon and Brice Halimi draw on Russell's 1906 substitutional theory to reflect on generality and variables. Owing to Russell's universalism (his view that variables are ultimately unrestricted), the main problem is how to get a genuinely restricted range of variation. Gandon argues that structuring the variables is the Russellian answer. He goes on to cast a new light on the Julius Caesar problem and on the problem of absolute generality. Russell's substitutional approach, indeed, helps to explain why ‘2 is Julius Caesar’ is meaningless while avoiding any type distinction. It also gives rise to an ‘absolutely restricted generality’, i.e., restricted ranges of significance structurally emerging from within an absolute understanding of everything — whereas Tarski's semantics leans on relatively unrestricted domains (each domain representing a relative interpretation of ‘everything’). In connection with that point, Halimi uses the framework of syntactic fibrations as a platform to compare and partly conciliate Russell's and Tarski's respective conceptions of what it is to be a possible value for a variable. The aim pursued is first to reactivate Russell's substitutional insight in a fibrational setting. It is also to relativize Tarski's semantics as a degenerate case of what can be introduced as ‘Tarski's fibration’, namely the environment obtained when all assignments of values for variables of some language are not considered in one structure at a time, but taken upon the base space of all structures for the language. In Russell's as in Tarski's fibration, the structure of structured variables comes from the constraints set upon all the ways of picking out an item in each fiber. In other words, logical structure is construed as what restricts the available sections of some given logical fibration. The tool of fibrations, coming from category theory, thus allows one to put to the fore variables that are both structured as in Russell and interpreted as in Tarski.