On the methodology of informal rigour: set theory, semantics, and intuitionism∗ Walter Dean† & Hidenori Kurokawa‡ In his paper Kreisel adumbrates a crucial insight into the nature of mathematics and foundations of mathematics by focusing on the notion of “informal rigor.” It seems to me that philosophy of mathematics should pay much more attention to this notion than has been the case. (Isaacson, 2011, p. 13) 1 Introduction Georg Kreisel is best known amongst philosophers for his defense of the model theoretic analysis of first-order logical validity and his argument that the Continuum Hypothesis (CH) possesses a definite truth value. The former – which is often referred to as Kreisel’s squeezing argument – continues to play a role in debates about logical consequence and validity after its popularization by authors such as Field (1989, 2008), Boolos (1985), and Etchemendy (1990). The latter has been influential in subsequent debates in philosophy of mathematics regarding the role of second-order logic in securing the determinacy of set theoretic truth – e.g. Weston (1976), Shapiro (1985, 1991), Potter (2004), Isaacson (2011). The locus classicus for both of Kreisel’s arguments is his paper ‘Informal rigour and complete- ness proofs’ (1967b). It is fair to say that this paper is well-known. But it is also not a substantial exaggeration to say that the sections of (1967b) which contain Kreisel’s validity and CH arguments are the only part of his otherwise substantial output which has been widely read or appreciated by philosophers. This is to some extent understandable. For not only is Kreisel’s style sometimes obscure, even the papers which he directed at philosophical audiences are typically informed by mathematical and historical considerations which are not readily appreciated without an under- standing of the milieu in which they were composed. This is particularly true of (1967b). For although this paper is one of several places in which Kreisel juxtaposed his validity and CH arguments, it also provides the most complete exposition of a technique which he referred to as informal rigour. Kreisel presented these arguments as examples arXiv:2104.14887v1 [math.LO] 30 Apr 2021 of the technique. But it is not entirely straightforward to discern what they have in common as applications of a more general method. Thus although ‘informal rigour’ is often taken to be a hallmark of Kreisel’s work, there is little consensus on how this method should be characterized or the sorts of questions it is intended to address. The primary goals of this paper will thus be to distill from Kreisel’s various expositions a general schema for what we will refer to as an informally rigorous argument and to illustrate in detail how ∗Forthcoming in Intuitionism, Computation, and Proof: Selected themes from the research of G. Kreisel, M. Antonutti Marfori and M. Petrolo (editors), Springer. Please do not cite without permission. †University of Warwick, [email protected]. ‡Kanazawa University, [email protected]. 1 Kreisel’s examples conform to it. In this context we will additionally demarcate what has come to be called a squeezing argument – as paradigmatically illustrated by Kreisel’s validity argument – and illustrate how such arguments can be understood as a subspecies of Kreisel’s more general method. We will present our proposed schematization of informal rigour in §3. In§4 we will then present detailed reconstructions of the three primary examples which Kreisel considered in (1967b). In addition to his validity and CH arguments, this includes his analysis of the notion of a creating subject which figures in Brouwer’s development of intuitionistic analysis. Little subsequent attention has been paid to this case as an example of informal rigour. But it is of particular interest not only because Kreisel’s exposition contains a novel mathematical result (whose proof we will reconstruct in §4.2.3) but also because it illustrates the central role which Kreisel assigned to reflection on what he refers to as ‘new primitive notions’ in the operation of informal rigour. In Appendix A we will summarize more briefly Kreisel’s proposals about the applicability of informal rigour to the following additional concepts: set, nonstandard model, finitist proof, predicative definability, and intuitionistic validity. These examples are illustrative of the many trees in the forest we are about to enter. But before considering them individually, it will be useful to consider in §2 the context in which Kreisel introduced informal rigour, inclusive of his motives for promoting this methodology and his strategy for presenting it to his intended audiences. In addition to the complexity of Kreisel’s relationships with his interlocutors, many aspects of his exposition were also informed by specific mathematical developments to which he either directly contributed or was otherwise an interested party. Several of these relate to the other titular concern of (1967b) – i.e. the significance of complete- ness proofs in mathematical logic. About such proofs Kreisel begins by remarking [Q]uite generally, problems of completeness (of rules) involve informal rigour, at least when one is trying to decide completeness with respect to an intuitive notion of consequence. (1967b, p. 139) By the time he composed (1967b) Kreisel indeed had extensive experience not only with the tech- nical details of completeness proofs for a variety of logical systems but also with the prior task of distilling notions of consequence and validity from the foundational work of theorists such as Zermelo, Poincaré, Hilbert, Brouwer, Heyting, and Gödel. It is thus useful to flag the following por- tions of Kreisel’s mathematical work as antecedents to his introduction of the expression ‘informal rigour’: i) his refinement of Hilbert and Bernays’s (1939) arithmetization of Gödel’s (1929) original completeness proof for classical first order logic (1950; 1953); ii) his metamathematical analysis of the completeness of intuitionistic first-order logic (1961; 1958a; 1958f; 1970a); iii) his investigation on the role of reflection principles in the characterization of finitary mathematics (1958a; 1970b; 1968); iv) his analysis of predicativity in terms of the hyperarithmetical sets (1959a; 1960a; 1960b; 1962a); and v) his anticipation of the method of forcing in set theory (1958g; 1961).1 Such developments may seem far afield to contemporary readers of ‘Informal rigour and com- pleteness proofs’. But perhaps the most interesting aspect of this paper is how Kreisel understood results in mathematical logic to be related to debates in mid-20th century analytic philosophy – e.g. the possibility of conceptual analysis, the viability of logical positivism, and the tenability of realism, idealism, and formalism in the foundations of mathematics. But although in some instances Kreisel’s proposals have gone on to have significant downstream effects, it is still difficult to concisely 1The following sources contain discussion of these developments: i) (Smorynski, 1984), (Dean, 2020); ii) (Sundholm, 1983), (McCarty, 2008), (Dean and Kurokawa, 2016); iii) (Feferman, 1995), (Dean, 2015); iv) (Feferman, 2005), (Dean and Walsh, 2017, §2.3); v) (Kreisel, 1980, §III) as well as the other sources cited in note 69 below. 2 summarize his philosophical contributions. In fact Kreisel’s influence has mostly been felt at one level of remove via his associates and interpreters who have often described his views incompletely or out of mathematical and historical context (often out of expository necessity). In §5 we will discuss a source which helps to illustrate the relationship of Kreisel’s views with some of his contemporaries – i.e. his exchange with Yehoshua Bar-Hillel recorded in the comments and replies published along with (1967b). Much of this concerns the comparison of Kreisel’s concep- tion of informal rigour relative to the method which Carnap called explication. Kreisel was clearly dissatisfied that this discussion did not engage more deeply with the mathematical details of his examples. But it is still a useful waypoint in appreciating the enduring significance of informal rigour in relation to contemporary discussions of philosophical methodology. 2 Context We will see below that Kreisel took the validity argument to be the most paradigmatic of his case studies in informal rigour. But it is also clear that he understood the significance of this argument in relation to the case for the determinacy of CH which he which to impress on his audiences in the mid-1960s. This is attested by the fact that he juxtaposed the two arguments in at least the following four sources:2 - Mathematical logic. In T.L. Saaty, editor, Lectures on Modern Mathematics, Vol. III. Wiley, 1965. [(1965) / early 1964 / §1.8, pp. 111-112, pp. 114-118] - Mathematical logic: What has it done for the philosophy of mathematics. In R. Schoenman, editor, Bertrand Russell, Philosopher of the Century. Allen and Unwin, 1967. [(1967a) / December 1964 /§3, pp. 253-262] - Informal Rigour and Completeness Proofs. In I. Lakatos, editor, Problems in the philosophy of mathematics: Proceedings of the International Colloquium in the Philosophy of Science, London, 1965. North-Holland, 1967. [(1967b) / July 1966 / §1-2, pp. 147-157] - Elements of Mathematical Logic. Model theory (with Jean-Louis Krivine). Studies in Logic and the Founda- tions of Mathematics. North-Holland. [(1967c) / Krivine, chapters 0-5: 1960-1961; Kreisel chapters 6-7 and appendices: 1966 / Appendix A, §4, pp. 189-194] The task of reconstructing Kreisel’s methodology is made non-trivial by the factors which shaped his presentations in these sources. It will hence be useful to begin by recording some details about the circumstances of their publication. Kreisel was 42 years old in 1965 and had already published more than 30 papers in mathematical logic spanning a number of areas.3 This work forms much of the background to (1965) which corresponds to an entry on mathematical logic which Kreisel was invited to write for Thomas Saaty’s three volume series Lectures in Modern Mathematics.