Logic, Truth and Description. Essays on Chateaubriand's Logical Forms Jairo J. da Silva (ed.) CDD: 160 CHATEAUBRIAND’S LOGICISM ABEL LASSALLE CASANAVE Department of Philosophy Federal University of Santa Maria Campus Universitário, Km 9, Camobi 97150-900 SANTA MARIA, RS BRAZIL abel@ccsh.ufsm.br Abstract: In his doctoral dissertation, O. Chateaubriand favored Dedekind’s analysis of the notion of number; whereas in Logical Forms, he favors a fregean approach to the topic. My aim in this paper is to examine the kind of logicism he defends. Three aspects will be considered: the concept of analysis; the universality of arithmetical properties and their definability; the irreducibility of arithmetical objects. Key-words: Logicism. Analysis. Arithmetical truth. Dedekind. Frege. Chateau- briand. In Chapter IV of his doctoral dissertation, Oswaldo Chateaubriand asserts that Dedekind’s analysis of the notion of number was the deepest 1 . At the end of Chapter 9 of Logical Forms, he refers to Dedekind again, but only very briefly 2 . In this book, he favors a (kind of) fregean approach to the topic. What was Chateaubriand’s motivation to make this change? In what follows, our aim is to shed some light on this question by examining the version of logicism developed by him. First, we will consider Chateaubriand’s concept of logical analysis; second, his 1 2 See Chateaubriand (1971, p. 129). See Chateaubriand (2001, p. 319). Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 13-20, jan.-jun. 2004. 14 ABEL LASSALLE CASANAVE concept of universality, which is used to characterize a property as logical; third, the nature of arithmetical objects. In the above-mentioned chapter of his doctoral thesis, Dedekind’s analysis is called type-analysis, whereas the definitions given by Frege, Russell, Zermelo and von Neumann are tokens of this type. Chateaubriand points out, criticizing Quine, that with regard to Frege’s project it would not suffice to merely define an arbitrary progression, i.e., an arbitrary infinite series whose members have a finite number of predecessors, but it is necessary to define a progression whose elements have the correct properties. This criticism implies that Quine’s characterization of Frege’s project in Word and Object is not acceptable: even on the assumption that the numerals form a progression, they do not have, for Frege, the correct properties of numbers. The central point in this controversy about the notion of number concerns the nature of analysis: whereas Quine defends a conception of analysis as substitution (elimination), Chateaubriand adheres to a conception of analysis as clarification. Let us briefly recall, following A. Coffa, the main steps of an analysis according to the model of analysis as clarification 3 . These are, first, the identification of the obscure concept that will be made the object of clarification (analysandum) – this is, with regard to the analysis of number, the concept of number. Second, the establishment of the conditions of adequacy – these are the “essential” characteristics constituting the “real” meaning of the analysandum. And, third, the analysis in the proper sense (analysans) which has to satisfy the conditions of adequacy established by the second step – the definition of natural number. The crucial question is how we have to understand the conditions of adequacy. 3 For a discussion about the models of analysis to which I am referring here, see Coffa (1975) and Simpson (1975/1995). See also Seoane (2000). Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 13-20, jan.-jun. 2004. CHATEAUBRIAND’S LOGICISM 15 Actually, the proponent of the model of analysis as clarification cannot concede that the conditions of adequacy are confined to the arithmetical propositions which are acknowledged as true, because this requirement amounts to nothing but the quinean requirement that an analysis should fulfill all “useful contexts” in which the analysandum is used, i.e., to those “that deserve to be preserved”. The notion of useful context justifies the claim that any arbitrarily defined progression satisfying the arithmetical truths constitutes an analysis (substitute) of the concept of number. The definitions given by Zermelo and von Neumann are good candidates for illustrating Quine’s thesis. The aim of Dedekind 4 (and Frege and Russell) was considerably different, namely, to give an analysis in the sense of clarification. The distinction between type-analysis and token-analysis introduces a complication with regard to the description of the model, but it seems that we can say that the type-analysis establishes the conditions of adequacy, and that the remaining definitions, as instances of the type, can be understood as the ultimate step of Coffa’s model. Frege’s tokenanalysis would consist in characterizing the numbers individually and showing that they form a progression; in Dedekind’s structural characterization (type-analysis) the “individuation” of numbers is given by structural relations 5 . In this, it seems to me, consisted the depth of 4 About his own essay Dedekind (1890, p. 99-100) writes: “it is a synthesis constructed after protracted labor, based upon a prior analysis of the sequence of natural numbers just as it presents itself, in experience, so to speak, for our consideration. What are the mutually independent fundamental properties of the sequence N, that is, those properties that are not derivable from one another but from which all others follow?” 5 Unlike Zermelo’s and von Neumann’s definitions, it is difficult to regard Frege’s and Russell’s definitions as mere implementations of Dedekind’s characterization of number. I do not think that this consideration is alien to the change of positions mentioned above. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 13-20, jan.-jun. 2004. 16 ABEL LASSALLE CASANAVE Dedekind’s analysis for Chateaubriand at the time of his doctoral dissertation. But, the change of position in Logical Forms cannot be understood solely as a simple inversion of the following kind: in place of Dedekind’s structural characterization he puts numbers considered as self-subsistent objects à la Frege. On the contrary, Chateaubriand sees in this last thesis the fountain of the difficulties which the fregean program has to face – a thesis derived from the nature of mathematical discourse: it seems that it does not make sense to treat numbers as concepts; they should be regarded as objects. In the context of the program of reducing arithmetic to logic in Frege’s sense, numbers should be conceived of not only as objects but also as logical objects. This is accomplished by Frege by means of the method of abstraction and by objectifying the resulting abstractions to extensions, conceived of as logical objects: this is the way to paradox. Chateaubriand, on the other hand, asserts in Logical Forms that Frege succeeded in showing that the concepts of arithmetic are logical concepts. The aspect stressed by Chateaubriand – that the concepts of arithmetic are logical concepts – agrees perfectly with his own reconstruction of logic and arithmetic. For, firstly, he structures reality as a hierarchy consisting of objects of level 0, properties (which are the counterparts to Frege’s concepts) and states of affairs of level ≥ 1, where logical properties are conceived of as follows: What seems to characterize a logical notion is universality. Notions such as instantiation, existential quantification, etc., are logical notions in this sense because they are significant for all properties. Since it does not seem to be possible to treat them ontologically as units, they are divided into to an indefinite number of “parts” which are treated as ontologically units. Each of these parts is a logical property with a limited range of significance (or applicability) given by its type, but together they are significant all over the hierarchy above a certain level. (Chateaubriand 2001, p. 302) Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 13-20, jan.-jun. 2004. CHATEAUBRIAND’S LOGICISM 17 And, secondly, numerical properties are, according to Chateaubriand, logical properties in this sense of universality. Now, to introduce the properties Nullness, Oneness, Twoness, Threeness, etc., as logical properties of level 2 or above seems to reflect the option chosen by Chateaubriand to give Frege’s cardinal intuition (without objectification) priority over Dedekind’s ordinal intuition. But, to be sure, to show that arithmetical properties are logical properties in the way proposed by Chateaubriand was not what Frege had in mind; this was rather, to define arithmetical concepts in terms of logical ones, which are considered by him as more basic. This is recognized by Chateaubriand at another place where he writes: What Frege did was to start with certain logical properties, that seemed clearly logical and go on to show that certain mathematical properties, which did not seem to be clearly logical to many people, could be defined in terms of those purely logical properties. (Chateaubriand 2000, p. 179) The preceding observation was not meant to accuse Chateaubriand that his interpretation of Frege is inaccurate, but only to bring into focus that his notion of arithmetical property as logical property allows (partially, as we will see below) to dissociate such properties from any intuitive content, which was one goal of Frege’s program. But this goal was also pursued by other projects aiming to eliminate the reference to intuition in the characterization of arithmetic which is to be found in Kant’s foundation of mathematics. Thus, for example, Dedekind claims that his foundation of the notion of natural number is purely logical, i.e., it does not make any reference to spatial or temporal intuition 6 . With this I want to stress that the question of definability (and the conception of 6 Dedekind (1888, p. 790-91) writes: “In speaking of arithmetic (algebra, analysis) as merely part of logic I mean to imply that I consider the numberconcept entirely independent of the notions or intuitions of space and time – that I rather consider it an immediate product of the pure laws of thought”. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 13-20, jan.-jun. 2004. ABEL LASSALLE CASANAVE 18 definition) is not external to the model of analysis as clarification, neither in Frege nor in Dedekind. From this point of view Dedekind’s structural solution, in which there are no numbers as proper objects, also eliminates any intuitive component associated to the “objects” of mathematics. This thesis is usually attributed to Kant, and it implies that mathematical truths are synthetic a priori. But it seems that Chateaubriand’s thesis has as a consequence that arithmetical truths do not have the same status as logical truths, which consist exclusively of logical notions 7 . He writes: Frege showed that mathematics is a part of logic in the sense that all arithmetical properties are logical properties, but he did not succeed in showing that mathematics is a part of logic in the sense that arithmetical objects are logical objects. One could hold that in this respect mathematics has its own specificity and that these objects must be postulated. (Chateaubriand 2001, p. 315) However, it does not seem that the fregean project, even when it is understood in a very broad sense, could leave aside the characterization of numbers as logical objects: although the analysis showing that numerical concepts (or properties) are logical concepts (or properties) is independent of the conception of extensions as logical objects, as Chateaubriand claims, it does not follow from this that the analysis could refrain from conceiving numbers as logical objects. And, the point is, then, that if we accept that arithmetical objects have a residual irreducible character, as Chateaubriand conjectures, then it is the analytic character of arithmetical truths which appears to be problematic. My intention here is not to suggest a return to Kant, but to point out that Chateaubriand’s conjecture would explain a large variety of “postulations” of “objects” such as exhibitions a priori in intuition, mental constructions, inscriptions, etc., and also numbers considered as sets à la Zermelo or von Neumann. And Chateaubriand’s examination of 7 See Chateaubriand (2001, p. 28). Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 13-20, jan.-jun. 2004. CHATEAUBRIAND’S LOGICISM 19 pure set structures, which are his potential candidates for “mathematical objects”, seems also to lead to the same difficulty: the necessity to objectify in some way or other the positions in pure structures. And, if the conjecture is considered as a thesis, then Chateaubriand’s logicism does not imply, prima facie, that the arithmetical truths are analytic. I said “prima facie” because one observation that Chateaubriand makes at the end of Chapter 12 of Logical Forms presumably takes this into account: he observes that in general numerical properties can occur in mathematical discourse as “subjects” or “predicates”, which does not imply an ontological distinction between objects and properties 8 . The preceding observations amount less to a criticism than to some questions: Are the numerical properties, considered as logical properties, indefinable, or can they be defined in terms of logical properties of a higher level? In the latter case: What is the conceptual import of such definitions? And finally: What class of truths are the arithmetical truths, given that we need to postulate objects that are irreducible? These questions could perhaps be synthesized as follows: In what does the logical analysis of the concept of number consist, for Oswaldo Chateaubriand? I hope that this question does justice to his extraordinary book, if only in a minimal way. REFERENCES CHATEAUBRIAND, O. Ontic Commitment, Ontological Reduction, and Ontology. Berkeley. Doctoral Dissertation. University of California, 1971. ———. “Ockham’s Razor”. O Que Nos Faz Pensar, 3, pp. 51-75, 1990. ———. “Logical Forms”. The Proceedings of the Twentieth World Congress of Philosophy, 6, pp. 161-182, 2000. 8 See Chateaubriand (2001, p. 425). Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 13-20, jan.-jun. 2004. 20 ABEL LASSALLE CASANAVE CHATEAUBRIAND, O. Logical Forms. Part I. Truth and Description. Campinas: Centro de Lógica, Epistemologia e História da Ciência/ UNICAMP, 2001. (Coleção CLE, 34) COFFA, A. “Dos concepciones de la elucidación filosófica”. Crítica, VII(21), pp. 43-65, 1975. DEDEKIND, J.W.R. Was sind und was sollen die Zahlen?, 1888. In: W. Ewald (ed.) pp. 790-833, 1999. ———. Letter to Keferstein, 1890. In: J. van Heijenoort (ed.), pp. 99-103, 1967. EWALD, W. (ed). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford: Clarendon Press, 1999. SEOANE, J. “Modalidades elucidatórias”. Filósofos, V(1), pp. 119-137, 2000. SIMPSON, T.M. “Análisis y eliminación: una módica defensa de Quine”. Crítica, VII(21): 69-80, 1975. ———. “Elucidaciones filosóficas”. Crítica, XXVII(79): 86-91, 1995. VAN HEIJENOORT, J. (ed.). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge, MA: Harvard University Press, 1967. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 13-20, jan.-jun. 2004. Logic, Truth and Description. Essays on Chateaubriand's Logical Forms Jairo J. da Silva (ed.) CDD: 160 THE LOGICAL CHARACTER OF NUMBER: REPLY TO ABEL LASALLE CASANAVE OSWALDO CHATEAUBRIAND Department of Philosophy Pontifical Catholic University of Rio de Janeiro Rua Marquês de São Vicente, 225, Gávea 22453-900 RIO DE JANEIRO, RJ BRAZIL oswaldo@fil.puc-rio.br Abstract: In §1 I discuss Dedekind and Frege on the logical and structural analysis of natural numbers and present my view that the logical analysis of the notion of number involves a combination of their analyses. In §2 I answer some of the specific questions that Abel raises in connection with Chapter 9 of Logical Forms. Key-words: Number structure. Cardinality. Logic. Hume’s Principle. Dedekind. Frege. Plato. Abel rightly says that I favor a conception of analysis as clarification. In fact, I agree that analysis involves a search for essence – at least in most cases. The question1 that Abel raises at the end of his comments is quite pertinent to my aims and I will try to answer it here. Before I do, however, I should point out that my book is not specifically about problems in the philosophy of mathematics, but has a more general nature. So even though I discuss various issues related to 1 “In what does the logical analysis of the concept of number consist, for Oswaldo Chateaubriand?” Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 21-30, jan.-jun. 2004. 22 OSWALDO CHATEAUBRIAND mathematics, these are not sustained discussions in which I am trying to present a systematically worked out view. This is reflected in my discussion in Chapter 9, which is the basis for most of Abel’s comments. Parts of my discussion in that chapter are comments on Frege’s views; other parts are comments on Plato’s views; yet other parts are speculations to which I am not committed; and still other parts are actual views that I hold. It is not always easy to tell what is one thing and what is another, and the tensions that Abel discerns in my discussion are due to this. I will try to clarify some of the ambiguities below 2 . 1. THE NATURE OF NUMBER I still consider Dedekind’s essay the deepest analysis of the structure of the natural numbers. It is, in my view, a paradigm of conceptual analysis, and I never cease to be amazed by its elegance and by how much is accomplished in the space of mere 50 or 60 pages. Nothing that has been written on the nature of number gets even close to Dedekind’s essay – especially if we join it to his letter to Keferstein – as a mathematical conceptual analysis of the structure of the number series. In which way does it fall short of being a complete logical analysis? I think that there are three main weaknesses in this regard. One is the account of what is logical in the preface to the first edition. Dedekind says at the outset: 3 In speaking of arithmetic (algebra, analysis) as merely a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time – that I rather consider it an immediate product of the pure laws of thought. 2 In connection with the subjects that I will discuss here, including the relation between Dedekind and Frege, there is a very detailed and interesting discussion in Dummett (1991). 3 The quotations from Dedekind Was sind und was sollen die Zahlen? are from the translation in Ewald (1996). Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 21-30, jan.-jun. 2004. REPLY TO ABEL LASALLE CASANAVE 23 And a little later: If we scrutinize closely what is done in counting a set or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispensable foundation ... the whole science of numbers, must, in my opinion, be established. Although I quite agree with these ideas of Dedekind – and he does a masterful presentation of the concepts of set, relation, function, etc. – his book does not contain a deep analysis of the logical as such. His attitude is essentially that of a mathematician: given these basic ideas, let us develop the theory. And he does. Another related point, it seems to me, is that there isn’t a clear logical account of cardinality attributions. There is a theory of cardinality, to be sure, but it is not quite clear what is the logical character of a cardinality attribution. If I say that there are ten people in the room, what is logical about that? The answer that we derive from Dedekind is that we can count the set of people in the room by establishing a one-one correspondence with Z10 – the set of positive integers smaller than or equal to 10. This is fine as far as it goes, but in my view something is missing. The third weakness is Dedekind’s proof of Theorem 66 that there are infinite sets. Although this proof has been much criticized, I am actually rather sympathetic to it. I think that it, or something like it, can be used as an argument for the existence of potential infinities, but I do not think that it can be used to establish the existence of actual infinities. I think, for example, that an intuitionist can argue more or less à la Dedekind for the potential infinity of the natural numbers, but the argument will depend essentially on his intuition of time. Logical infinities, on the other hand, independent of intuitions of space and time, must in my view be actual infinities. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 21-30, jan.-jun. 2004. 24 OSWALDO CHATEAUBRIAND It is in connection with the first two of these points that I emphasize Frege’s ideas. The depth of Frege’s analysis of the logical is as impressive as the depth of Dedekind’s analysis of the number structure; especially because Frege went at it from so many different directions – ontological, epistemological, linguistic, formal, axiomatic, etc. One might disagree here and there but there is no question that he did bring an enormous amount of light into the nature of the logical. In particular, he had a very deep insight into the logical structure of reality and the logical nature of cardinality. This insight is his distinction between properties of things and properties of properties that he claims in “Function and Concept” to be “founded deep in the nature of things” (p. 31). He realized that cardinality attributions are attributions to properties of things and not to the things themselves and that these higher-order cardinality properties are purely logical. These are the properties Nullness, Oneness, Twoness, etc. that I emphasize. They are essentially the Platonic forms that are structured as the natural numbers – and according to (Wedberg’s interpretation of) Plato they are the (pure) numbers. But Frege had the idea that numbers are objects rather than properties, because they can be referred to by definite descriptions and names that appear in subject position in sentences. This is an idea that I criticize at length in my book, and it leads Frege to introduce extensions as objects and to characterize numbers as those objects that are the extensions of the cardinality properties. This is one place where I see Frege’s analysis as going awry. Although Frege already had a very insightful logical analysis of the structure of the number series in Begriffsschrift using properties (functions), the decision in Grundlagen to bring in extensions as objects – on the ground that numbers must be objects – compromised his whole enterprise. As I see it, the problem is not so much that Basic Law V leads to contradiction, but the character of this law – and of Hume’s Principle, which is essential for Frege’s proof that there are infinitely many numbers. When we read them intuitively, both of these principles seem so simple and so clearly true that it is hard to see how the first could lead Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 21-30, jan.-jun. 2004. REPLY TO ABEL LASALLE CASANAVE 25 to contradiction and the second could lead to the existence of infinitely many objects. The basic insights behind these principles are: (SE) The concept F has the same extension as the concept G iff ∀x(Fx ↔ Gx) (SN) The concept F applies to the same number of things as the concept G iff F ~ G, where ‘F ~ G’ means that F can be put into one-one correspondence with G. What could be more natural than that?4 The problem arises, in both cases, from introducing singular terms ‘the extension of F’, ‘the extension of G’, ‘the number that belongs to F’, ‘the number that belongs to G’ and formulating the principles as: (V) the extension of F = the extension of G iff ∀x(Fx ↔ Gx). (HP) the number that belongs to F = the number that belongs to G iff F ~ G. For now, as if by magic, it turns out that these singular terms denote objects, and extensions and numbers begin to appear in all their infinite 4 (S ) has the air of a mere terminological convention: when the concepts E F and G apply to the same things, then we will say that they have the same extension. (SN) makes a more interesting statement, especially when we consider concepts that apply to infinitely many things, but still seems ontologically very innocuous. If one reads ‘has the same number as’ or ‘the number of F’s is the same as the number of G’s’ for ‘applies to the same number of things as’, then there is a stronger ontological suggestion. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 21-30, jan.-jun. 2004. 26 OSWALDO CHATEAUBRIAND multiplicity. I do not deny that there are extensions and numbers, but I disagree that they can be shown to exist by such linguistic maneuvers 5 . So just as I think that the considerations in Dedekind’s theorem 66 do not prove the existence of an actual infinity of objects, I also think that the considerations in the formulation of Hume’s Principle do not prove the existence of an actual infinity of objects. Hence, I do not consider that the reconstruction of Frege’s arithmetic in second-order logic by means of Hume’s Principle establishes that the concept of number has a purely logical character 6 . My view is that one should work with the properties themselves and that it is a logical axiom that there are infinitely many logical properties – including the properties Nullness, Oneness, Twoness, etc. These are the numbers, and their logical character was clearly revealed by Frege’s work. With my argument in p. 425 that properties can be both subject and predicate I try to counteract Frege’s claim that since numbers are referred to by expressions appearing in subject position, then they must be objects. I think, therefore, that the logical analysis of number is the combination of Frege’s analysis and Dedekind’s 7 . A problem with this approach is that a cardinality property such as Nullness, or Oneness, is not an ontological unit, but will appear all over the hierarchy of properties. And even though in my account properties can accumulate, it is not possible to accumulate all the Nullness properties 5 In other words, although one may postulate that for every concept F there is an object that is the extension of F and that there is an object that is the number that belongs to F, this does not seem to follow from our intuitive understanding of (SE) and (SN) – nor does it follow that these postulations have a logical character. 6 This is not to deny the logical interest of the work that has been done in this direction both by the defenders and by the opponents of this view (Wright, Boolos, Hale, Heck, etc.). 7 To these I would also add Peano, whose (second-order) axiomatization gives a more intrinsic characterization of the number structure. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 21-30, jan.-jun. 2004. REPLY TO ABEL LASALLE CASANAVE 27 into a single Nullness property. It is in this connection that I make a reference to Dedekind in p. 319. After the first passage I quoted above Dedekind continues: My answer to the problems propounded in the title of this paper is, then, briefly this: numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. My suggestion in p. 339 (note 38) – admittedly somewhat enigmatic and certainly not worked out – is that in the most general sense numbers might have the character of intersubjective abstractions based on the nature of things. And I do not think that it follows from this that the number concept is not a logical concept. 2. SOME REMARKS ON CHAPTER 9 I was never completely satisfied with Chapter 9 because too many different things are discussed in it. I tried to revise it in various ways, even to split it up into different discussions, but I did not manage to restructure it to my satisfaction. So I let it stand even though it does not have a main line of argument. I will not discuss all the different aspects of Chapter 9 here, but I will comment on those that are related to Abel’s questions. One of his main questions is whether I view mathematical objects as having an irreducible non-logical character. I do not, but I indulge in a fair amount of speculation about it. At the very beginning I say that for Frege “a proper account of the ontology of arithmetic must have the numbers as objects” 8 . In p. 313 I go back to this issue and start the long speculative discussion about it that goes on until p. 317. The way I thought about this when I wrote the chapter was as follows. Frege 8 P. 297. In p. 314 I refer back to this remark in a way that might perhaps suggest that I am agreeing with Frege. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 21-30, jan.-jun. 2004. 28 OSWALDO CHATEAUBRIAND postulates extensions at level 0 and the interaction between these extensions and the properties in the hierarchy led to the contradictions that affected his system. So what should one do if one wants numbers to be objects? My initial idea was along the lines of Columbus’ solution to the problem of getting an egg to stand on end: if one wants to have sets as level 0 objects, why not postulate something like (the objects of) Zermelo-Fraenkel set theory and be done with it? This led me to the discussion of the pure set structures and of the difficulty of talking about structures without objectifying their content in some way. Which led, in turn, to the discussion of Plato and Gödel in notes 23-30. Let me comment briefly on some of these issues. I think that Plato’s distinction between ideal numbers and mathematical numbers (as interpreted by Wedberg) is very interesting and connects in several ways with what I was saying above about the nature of numbers. The mathematician uses exemplars of the structure of ideal numbers, and for the purposes of mathematics any exemplar will do as a representation of the structure of numbers. That is the reason why we find the number structure imbedded in so many mathematical structures and theories. Yet it is hard to maintain that these exemplars are the numbers, or that there is nothing to number aside from these exemplars. This is essentially Quine’s position which I reject 9 . Although the structure consisting of the ideal numbers Nullness, Oneness, Twoness, etc. may not be the number structure either, it captures the fundamental character of numbers as individual cardinality properties – and each element of this structure may be considered a self-subsistent entity in a sense possibly akin to Frege’s. But what is the structure as such? Now I think that it is the successor relation itself – i.e., considered in intension – 9 As Abel mentions, I discuss Quine’s position at some length in my thesis. Some of this discussion is also in my paper “Ockham’s Razor”, which is a preliminary version of Chapter 24. Manuscrito – Rev. Int. Fil., Campinas, v. 27, n. 1, p. 21-30, jan.-jun. 2004. REPLY TO ABEL LASALLE CASANAVE 29 although when I wrote those remarks on structures I was still trying to find a way to come to grips with them10 . Another aspect of the discussion in Chapter 9 is to understand better the notions of extension and of set. The notion of extension that I characterize in terms of states of affairs (pp. 311-13) is a quite adequate representation of many intuitions that we have about the extension of properties. The pure set structures discussed in pp. 313-17, on the other hand, are not at all good candidates for extensions, but they may be good candidates for mathematical “objects”. And since these pure set structures are pure cardinality structures or iterations of pure cardinality structures, one could say that in this sense all mathematics can be reduced to the study of number 11 . Moreover, if cardinality is a logical notion, this gets us very close to a logical account of mathematics. In fact, in p. 314 I say that these cardinality structures could be taken to be logical properties rather than objects, but since there is “a generalized feeling that a structure is something like an object” I go on to examine them from this point of view. As I said at the beginning many of these ideas are preliminary ones that I did not attempt to develop systematically in the book, but I hope to develop them in a not too distant future. 10 As I mention in several places (e.g., note 37 p. 40) Frege argued that the axioms of Geometry define a higher-order concept. In the same sense, we may formulate the Peano axioms as a big predicate that characterizes the successor relation along the lines suggested in (c) of note 5 (p. 207) for the axiom of induction (but with only ‘Sxy’ as argument). For this to be a purely intensional characterization we must interpret the quantifiers intensionally rather than objectually, but this is not something that I develop in the book. (Though I did give a talk about it in the VI Colóquio Conesul de Ciências Formais in 2002.) 11 There are some remarks along these lines in Tarski (1986, p. 151). In this connection see also §1 of my reply to Frank Sautter. Manuscrito – Rev. Int. 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