Discusses the metaphysics of the iterative conception of set.

George Boolos has argued that the iterative conception of set justifies most, but not all, the ZFC axioms, and that a second conception of set, the Frege-von Neumann conception (FN), justifies the remaining axioms. This article challenges Boolos's claim that FN does better than the iterative conception at justifying the axioms in question.

This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics.

The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the (...) existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”. (shrink)

The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in ...

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

The basic relations and functions that mathematicians use to identify mathematical objects fail to settle whether mathematical objects of one kind are identical to or distinct from objects of an apparently different kind, and what, if any, intrinsic properties mathematical objects possess. According to one influential interpretation of mathematical discourse, this is because the objects under study are themselves incomplete; they are positions or akin to positions in patterns or structures. Two versions of this idea are examined. It is argued (...) that the evidence adduced in favor of the incompleteness of mathematical objects underdetermines whether it is the objects themselves or our knowledge of them that is incomplete. Also, holding that mathematical objects are incomplete conflicts with the practice of mathematics. The objection that structuralism is committed to the identity of indiscernibles is evaluated and it is also argued that the identification of objects with positions is metaphysically suspect. (shrink)

Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set?

An engaging account of the origins of the modern mathematical theory of the infinite, his book is also a spirited defense against the attacks and misconceptions ...

In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor’s theorem, its proof, how it can be used to manufacture (...) paradoxes, Frege’s diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes. (shrink)

I trace changes to Frege's understanding of numbers, arguing in particular that the view of arithmetic based in geometry developed at the end of his life (1924–1925) was not as radical a deviation from his views during the logicist period as some have suggested. Indeed, by looking at his earlier views regarding the connection between numbers and second-level concepts, his understanding of extensions of concepts, and the changes to his views, firstly, in between Grundlagen and Grundgesetze, and, later, after learning (...) of Russell's paradox, this position is a natural position for him to have retreated to, when properly understood. (shrink)

According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical literature is that sets are somehow constituted by their members. In the first part of the paper, I present a number of problems for this answer, paying special attention to the view that sets are metaphysically dependent upon their members. In the second part of the paper, (...) I outline a different approach, which circumvents these problems by dispensing with the priority or dependence relation altogether. Along the way, I show how this approach enables the mathematical structuralist to defuse an objection recently raised against her view. (shrink)

In this book, Eli Hirsch focuses on identity through time, first with respect to ordinary bodies, then underlying matter, and eventually persons.

This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle (...) which pluralities form sets, the paper concludes by affirming a naturalistic approach to the philosophy of set theory. (shrink)

After briefly reviewing the standard set-theoretic resolutions of the Burali-Forti paradox, we examine how the paradox arises in set theory formalized with plural quantifiers. A significant choice emerges between the desirable unrestricted availability of ordinals to represent well-orderings and the sensibility of attempting to refer to ‘absolutely all ordinals’ or ‘absolutely all well-orderings’. This choice is obscured by standard set theories, which rely on type distinctions which are obliterated in the setting with plurals. Zermelo's attempt ( 1930 ) to secure (...) ordinal representability of arbitrary well-orderings through relativization of quantification to set-theoretic models is reviewed and found wanting. The natural modal-structural recasting provides, it is claimed, a good repair. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

This paper argues for the thesis that, roughly put, it is impossible to talk about absolutely everything. To put the thesis more precisely, there is a particular sense in which, as a matter of semantics, quantifiers always range over domains that are in principle extensible, and so cannot count as really being ‘absolutely everything’. The paper presents an argument for this thesis, and considers some important objections to the argument and to the formulation of the thesis. The paper also offers (...) an assessment of just how implausible the thesis really is. It argues that the intuitions against the thesis come down to a few special cases, which can be given special treatment. Finally, the paper considers some metaphysical ideas that might surround the thesis. Particularly, it might be maintained that an important variety of realism is incompatible with the thesis. The paper argues that this is not the case. (shrink)

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

Kit Fine; XIV*—Ontological Dependence, Proceedings of the Aristotelian Society, Volume 95, Issue 1, 1 June 1995, Pages 269–290, https://doi.org/10.1093/aristote.

It is my aim in this paper to show that the contemporary assimilation of essence to modality is fundamentally misguided and that, as a consequence, the corresponding conception of metaphysics should be given up. It is not my view that the modal account fails to capture anything which might reasonably be called a concept of essence. My point, rather, is that the notion of essence which is of central importance to the metaphysics of identity is not to be understood in (...) modal terms or even to be regarded as extensionally equivalent to a modal notion. The one notion is, if I am right, a highly refined version of the other; it is like a sieve which performs a similar function but with a much finer mesh. (shrink)

In this paper I outline an alternative to hermeneutic fictionalism, an alternative I call indifferentism, with the same advantages as hermeneutic fictionalism with respect to ontological issues but avoiding some of the problems that face fictionalism. The difference between indifferentism and fictionalism is this. The fictionalist about ordinary utterances of a sentence S holds, with more orthodox views, that the speaker in some sense commits herself to the truth of S. It is only that for the fictionalist this is truth (...) in the relevant fiction. According to the indifferentist, by contrast, we are simply non-committal – or indifferent – with respect to some aspects of what is literally said in our assertive utterances. (shrink)

Some novel solutions to problems in mathematics and philosophy involve employing schemas rather than quantified expressions to formulate certain propositions. Crucial to these solutions is an insistence that schematic generality is distinct from quantificational generality. Although many concede that schemas and quantified expressions function differently, the dominant view appears to be that the generality expressed by the former is ultimately reducible to the latter. In this paper, I argue against this view, which I call the 'Reductionist view'. But instead of (...) focusing on the difference between schemas and quantified expressions in formal systems, as most proponents of schemas have done, I focus on the difference between their supposed linguistic correlates, i.e., any statements and every-statements. Drawing insights from prominent accounts of the difference between these kinds of statements, I develop an alternative account. I argue that this account is not only preferable in its own right but it also supplies a response to the Reductionist view. (shrink)

The year 1897 saw the publication of the first of the modern logical paradoxes. It was published by Cesare Burali-Forti, the Italian mathematician whose name it has come to bear. Burali-Forti's own formulation of the paradox was not altogether satisfactory, as he had confused well-ordered sets as defined by Cantor with what he himself called “perfectly ordered sets”. However, he soon realized his mistake, and published a note admitting the error and making the correction. He concluded the note with the (...) observation that his result could be established on the basis of the correct definition of well-ordered set as easily as for the “perfectly ordered sets” for which it had first been obtained. We shall reproduce his results in their corrected form. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The humble origins of Russell's paradox by J. Alberto Coffa ON SEVERAL OCCASIONS Russell pointed out that the discovery of his celebrated paradox concerning the class of all classes not belonging to themselves was intimately related to Cantor's proof that there is no greatest cardinal. lOne of the earliest remarks to that effect occurs in The Principles ofMathematics where, referring to the universal class, the class of all classes (...) and the class of all propositions, he notes that when we apply the reasoning of his [Cantor's] proofto the cases in question we find ourselves met by definite contradictions, of which the one discussed in Chapter x is an example. (P. 362) And in a footnote he adds: "It was in this way that I discovered this contradiction". Throughout his writings Russell left a number ofhints concerning the sort of connection he had drawn between Cantor's proof and his own discovery. In fact, his suggestions are so specific that there would seem to be little room left for speculation concerning how the discovery took place.2 The picture that emerges almost immediately from Russell's observations is the following. For reasons which are 1 See, e.g., Bertrand Russell, The Principles of Mathematics, 2nd ed. (London: Allen & Unwin, 1937), §§100, 344-9; G. Frege, Wissenschaftlicher Briefwechsel (Hamburg: Felix Meiner Verlag, 1976), pp. 215-16; B. Russell, Essays in Analysis, ed. D. Lackey (New York: George Braziller, 1973), p. 139; B. Russell, Introducrion to Mathemarical Philosophy (New York: Simon and Schuster, 1971), p. 136; B. Russell, My Philosophical Development (New York: Simon and Schuster, 1959), pp. 75-6; The Autobiography of Bertrand Russell, I (New York: Bantam Books, 1968), 195. 2 See, e.g., Ch. Thiel's "Einleitung des Herausgebers" in Frege, Wissenschaftlicher Briefwechsel, pp. 203 and 216 (footnote), and 1. Grattan-Guinness, "How Bertrand Russell Discovered his Paradox", Historia Marhematica, 5 (1978),127-37. 31 32 Russell, nos. 33-4 (Spring-Summer 1979) never made clear, Russell decided to analyze Cantor's argument applying it to "large" classes such as the universal class V, and the class of all classes. When, for example, we consider V, its power set and the correlationf(x) = {x} if x is not a class, f(x) =. x otherwise, then Cantor's diagonal class D turns out to be the class of all classes not belonging to themselves. Moreover, since the element ofV which is taken toD byfisD itself(i.e., sincef(D) = D), Cantor's reasoning invites us to raise the question whether D belongs tof(D) (i.e., toD) or not; and it establishes that it does precisely if it doesn't.3 The purpose of this note is to present recently uncovered information that complements and corrects our present understanding of Russell's discovery of his paradox. As it turns out, far from originating from his desire to apply the ideas in Cantor's theorem, a version of Russell's paradox first occurred in an argument that Russell had devised, late in 1900, in order to establish the invalidity of that theorem. An appeal to "large" classes such as V or Class was, as we shall see, essential to Russell's attempted refutation. As he displayed the details of his counterexample Russell's paradox emerged, at first unrecognized, as the by-product of a project that the paradox itself would eventually undermine. In a paper written in 1901 and largely devoted to a popular exposition and defence of Cantor's theory of the infinite, Russell expressed what appeared to be a rpinor reservation to Cantor's treatment : There is a greatest of all infinite numbers, which is the number of things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because, if everything has been taken, there is nothing left to add. Cantor has a proof that there is no greatest number, and if this proof were valid, the contradictions of infinity would reappear in a sublimated form. But in this one point, the master has been guilty of a very subtle fallacy, which I hope to explain in some future work.4 3 See Principles, §349. Perhaps I should remind... (shrink)

Can we quantify over everything: absolutely, positively, definitely, totally, every thing? Some philosophers have claimed that we must be able to do so, since the doctrine that we cannot is self-stultifying. But this treats restrictivism as a positive doctrine. Restrictivism is much better viewed as a kind of militant quietism, which I call dadaism. Dadaists advance a hostile challenge, with the aim of silencing everyone who holds a positive position about ‘absolute generality’.

Mathematics tells us there exist infinitely many prime numbers. Nominalist philosophy, introduced by Goodman and Quine, tells us there exist no numbers at all, and so no prime numbers. Nominalists are aware that the assertion of the existence of prime numbers is warranted by the standards of mathematical science; they simply reject scientific standards of warrant.