Michael D. RESNIK (Ed.): Mathemati­ dispensability argument), Gödel (mathe­ cal Objects and Mathematical Know­ matical reaIism) and the (later) Witttgen­ ledge. Aldershot / Broockfield, USA / stein (his ideas were transmitted largely Singapore / Sydney: Dartmouth 1995, through the work ofM. Dummett who ap­ xxi + 647 pp. (= The International Re­ plied them to intuitionistic logic) began to search Library of Philosophy 13). influence the course ofthe contemporary philosophy of mathematics. The book under review is an anthology of The papers collected in the volume texts in the contemporary philosophy of concentrate on two fundamental ques­ mathematics. It was published in the se­ tions wh ich have been in fact considered ries ofbooks "The International Research since Plato: What are mathematical ob­ Library of Philosophy". The aim of this jects? and How can we know them? series is to collect "in book form a wide Resnik claims that most of the collected range of important and influential essays essays were strongly influenced by arti­ in philosophy, drawn predominantly cles by Paul Benacerraf and Hilary Put­ from English-language journals" (p. xi). nam which had appeared in the decade Each volume deals with a field of inquiry 1965-1975, especially by Benacerrafs which has received significant attention "What Numbers Could Not Be" and in philosophy in the last 25 years. The Li­ "Mathematical Truth" and Putnam' s brary is divided into four series of vol­ "Mathematics without Foundations". umes: (1) Metaphysics and Epistemol­ The texts in the first part of the book ogy, (2) The Philosophy ofMathematics are collected under the label "Field's re­ and Science, (3) The Philosophy of sponse" and are devoted to Hartry Field's Logic, Language and Mind, (4) The Phi­ anti-realist ideas claiming that there are losophy ofValue. no mathematical objects and that The book under review was published mathematics is theoretically dispensable. as the second volume of the series "The Field's programme was illustrated by de­ Philosophy of Mathematics and Sc i­ veloping aversion ofNewtonian gravita­ ence". It was edited by Michael D. Resnik tion theory in a nominaIistic style avoid­ from University of North Carolina at ing abstract mathematical concepts. Chapel Hill. The book consists of 24 pa­ (Resnik claims in the Introduction that pers collected in 7 parts as weil as an In­ "probably no other recent work in the phi­ troduction written by the editor and a losophy of mathematics has generated short name index. more interest, admiration and discussion Introduction provides an overview of than Field's" (p. xvii).) One finds in the the contemporary philosophy of mathe­ first part two papers by Hartry Field ("Re­ matics and a survey of the positions de­ alism and Anti-ReaIism about Mathemat­ veloped in the volume. Resnik argues ics" and "Is Mathematical Knowledge here that "philosophy ofmathematics be­ Just Logical Knowledge?"), David Mala­ came a speciality only at the beginning of ment's review of Field's book Science the 20th century - thanks to the work of without Numbers, Stewart Shapiro's pa­ Frege and RusselI" (p. xiii). This epoch per "Conservativeness and Incomplete­ brought logicism Gust Frege and Rus­ ness" in which a serious ambiguity in the seil), intuitionism (Brouwer) and formal­ notion of conservativeness - a key notion ism (Hilbert). It gradually drew to a close in Field's conception - is highlighted and in the late 1940s and early 1950s. At the Field' s ability to dispense with the deduc­ same time the writings of Quine (the in- tive use of mathemtaics is questioned as 258 weIl as John Burgess' paper "Synthetic paper by Michael D. Resnik ( "Second­ Mechanics" in which a discussion ofthe Order Logic Still Wild" ) and a paper by relationship between intrinsic and extrin­ Stewart Shapiro ("Second-Order Logic, sic foundations of physical theories is Foundations, and Rules"). Boolos pro­ considered. posed to interprete second-order logic Papers coIlected in Part 11 provide an­ quantifiers as plural quantifiers ranging other anti-realist response, namely the over individuals (and to avoid in this way claim that mathematics is reaIly talk the necessity of engaging sets, classes or about possibilities involving concrete ob­ Fregean concepts in concepts based on jects. One can find here a paper by Char­ second-order logic). Papers by Resnik les S. Chichara ("A Simple Type Theory and Shapiro discuss Boolos' programme. Without Platonic Domains") and a paper Part VI contains papers devoted to the by Philip Kitcher ("Arithmetic for the structuralism - the claim that mathemat­ Millian"). ics studies structures and that mathemati­ Papers in Parts III and IV represent cal objects are featureless positions in the realism. In particular in Part III one these structures. This view was in fact finds two papers by Penelope Maddy: proclaimed already by Dedekind. Struc­ "Physicalistic Platonism" and "Percep­ turalist ideas can be found in the writings tion and Mathematical Intuition". The re­ of Hilbert, Bernays, Bourbaki and cate­ alism ofMaddy is a naturalistic version of gory theorists. In the volume one finds Gödel's realism. She claims that we can papers by contemporary representatives see sets, more exactly, sets of concrete of structuralism, namely by Michael D. objects whose members are before our Resnik ("Mathematics as a Science of eyes, and that mathematics is ultimately Patterns: Ontology and Reference" and about sets. A paper by Charles S. "A Naturalized Epistemology for a Pla­ Chichara included into Part III ("A Gö• tonist Mathematical Ontology"), Stewart delian Thesis Regarding Mathematical Shapiro ("Structure and Ontology") and Objects: Do They Exist? And Can We Geoffrey HeIlman ("Modal-Structural Perceive Them?") is a critical discussion Mathematics"). A critical survey by ofboth Gödel and Maddy. Charles Parsons ("The Structuralist View The papers by Crispin Wright ("Why of Mathematical Objects") was included Numbers Can Believably Be: A Reply to here as weil. Hartry Field") and W. W. Tait ("Truth and The book is closed by Part VII where a Proof: The Platonism of Mathematics") paper by Charles Parsons ("Mathematical included into Part IV represent another Intuition") and a paper by Richard Tie­ approach to realism. In particular Wright szen ("Phenomenology and Mathemati­ argues that to establish that numbers exist cal Knowledge") devoted to mathemati­ it suffices to establish that number words cal intuition were included. Parsons function as singular terms. claims in his essay that some mathemati­ Part V deals with some problems con­ cal objects are not featureless positions in nected with the usage ofthe second-order patterns and that consequently we may logic. The main question considered in grasp some ofthem through sensuous in­ papers coIlected in this part is whether tuition. Tieszen expounds a theory of second-order logic is a new tool for logis­ mathematical intuition that combines tic and other mathematical reductionists. themes from both Parsons and Edmud One finds here three papers by George Husserl. Boolos ( "To Be is To Be a Value of a So far the contents of the book. One Variable (Or To Be Some Values ofSome should add that the volume was published Variables)", "Nominalist Platonism" and in a careful way - aIl papers are repro­ "Saving Frege from Contradiction"), a duced in an genuine setting, with genuine