{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,12,5]],"date-time":"2022-12-05T15:26:22Z","timestamp":1670253982873},"reference-count":57,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":3058,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[2005,9]]},"abstract":"Despite some discomfort with this grandly philosophical topic, I do in fact hope to address a venerable pair of philosophical chestnuts: mathematical truth and existence. My plan is to set out three possible stands on these issues, for an exercise in compare and contrast. A word of warning, though, to philosophical purists (and perhaps of comfort to more mathematical readers): I will explore these philosophical positions with an eye to their interconnections with some concrete issues of set theoretic method.<\/jats:p>Let me begin with a brief look at what to count as \u2018philosophy\u2019. To some extent, this is a matter of usage, and mathematicians sometimes classify as \u2018philosophical\u2019 any considerations other than outright proofs. So, for example, discussions of the propriety of particular mathematical methods would fall under this heading: should we prefer analytic or synthetic approaches in geometry? Should elliptic functions be treated in terms of explicit representations (as in Weierstrass) or geometrically (as in Riemann)? Should we allow impredicative definitions? Should we restrict ourselves to a logic without bivalence or the law of the excluded middle? Also included in this category would be the trains of thought that shaped our central concepts: should a function always be defined by a formula? Should a group be required to have an inverse for every element? Should ideal divisors be defined contextually or explicitly, treated computationally or abstractly? In addition, there are more general questions concerning mathematical values, aims and goals: Should we strive for powerful theories or low-risk theories? How much stress should be placed on the fact or promise of physical applications? How important are interconnections between the various branches of mathematics? These philosophical questions of method naturally include several peculiar to set theory: should set theorists focus their efforts on drawing consequences for areas of interest to mathematicians outside mathematical logic? Should exploration of the standard axioms of ZFC be preferred to the exploration and exploitation of new axioms? How should axioms for set theory be chosen? What would a solution to the Continuum Problem look like?<\/jats:p>","DOI":"10.2178\/bsl\/1122038992","type":"journal-article","created":{"date-parts":[[2005,7,22]],"date-time":"2005-07-22T18:50:46Z","timestamp":1122058246000},"page":"351-376","source":"Crossref","is-referenced-by-count":13,"title":["Mathematical Existence"],"prefix":"10.1017","volume":"11","author":[{"given":"Penelope","family":"Maddy","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600003115_ref038","doi-asserted-by":"publisher","DOI":"10.1023\/A:1010604808920"},{"key":"S1079898600003115_ref013","volume-title":"Georg Cantor: His mathematics and philosophy of the infinite","author":"Dauben","year":"1979"},{"key":"S1079898600003115_ref048","volume-title":"Philosophy of mathematics: Structure and ontology","author":"Shapiro","year":"1997"},{"key":"S1079898600003115_ref040","volume-title":"Oxford handbook for the philosophy of mathematics and logic","author":"Maddy"},{"key":"S1079898600003115_ref049","volume-title":"Thinking about mathematics","author":"Shapiro","year":"2000"},{"key":"S1079898600003115_ref034","volume-title":"Realism in mathematics","author":"Maddy","year":"1990"},{"key":"S1079898600003115_ref050","first-page":"1","article-title":"The future of set theory","volume":"6","author":"Shelah","year":"1993","journal-title":"Israel Mathematical Conference Proceedings"},{"key":"S1079898600003115_ref036","unstructured":"Maddy Penelope [2000], Does mathematics need new axioms?, pp. 413\u2013422, in Feferman et al. [2000]."},{"key":"S1079898600003115_ref032","volume-title":"Mathematical thought from ancient to modern times","author":"Kline","year":"1972"},{"key":"S1079898600003115_ref029","volume-title":"Collected works, volume II","author":"G\u00f6del","year":"1990"},{"key":"S1079898600003115_ref008","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/12.1.18"},{"key":"S1079898600003115_ref053","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9281-1"},{"key":"S1079898600003115_ref019","volume-title":"In the light of logic","author":"Feferman","year":"1998"},{"key":"S1079898600003115_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-7731-1"},{"key":"S1079898600003115_ref033","volume-title":"Mathematics: the loss of certainty","author":"Kline","year":"1980"},{"key":"S1079898600003115_ref027","unstructured":"G\u00f6del Kurt [1944], Russell's mathematical logic, reprinted in G\u00f6del [1990], pp. 119\u2013141."},{"key":"S1079898600003115_ref051","unstructured":"Steel John [2000], Mathematics needs new axioms, pp. 422-433, in Feferman et al. [2000]."},{"key":"S1079898600003115_ref004","doi-asserted-by":"publisher","DOI":"10.1093\/0195159888.001.0001"},{"key":"S1079898600003115_ref025","volume-title":"Foundations of arithmetic","author":"Frege","year":"1884"},{"key":"S1079898600003115_ref057","first-page":"681","article-title":"The continuum hypothesis, Parts I and II","volume":"48","author":"Woodin","year":"2001","journal-title":"Notices of the American Mathematical Society"},{"key":"S1079898600003115_ref016","volume-title":"The evolution of physics","author":"Einstein","year":"1938"},{"key":"S1079898600003115_ref017","unstructured":"Feferman Solomon [1987], Infinity in mathematics: is Cantor necessary?, reprinted in Feferman [1998], pp. 28\u201373."},{"key":"S1079898600003115_ref020","doi-asserted-by":"crossref","unstructured":"Feferman Solomon [2000], Why the programs for new axioms need to be questioned, pp. 401\u2013413, in Feferman et al. 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