{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,28]],"date-time":"2025-03-28T00:53:58Z","timestamp":1743123238186,"version":"3.40.3"},"publisher-location":"Dordrecht","reference-count":59,"publisher":"Springer Netherlands","isbn-type":[{"type":"print","value":"9789400744349"},{"type":"electronic","value":"9789400744356"}],"license":[{"start":{"date-parts":[[2012,1,1]],"date-time":"2012-01-01T00:00:00Z","timestamp":1325376000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2012,1,1]],"date-time":"2012-01-01T00:00:00Z","timestamp":1325376000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012]]},"DOI":"10.1007\/978-94-007-4435-6_8","type":"book-chapter","created":{"date-parts":[[2012,7,10]],"date-time":"2012-07-10T14:08:37Z","timestamp":1341929317000},"page":"161-180","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Primitive Recursive Arithmetic and Its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections"],"prefix":"10.1007","author":[{"given":"William W.","family":"Tait","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2012,6,7]]},"reference":[{"key":"8_CR1","unstructured":"Bernays, P. 1930\u20131931. Die Philosophie der Mathematik und die Hilbertsche Beweistheorie. Bl\u00e4tter f\u00fcr deutsche Philosophie 4:\u00a0326\u2013367. Reprinted in Bernays\u00a0(1976). A translation by P.\u00a0Mancosu appears in Mancosu\u00a0(1998), 234\u2013265."},{"key":"8_CR2","volume-title":"Abhandlungen zur philosophie der mathematik","author":"P Bernays","year":"1976","unstructured":"Bernays, P. 1976. Abhandlungen zur philosophie der mathematik. Darmstadt: Wissenschaftliche Buchgesellschaft."},{"key":"8_CR3","unstructured":"Cantor, G. 1883a. Grundlagen einer allgemeinen Mannigfaltigheitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen. Leipzig: Teubner. A separate printing of Cantor\u00a0(1883b), with a subtitloe, preface and some footnotes added. A translation Foundations of a general theory of manifolds: a mathmatico-philosophical investigation into the theory of the infinite by W. Ewald is in (Ewald,\u00a01996, 639\u2013920)."},{"key":"8_CR4","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1007\/BF01446819","volume":"21","author":"G. Cantor","year":"1883b","unstructured":"Cantor, G. 1883b. \u00dcber unendliche, lineare Punktmannigfaltigkeiten, 5. Mathematische Annalen 21:\u00a0545\u2013586. In Cantor\u00a0(1932).","journal-title":"Mathematische Annalen"},{"key":"8_CR5","volume-title":"In Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed","author":"G Cantor","year":"1932","unstructured":"Cantor, G. 1932. In Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. E. Zermelo. Berlin: Springer."},{"key":"8_CR6","doi-asserted-by":"crossref","unstructured":"Curry, H. 1940. A formalization of recursive arithmetic. Bulletin of the American Mathematical Society 263\u2013282.","DOI":"10.2307\/2371522"},{"key":"8_CR7","unstructured":"Dedekind, R. 1872. Stetigkeit und irrationale Zahlen. Braunschweig: Vieweg. In Dedekind\u00a0(1932). Republished in 1969 by Vieweg and translated in Dedekind\u00a0(1963)."},{"key":"8_CR8","unstructured":"Dedekind, R. 1888. Was sind und was sollen die Zahlen? Braunschweig: Vieweg. In Dedekind\u00a0(1932). Republished in 1969 by Vieweg and translated in Dedekind\u00a0(1963)."},{"key":"8_CR9","unstructured":"Dedekind, R. 1890. Letter to Keferstein. Translated in van Heijenoort\u00a0(1967), 99\u2013103. Cambridge: Harvard University Press."},{"key":"8_CR10","volume-title":"In Gesammelte Werke","author":"R Dedekind","year":"1932","unstructured":"Dedekind, R. 1932. In Gesammelte Werke, vol. 3, ed. R. Fricke, E. Noether, and O. Ore. Braunschweig: Vieweg."},{"key":"8_CR11","unstructured":"Dedekind, R. 1963. Essays on the theory of numbers. New York: Dover. English translation by W.W. Berman of Dedekind\u00a0(1872) and Dedekind\u00a0(1888)."},{"key":"8_CR12","volume-title":"Richard Dedekind et les fondements des mathematiques (avec de nombeux textes in\u00e9dits)","author":"P Dugac","year":"1976","unstructured":"Dugac, P. 1976. Richard Dedekind et les fondements des mathematiques (avec de nombeux textes in\u00e9dits). Paris: Librairie Philosophique J. Vrin."},{"volume-title":"From Kant to Hilbert: a source book in the foundations of mathematics","year":"1996","key":"8_CR13","unstructured":"Ewald, W. (ed.). 1996. From Kant to Hilbert: a source book in the foundations of mathematics. Oxford: Oxford University Press. Two volumes."},{"key":"8_CR14","volume-title":"Begriffsschrift, eine der arithmetischen nachgebildete Formalsprache des reinen Denkens","author":"G Frege","year":"1879","unstructured":"Frege, G. 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formalsprache des reinen Denkens. Halle: L. Nebert."},{"key":"8_CR15","unstructured":"Frege, G. 1893. Grundgesetze der Arithmetik: Begriffsschriftlich abgeleitet, Band I, Jena: H. Pohle. Reprinted in 1962 along with Frege\u00a0(1903) by Hildesheim: Georg Olms."},{"key":"8_CR16","volume-title":"Grundgesetze der Arithmetik: Begriffsschriftlich abgeleitet, Band II","author":"G Frege","year":"1903","unstructured":"Frege, G. 1903. Grundgesetze der Arithmetik: Begriffsschriftlich abgeleitet, Band II. Jena: H.\u00a0Pohle."},{"key":"8_CR17","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1111\/j.1746-8361.1958.tb01464.x","volume":"12","author":"K. G\u00f6del","year":"1958","unstructured":"G\u00f6del, K. 1958. \u00dcber eine bisher noch nicht ben\u00fctzte Erweiterung des finiten Standpunktes. Dialectica 12:\u00a0280\u2013287. Reprinted with an Englsh translation in (G\u00f6del,\u00a01990, 240\u2013252). G\u00f6del\u00a0(1972) is a revised version.","journal-title":"Dialectica"},{"key":"8_CR18","doi-asserted-by":"crossref","unstructured":"G\u00f6del, K. 1972. On an extension of finitary mathematics which has not yet been used. In Collected works, vol. II G\u00f6del\u00a0(1990), 271\u2013280. Revised version of G\u00f6del\u00a0(1958).","DOI":"10.1093\/oso\/9780195147216.003.0022"},{"key":"8_CR19","volume-title":"Collected works","author":"K G\u00f6del","year":"1990","unstructured":"G\u00f6del, K. 1990. Collected works, vol. II. Oxford: Oxford University Press."},{"key":"8_CR20","first-page":"61","volume-title":"History and philosophy of modern mathematics: minnesota studies in the philosophy of science, vol. XI","author":"W Goldfarb","year":"1988","unstructured":"Goldfarb, W. 1988. Poincar\u00e9 against the logicists. In History and philosophy of modern mathematics: minnesota studies in the philosophy of science, vol. XI, ed. W.\u00a0Aspray and P.\u00a0Kitcher, 61\u201381. Minneapolis: The University of Minnesota Press."},{"key":"8_CR21","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1112\/plms\/s2-48.1.401","volume":"48","author":"R Goodstein","year":"1945","unstructured":"Goodstein, R. 1945. Function theory in an axiom-free equation calculus. Proceedings of the London Mathematical Society 48:\u00a0401\u201334.","journal-title":"Proceedings of the London Mathematical Society"},{"key":"8_CR22","volume-title":"Recursive number theory","author":"R Goodstein","year":"1957","unstructured":"Goodstein, R. 1957. Recursive number theory. Amsterdam: North-Holland."},{"key":"8_CR23","volume-title":"Gesammelte mathematische und physikalische Werke","author":"H Grassmann","year":"1904","unstructured":"Grassmann, H. 1904. Gesammelte mathematische und physikalische Werke, vol.\u00a02. Leipzig: Druck und Verlag von B.G. Teubner."},{"key":"8_CR24","unstructured":"Hilbert, D. 1905. \u00dcber die Grundlagen der Logik und der Arithmetik. In Verhandlungen des Dritten Internationalen Mathematiker-Kongress. Leipzig: Teubner."},{"key":"8_CR25","doi-asserted-by":"crossref","unstructured":"Hilbert, D. 1918. Axiomatisches denken. Mathematishe Annalen 78:\u00a0405\u201315. Reprinted in (Hilbert,\u00a01932-9325, vol. 3, 1105\u20131115). Translated by W. Ewald in (Ewald,\u00a01996, vol. 2).","DOI":"10.1007\/BF01457115"},{"key":"8_CR26","doi-asserted-by":"crossref","unstructured":"Hilbert, D. 1922. Neubegr\u00fcndung der Mathematik: Erste Mitteilung. Abhandlungen aus dem Seminar der Hamburgischen Universit\u00e4t 1:\u00a0157\u2013177. English translation in (Mancosu,\u00a01998, 198\u2013214) and (Ewald,\u00a01996, 1115\u20131134).","DOI":"10.1007\/BF02940589"},{"key":"8_CR27","doi-asserted-by":"crossref","unstructured":"Hilbert, D. 1923. Die logischen Grundlagen der Mathematik. Mathematische Annalen 88: 151\u2013165. English translation in (Ewald,\u00a01996, 1134\u20131148).","DOI":"10.1007\/BF01448445"},{"key":"8_CR28","doi-asserted-by":"crossref","unstructured":"Hilbert, D. 1926. \u00dcber das Unendliche. Mathematische Annalen 95:\u00a0161\u201390. Translated by Stefan Bauer-Mengelberg in From Frege to G\u00f6del: a source book in mathematical logic, 367\u201392.","DOI":"10.1007\/BF01206605"},{"key":"8_CR29","doi-asserted-by":"crossref","unstructured":"Hilbert, D. 1932\u20139325. Gesammelte Abhandlungen. Bedrlin: Springer. 3 volumes.","DOI":"10.1007\/978-3-642-50831-8"},{"key":"8_CR30","unstructured":"Hilbert, D. 2011. In David Hilbert\u2019s Lectures on the Foundations of Arithmetic and Logic 1917\u20131933, ed. M. Hallett, W. Ewald, W. Sieg and U. Majer. Berlin: Springer."},{"key":"8_CR31","unstructured":"Kronecker, L. 1881. Grundz\u00fcge einer arithmetischen Theorie der algebraischen Gr\u00f6ssen. In Leopold Kronecker\u2019s Werke, vol.\u00a02, ed. K.\u00a0Hensel, 236\u2013387 New York: Chelsea."},{"key":"8_CR32","first-page":"147","volume-title":"Leopold Kronecker\u2019s Werke","author":"L Kronecker","year":"1886","unstructured":"Kronecker, L. 1886. \u00dcber einige Anwendungen der Modulsysteme auf elementare algebraische Fragen. In Leopold Kronecker\u2019s Werke, vol.\u00a03, ed. K.\u00a0Hensel, 147\u2013208. New York: Chelsea."},{"key":"8_CR33","doi-asserted-by":"crossref","unstructured":"Kronecker, L. 1887. ber den zahlbegriff. In Leopold Kronecker\u2019s Werke, ed. K.\u00a0Hensel, 251\u2013274, New York: Chelsea.","DOI":"10.1515\/crll.1887.101.337"},{"volume-title":"From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920\u2019s","year":"1998","key":"8_CR34","unstructured":"Mancosu, P. (ed.). 1998. From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920\u2019s. Oxford: Oxford University Press."},{"key":"8_CR35","doi-asserted-by":"publisher","first-page":"303","DOI":"10.2307\/421183","volume":"5","author":"P Mancosu","year":"1999","unstructured":"Mancosu, P. 1999. Between russell and hilbert: Behmann on the foundations of mathematics. Bulletin of Symbolic Logic 5: 303\u2013330.","journal-title":"Bulletin of Symbolic Logic"},{"key":"8_CR36","volume-title":"Logic colloqium \u201973","author":"P Martin-L\u00f6f","year":"1973","unstructured":"Martin-L\u00f6f, P. 1973. An intuitionistic theory of types: predicative part. In Logic colloqium \u201973, ed. H.\u00a0E. Rose and J.\u00a0C. Shepherdson. Amsterdam: North-Holland."},{"key":"8_CR37","first-page":"221","volume-title":"Twenty-five years of constructive type theory","author":"P Martin-L\u00f6f","year":"1998","unstructured":"Martin-L\u00f6f, P. 1998. An intuitionistic theory of types. In Twenty-five years of constructive type theory, ed. G.\u00a0Sambin and J.\u00a0Smith, 221\u2013244. Oxford: Oxford University Press."},{"key":"8_CR38","volume-title":"Lehrbuch der Combinatorik","author":"E Netto","year":"1901","unstructured":"Netto, E. 1901. Lehrbuch der Combinatorik. Leipzig: Verlag von B.G. Teubner."},{"key":"8_CR39","unstructured":"Poincar\u00e9, H. 1894. Sur la Nature du Raisonnement math\u00e9matique. Revue de m\u00e9taphysique et de morale 2:\u00a0371\u201384. Translation by George Bruce Halsted in Ewald\u00a0(1996), vol. 2, 972\u2013982."},{"key":"8_CR40","unstructured":"Poincar\u00e9, H. 1900. Du r\u00f4le de l\u2019intuition et de la logique en math\u00e9matiques. In Compte rendu du Deuxi\u00e9me congr\u00e8s international des math\u00e9maticiens tenu \u00e0 Paris du 6 au 12 ao\u00fbt 1900, 210\u201322. Pais: Gauthier-Villars. Translation by George Bruce Halsted, reprinted Ewald\u00a0(1996), vol. 2, 1021\u20131038."},{"key":"8_CR41","unstructured":"Poincar\u00e9, H. 1905. Les math\u00e9matiques et la logique. Revue de m\u00e9taphysique et de morale 13: 815\u201335. Translation by George Bruce Halsted in Ewald\u00a0(1996), vol. 2, 1021\u20131038."},{"key":"8_CR42","unstructured":"Poincar\u00e9, H. 1906a. Les math\u00e9matiques et la logique. Revue de m\u00e9taphysique et de morale 14: 17\u201334. Translation by George Bruce Halsted in Ewald\u00a0(1996), vol. 2, 1038\u20131052."},{"key":"8_CR43","unstructured":"Poincar\u00e9, H. 1906b. Les math\u00e9matiques et la logique. Revue de m\u00e9taphysique et de morale 14: 294\u2013317. Translation by George Bruce Halsted in Ewald\u00a0(1996), vol. 2, 1052\u20131071."},{"key":"8_CR44","first-page":"866","volume":"14","author":"H Poincar\u00e9","year":"1906","unstructured":"Poincar\u00e9, H. 1906c. A propos de la logistique. Revue de m\u00e9taphysique et de morale 14:\u00a0866\u2013868.","journal-title":"Revue de m\u00e9taphysique et de morale"},{"key":"8_CR45","unstructured":"Poincar\u00e9, H. 1909. Le llogique de l\u2019infin i. Revue de m\u00e9taphysique et de morale 17:\u00a0461\u201382. Translation by George Bruce Halsted in Ewald\u00a0(1996), vol. 2, 1038\u20131052."},{"key":"8_CR46","doi-asserted-by":"publisher","first-page":"925","DOI":"10.1090\/S0002-9904-1947-08911-4","volume":"53","author":"J Robinson","year":"1947","unstructured":"Robinson, J. 1947. Primitive recursive functions. Bulletin of the American Mathematical Society 53:\u00a0925\u2013942.","journal-title":"Bulletin of the American Mathematical Society"},{"key":"8_CR47","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/421139","volume":"5","author":"W Sieg","year":"1999","unstructured":"Sieg, W. 1999a. Hilbert\u2019s programs: 1917\u20131922. Bulletin of Symbolic Logic 5:\u00a01\u201344.","journal-title":"Bulletin of Symbolic Logic"},{"key":"8_CR48","unstructured":"Skolem, T. 1923. Einige Bemerkungen zur axiomatischen Begr\u00fcndung der Mengenlehre. Matematikerkongressen in Helsingfors 4\u20137 Juli 1922, Den femte skandinaviske matematikerkongressen, Redog\u00f6relse, 217\u2013232. Helsingfors: Akademiska Bokhandeln."},{"key":"8_CR49","unstructured":"Skolem, T. 1947. The development of recursive arithmetic. In Copenhagen: proceedings of the tenth congress of scandinavian mathematicians, 1\u201316. Reprinted in Skolem\u00a0(1970), 499\u2013514."},{"key":"8_CR50","first-page":"10","volume":"XXIX","author":"T. Skolem","year":"1956","unstructured":"Skolem, T. 1956. A version of the proof of equivalence between complete induction and the uniqueness of primitive recursions. Kongelige Norske Videnskabsselskabs Forhandlinger XXIX:\u00a010\u201315.","journal-title":"Kongelige Norske Videnskabsselskabs Forhandlinger"},{"key":"8_CR51","volume-title":"In Selected works in logic, ed","author":"T Skolem","year":"1970","unstructured":"Skolem, T. 1970. In Selected works in logic, ed. J.E. Fenstad. Oslo: Universitetsforlaget."},{"key":"8_CR52","first-page":"1","volume-title":"Recursive function theory, proceedings of symposia in pure mathematics","author":"C Spector","year":"1962","unstructured":"Spector, C. 1962. Provably recursive functionals of analysis: a consistency proof of analysis by an extension of the principles formulated in current intuitionistc mathematics. In Recursive function theory, proceedings of symposia in pure mathematics, vol.\u00a05, ed. J.\u00a0Dekker, 1\u201327. Providence: American Mathematical Society."},{"key":"8_CR53","doi-asserted-by":"publisher","first-page":"524","DOI":"10.2307\/2026089","volume":"78","author":"W Tait","year":"1981","unstructured":"Tait, W. 1981. Finitism. Journal of Philosophy 78:\u00a0524\u2013556.","journal-title":"Journal of Philosophy"},{"key":"8_CR54","doi-asserted-by":"crossref","unstructured":"Tait, W. 2000. Cantor\u2019s Grundlagen and the paradoxes of set theory, 269\u201390. Reprinted in Tait\u00a0(2005b), 252\u2013275.","DOI":"10.1017\/CBO9780511570681.014"},{"key":"8_CR55","doi-asserted-by":"crossref","unstructured":"Tait, W. 2005a. Proof-theoretic semantics for classical mathematics. In Proof-theoretic semantics for classical mathematics, ed. R.\u00a0Kahle and P.\u00a0Schroeder-Heister. Special edition of Synthese.","DOI":"10.1007\/s11229-004-6271-x"},{"key":"8_CR56","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195141924.001.0001","volume-title":"The provenance of pure reason: essays in the philosophy of mathematics and its history","author":"W Tait","year":"2005","unstructured":"Tait, W. 2005b. The provenance of pure reason: essays in the philosophy of mathematics and its history. Oxford: Oxford University Press."},{"key":"8_CR57","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1093\/philmat\/nkj004","volume":"14","author":"W Tait","year":"2006","unstructured":"Tait, W. 2006. G\u00f6del\u2019s interpretation of intuitionism. Philosophia Mathematica 14: 208\u2013228.","journal-title":"Philosophia Mathematica"},{"volume-title":"From Frege to G\u00f6del: A Source Book in Mathematical Logic","year":"1967","key":"8_CR58","unstructured":"van Heijenoort, J. (ed.). 1967. From Frege to G\u00f6del: A Source Book in Mathematical Logic. Cambridge: Harvard University Press."},{"key":"8_CR59","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/BF02102305","volume":"10","author":"H. Weyl","year":"1921","unstructured":"Weyl, H. 1921. \u00dcber die neue Grundlagenkrise der Mathematik. Mathematische Zeitschrift 10: 39\u201379. Translated by P. Mancosu in Mancosu\u00a0(1998).","journal-title":"Mathematische Zeitschrift"}],"container-title":["Epistemology versus Ontology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-94-007-4435-6_8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,26]],"date-time":"2024-04-26T17:29:07Z","timestamp":1714152547000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-94-007-4435-6_8"}},"subtitle":["In Honor of Per Martin-L\u00f6f on the Occasion of His Retirement"],"short-title":[],"issued":{"date-parts":[[2012]]},"ISBN":["9789400744349","9789400744356"],"references-count":59,"URL":"https:\/\/doi.org\/10.1007\/978-94-007-4435-6_8","relation":{},"subject":[],"published":{"date-parts":[[2012]]},"assertion":[{"value":"7 June 2012","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}