Segre, Michael Peano’s axioms in their historical context. (English) Zbl 0835.01006 Arch. Hist. Exact Sci. 48, No. 3-4, 201-342 (1994). The paper is a full-scale study on Giuseppe Peano’s philosophy of mathematics, published with several rare photographs and a list of Peano’s writings (pp. 322-335) taken (with corrections) basically from H. C. Kennedy’s pioneer work ‘Life and work of Giuseppe Peano’ [Dordrecht, Reidel (1980; Zbl 0429.01015)]. It concentrates on the mathematical and logical contexts of Peano’s axiomatization of arithmetic (1893). The author shows convincingly that Peano’s axiomatization was not intended to contribute to the logical foundations of mathematics, but that it was a reaction to the 19th century mathematics’ quest for rigor caused by the discovery of non-Euclidean geometry which undermined the trust in Euclidean geometry as guiding mathematical doctrine, and by the discovery of monster formulae in analysis. In particular the author tries to defend three theses (cf. p. 205): (1) Peano formulated his axioms to give a clear and rigorous presentation of arithmetic, and, thus, of mathematics in general. (2) Peano was not interested in reducing mathematics to logic. (3) Even limited to the rigor in symbolism, Peano’s sort of “foundationism” was a failure.The material is organized in eight chapters. Chapter 1 (pp. 208-224) gives an outline of the history of the concept of mathematical rigor from ancient times up to Cauchy. Chapter 2 (pp. 225-241) is devoted to Cauchy’s rigorous calculus and Weierstrass’s refinements. Chapter 3 (pp. 242-250) gives a brief presentation of Peano’s life and work following Kennedy’s biography. Chapter 4 (pp. 251-265) sketches the state of art in symbolic logic at Peano’s time and its emergence since Leibniz. Chapter 5 (pp. 266-280) gives a detailed presentation of the stages in Peano’s work which led him to the formulation of his axioms. Chapter 6 treats his mathematical work following the axiomatization of arithmetic, especially his discovery of “Peano’s curve” which showed the limited reliability of intuition in mathematics. Chapter 7 (pp. 287-300) discusses Peano’s axioms in detail. The final Chapter 8 relates Peano’s conception to found mathematics to formalism, logicism and intuitionism.The author stresses that Peano tried to use logic as an instrument in mathematics (p. 206). For him this indicates the originality of Peano’s approach compared with the logical systems of his predecessors and fellow logicians. This judgement can be doubted, since almost every mathematician doing logic up to the publication of Whitehead’s and Russell’s ‘Principia mathematica’ (1910/13) regarded the mathematization of logic as a conditio sine qua non for its applicability in mathematics. The best example might be Ernst Schröder who thought to use his algebra of relatives as a universal language being a device to reformulate all branches of mathematics and physics. The paper under review is an impressive example of contextual research in the history of mathematics. It proves the need to investigate the history of logic and foundations within a broader perspective on the development of mathematics and philosophy in general. Reviewer: V.Peckhaus (Erlangen) Cited in 5 Documents MSC: 01A55 History of mathematics in the 19th century Keywords:Peano arithmetic; rigor; foundations; axiomatics Biographic References: Peano, G. Citations:Zbl 0429.01015 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Abrusci, V. Michele. ?Peano e Hilbert: G. Peano nelle prime fasi delle ricerche fondazionali di D. Hilbert.? In Peano e i fondamenti della matematica, 185-195. [2] Agassi, Joseph. Towards an Historiography of Science. The Hague: Mouton, 1963; Middletown, Conn.: Wesleyan Univ. Press, 1967. [3] Agassi, Joseph. ?Logic and Logic of,? Pozna? Studies 4 (1978): 1-11. [4] Agassi, Joseph. ?Presuppositions for Logic.? The Monist 65 (1982): 465-480. [5] Aimonetto, Italo. ?Il concetto di numero naturale in Frege, Dedekind e Peano.? Filosofia (1969) 20: 579-606. [6] Archimedes. The Method of Archimedes, Recently Discovered by Heiberg: A Supplement to The Works of Archimedes 1897. Thomas L. Heath, ed. Cambridge: At the University Press, 1912. [7] Ashurst, Gareth F. Founders of Modern Mathematics. London: Frederick Muller, 1982. [8] Aspray, William, & Kitcher, Philip, eds. History and Philosophy of Modern Mathematics. Minneapolis: University of Minnesota Press, 1988. · Zbl 1367.00029 [9] Baker, G. P., & Hacker, P. M. S. Frege: Logical Excavations. New York; Oxford: Oxford Univ. Press; Basil Blackwell, 1984. [10] Becker, Oskar. Grundlagen der Mathematik: in geschichtlicher Entwicklung. 4th ed. Frankfurt a. M.: Suhrkamp, 1990. · Zbl 0718.01002 [11] Beth Evert W. The Foundations of Mathematics. Amsterdam: North-Holland, 1959. · Zbl 0085.24104 [12] Berkeley, George. Works. 9 vols. A. A. Luce & T. E. Jessop, eds. London: Nelson, 1948-1957. [13] Black, Max. The Nature of Mathematics: A Critical Survey. London: Kegan Paul, Trench, Trubner & Co., 1933. [14] Bolzano, Bernard. Schriften. Vol 1: Functionenlehre, Karel Rychlík, ed. Prague: Königliche Böhmische Gesellschaft der Wissenschaften, 1930. [15] Bolzano, Bernard. Rein analytischer Beweis des Lehrsatzes..., Prague, 1814-17. Ostwald’s Klassiker no. 153. Leipzig: Engelmann, 1905. [16] Bolzano, Bernard. Paradoxien des Unendlichen. F. P?ihonsky, ed. Berlin, 1889. · Zbl 0134.24601 [17] Boole, George. The Mathematical Analysis of Logic. Cambridge, 1847. Reprint. Oxford: Blackwell, 1965. [18] Boole, George. An Investigation of the Laws of Thought. Reprint. New York: Dover, 1958. · Zbl 0084.07701 [19] Borga, Marco. ?La logica, il metodo assiomatico e la problematica metateorica?, in Borga, Freguglia & Palladino, I contributi fondazionali della scuola di Peano, pp. 11-75, 1985. [20] Borga, M., Freguglia, P., & Palladino, D. I contributi fondazionali della scuola di Peano. Milan: Franco Angeli, 1985. [21] Bottazzini, Umberto. Il calcolo sublime: storia dell’analisi matematica da Euler a Weierstrass. Turin: Boringhieri, 1981. · Zbl 0507.01001 [22] Bottazzini, Umberto. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Warren Van Egmond, trans. New York: Springer, 1986. · Zbl 0597.01011 [23] Bottazzini, Umberto. ?Peano e la logica dei controesempi.? In Peano e i fondamenti della matematica, 237-253. [24] Boyer, Carl B. The History of the Calculus and its Conceptual Development (The Concepts of the Calculus). New York: Dover, 1959. · Zbl 0095.00302 [25] Breidert, Wolfgang. George Berkeley. Basel: Birkhäuser, 1989. [26] Brouwer, Luitzen Egbertus Jan. ?Intuitionism and Formalism?. Bulletin of the American Mathematical Society 20 (October 1913?July 1914): 81-96. · Zbl 0761.03001 [27] Brouwer, Luitzen Egbertus Jan. Collected Works. 2 vols. Vol 1: A. Heyting, ed. Vol. 2: Hans Freudenthal, ed. Amsterdam: North-Holland, 1975, 1976. [28] Cajori, Florian. A History of the Conceptions of Limits and Fluxions in Great Britain. Chicago: The Open Court, 1919. · JFM 47.0035.12 [29] Cajori, Florian. A History of Mathematical Notations. 2 vols. Chicago: The Open Court, 1928-1929. [30] Cassina, Ugo. ?L’Opera scientifica di Giuseppe Peano.? Rendiconti del Seminario Matematico e Fisico di Milano 7 (1933): 323-389. · Zbl 0009.38901 · doi:10.1007/BF02923786 [31] Cassina, Ugo. ?L’Oeuvre philosophique de G. Peano.? Revue de Métaphysique et de Morale 40 (1933): 481-491. [32] Cassina, Ugo. ?Storia ed analisi del ?Formulario completo? di Peano.? Bollettino della Unione Matematica Italiana 10 (1955): 244-265, 544-574. · Zbl 0066.24502 [33] Cassina, Ugo. ?Sul ?Formulario Matematico? di Peano,? in Terracini (ed.), In memoria di Giuseppe Peano, pp. 71-102, 1955. · Zbl 0066.00607 [34] Cauchy, Augustin. Oeuvres complètes d’Augustin Cauchy. Series 1, 12 vols., series 2, 15 vols. Paris: Gauthier-Villars, 1882-. [35] Celebrazioni in memoria di Giuseppe Peano nel cinquantenario della morte. Atti del Convegno organizzato dal Dipartimento di Matematica dell’Università di Torino, 27-28 ottobre 1982. Turin, 1982. [36] Cellucci, Carlo, ?Gli scopi della logica matematica.? in Peano e i Fondamenti della Matematica: 73-138. [37] Cohen, I. Bernard. Introduction to Newton’s ?Principia?. Cambridge (Mass.): Harvard University Press, 1971. · Zbl 0227.01011 [38] Couturat, Louis. ?La logique mathématique de M. Peano.? Revue de Métaphysique et de Morale 7 (1899): 616-646. [39] Couturat, Louis. La logique de Leibniz. Paris, 1901. Reprint. Hildesheim: Olms, 1969. [40] Crowe, Michael J. A History of Vector Analysis. Notre Dame: University of Notre Dame Press, 1967. · Zbl 0165.00303 [41] Davis, Philip, J., & Hersh, Reuben. The Mathematical Experience. Penguin Books 1990. [42] Dedekind, Richard. Gesammelte mathematische Werke. 3 vols. Robert Fricke, Emmy Noether & Öystein Ore, eds. Braunschweig: Friedr. Vieweg & Sohn, 1930-1932. [43] Dedekind, Richard. Essays on the Theory of Numbers. Wooster Woodruff Beman, trans. New York: Dover Publications, 1963. · Zbl 0112.28101 [44] Dictionary of Scientific Biography. Charles Coulston Gillispie, ed. 18 vols. New York: Charles Scribner’s Sons, 1970-1990. [45] Dieudonné, Jean A. ?The Work of Nicholas Bourbaki.? The American Mathematical Monthly 77 (1970): 134-145. · Zbl 0188.00701 · doi:10.2307/2317325 [46] Dugac, Pierre, ?Eléments d’analyse de Karl Weierstrass.? Archive for History of Exact Sciences 10 (1973): 41-176. · Zbl 0259.01012 · doi:10.1007/BF00343406 [47] Dummett, Michael. The Interpretation of Frege’s Philosophy. Cambridge, Mass.: Harvard Univ. Press, 1981. [48] Edwards, Charles Henry. The Historical Development of the Calculus. New York: Springer, 1979. [49] Euler, Leonhard. Opera omnia, 3 series. Leipzig, Berlin, Zurich, etc.: Teubner, 1911. · JFM 42.0008.01 [50] Fauvel, John, & Gray, Jeremy, eds. The History of Mathematics: A Reader. London: MacMillan, Milton Keynes: The Open University 1987. · Zbl 0931.01001 [51] Fourier, Joseph. Oeuvres de Fourier. 2 vols. Gaston Darboux, ed. Paris, 1888-1890. [52] Fourier, Joseph. The Analytical Theory of Heat. Alexandre Freeman, trans. New York: Dover, 1955. · Zbl 0066.07801 [53] Fraenkel, Abraham A., Bar-Hillel, Yehoshua, & Levy, Azriel. Foundations of Set Theory. 2nd ed. Amsterdam: North-Holland, 1984. · Zbl 0623.03047 [54] Frege, Gottlob. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle, 1879. Reprint. Hildesheim: Olms, 1964. [55] Frege, Gottlob. Grundgesetze der Arithmetik. Jena, 1893. Reprint. Hildesheim: Olms, 1962. [56] Frege, Gottlob. Die Grundlagen der Arithmetik: Eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau, 1884. · Zbl 0654.03005 [57] Frege, Gottlob. The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number. J. L. Austin, trans. Oxford: Blackwell, 1950. · Zbl 0037.00602 [58] Freguglia, Paolo, ?La logica matematica di Peano: un’analisi,? Physis, 23 (1981): 325-336. [59] Freudenthal, Hans. ?Die Grundlagen der Geometrie um die Wende des 19. Jahrhunderts.? Mathematisch-Physikalische Semester berichte 7 (1960): 2-25. · Zbl 0104.14601 [60] Freudenthal, Hans. ?The Main Trends in the Foundations of Geometry in the 19th Century.? In Logic, Methodology and Philosophy of Science, edited by E. Nagel, P. Suppes & A. Tarski, 613-621. Stanford: Stanford Univ. Press, 1962. [61] Galuzzi, Massimo. ?Geometria algebrica e logica tra Otto e Novecento.? In Gianni Micheli (ed.) Storia d’Italia. Annali 3. Storia e tecnica nella cultura e nella società dal Rinascimento a oggi. Turin: Einaudi, 1980, pp. 1004-1105. [62] Gericke, Helmuth. Geschichte des Zahlbegriffs. Mannheim: Bibliographisches Institut, 1970. · Zbl 0254.01003 [63] Gillies, Donald A. Frege, Dedekind, and Peano on the Foundations of Arithmetic. Assen: Van Gorcum, 1982. · Zbl 0514.03004 [64] Gillies, Donald A, ed. Revolutions in Mathematics. Oxford: Clarendon Press, 1992. · Zbl 0758.01019 [65] Giusti, Enrico. ?Gli ?errori? di Cauchy e i fondamenti dell’analisi.? Bollettino di Storia delle Scienze Matematiche 4, fasc. 2 (1984): 24-54. · Zbl 0573.01004 [66] Grabiner, Judith V. ?Is Mathematical Truth Time-Dependent?? The American Mathematical Monthly 81 (1974): 354-365. · Zbl 0284.01013 · doi:10.2307/2318997 [67] Grabiner, Judith V. The Origins of Cauchy’s Rigorous Calculus. Cambridge, Mass: MIT Press, 1981. · Zbl 0517.01002 [68] Grabiner, Judith V. ?Changing Attitudes toward Mathematical Rigor: Lagrange and Analysis in the Eighteenth and Nineteenth Centuries.? In H. N. Jahnke & M. Otte, eds. Epistemological and Social Problems of the Sciences in the Early Nineteenth Century (Dordrecht: Reidel, 1981), pp. 311-330. [69] Grassmann, Hermann. Gesammelte mathematische und physikalische Werke. 3 vols. S. Study, G. Scheffers & F. Engel, eds. Leipzig: Teubner, 1894-1911. Reprint. New York: Johnson Reprint, 1972. [70] Grassmann, Robert. Die Formenlehre oder Mathematik. Stettin, 1872. Reprint. Hildesheim: Olms, 1966. [71] Grattan-Guinness, Ivor. The Development of the Foundations of Mathematical Analysis from Euler to Riemann. Cambridge, Mass: MIT Press, 1970. · Zbl 0215.04401 [72] Grattan-Guinness, Ivor. ?From Weierstrass to Russell: A Peano Medley.? Rivista di storia della scienza, 2 (1) (1985): 1-16. [73] Grattan-Guinness, Ivor. ?Living Together and Living Apart. On the Interactions between Mathematics and Logics from the French Revolution to the First World War.? S. Afr. J. Philos. 7 (2), (1988): 73-82. [74] Grattan-Guinness, Ivor. ?Bertrand Russell (1872-1970) After Twenty Years.? Notes Rec. R. Soc. Lond. (1990) 44: 280-306. · Zbl 0978.01513 · doi:10.1098/rsnr.1990.0024 [75] Grattan-Guinness, Ivor. ed. Joseph Fourier: 1768-1830. Cambridge, Mass.: MIT Press, 1972. · Zbl 0245.01008 [76] Grattan-Guinness, Ivor. ed. From the Calculus to Set Theory: 1630-1919. London: Duckworth, 1980. · Zbl 0439.01005 [77] Guicciardini, Niccolò. The Development of Newtonian Calculus in Britain, 1700-1800. Cambridge: Cambridge Univ. Press, 1987. [78] Heijenoort, Jean Van, ed. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge, Mass: Harvard University Press, 1971. · Zbl 1001.03005 [79] Hilbert, David. Grundlagen der Geometrie. Leipzig, 1899. [80] Hilbert, David. The Foundations of Geometry. E. J. Townsend, trans. La Salle: The Open Court, 1947. [81] Hilbert, David. Gesammelte Abhandlungen. 3 vols. Berlin, 1932-1935. Reprint. New York: Chelsea, 1965. [82] Hilbert, David. ?Über den Zahlbegriff.? Jahresbericht der Deutschen Mathematiker-Vereinigung (1900) 8: 180-184. · JFM 31.0165.02 [83] Hilbert, David. ?Mathematical Problems?. Bulletin of the American Mathematical Society 8 (October 1901?July 1902): 437-479. [84] Hilbert, David. ?Probleme der Grundlegung der Mathematik?. Atti del Congresso Internazionale dei Matematici, Bologna 3-10 Settembre 1928, Vol. 1. Bologna: Zanichelli, 1929. Pp. 135-141. [85] Hilbert, David. Ricerche sui fondamenti della matematica. V. Michele Abrusci, ed. Naples: Bibliopolis, 1978. [86] Israel, Giorgio. ??Rigore? ed ?assiomatica? nella matematica moderna.? Scienza e storia: analisi critica e problemi attuali. Rome: Editori Riuniti, 1980. Pp. 427-450. [87] Israel, Giorgio. ??Rigor? and ?Axiomatica? in Modern Mathematics.? Fundamenta Scientiae 2 (1981): 205-219. [88] Jevons, Stanley W. The Principles of Science: a Treatise on Logic and Scientific Method. London, 1874, 1877, 1879. Reprint. New York: Dover, 1958. · Zbl 0099.00503 [89] Jevons, Stanley W. Pure Logic and Other Minor Works. London, 1980. Robert Adamson, ed. New York: Franklin, 1971. [90] Jourdain, Philip E. B. ?The Development of the Theories of Mathematical Logic and the Principles of Mathematics.? The Quarterly Journal of Pure and Applied Mathematics 41 (1910): 324-352; 43 (1912): 219-314; 44 (1913): 113-128. · JFM 41.0050.03 [91] Kennedy, Hubert C. ?The Mathematical Philosophy of Giuseppe Peano.? Philosophy of Science 30 (1963): 262-266. · doi:10.1086/287940 [92] Kennedy, Hubert C. ?Giuseppe Peano at the University of Turin.? The Mathematics Teacher 61 (November 1968): 703-706. [93] Kennedy, Hubert C. ?The Origins of Modern Axiomatics: Pasch to Peano.? The American Mathematical Monthly 79 (1972): 133-136. · Zbl 0227.50003 · doi:10.2307/2316533 [94] Kennedy, Hubert C. ?Peano’s Concept of Number.? Historia Mathematica 1 (1974): 387-408. · Zbl 0288.01014 · doi:10.1016/0315-0860(74)90031-7 [95] Kennedy, Hubert C. ?Nine Letters from Giuseppe Peano to Bertrand Russell.? Journal of the History of Philosophy 13 (1975): 205-220, p. 207. [96] Kennedy, Hubert C. Peano: Life and Works of Giuseppe Peano. Dordrecht: Reidel, 1980. · Zbl 0429.01015 [97] Kennedy, Hubert C. Peano: Storia di un matematico. Paolo Pagli, trans. Turin: Boringhieri, 1983. [98] Kitcher, Philip. ?Mathematical Rigor ? Who Needs It?? Noûs 15 (March 1981): 469-493. · Zbl 1366.00032 [99] Kitcher, Philip. ?The Foundations of Mathematics.? In Companion to the History of Modern Science, edited by R. C. Olby, G. N. Cantor, J. R. R. Christie & M. J. S. Hodge, 677-689. London: Routledge, 1990. [100] Kleene, Stephen Cole. Introduction to Metamathematics. New York / Amsterdam / Groningen: Van Nostrand / North-Holland / P. Noordhoff, 1952. · Zbl 0047.00703 [101] Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. · Zbl 0277.01001 [102] Kline, Morris. Mathematics: The Loss of Certainty. Oxford: Oxford University Press, 1982. · Zbl 0458.03001 [103] Knobloch, Eberhard. ?Einfluß der Symbolik und des Formalismus auf die Entwicklung des mathematischen Denkens.? Berichte zur Wissenschaftsgeschichte 3 (1980): 77-94. [104] Kronecker, Leopold. Werke. 5 vols. Kurt Hensel, ed. Reprint. New York: Chelsea, 1968. [105] Lagrange, Joseph Louis. Oeuvres. 14 vols. J.-A. Serret & G. Darboux, eds. Paris: Gauthier-Villars, 1867-1892. [106] Lakatos, Imre. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge University Press, 1976. · Zbl 0334.00022 [107] Lakatos, Imre. ?Cauchy and the Continuum: the Significance of Non-Standard Analysis for the History and Philosophy of Mathematics?. in John Worrall & Gregory Currie, eds., Mathematics, Science and Epistemology, Vol. 2. Cambridge: Cambridge University Press, 1978. Pp. 43-60. · Zbl 0398.01009 [108] Leibniz, Gottfried Wilhelm. Mathematische Schriften. 7 vols. C. I. Gerhardt, ed. Berlin 1849-1864. Reprint. Hildesheim: Olms, 1961-1962. [109] Leibniz, Gottfried Wilhelm. Opuscules et fragments inédits. Louis Couturat, ed. Paris, 1903. Reprint. Hildesheim: Olms, 1961. [110] Levi, Beppo. ?L’opera matematica di Giuseppe Peano.? Bollettino della Unione Matematica Italiana, 11 (1932): 253-262. [111] Levi, Beppo. ?Intorno alle vedute di G. Peano circa la logica matematica.? Bollettino della Unione Matematica Italiana 12 (1933): 65-68. · Zbl 0006.38603 [112] Maclaurin, Colin. A Treatise of Fluxions. Edinburgh, 1742. · Zbl 0516.01024 [113] Mehrtens, Herbert. ?Anschauungswelt versus Papierwelt ? Zur historischen Interpretation der Grundlagenkrise der Mathematik.? In H. Poser & H.-W. Schütt (eds.) Ontologie und Wissenschaft. Berlin: TU-Berlin, 1984, pp. 231-276. [114] Mehrtens, Herbert. Moderne Sprache Mathematik. Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme. Frankfurt: Suhrkamp, 1990. [115] Mehrtens, H., Bos, H., & Schneider, I. Social History of Nineteenth Century Mathematics. Boston: Birkhäuser, 1981. · Zbl 0467.01009 [116] Nagel, Ernst, & Newman, James R. Gödel’s Proof. New York: University Press, 1958. · Zbl 0086.24602 [117] Newton, Isaac. Philosophiae naturalis principia mathematica. London, 1687. · Zbl 0732.01044 [118] Newton, Isaac. Mathematical Principles. 2 vols. Motte’s translation revised by Florian Cajori. Berkeley: University of California Press, 1962. [119] Padoa, Alessandro. ?Il contributo di G. Peano all’ideografia logica.? Periodico di Matematiche 13 (January 1933): 15-22. · JFM 59.0853.01 [120] Padoa, Alessandro. ?Ce que la logique doit à Peano.? Actualités scientifiques et industrielles (1936): 31-37. · JFM 62.1038.04 [121] Palladino, Franco. ?Le lettere di Giuseppe Peano nella corrispondenza di Ernesto Cesàro.? Nuncius, Anno 8 (1993), fasc. 1, pp. 249-273. [122] Peano e i Fondamenti della Matematica. Modena: Mucchi, 1992. [123] Peano, Giuseppe. Opere scelte. 3 vols. Rome: Cremonese, 1957-1959. [124] Peano, Giuseppe. Selected Works. Hubert C. Kennedy, trans. and ed. Toronto: University of Toronto Press, 1973. · Zbl 0926.01013 [125] Peano, Giuseppe. Arbeiten zur Analysis und zur mathematischen Logik. Edited by G. Asser. Leipzig: Teubner, 1990. · Zbl 0684.01018 [126] Peckhaus, Volker. Hilbertprogramm und Kritische Philosophie. Göttingen: Vandenhoek & Ruprecht, 1990. [127] Peirce, Charles S. Collected Papers. Charles Hartshorne & Paul Weiss, eds. Cambridge, Mass.: Harvard University Press, 1931-1938. [128] Pincherle, Salvatore. ?Saggio di una introduzione alla teoria delle funzioni analitiche secondo i principii del prof. C. Weierstrass.? Giornale di Matematiche 18 (1880): 178-254, 317-357. · JFM 12.0307.01 [129] Poincaré, Henri. ?L’Oeuvre mathématique de Weierstrass.? Acta Mathematica 22 (1899): 1-18. · JFM 29.0017.03 · doi:10.1007/BF02417867 [130] Poincaré, Henri. ?Du role de l’intuition et de la logique en mathématiques?. Compte rendu du Deuxième Congrès International des Mathématiciens, edited by E. Duporcq, 115-130. Paris: Gauthier-Villars, 1902. Reprint. Nendeln/Liechtenstein: Kraus, 1967. [131] Poincaré, Henri. Oeuvres. 11 vols. Paris: Gauthier-Villars, 1950-1956. [132] Pringsheim, Alfred. ?Grundlagen der Allgemeinen Funktionenlehre?. Enzyklopädie der mathematischen Wissenschaften, Vol. 2, Part 1, First half: Analysis (Leipzig: Teubner, 1899-1916): 1-53. [133] Quine, Willard van Orman. ?Peano as a Logician.? History and Philosophy of Logic 8 (1987): 15-24. · Zbl 0634.03001 · doi:10.1080/01445348708837105 [134] Reid, Constance. Hilbert. Berlin: Springer, 1970. [135] Robinson, Abraham. Non-Standard Analysis. Revised edition. Amsterdam: North-Holland; New York: American Elsevier, 1974. [136] Rodríguez-Consuegra, Francisco. ?Elementos logicistas en la obra de Peano y su escuela.? Mathesis 4 (1988): 221-299. [137] Rodríguez-Consuegra, Francisco. The Mathematical Philosophy of Bertrand Russell: Origins and Development. Basel: Birkhäuser, 1991. · Zbl 0747.03002 [138] Romano, Lalla. ?Lo spirito creativo è leggero.? Spirali, Anno. 3 (June 1980), n. 6, pp. 5-6. [139] Russell, Bertrand. ?Sur la logique des relations, avec des applications à la théorie de séries.? Revue de Mathématiques (Rivista di Matematica) 7 (1900-1901): 115-148. [140] Russell, Bertrand. The Principles of Mathematics. Cambridge: At the University Press, 1903. 2nd ed. London: Allen & Unwin, 1937. [141] Russell, Bertrand. Introduction to Mathematical Philosophy London: George Allen & Unwin, New York: Macmillan, 1919. · JFM 47.0036.12 [142] Russell, Bertrand. The Autobiography of Bertrand Russell. 2 vols. London: George Allen and Unwin, 1967-1968. [143] Russell, Bertrand, & Whitehead, Alfred N. Principia Mathematica. 3 vols. Cambridge: At the University Press, 1910-1913. [144] Sageng, Erik L. Colin MacLaurin and the Foundations of the Method of Fluxions. Ph.D. dissertation. Princeton University, 1989. [145] Schneider, Ivo. Archimedes: Ingenieur, Naturwissenschaftler und Mathematiker. Darmstadt: Wissenschaftliche Buchgesellschaft, 1979. · Zbl 0433.01007 [146] Schneider, Ivo. Isaac Newton. Munich: Beck, 1988. [147] Segre, Michael. In the Wake of Galileo. New Brunswick: Rutgers University Press, 1991. [148] Segre, Michael. ?Peano, Logicism and Formalism.? In I. C. Jarvie & N. Laor, (eds.) The Enterprise of Critical Rationalism. Festschrift Agassi. Vol. I: Critical Rationalism, Metaphysics and Science. Dordrecht: Kluwer, forthcoming. [149] Spalt, Detlef D. Vom Mythos der Mathematischen Vernunft. Darmstadt: Wissenschaftliche Buchgesellschaft, 1987. [150] Stigt, Walter P. van. Brouwer’s Intuitionism. Amsterdam: North-Holland, 1990. · Zbl 0707.03001 [151] Struik, Dirk J. A Concise History of Mathematics. 2 vols. New York: Dover, 1948, 1967. · Zbl 0032.09701 [152] Styazhkin, N. I. History of Mathematical Logic from Leibniz to Peano. Cambridge; Mass: MIT Press, 1969. · Zbl 0196.00602 [153] Terracini, Alessandro, ed. In memoria di Giuseppe Peano. Cuneo: Liceo Scientifico Statale, 1955. [154] Toepell, Michael-Markus. Über die Entstehung von David Hilberts ?Grundlagen der Geometrie?. Göttingen: Vandenhoeck & Ruprecht, 1986. · Zbl 0602.01013 [155] Toepell, Michael-Markus. ?On the Origins of Hilbert’s ?Grundlagen der Geometrie?.? Archive for History of Exact Sciences, 35 (1986): 329-344. · Zbl 0597.01012 · doi:10.1007/BF00357305 [156] Torretti, Robert. Philosophy of Geometry from Riemann to Poincaré. Dordrecht: Reidel, 1978. [157] Tropfke, Johannes. Geschichte der Elementarmathematik. 4th edition, Vol. 1: Arithmetik und Algebra. Revised by Kurt Vogel, Karin Reich, & Helmuth Gericke. Berlin: Walter de Gruyter, 1980. · Zbl 0016.14501 [158] Vailati, Giovanni, ?La logique mathématique et sa nouvelle phase de développement dans les écrits de M. J. Peano.? Revue de Métaphysique et de Morale 7 (1899): 86-102. [159] Volkert, Klaus T. Die Krise der Anschauung. Göttingen: Vandenhoeck & Ruprecht, 1986. [160] Wang, Hao. ?The Axiomatization of Arithmetic?. The Journal of Symbolic Logic, 22 (June 1957): 145-158. · Zbl 0078.00503 · doi:10.2307/2964176 [161] Weierstrass, Karl. Mathematische Werke. 7 vols. Reprint. Hildesheim: Olms, New York: Johnson, 1967. [162] Weyl, Hermann. Gesammelte Abhandlungen. 4 vols. Berlin: Springer, 1968. · Zbl 0164.30103 [163] Wisdom, J. O. ?The Analyst Controversy: Berkeley’s Influence on the Development of Mathematics.? Hermathena, 54 (1939): 3-29. · Zbl 0063.08296 [164] Wisdom, J. O. ?The Analyst Controversy: Berkeley as a Mathematician.? Hermathena, 59 (1942): 111-128. · Zbl 0063.08298 [165] Wisdom, J. O. ?Berkeley’s Criticism of the Infinitesimal.? The British Journal for the Philosophy of Science, 4 (May 1953?February 1954): 22-25. [166] Youschkevitch, A. P. ?Lazare Carnot and the Competition of the Berlin Academy in 1786 on the Mathematical Theory of the Infinite.? In C. C. Gillispie, Lazare Carnot Savant, 147-168. Princeton: Princeton University Press, 1971. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.