Dolecki, Szymon; Greco, Gabriele H. The forgotten mathematical legacy of Peano. (English) Zbl 1429.01012 Diss. Math. 537, 1-77 (2019). The paper provides a nice survey of the mathematical legacy of Giuseppe Peano (1858–1932). Peano is perhaps best known for his axiomatization of the natural numbers, the construction of a space-filling curve, and the existence theorem for ordinary differential equations. The goal of the paper is to summarize Peano’s other important contributions to analysis, topology, measure theory, geometry, and the foundations of mathematics. Peano’s ideas were surprisingly modern, but many of them are now forgotten or attributed to other mathematicians. The authors of the present paper have already published numerous articles on various aspects of Peano’s work, making it possible for the modern reader to appreciate Peano’s remarkable discoveries. This paper will be useful to all readers interested in the history of 19th- and 20th-century mathematics. Reviewer: Antonín Slavík (Praha) (MR 3932611) MSC: 01A55 History of mathematics in the 19th century 01A60 History of mathematics in the 20th century 03-03 History of mathematical logic and foundations 26-03 History of real functions 28-03 History of measure and integration 34-03 History of ordinary differential equations 54-03 History of general topology Biographic References: Peano, Giuseppe PDFBibTeX XMLCite \textit{S. Dolecki} and \textit{G. H. Greco}, Diss. Math. 537, 1--77 (2019; Zbl 1429.01012) Full Text: DOI References: [1] Aristotle, Topics, English translation by W. A. Pickard-Cambridge, Univ. of Adelaide, 2015; https://ebooks.adelaide.edu.au/a/aristotle/a8t/index.html. [2] F. Arzarello and C. S. Roero, Un inedito di Peano sulla sua celebre curva. Le radici logicoaritmetiche di un oggetto geometrico, in: Giuseppe Peano: matematica, cultura e società, Edizioni l’Artistica di Savigliano, 2001, 8–26 [3] M. Baake and U. Schlagel, The Peano–Baker series, Proc. Steklov Inst. Math. 275 (2011), 167–171. [7] M. Barnabei, A. Brini, and G. Rota, On the exterior calculus of invariant theory, J. Algebra 96 (1985), 120–160. [8] A. Besenyei, Peano’s unnoticed proof of Borel’s theorem, Amer. Math. Monthly 121 (2014), 69–72. [9] F. Bigolin and G. H. Greco, Geometric characterization of C1manifolds in Euclidean spaces by tangent cones, J. Math. Anal. Appl. 396 (2012), 145–163. [4] G. Birkhoff and U. Merzbach, A Sourcebook in Classical Analysis, Harvard Univ. Press, Cambridge, 1973. [11] E. Borel, Sur quelques points de la théorie des fonctions, Ann. Sci. École Norm. Sup. 12 (1895), 9–55. [5] G. Bouligand, Sur quelques points de la topologie restreinte du premier ordre, Bull. Soc. Math. France 56 (1928), 407–420. · JFM 54.0099.01 [6] G. Bouligand, Introduction à la géométrie infinitésimale directe, Vuibert, Paris, 1932. · JFM 58.0086.03 [7] C. Burali-Forti, Logica matematica, Hoepli, Milano, 1894 (2nd ed., 1919). [8] C. Burali-Forti, Sur l’égalité et sur l’introduction des éléments dérivés dans la science, Enseign. Math. 1 (1899), 246–261. · JFM 30.0075.01 [9] C. Burali-Forti, Sur les différentes méthodes logiques pour la définition du nombre réel, in: Bibliothèque du congrès international de philosophie, Paris 1900, Vol. III, Colin, Paris, 1901, 289–308. [10] C. Burali-Forti, Gli enti astratti definiti come enti relativi ad un campo di nozioni, Atti R. Accad. Lincei Rend. Accad. Lincei Sci. Fis. Mat. Nat. 21 (1912), 677–682. [11] C. Burali-Forti, Nuove applicazioni degli operatori, Atti R. Accad. Sci. Torino 50 (1915), 669–684. [12] C. Burali-Forti, A proposito dell’articolo di E. Maccaferri, Bollettino di Matematica 20 (1924), 128–129. [13] 70S. Dolecki and G. H. Greco [14] C. Burali-Forti, A proposito di una lettera di Mario Pieri, Bollettino di Matematica 21 (1925), 136–137. [15] G. Cantor, Über unendliche lineare Punktmannigfaltigkeiten, Math. Ann. 23 (1884), 454– 488. [22] G. Cantor, Gesammelte Abhandlungen, Springer, Berlin, 1932. [16] H. Cartan, Théorie des filtres, C. R. Acad. Sci. Paris 205 (1937), 595–598. · JFM 63.0569.02 [17] H. Cartan, Filtres et ultrafiltres, C. R. Acad. Sci. Paris 205 (1937), 777–779. · JFM 63.0569.03 [18] U. Cassina, Linee, superficie, solidi, Rend. Sem. Mat. Fis. Univ. Milano 4 (1931), 18–37. [26] U. Cassina, L’opera scientifica di Giuseppe Peano, Rend. Sem. Mat. Fis. Univ. Milano 7 (1933), 323–389. Reprinted (partially modified) in [31], 397–468. [19] U. Cassina, Curva di Peano in base due, Periodico Matematiche 19 (1939), 113–125. · Zbl 0021.11603 [20] U. Cassina, Il concetto di linea piana e la curva di Peano, Riv. Mat. Univ. Parma 1 (1950), 275–292. Reprinted in [30], 112–136. · Zbl 0040.01803 [21] U. Cassina, Sulle definizioni per astrazione, in: Atti I Congresso Studi Metodologici, Torino, 1952. Reprinted in [30], 283–290. [22] U. Cassina, Critica dei principii della matematica e questioni di logica, Cremonese, Roma, 1961. · Zbl 0091.00705 [23] U. Cassina, Dalla geometria egiziana alla matematica moderna, Cremonese, Roma, 1961. · Zbl 0102.24306 [24] A. Cauchy, Mémoire sur le rapport différentiel de deux grandeurs qui varient simultanément, Exercices d’analyse et de physique mathématique 2 (1841), 188–129. [25] G. Choquet, Sur les notions de filtre et de grille, C. R. Acad. Sci. Paris 224 (1947), 171–173. · Zbl 0029.07602 [26] W. A. Coppel, Foundations of Convex Geometry, Cambridge Univ. Press, Cambridge, 1998. [27] G. Darboux, Sur un théorème relatif à la continuité des fonctions, Bull. Sci. Math. Astronom. 3 (1872), 307–313. · JFM 04.0192.01 [28] P.J. Davis, Interpolation and Approximation, Dover Publ., New York, 1975. · Zbl 0329.41010 [29] R. Dedekind, Was sind und was sollen die Zahlen?, Vieweg, Braunschweig, 18881, 18932, 19113. English transl. of the second edition in [38] by W. W. Beman. [30] R. Dedekind, Essays of the Theory of Numbers, Dover Publ., New York, 1975. [31] S. Dickstein, Considerationes de Hoene Wroński super methaphysica de calculo infinitesimale, in: In occasione de septuagesimo anno de Giuseppe Peano, Schola et Vita (supplement 27 August 1928), 79–82. [32] S. Dickstein, Peano jako historyk matematyki, Wiadomości Mat. 36 (1934), 65–70. [33] P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford Univ. Press, London, 1958 (4th ed.). [42] S. Dolecki, Tangency and differentiation: some applications of convergence theory, Ann. Mat. Pura Appl. 80 (1982), 223–255. [34] S. Dolecki and G. H. Greco, Towards historical roots of necessary conditions of optimality: Regula of Peano, Control Cybernet. 36 (2007), 491–518. · Zbl 1166.49020 [35] S. Dolecki and G. H. Greco, Tangency vis-à-vis differentiability in the works of Peano, Severi and Guareschi, J. Convex Anal. 18 (2011), 301–339. · Zbl 1236.26004 [36] S. Dolecki and G. H. Greco, Niezrozumiała niepamięć o dziedzictwie Peany, in: W. Więsław (ed.), Dzieje Matematyki Polskiej II, Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, 2013, 39–51. [46] T. M. Flett, Differential Analysis. Differentiation, Differential Equations and Differential Inequalities, Cambridge Univ. Press, Cambridge, 1980. [37] M. Fréchet, Sur la notion de différentielle. C. R. Acad. Sci. Paris 152 (1911), 845–847. The forgotten mathematical legacy of Peano71 [38] M. Fréchet, Sur la notion de différentielle, C. R. Acad. Sci. Paris 152 (1911), 1950–1951. [39] A. Genocchi, Calcolo differenziale e principii di calcolo integrale pubblicato con aggiunte dal Dr. Giuseppe Peano, Fratelli Bocca, Torino, 1884. [40] A. Genocchi, Differentialrechnung und Grundzüge der Integralrechnung, herausgegeben von Giuseppe Peano, Teubner, Leipzig 1899. German transl. of [49, (1884)] by G. Bohlmann and A. Schepp with a preface by A. Mayer. [41] A. Genocchi, Differenc al noe isqislen e i osnovy integral nago isqislen , izdanny prof. Giuseppe Peano, no-Russkoe Knigoizdatel stvo F. A. ogansona, Kiev-Peterburg-Har kov, 1903. Russian transl. of [49, (1884)] by N. S. Sineokov. [42] A. Genocchi, Differencial noe isqislenie i naqala integral nogo isqisleni . Izdannoe s dopolneni mi i primeqani mi prof. Giuseppe Peano, Academia, Peterburg, 1922. Russian transl. of [49, (1884)] by K. A. Posse. [43] Ph. Gilbert, Correspondance, Nouvelles Annales de Mathématiques 3 (1884), 153–155. [44] H. G. Grassmann, A New Branch of Mathematics, English transl. by L. C. Kannenberg of “Die Ausdehnungslehre” (1844), Open Court, Chicago, 1995. [45] H. G. Grassmann, Lehrbuch der Arithmetik für höhere Lehranstalten, Enslin, Berlin, 1861. [46] H. G. Grassmann, Extension Theory, English transl. by L. C. Kannenberg of “Die Ausdehnungslehre” (1862), Amer. Math. Soc., Providence, 2000. [47] G. H. Greco, Analisi Matematica Uno: Funzioni di una variabile (calcolo differenziale e integrale). In occasione del 150oanniversario della nascita di Giuseppe Peano (1858– 1932), UniversityATtn, Trento, 2012. [48] G. H. Greco, Analisi Matematica Due: Funzioni di più variabili (calcolo differenziale). In occasione del 150oanniversario della nascita di Giuseppe Peano (1858–1932), forthcoming, 2018. [49] G. H. Greco, Analisi Matematica Tre: Funzioni di più variabili (calcolo integrale). In occasione del 150oanniversario della nascita di Giuseppe Peano (1858–1932), forthcoming, 2018. [50] G. H. Greco and S. Mazzucchi, Peano’s 1886 existence theorem on first-order scalar differential equations: a review, Boll. Un. Mat. Ital. 9 (2016), 375–389. · Zbl 1354.01011 [51] G. H. Greco and S. Mazzucchi, The originality of Peano’s 1886 existence theorem on scalar differential equations, J. Convex Anal. 23 (2016), 649–659. · Zbl 1351.01012 [52] G. H. Greco and S. Mazzucchi, Peano’s 1890 existence theorem for systems of differential equations: a revisited proof, forthcoming, 2018. [53] G. H. Greco, S. Mazzucchi, and E. Pagani, Peano on derivative of measures: strict derivative of distributive set functions, Atti Accad. Lincei Rend. Lincei Mat. Appl. 21 (2010), 305–339. · Zbl 1198.01024 [54] G. H. Greco, S. Mazzucchi, and E. Pagani, Peano on definition of surface area, Atti Accad. Lincei Rend. Lincei Mat. Appl. 27 (2016), 251–286. · Zbl 1347.01006 [55] G. H. Greco and E. M. Pagani, Reworking on affine exterior algebra of Grassmann: Peano and his school, Ist. Lombardo Rend. Cl. Sci. Mat. Nat. 144 (2010), 17–52. [56] G. H. Greco and E. Pagani, A reworking on Grassmann regressive product in an exterior algebra: Peano and his school, forthcoming, 2018. [67] T. H. Grönwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. 20 (1919), 292–296. [57] H. Hankel, Untersuchungen über die unendlich oft oscillirenden und unstetigen Functionen, Abdruck aus dem Gratulationsprogramm der Tübinger Universität vom 6. März 1870, Math. Ann. 20 (1882), 63–112. 72S. Dolecki and G. H. Greco [58] H. Hankel, Untersuchungen über die unendlich oft oszillierenden und unstetigen Funktionen, reprint of [68], annotated by P. E. B. Jourdain, Ostwalds Klassiker der exakten Wissenschaften 153 (1905), 44–115. [70] A. Harnack, Über den Inhalt von Punktmengen, Math. Ann. 25 (1885), 241–250. [59] F. Hausdorff, Grundzüge der Mengenlehre, Verlag von Veit, Leipzig, 1914. [60] T. Hawkins, Lebesgue’s Theory of Integration. Its Origin and Developments, AMS Chelsea, Providence, 1975. [61] C. Hermite, Cours de M. Hermite professé pendant le 2e semestre 1881–82, Hermann, Paris, 1883. [74] D. Hilbert, Über die stetige Abbildung einer Linie auf ein Flächenstück, Math. Ann. 38 (1891), 459–460. [62] D. Hilbert, Grundlagen der Geometrie, 1st ed., Teubner, Leipzig, 1899. [63] D. Hilbert, Über den Zahlbegriff, Jahresber. Deutsch. Math.-Verein. 8 (1900), 180–183. [77] D. Hilbert, Les principes fondamentaux de la géométrie, Ann. Sci. École Norm. Sup. 17 (1900), 103–209. [64] D. Hilbert, The Foundations of Geometry, 1st ed., Open Court Publ., La Salle, 1902. · JFM 33.0082.10 [65] D. Hilbert, The Foundations of Geometry, 2nd ed., Open Court Publ., La Salle, 1971. · Zbl 0228.50002 [66] E.W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Vol. I, Cambridge Univ. Press, Cambridge, 1921. · JFM 48.1206.01 [67] H. Ishiguro, Leibniz’s Philosophy of Logic and Language, 2nd ed., Cambridge Univ. Press, Cambridge, 1990. [82] C. Jacobi, De determinantibus functionalibus, J. Reine Angew. Math. 22 (1841), 319–359. [68] C. Jordan, Cours d’Analyse de l’École Polytechnique (3 vols.), Gauthier-Villars, Paris, 1882–87. [69] C. Jordan, Extrait d’une Lettre de M. C. Jordan, Nouvelles Annales de Mathématiques 3 (1884), 47. [70] C. Jordan, Remarques sur les intégrales définies, J. Math. Pures Appl. 8 (1892), 69–99. [71] C. Jordan, Cours d’Analyse de l’École Polytechnique (3 vols.), 2nd ed., Gauthier-Villars, Paris, 1893–96. [72] C. Jordan, Question n. 60, L’intermédiare des mathématiciens 1 (1894), 23. [88] H. C. Kennedy, Is there an elementary proof of Peano’s existence theorem for first order differential equations?, Amer. Math. Monthly 76 (1969), 1043–1045. [73] H. C. Kennedy, Life and Works of Giuseppe Peano, Reidel, Dordrecht, 1980. · Zbl 0429.01015 [74] H. C. Kennedy, Twelve Articles on Giuseppe Peano, Peremptory Publ., San Francisco, 2002. [75] H. C. Kennedy, Life and Works of Giuseppe Peano, Peremptory Publ., Concord, 2006. · Zbl 0429.01015 [76] W. M. Kozłowski, Ingeniositate de idea de neo-latino, in: In occasione de septuagesimo anno de Giuseppe Peano, Schola et Vita (supplement 27 August 1928), 29–33. [77] W. M. Kozłowski, Wspomnienie o Józefie Peano, Wiadomości Mat. 36 (1934), 57–64. [94] H. Lebesgue, Intégrale, longueur, aire, Ann. Mat. Pura Appl. 7 (1902), 231–359. [78] H. Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives, GauthierVillars, Paris, 1904. [79] J. P. G. Lejeune-Dirichlet, Werke, Vol. I, Reimer, Berlin, 1889. [80] A. Lotze, Die Grassmannsche Ausdehnungslehre, in: Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Vol. III.1.2, Teubner, Leipzig, 1914– 1931, 1425–1550. [98] E. Maccaferri, Le definizioni per astrazione e la classe di Russell, Rend. Circ. Mat. Palermo 35 (1913), 165–171. The forgotten mathematical legacy of Peano73 [81] J. Mawhin, Problèmes de Cauchy pour les équations différentielles et théories de l’intégration: influences mutuelles, Cahiers du séminaire d’histoire des mathématiques 9 (1988), 231–246. [82] J. Mawhin, Analyse. Fondements, techniques, évolution, De Boeck Université, Bruxelles, 1992. · Zbl 0759.26004 [83] J. Mawhin, De La Vallée Poussin’s contributions to the fundamental theory of ordinary differential equations, in: De La Vallée Poussin, Charles-Jean: Collected Works, Académie Royale de Belgique, Vol. 2, 2001, 301–314. [84] H. Minkowski, Geometrie der Zahlen, Teubner, Leipzig, 1896. · Zbl 0050.04807 [85] M. A. Mnatsakanian, On the area of the region on a developable surface, Dokl. Akad. Nauk Armen. 73 (1981), 97–102 (in Russian). [104] E. H. Moore, On certain crinkly curves, Trans. Amer. Math. Soc. 1 (1900), 72–90. [105] G. H. Moore, Zermelo’s Axiom of Choice. Its Origins, Development, and Influence, Springer, New York, 1982. [86] R. Murawski, Giuseppe Peano a rozwój logiki symbolicznej, Wiadomości Mat. 27 (1987), 261–277. [87] R. Murawski, Giuseppe Peano and symbolic logic, in: R. Murawski, Essays in the Philosophy and History of Logic and Mathematics, Rodopi, Amsterdam, 2010, 169–182. · Zbl 1216.01007 [88] F. J. Murray and K. S. Miller, Existence theorems for ordinary differential equations, New York Univ. Press, New York, 1954. [109] O. Nikodym, Sur une généralisation des intégrales de M. J. Radon, Fund. Math. 15 (1930), 131–179. [89] A. Padoa, Essai d’une théorie algébrique des nombres entiers, précédé d’une introduction logique à une théorie déductive quelconque, in: Bibliothèque du congrès international de philosophie, Paris 1900, Vol. III, Colin, Paris, 1901, 309–365. [90] A. Padoa, Dell’astrazione matematica, in: Questioni filosofiche (Soc. Fil. Italiana, ed.), Formiggini, Bologna, 1908, 91–104. [112] V. Pambuccian, The axiomatics of order geometry, I. Ordered incidence spaces, Expo. Math. 29 (2011), 24–66. [91] E. Pascal, Esercizi e note critiche di calcolo infinitesimale, Hoepli, Milano, 1895 (2nd ed., 1909). · JFM 26.0303.01 [92] M. Pasch, Vorlesungen über neuere Geometrie, 1st ed., Teubner, Leipzig, 1882. · JFM 14.0498.01 [93] M. Pasch, Vorlesungen über neuere Geometrie, 2nd ed., Teubner, Leipzig, 1926. [94] G. Peano, Sull’integrabilità delle funzioni, Atti R. Accad. Sci. Torino 18 (1883), 439–446. [95] G. Peano, Extrait d’une lettre, Nouvelles Annales de Mathématiques 3 (1884), 45–47. [96] G. Peano, Lettre à M. C. Jordan du 16 février 1884, Archives de la Bibliothèque Centrale de l’École Polytechnique, Paris, Dossier C. Jordan, Promotion 1855, Art.VI §2 Sect. a2 N. 69. [97] G. Peano, Correspondance, Nouvelles Annales de Mathématiques 3 (1884), 252–256. [98] G. Peano, Sull’integrabilità delle equazioni differenziali di primo ordine, Atti R. Accad. Sci. Torino 21 (1886), 677–685. English transl. in [167, (1973), pp. 51–57]. [99] G. Peano, Applicazioni geometriche del calcolo infinitesimale, Fratelli Bocca, Torino, 1887. · JFM 19.0248.01 [100] G. Peano, Calcolo geometrico secondo Ausdehnungslehre di H. Grassmann, Fratelli Bocca, Torino, 1888. [123] G. Peano, Intégration par séries des équations différentielles linéaires, Math. Ann. 32 (1888), 450–456. English transl. in [167, (1973), pp. 58–66]. [101] G. Peano, Arithmetices principia, nova methodo exposita, Fratelli Bocca, Torino, 1889. English transl. in [167, (1973), pp. 101–134]. 74S. Dolecki and G. H. Greco [102] G. Peano, I principii di geometria logicamente esposti, Fratelli Bocca, Torino, 1889. [103] G. Peano, Une nouvelle forme du reste dans la formule de Taylor, Mathesis 9 (1889), 182–183. [127] G. Peano, Démonstration de l’intégrabilité des équations différentielles ordinaires, Math. Ann. 37 (1890), 182–228. · JFM 22.0302.01 [104] G. Peano, Sulla definizione dell’area d’una superficie, Atti R. Accad. Lincei Rend. Accad. Lincei Sci. Fis. Mat. Nat. 6 (1st sem.) (1890), 54–57. [129] G. Peano, Sur une courbe, qui remplit toute une aire plane, Math. Ann. 36 (1890), 157–160. English transl. in [167, (1973), pp. 143–148]. [105] G. Peano, Sopra alcune curve singolari, Atti R. Accad. Sci. Torino 26 (1890-91), 299–302. English transl. in [167, (1973), pp. 150–152]. [106] G. Peano, Gli elementi di calcolo geometrico, Candeletti, Torino, 1891. · JFM 23.0735.01 [107] G. Peano, Die Grundzüge des geometrischen Calculs, Teubner, Leipzig, 1891. Transl. of [131] by A. Schepp. [108] G. Peano, Sulla formula di Taylor, Atti R. Accad. Sci. Torino 27 (1892), 40–46. [109] G. Peano, Esempi di funzioni sempre crescenti e discontinue in ogni intervallo, Rivista di Matematica 2 (1892), 41–42. · JFM 24.0353.03 [110] G. Peano, Sur la définition de la dérivée, Mathesis 2 (1892), 12–14. · JFM 24.0248.05 [111] G. Peano, Sur le théorème général relatif à l’existence des intégrales des équations différentielles ordinaires, Nouvelles Annales de Mathématiques 11 (1892), 79–82. · JFM 24.0279.01 [112] G. Peano, Lezioni di analisi infinitesimale (2 vols.), Candeletti, Torino, 1893. [113] G. Peano, Lettre à M. C. Jordan du 6 novembre 1894, Archives de la Bibliothèque Centrale de l’École Polytechnique, Paris, Dossier C. Jordan, Promotion 1855, Art.VI §2 Sect. a2 N. 160. [114] G. Peano, Sui fondamenti della geometria, Rivista di Matematica 4 (1894), 51–90. · JFM 25.0854.01 [115] G. Peano, Notations de logique mathématique (Introduction au Formulaire de Mathématiques), Turin, 1894. [116] G. Peano, Notations de logique mathématique, Formulaire de Mathématique, Guadagnini, Torino, 1894, 3–52. [117] G. Peano, Saggio di calcolo geometrico. Atti R. Accad. Sci. Torino 31 (1895-96), 952–975. English transl. in [167, (1973), pp. 169–188]. · JFM 27.0464.01 [118] G. Peano, Réponse n. 60 (C. Jordan), Courbe dont l’aire soit indéterminée, L’intermédiaire des mathématiciens 3 (1896), 39. [119] G. Peano, Generalità sulle equazioni differenziali ordinarie, Atti R. Accad. Sci. Torino 33 (1897), 9–18. [120] G. Peano, Analisi della teoria dei vettori, Atti R. Accad. Sci. Torino 33 (1897-98), 513–534. · JFM 30.0505.08 [121] G. Peano, Entwicklung der Grundbegriffe des geometrischen Calculs, Salzburg, 1897. German transl. of [142] by A. Lanner. [122] G. Peano, Zarys rachunku geometrycznego, Wydawnictwo Redakcyi „Prac matematycznofizycznych”, Warszawa, 1897. Polish transl. of [142] by S. Dickstein. [123] G. Peano, Formulaire de Mathématiques, tome II §1 (Logique mathématique), Fratelli Bocca, Torino, 1897. [124] G. Peano, Formulaire de Mathématiques, tome II §2 (Arithmétique), Fratelli Bocca, Torino, 1898. [125] G. Peano, Les définitions mathématiques, in: Bibliothèque du congrès international de philosophie, Paris 1900, Vol. III, Colin, Paris, 1901, 279–288. [126] G. Peano, Definicye w matematyce, Wiadomości Mat. 6 (1902), 174-181. Polish transl. of [150] by Z. Krygowski. The forgotten mathematical legacy of Peano75 [127] G. Peano, Aritmetica generale e algebra elementare, Paravia, Torino, 1902. · JFM 33.0190.14 [128] G. Peano, Formulaire Mathématique, Fratelli Bocca, Torino, 4th ed., 1902-3. [129] G. Peano, De latino sine flexione: lingua auxiliare internationale, Revue de Mathématiques 8 (1903), 74–83. · JFM 34.0078.03 [130] G. Peano, Additione super theorema de Cantor–Bernstein, Rivista Matematica 8 (1906), 143–157. [131] G. Peano, Formulario Mathematico, 5th ed., Fratelli Bocca, Torino, 1908. [132] G. Peano, Sulla definizione di funzione, Atti R. Accad. Lincei Rend. Accad. Lincei Sci. Fis. Mat. Nat. 20 (1911), 3–5. [133] G. Peano, Le definizioni in matematica, Arxivs de l’Institut de ciencies, Barcelona, Institut d’Estudis Catalans 1 (1911), 49–70. [134] G. Peano, Resto nelle formule di quadratura espresso con un integrale definito, Atti R. Accad. Lincei Rend. Acc. Lincei Sci. Fis. Mat. Nat. 22 (1913), 562–569. [135] G. Peano, Residuo in formulas de quadratura, Mathesis 4 (1914), 5–10. · JFM 45.0451.04 [136] G. Peano, Le definizioni per astrazione, Bollettino della Mathesis, Società italiana di Matematica 1915, 106–120. [137] G. Peano, Sul principio di identità, Bollettino della Mathesis, Società italiana di Matematica 1916, 40–41. [138] G. Peano, Pubblicazioni di G. Peano prof. ord. di Calcolo infinitesimale nella R. Università di Torino, 1916, 7 pp. See file 1916e in [170]. [139] G. Peano, Eguale, Bollettino di matematica 15 (1917–18), 195–198. [140] G. Peano, Letter to Tommaso Boggio, 1919, see Appendix of [5]. [141] G. Peano, Le definizioni in matematica, Periodico di Matematiche 1 (1921), 175–189. English transl. in [167, (1973), pp. 235–246]. · JFM 48.1128.20 [142] G. Peano, Selected Works of Giuseppe Peano, translated and edited, with a biographical sketch and bibliography, by H. C. Kennedy, George Allen & Unwin, London, 1973. [168] G. Peano, Arbeiten zur Analysis und zur mathematischen Logik (G. Asser, ed.), Teubner, Leipzig, 1990. [169] G. Peano, Geometric Calculus, according to the Ausdehnungslehre of H. Grassmann. English transl. of [122] by L. C. Kannenberg, Birkhäuser, Boston, 2000. · Zbl 0938.01051 [143] G. Peano, Opera Omnia, CD-ROM (C. S. Roero, ed.), Dipartimento di Matematica, Università di Torino, 2002. [144] E. Picard, Sur le théorème général relatif à l’existence des intégrales des équations différentielles ordinaires, Nouvelles Annales de Mathématiques 10 (1891), 197–201. [172] M. J. D. Powell, Approximation Theory and Methods, Cambridge Univ. Press, Cambridge, 1981. [145] W. Prenowitz and J. Jantosciak, Join Geometries. A Theory of Convex Sets and Linear Geometry, Springer, New York, 1979. · Zbl 0421.52001 [146] A. Pringsheim, Grundlagen der allgemeinen Funktionenlehre, in: Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Vol. II.1.1, Teubner, Leipzig, 1899, 1–53. [147] A. Pringsheim and J. Molk, Principes fondamentaux de la théorie des fonctions, in: Encyclopédie des Sciences Mathématiques Pures et Appliquées, Vol. II.1, Gauthier-Villars, Paris, 1909, 1–112. [148] J. Radon, Theorie und Anwendungen der absolut additiven Mengenfunktionen, Sitzungsberichte der Akademie der Wissenschaften in Wien, Mathematisch-naturwissenschaftliche Klasse, Abteilung IIa 112 (1913), 1295–1438. 76S. Dolecki and G. H. Greco [177] W. T. Reid, Review: F. J. Murray and K. S. Miller, “Existence theorems for ordinary differential equations”, Bull. Amer. Math. Soc. 61 (1955), 353–355. [149] W. Retter, Topics in abstract order geometry, PhD thesis, Technische Universität Hamburg-Harburg, 2013. [150] J. Richard, Les Principes des Mathématiques et le Problème des Ensembles, Revue générale des sciences pures et appliquées 16 (1905), 541–543. [151] B. Russell, The Collected Papers, Vol. III (Toward the “Principles of Mathematics” 1900– 1902), Routledge, London, 1993. [152] B. Russell, The Principles of Mathematics, Cambridge Univ. Press, Cambridge, 1903. · JFM 34.0062.14 [153] B. Russell, On denoting, Mind 14 (1905), 479–493. [154] B. Russell, Autobiography (1872–1914), Atlantic Monthly Press, Boston, 1967. [155] H. Sagan, Space-Filling Curves, Springer, New York, 1994. [185] A. Sard, Integral representation of remainders, Duke Math. J. 15 (1948), 333–345. [186] A. Sard, Linear Approximation, Amer. Math. Soc., Providence, 1963. [156] J. Schoenflies, Die Entwickelung der Lehre von den Punktmannigfaltigkeiten, Jahresber. Deutsch. Math.-Verein. 8 (1900), 1–251. [188] L. Schwartz, A Mathematician Grappling with his Century, Birkhäuser, Basel, 2001. [189] M. Segre, Le lettere di Giuseppe Peano a Felix Klein, Nuncius 12 (1997), 109–122. [157] J.-A. Serret, Cours d’algèbre supérieure, Vol. I, Gauthier-Villars, Paris, 1866. [158] J.-A. Serret, Cours de calcul différentiel et intégral, 2 vols., 2nd ed., Gauthier-Villars, Paris, 1879–80. [159] F. Severi, Conferenze di geometria algebrica, Regia Università di Roma (Anno 1927–1928), Roma, 1928. · JFM 56.1152.09 [160] F. Severi, Su alcune questioni di topologia infinitesimale, Ann. Soc. Polonaise Math. 9 (1931), 97–108. · JFM 57.0754.01 [161] F. Severi, Sulla differenziabilità totale delle funzioni di più variabili, Ann. Mat. Pura Appl. 13 (1935), 1–35. · Zbl 0009.30801 [162] E. Stamm, Logica mathematico de G. Peano, in: In occasione de septuagesimo anno de Giuseppe Peano, Schola et Vita (supplement 27 August 1928), 33–35. [163] E. Stamm, Józef Peano, Wiadomości Mat. 36 (1934), 1–56. [164] G. K. C. von Staudt, Geometrie der Lage, Korn, Nürnberg, 1847. [198] O. Stolz, Über einen zu einer unendlichen Punktmenge gehörigen Grenzwerth, Math. Ann. 24 (1884), 152–156. [199] A. H. Stroud, Numerical Quadrature and Solutions of Ordinary Differential Equations, Springer, New York, 1974. [165] J. Tannery, Review of “Peano G.: Applicazioni geometriche del calcolo infinitesimale”, Bull. Sci. Math. 11 (1887), 237–239. [201] A. Tarski and S. Givant, Tarski’s system of geometry, Bull. Symbolic Logic 5 (1999), 175–214. [166] J. Thomae, Einleitung in die Theorie der bestimmten Integrale, Nebert, Halle, 1875. [167] St. Thomas Aquinas, Summa Theologica. English transl. by A. J. Freddoso, Univ. of Notre Dame, https://www3.nd.edu/∼afreddos/. [204] R. Torretti, Philosophy of Geometry from Riemann to Poincaré, Reidel, Dordrecht, 1978. [168] G. Vacca, Sui precursori della logica matematica I, II, Revue de Mathèmatiques 6 (1899), 121–125 and 183–186. · JFM 30.0076.01 [169] G. Vailati, La Teoria Aristotelica della Definizione, 1903, in [209], Vol. I, 317–328. [170] G. Vailati, Di un’opera dimenticata di P. Gerolamo Saccheri, Logica Demonstrativa, 1903, in [209], Vol. II, 212–219. The forgotten mathematical legacy of Peano77 [171] G. Vailati, La grammatica dell’algebra, 1908, in [209], Vol. I, 92–110. [172] G. Vailati, Scritti (M. Quaranta, ed.), Vol. I (Scritti di filosofia), Vol. II (Scritti di scienza), Vol. III (Scritti di scienze umane), Forni, 1987. [173] M. L. J. van de Vel, Theory of Convex Structures, North-Holland, Amsterdam, 1993. · Zbl 0785.52001 [174] A. Voss, Differential- und Integralrechnung, in: Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Vol. II.1.1, Teubner, Leipzig, 1899, 54– 134. [175] A. N. Whitehead, The Axioms of Descriptive Geometry, Cambridge Univ. Press, Cambridge, 1907. · JFM 38.0502.03 [176] W. Wunderlich, Irregular curves and functional equations, Ganita (Proc. Benares Math. Soc.) 5 (1954), 215–230. · Zbl 0058.38307 [177] W. Wunderlich, Über Peano-Kurven, Elemente Math. 28 (1973), 1–10. [215] E. Zermelo, Beweis, daß jede Menge wohlgeordnet werden kann, Math. Ann. 59 (1904), 514–516. [178] E. Zermelo, Über das Maß und die Diskrepanz von Punktmengen, J. Reine Angew. Math. 158 (1927), 154–167. 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