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Empirical Methods in Mathematics

A Case-Study: Goldbach’s Conjecture

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Spanish Studies in the Philosophy of Science

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 186))

Abstract

Philosophy of mathematics has changed considerably during the last few years. The classical distinction between logicism, formalism and intuitionism, a major topic a few decades ago, has become less outstanding. The search for the logical foundations of mathematics belongs to the history of logic and mathematics. There is no more crisis concerning the foundations of mathematics. Conversely, the “fundamentalist” program of research in philosophy of mathematics seems to be currently stagnant. In 1967, Putnam published a celebrated paper, “Mathematics without Foundations”, where he claimed vigorously that:

I don’t think mathematics is unclear; I don’t think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs ‘foundations’2.

Research supported by the Spanish Dirección General de Política Científica, Ministerio de Educación y Ciencia, PB92-0846-C06-01.

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References

  • R. Archibald (1935): “Goldbachs Theorem I, II”, Scripta Mathematica 3, 44–50 and 153–161.

    Google Scholar 

  • W. Aspray and P. Kitcher (eds.) (1988): History and Philosophy of Modern Mathematics, Minneapolis, University of Minnesota Press.

    Google Scholar 

  • V. Aubry (1903): “Nombre pair somme de deux nombres premiers,” L’intermédiaire des mathématiciens X, 61–62.

    Google Scholar 

  • W. Balzer, C.U. Moulines and J.D. Sneed (1987): An Architectonic for Science, Dordrecht, Reidei.

    Book  Google Scholar 

  • P. Benacerraf (1965): “What Numbers could not be”, Philosophical Review 74, 47–73.

    Article  Google Scholar 

  • P. Benacerraf and H. Putnam (eds.) (1964): Philosophy of Mathematics: Selected Readings, Englewood Cliffs, Prentice Hall.

    Google Scholar 

  • J.P. van Bendegem (1987):“Fermat’s Last Theorem seen as an Exercise in Evolutionary Epistemology”, in W. Callebaut and R. Pinsten (eds.), Evolutionary Epistemology, Dordrecht, Reidel, 337–365.

    Chapter  Google Scholar 

  • J.P. van Bendegem (1989a): “Non-Formal Properties of Mathematical Proofs”, PSA 1988, vol. I, pp. 249–254.

    Google Scholar 

  • J.P. van Bendegem (1989b): “Foundations of Mathematics or Mathematical Practice: is one forced to choose?”, Philosophica 43:1, 197–213

    Google Scholar 

  • J. Bigelow (1988): The Reality of Numbers, Oxford, Clarendon Press.

    Google Scholar 

  • J. Bohman and C.-E. Froberg (1975): “Numerical Results on the Goldbach Conjecture”, BIT 15, 239–243.

    Article  Google Scholar 

  • N.V. Bougaief (1885): “Sur une loi générale de la théorie de la partition des nombres”, Comptes Rendus de l’Académie de Sciences de Paris 100, 1124.

    Google Scholar 

  • V. Brun (1914): “Über Hypothesenbildung”, Archiv for Mathematik og Natur videnskab XXXIV, Nr. 2.

    Google Scholar 

  • V. Brun (1915): “Ueber das Goldbachsche Gesetz und die Anzahl der Primzahlpaare”, Archiv for Mathematik 34:8, 1–19.

    Google Scholar 

  • V. Brun (1917): “Sur les nombres premiers de la forme ap+b”, Ibid., 14, 1–9.

    Google Scholar 

  • V. Brun (1919): “Le crible d’Erathosthène et le théorème de Goldbach”, Comptes Rendus de l’Académie de Sciences de Paris 168, 544–546.

    Google Scholar 

  • V. Brun (1920): “Le crible d’Erathosthène et le théorème de Goldbach”, Skr. Norske Vid. Akad. 1:3, 1–36.

    Google Scholar 

  • G. Cantor (1894): “Vérification jusqu’à 1000 du théorème empirique de Goldbach”, Compte Rendu, Congrès de Caen, Association française pour l’avancement des sciences, Paris 1895, 117–134.

    Google Scholar 

  • G. Cantor (1895): “Note 574”, L’intermédiaire des tnathématiciens 2, 179.

    Google Scholar 

  • R. Carnap (1935): “Formalwissenschaft und Realwissenschaft”, Erkenntnis 5, 29–35.

    Google Scholar 

  • R. Carnap (1939): Foundations of Logic and Mathematics, Chicago, Univ. of Chicago Press.

    Google Scholar 

  • E.C. Catalan (1888): “Sur le postulat de Bertrand”, Mém. Soc. R. Se. Liège (2), 15, 108–110.

    Google Scholar 

  • Chen Jing Run (1973): “On the representation of a large even integer as the sum of a prime and the product of at most two primes”, Sci. Sínica 16, 157–176 (previously in Kexue Tongbao 17, 1966, 385–386).

    Google Scholar 

  • Chen Jing Run (1978): “On the Goldbach’s problem and the sieve methods”, Sci. Sinica 21, 701–739.

    Google Scholar 

  • Chen Jing Run and Pan Cheng Dong (1980): “On the exceptional set of Goldbach numbers”, Sci. Sinica 23, 219–232.

    Google Scholar 

  • Ch. Chihara (1990): Constructibility and Mathematical Existence, Oxford, Clarendon Press.

    Google Scholar 

  • R.F. Churchhouse (1989): “Presidential Address”, Bulletin of the Institute of Mathematics and its Applications 25:3–4, 40–49.

    Google Scholar 

  • J.G. van der Corput (1937): “Sur l’hypothèse de Goldbach pour presque tous les nombres pairs”, Acta Arithmetica 2, 266–290.

    Google Scholar 

  • J.G. van der Corput (1938): “Sur l’hypothèse de Goldbach”, Proc. Akad. Wet. Amsterdam 41, 76–80.

    Google Scholar 

  • L. Corry (1992): “Nicolas Bourbaki and the concept of mathematical structure”, Synthese 92, 315–348.

    Article  Google Scholar 

  • M.J. Crowe (1975): “Ten ‘Laws’ Concerning Patterns of Change in the History of Mathematics”, Historia Mathematica 2, 161–166.

    Article  Google Scholar 

  • N.G. Cudakov (1947): “On Goldbach-Vinogradov’s Theorem”, Ann. of Math. 48, 515–545.

    Article  Google Scholar 

  • A. Cunningham (1906): “Evidence of Goldbach’s Theorem”, The Messenger of Mathematics 5:36, 17–30.

    Google Scholar 

  • J.W. Dauben (1984): “Conceptual Revolutions and the History of Mathematics”, in E. Mendehlsohn (ed.), Transformation and Tradition in the Sciences, Cambridge, Cambridge Univ. Press. 81–103.

    Google Scholar 

  • J.W. Dauben (1992): “Are There Revolutions in Mathematics?”, in Echeverría, Ibarra and Mormann (eds.), The Space of Mathematics, 205–229.

    Google Scholar 

  • P.J. Davis and R. Hersch (1981): The Mathematical Experience, Boston, Birkhäuser.

    Google Scholar 

  • A. Desboves (1855): “Sur un théorème de Legendre et son application à la recherche de limites qui comprennent entre elles des nombres premiers”, Nouvelles Annales de Mathématiques 14, 281–295.

    Google Scholar 

  • M. Detlefsen and Mark Luker (1980): “The Four-color Theorem and Mathematical Proof”, Journal of Philosophy 77:12, 803–819.

    Article  Google Scholar 

  • M. Detlefsen (ed.) (1992): Proof and Knowledge in Mathematics, London, Routledge.

    Google Scholar 

  • H.G. Diamond (1982): “Elementary Methods in the Study of the Distribution of Prime Numbers”, Bulletin of the American Mathematical Society 7:3, 553–589.

    Article  Google Scholar 

  • L.E. Dickson (1919): History of the Theory of Numbers, Washington, Carnegie Inst., 3 vols, (reimpr. in New York, Chelsea 1966).

    Google Scholar 

  • J. Echeverría (1992): “Observations, Conjectures and Problems in Number Theory: the History of the Prime Number Theorem”, in J. Echeverría, A. Ibarray Th. Mormann (eds.), The space of Mathematics, 250–272.

    Google Scholar 

  • J. Echeverría, A. Ibarra and Th. Mormann (eds.) (1992): The Space of Mathematics, Berlin, De Gruyter.

    Google Scholar 

  • J. Echeverría (1994): “Métodos empíricos en Matemáticas: la Conjetura de Riemann como ejemplo”, forthcoming in Arbor (Madrid, CSIC).

    Google Scholar 

  • H.M. Edwards (1974): Riemann’s Zeta Function, New York, Academic Press.

    Google Scholar 

  • P. Erdös (1936): “On the arithmetical density of the sum of two sequences which forms a basis for the integers”, Acta Arithmetica 1, 197–200.

    Google Scholar 

  • P. Erdös (1937): “On the sum and difference of squares of primes”, Journal London Math. Soc. 12, 133.

    Article  Google Scholar 

  • T. Estermann (1938): “On Goldbach Problem”, Proc. London Math. Soc. (2), 44, 307–314.

    Article  Google Scholar 

  • L. Euler (1911–36):Opera Omnia, Leipzig, Teubner.

    Google Scholar 

  • S. Feferman (1985): “Working Foundations”, Synthese 62, 229–254.

    Article  Google Scholar 

  • H. Field (1980): Science without Numbers, Princeton, Princeton University Press.

    Google Scholar 

  • H. Field (1989): Realism, Mathematics and Modality, Oxford, Blackwell.

    Google Scholar 

  • H.F. Fliegel and D.S. Robertson (1989): “Goldbach’s Comet: The Numbers related to Goldbach’s Conjecture”, Journal of Recreational Mathematics 21, 1–7.

    Google Scholar 

  • J. Franklin (1987): “Non-deductive Logic in Mathematics”, British Journal for the Philosophy of Science 38, 1–18.

    Article  Google Scholar 

  • D. Gillies (ed.) (1992): Revolutions in Mathematics, Oxford, Clarendon.

    Google Scholar 

  • E. Glas (1989): “Testing the Philosophy of Mathematics in the History of Mathematics”, I, II, Stud. Hist. Phil. Sei. 20:1, 115–131 and 20:2, 157–174.

    Article  Google Scholar 

  • L.J. Goldstein (1973): “A History of the Prime Number Theorem”, American Math. Monthly 80, 599–615.

    Article  Google Scholar 

  • W.A. Golubiew (1956): “Abzählung von Vierlingen von 2,000,000 bis 3.000.000 und von Fünflingen von 0 bis 2,000,000”, Anzeiger der mathnaturw. Klasse der österreichischen Akademie der Wissenschaften, 153–157.

    Google Scholar 

  • N.D. Goodman (1981): “The Experiential Foundation of Mathematical Knowledge“, History and Philosophy of Logic 2, 55–65.

    Article  Google Scholar 

  • N.D. Goodman (1990): “Mathematics as Natural Science”, Journal of Symbolic Logic 55:1, 182–193.

    Article  Google Scholar 

  • J.V. Grabiner (1974): “Is mathematical truth time-dependent?”, American Mathematical Monthly 81, 354–365.

    Article  Google Scholar 

  • A. Granville, J. van de Lune and H.J.J. Te Riele (1989): “Checking the Goldbach Conjecture on a Vector Computer”, in R.A. Mollin (ed.), Number Theory and Applications, Dordrecht, Kluwer, 423–433.

    Google Scholar 

  • R.K. Guy (1981): Unsolved Problems in Number Theory, New York, Springer.

    Google Scholar 

  • J. Hadamard (1896): “Sur la distribution des zéros de la fonction (s) et ses conséquen cesarithmétiques”, Bulletin de la Société Mathématique de France 24, 199–220.

    Google Scholar 

  • H. Halberstam and H.E. Richert (1974): Sieve Methods, New York, Academic Press.

    Google Scholar 

  • M. Hallet (1979): “Towards a Theory of Mathematical Research Programmes” I, II, British Journal for the Philosophy of Science 30, 1–25 and 135–59.

    Article  Google Scholar 

  • G.H. Hardy (1922): “Goldbach’s Theorem”, Math. Tid. B, 1–16.

    Google Scholar 

  • G.H. Hardy (1966):Collected Papers, Oxford, Clarendon, 3 vols.

    Google Scholar 

  • Ch.J. Hargreave (1849): “Analytical researches concerning numbers”, Philosophical Magazine (3), XXXV, 49.

    Google Scholar 

  • R. Haussner (1896): “Über das Goldbachsche Gesetz”, Jahresbericht der Deutschen Mathema-tiker-Vereinigung 5, 62–66.

    Google Scholar 

  • R. Haussner (1899): “Tafeln für das Goldbach’sche Gesetz”, Abh. der Kaiserl. Leop. Deutschen Akad. der Naturforscherl, 214 pp.

    Google Scholar 

  • G. Hellman (1989): Mathematics without Numbers, Oxford, Oxford University Press.

    Google Scholar 

  • G. Hellman (1990): “Toward a Modal-Structural Interpretation of Set Theory”, Synthese 84, 409–443.

    Article  Google Scholar 

  • R. Hersch (1986): “Some Proposals for Reviving the Philosophy of Mathematics”, in Th. Tymoczko (ed), New Directions in the Philosophy of Mathematics, 9–28.

    Google Scholar 

  • D. Hilbert (1901): “Mathematische Probleme”, Archiv f. Math. u. Phys. 3:1, 44–63 and 213–237.

    Google Scholar 

  • R.A. Holland (1992): “Apriority and applied mathematics”, Synthese 92, 349–370.

    Article  Google Scholar 

  • Fr. Kambartel (1972): “Mathematics and the Concept of Theory”, in Y. Bar-Hillel (ed.), Logic, Methodology and Philosophy of Science, Amsterdam, North Holland, 210–219.

    Google Scholar 

  • J.G. Kemeny (1958): “Undecidable Problems of Elementary Number Theory”, Math. Annalen 135, 160–169.

    Article  Google Scholar 

  • P. Kitcher (1984): The Nature of Mathematical Knowledge, New York, Oxford University Press.

    Google Scholar 

  • M. Kline (1980): Mathematics: the loss of certainty, Oxford, Oxford Univ. Press.

    Google Scholar 

  • S. Körner (1964): “An Empiricist justification of mathematics”, in Y. Bar-Hillel (ed.), Logic, Methodology and Philosophy of Science, Amsterdam, North Holland, 222–227.

    Google Scholar 

  • T. Koetsier (1987): “Cauchy’s Rigorous Calculus: a Revolution in Kuhn’s Sense?”, Nieuw. Arch. Wisk. (4), 5, 335–353.

    Google Scholar 

  • S. Kripke (1980): Naming and Necessity, Cambridge, Mass., Harvard University Press.

    Google Scholar 

  • T.S. Kuhn (1970): The structure of scientific revolutions, 2nd ed., Chicago, Univ. of Chicago Press.

    Google Scholar 

  • C.-A. Laisant (1897): “Sur un procédé de vérification expérimentale du théorème de Goldbach”, Bulletin de la Société Mathématique de France XXV, 209–211.

    Google Scholar 

  • I. Lakatos (ed.) (1967): Problems in the Philosophy of Mathematics, Amsterdam, North Holland.

    Google Scholar 

  • I. Lakatos (1976a): “A Renaissance of Empiricism in the Recent Philosophy of Mathematics”, British Journal for the Philosophy of Science 27, 201–223.

    Article  Google Scholar 

  • I. Lakatos (1976b):Proofs and Refutations, Cambridge, Cambridge University Press.

    Google Scholar 

  • E. Landau (1900): “Über die zahlentheoretische Funktion (n) und ihre Beziehung zum Goldbachschen Satz”, Gött. Nachr„ 177–186.

    Google Scholar 

  • E. Landau (1912): “Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunction”, Proc. 5th. Int. Congr. Math, v. 1, 93–108.

    Google Scholar 

  • E. Landau (1930): “Die Goldbachsche Vermutung und der Schnirelmannsche Satz”, Nachr. Akad. Wiss. Math.-Phys. Kl., 255–276.

    Google Scholar 

  • D.H. Lehmer (1969): “Computer Technology applied to the Theory of Numbers”, in W.J. LeVeque (ed.), Studies in Number Theory, Englewood Cliffs, N.J., Prentice-Hall, 117–151.

    Google Scholar 

  • W.A. Light, T.J. Forres, N. Hammond y S. Roe (1980): “A note on Goldbach’s Conjecture”, BIT 20, 525.

    Article  Google Scholar 

  • F.J.E. Lionnet (1879): “Note sur la question Tout nombre pair est-il la somme de deux impairs premiers?”, Nouvelles Annales de Mathématiques 38, 356–360.

    Google Scholar 

  • B. Lucke (1962): “Zur Hardy-Littlewoodschen Behandlung des Goldbachschen Problems”, Diss. Math. Naturwiss. Göttingen.

    Google Scholar 

  • M. Machover (1983): “Towards a new philosophy of Mathematics”, Brit. J. Phil. Sei. 34, 1–11.

    Article  Google Scholar 

  • P. Maddy (1990): Realism in Mathematics, Oxford, Clarendon Press.

    Google Scholar 

  • P. Maddy (1991): “Philosophy of Mathematics: Prospects for the 1990’s”, Synthese 98, 155–164.

    Article  Google Scholar 

  • Y. Matijasevic (1971): “Diophantine representation of the set of Prime Numbers”, Soviet. Math. Doklady 12, 249–154.

    Google Scholar 

  • Y. Matijasevic (1990): “The Riemann Hypothesis from a Logician’s Point of View:”, in R.A. Mollin (ed.), Number Theory, Berlin, De Gruyter.

    Google Scholar 

  • H. Mehlberg (1962): “The Present Situation in the Philosophy of Mathematics”, in Logic and Language: Studies dedicated to R. Carnap, Dordrecht, Reidel, 69–103.

    Google Scholar 

  • H. Mehrtens (1976): “T.S. Kuhn’s Theories and Mathematics: a Discussion Paper on the ‘New Historiography’ of Mathematics”, Historia Mathematica 3, 297–320.

    Article  Google Scholar 

  • H. Mehrtens (1992): “Appendix 1992: revolutions reconsidered”, en D. Gillies (ed.), Revolutions in Mathematics, 42–48.

    Google Scholar 

  • J. Merlin (1911): “Sur quelques théorèmes d’Arithmétique et un énoncé qui les contient”, Comptes Rendus de l’Acad. de Sciences de Paris 153, 516–518.

    Google Scholar 

  • J. Merlin (1915): “Un travail sur les nombres premiers”, Bulletin des sciences mathématiques 39, 121–136.

    Google Scholar 

  • H.L. Montgomery and R.C. Vaughan (1975): “The exceptional set in Goldbach’s Problem”, Acta Arithmetica 27, 353–370.

    Google Scholar 

  • C. Parsons (1990): “The Structuralist View of Mathematical Objects”, Synthese 84, 303–346.

    Article  Google Scholar 

  • R. Phillips (ed.) (1987):Studies in the History of Mathematics, The Mathematical Association of America, vol. 26.

    Google Scholar 

  • N. Pipping (1927): “Neue Tafeln für das Goldbachsche Gesetz”, Commentationes Physico-Mathematicae, Societas Scientiarum Fennica, IV, 4.

    Google Scholar 

  • N. Pipping (1929): “Über Goldbachsche Spaltungen grosser Zahlen”, Comm. Phys.-Math. 10.

    Google Scholar 

  • N. Pipping (1938): “Die Goldbachsche Vermutung und der Goldbach-Vinogradovsche Satz”, Acta Acad. Aboensis, Math-Phys. 11.

    Google Scholar 

  • N. Pipping (1940): “Goldbachsche Spaltung der geraden X für X = 60,000 bis 99,998”, Ibid., 12.

    Google Scholar 

  • H.A. Pogorzelski (1969–70): “Goldbach Sentences in Abstract Arithmetics, I and II”, J. f.d. reine u. angew. Math. 237, 65–96 and 243, 32–54.

    Article  Google Scholar 

  • H.A. Pogorzelski (1974): “On the Goldbach conjecture and the consistency of general recursive arithmetic”, Ibid. 268/9, 1–16.

    Google Scholar 

  • H.A. Pogorzelski (1976): “Dirichlet theorems and prime number hypotheses of a conditional Goldbach theorem”, Ibid. 286/7, 33–45.

    Google Scholar 

  • H.A. Pogorzelski (1977a): “Semisemiological structure of the prime numbers and conditional Goldbach theorems”, Ibid. 290, 77–92.

    Google Scholar 

  • H.A. Pogorzelski (1977b): “Goldbach conjecture”, Ibid. 292, 1–12.

    Google Scholar 

  • Polignac (1859): “Recherches nouvelles sur les nombres premiers”, Comptes Rendus de l’Académie des Sc. XLIX, 350.

    Google Scholar 

  • G. Polya (1954): Mathematics and Plausible Reasoning, 2 vols., Princeton, Princeton University Press.

    Google Scholar 

  • G. Polya (1959): “Heuristic reasoning in the Theory of Numbers”, American Mathematical Monthly 66, 375–384.

    Article  Google Scholar 

  • K.R. Popper (1959): The Logic of Scientific Discovery, London, Hutchinson.

    Google Scholar 

  • H. Putnam (1960): “An unsolvable problem in number theory”, Journal of Symbolic Logic 25, 220–232.

    Article  Google Scholar 

  • G. Polya (1967a): “Mathematics without Foundations”, in Mathematics, Matter and Method, vol. I, 43–59.

    Google Scholar 

  • G. Polya (1967b): “The Thesis that Mathematics is Logic”, in Mathematics, Matter and Method: Philosophical Papers, vol. I. 12–42.

    Google Scholar 

  • G. Polya (1975): Philosophical Papers, 2 vols., Cambridge, Cambridge University Press.

    Google Scholar 

  • W.V. Quine (1963): From a Logical Point of View, New York, Harper & Row.

    Google Scholar 

  • M. Resnik (1981): “Mathematics as a Science of Patterns”, Nous 15, 529–550.

    Article  Google Scholar 

  • M. Resnik (1989): “Computation and Mathematical Empiricism”, Philosophical Topics 17:2, 129–144.

    Google Scholar 

  • L. Ripert (1903): “Vérification du théorème de Goldbach”, L’intermédiaire des mathématiciens 10, 74–77 and 166–167.

    Google Scholar 

  • G. de Rocquigny (1898): “Problème 1408”, L’intermédiaire des mathématiciens 5, 268.

    Google Scholar 

  • F. Romani (1985): “Uso heuristico degli elaboratori nella ricerca matematica”, Bolletino U.M.I. (6), 4-B, 627–638.

    Google Scholar 

  • G-C. Rota (1990): “The Concept of Mathematical Truth”, Math. Scientist 15, 65–73.

    Google Scholar 

  • A. Schinzel (1959): “Démonstration d’une conséquence de l’hypothèse de Goldbach”, Compositio Mathematica 14, 74–76.

    Google Scholar 

  • A. Schinzel and W. Sierpinski (1958):“Sur certaines hypothèses concernant les nombres premiers”, Acta Arithmetica 4, 185–208+ “Erratum”, Ibid. 259.

    Google Scholar 

  • L. Schnirelmann (1930): “Ueber additive Eigenschaften von Zahlen”, Math. Ann. 107 (1933), 649–690.

    Google Scholar 

  • A.H. Schoenfeld (1985): Mathematical Problem Solving, San Diego, Academic Press.

    Google Scholar 

  • A. H. Schoenfeld (1987): “Polya, Problem Solving and Education”, Mathematics Magazine 60:5, 283–291.

    Article  Google Scholar 

  • Ch. J. Scriba (1970): “Zur Entwicklung der additiven Zahlentheorie von Fermat bis Jacobi”, Jber. Deutsch. Math.-Verein. 72, 122–142.

    Google Scholar 

  • A. Seiberg (1950):“The general sieve method and its place in prime number theory”, Proc. Int. Congr. Math., vol. I, 286–292.

    Google Scholar 

  • E. Selmer and G. Nesheim (1942a): “Tafel der Zwillingsprimzahlen bis 200.000”, Norske Vid. Selsk. Forh. Trondhjem XV, 25.

    Google Scholar 

  • E. Selmer and G. Nesheim (1942b): “Die Goldbachschen Zwillingsdarstellungen der durch 6 teilbaren Zahlen 196.302–196.596”, Norske Vid. Selsk. Forh. Trondhjem XV, 28.

    Google Scholar 

  • E. Selmer (1943):“Eine neue asymptotische Formel für die Anzahl der Goldbachschen Spaltungen einer geraden Zahl und eine numerische Kontrolle”, Arch. Math. Naturvid. 46:1, 1–18.

    Google Scholar 

  • N. M. Shah and B. M. Wilson (1919): “On an empirical formula connected with Goldbach’s Theorem”, Proc. of the Cambridge Philosophical Society 19, 238–244 and 245–254.

    Google Scholar 

  • D. Shanks (1978): Solved & Unsolved Problems in Number Theory, New York, Chelsea.

    Google Scholar 

  • Shen Mok Kong (1964): “On Checking the Goldbach Conjecture”, Nor disk Tidskr. Imfor. Behand. 4.

    Google Scholar 

  • M.K. Sinisalo (1993): “Checking the Goldbach Conjecture up to 4.1011”, Mathematics of Computation, 61:204, 931–935. P.

    Google Scholar 

  • P. Stäckel (1896): “Über Goldbachs empirisches Theorem”, Nachrichten der K. Gesells. der Wiss. zu Göttingen, 292–299.

    Google Scholar 

  • P. Stäckel (1916): “Die Darstellung der geraden Zahlen als Summen von zwei Primzahlen”, Sitz.-ber. der Heidelb. Akad. der Wiss„ Math.-Naturwiss. Klasse, 8 August 1916.

    Google Scholar 

  • P. Stäckel (1917–18): “Die Lückenzahlen r-ter Stufe und die Darstellung der geraden Zahlen als Summen und Differenzen ungerader Primzahlen”, Ibid., 27 dez. 1917, 19 Jan. 1918 y 19 Jul. 1918.

    Google Scholar 

  • M.L. Steiny P.R. Stein (1965a): “New experimental results on the Goldbach Conjecture”, Math. Mag. 38, 72–80.

    Article  Google Scholar 

  • M.L. Steiny P.R. Stein (1965b): “Experimental results on additive 2-bases”, Math. Comp. 19, 427–434.

    Article  Google Scholar 

  • M.L. Steiny P.R. Stein (1965c): “Tables of the number of binary decompositions of all even numbers 0<2n<200,000 into prime numbers and lucky numbers”, Los Alamos Report LA-3106, vol. I-II.

    Google Scholar 

  • M. Steiner (1983): “The Philosophy of Mathematics of Imre Lakatos”, Journal of Philosophy 80:9, 502–521.

    Article  Google Scholar 

  • M.A. Stern (1858): “Über eine zahlentheoretische Funktion”, J. f. d. reine u. angew. Math. 55, 193–220.

    Article  Google Scholar 

  • H. Struve (1989): “Empirische Geometrie”, Zeitschrift für allgemeine Wissenschaftstheorie 20/2, 325–339.

    Article  Google Scholar 

  • J.J. Sylvester (1871): “On the partition of an even number into two primes”, Proc. London Math. Soc. 1:4, 4–6.

    Google Scholar 

  • J. J. Sylvester (1896–7): “On the Goldbach-Euler Theorem regarding prime numbers”, Nature 55, 196–7 y 269.

    Article  Google Scholar 

  • Tchebychef (1851): “Sur la fonction qui détermine la totalité des nombres premiers inférieures à une limite donnée”, Mém. de VAcad. de St. Petersbourg VI, 141–157 (reprint in Liouville’s Journal XVII, 1852, 341–365).

    Google Scholar 

  • P. Teller (1980): “Computer Proof”, Journal of Philosophy 77:12, 797–803.

    Article  Google Scholar 

  • N. G. Tschudakoff (1938): “On the density of the set of even integers which are not representable as a sum of two odd primes (in Russian)”, Izv. Akad. Nauk SSSR, Ser. Mat., 1, 25–40.

    Google Scholar 

  • P. Turän (1990): Collected Papers (ed. P. Erdös), Budapest, Akademiai Kiado.

    Google Scholar 

  • Th. Tymoczko (1979): “The Four-Color Problem and its Philosophical Significance”, Journal of Philosophy 76, 57–83.

    Article  Google Scholar 

  • Th. Tymoczko (ed.) (1986): New Directions in the Philosophy of Mathematics, Boston, Birkhäuser.

    Google Scholar 

  • Ch. de la Vallée Poussin (1898): “Sur la fonction (s) de Riemann et le nombre des nombres premiers inférieurs à une limite donnée”, Mémoire couronnée de VAcad. R. des Se. de Belgique LIX, Nr. 1.

    Google Scholar 

  • W.A. Verloren van Themaat (1989): “The Own Character of Mathematics Discussed with Consideration of the Proof of the Four-Color Theorem”, Zeitschrift für allgemeine Wissenschaftstheorie 20/2, 340–350.

    Article  Google Scholar 

  • I.M. Vinogradov (1937): “Representation of an odd number as a sum of three primes”, URSS Akad. Doklady 15, 191–294.

    Google Scholar 

  • Wang Yuan (ed.) (1984): Goldbach Conjecture, Singapore, World Scientific Publ.

    Google Scholar 

  • Hao Wang (1974): From Mathematics to Philosophy, London, Routledge & Kegan Paul.

    Google Scholar 

  • Hao Wang (1986): “Theory and Practice in Mathematics”, in Th. Tymoczko (ed.), New Directions in the Philosophy of Mathematics, 129–152.

    Google Scholar 

  • R. L. Wilder (1952): Introduction to the Foundations of Mathematics, New York.

    Google Scholar 

  • R.L. Wilder (1968): Evolution of Mathematical Concepts, New York, Wiley.

    Google Scholar 

  • R.L. Wilder (1981): Mathematics as a Cultural System, Oxford, Pergamon.

    Google Scholar 

  • H.C. Williams (1982): “The influence of computers in the development of Number Theory”, Comp. & Math, with Applications 8:2, 75.

    Article  Google Scholar 

  • L. Wittgenstein (1953): Philosophical Investigations, New York, Macmillan.

    Google Scholar 

  • L. Wittgenstein (1956): Remarks on the Foundations of Mathematics, Oxford, Blackwell.

    Google Scholar 

  • L. Wittgenstein (1975): Lectures on the Foundations of Mathematics, Cambridge 1939 (ed. C. Diamond), Chicago, University of Chicago Press.

    Google Scholar 

  • Z. Yuxin (1990): “From the logic of mathematical discovery to the methodology of scientific research programmes”, British J. for the Phil, of Sc. 41, 377–399.

    Article  Google Scholar 

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Authors and Affiliations

  1. Universidad del Pais Vasco, Spain

    Javier Echeverría

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  1. Javier Echeverría
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  1. The Evergreen State College, Olympia, Washington, USA

    Gonzalo Munévar

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© 1996 Kluwer Academic Publishers

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Echeverría, J. (1996). Empirical Methods in Mathematics. In: Munévar, G. (eds) Spanish Studies in the Philosophy of Science. Boston Studies in the Philosophy of Science, vol 186. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0305-0_2

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  • DOI: https://doi.org/10.1007/978-94-009-0305-0_2

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