Skip to main content
SpringerLink
Log in

Other Rules of Discovery

  • Chapter
  • First Online:
Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method

Part of the book series: Logic, Argumentation & Reasoning ((LARI,volume 1))

  • 1356 Accesses

Abstract

In Chapter 20 two of the oldest and better known rules of discovery have been considered: induction and analogy.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Newton (1972, II, 576).

  2. 2.

    Sartorius von Wantershausen (1856, 12).

  3. 3.

    Ibid.

  4. 4.

    See, for example, Graham et al. (1994, 6–7).

  5. 5.

    Words are stones is the title of Levi (2005).

  6. 6.

    Newton (1967–1981, VIII, 93).

  7. 7.

    Ibid., III, 73.

  8. 8.

    Ibid., VIII, 125.

  9. 9.

    Ibid., VIII, 129.

  10. 10.

    Ibid., III, 73.

  11. 11.

    Aristotle, Poetica, 1457 b, 6–7.

  12. 12.

    Aristotle, Topica, Z 2, 140 a 9–12.

  13. 13.

    Proclus, In Primum Euclidis Elementorum Librum Commentarii (Friedlein), 379.5–6.

  14. 14.

    Ibid., 379.15–16.

  15. 15.

    Kant (1997a, 631, B 744).

  16. 16.

    Ibid., 631–632, B 744.

  17. 17.

    Ibid., 632, B 744–745.

  18. 18.

    Pascal (1904–1914, IX, 242–243).

  19. 19.

    Ibid., IX, 244.

  20. 20.

    Ibid., IX, 243–244.

  21. 21.

    Ibid., IX, 278.

  22. 22.

    Ibid., IX, 243.

  23. 23.

    Frege (1979, 211).

  24. 24.

    Frege (1967, 55).

  25. 25.

    Frege (1979, 179).

  26. 26.

    Ibid., 208.

  27. 27.

    Frege (1984, 274).

  28. 28.

    Frege (1979, 208).

  29. 29.

    Hilbert and Bernays (1968–1970, I, 292).

  30. 30.

    Popper (1945, II, 13).

  31. 31.

    Pascal (1904–1914, II, 103).

  32. 32.

    Eilenberg and MacLane (1945, 237).

  33. 33.

    Whitehead and Russell (1925–1927, I, 11).

  34. 34.

    Ibid.

  35. 35.

    Ibid., I, 12.

  36. 36.

    Diogenes Laertius, Vitae Philosophorum, I.24.

  37. 37.

    Clagett (1999, 163). ‘Khet’ is an ancient Egyptian unit of measure for length, equal to 100 cubits.

  38. 38.

    Galilei (1968, VIII, 269).

  39. 39.

    Ibid., VIII, 209.

  40. 40.

    Ibid., VIII, 273.

  41. 41.

    Grosholz (2007, 15).

  42. 42.

    Galilei (1968, VIII, 268).

  43. 43.

    Hilbert (1996b, 1100).

  44. 44.

    Hilbert (2004a, 75).

  45. 45.

    Ibid., 541.

  46. 46.

    Hadamard (1954, 88).

  47. 47.

    Ibid.

  48. 48.

    For details, see Cellucci (2008a), Chapter 9.

  49. 49.

    Hilbert (1962, 1).

  50. 50.

    Tennant (1986, 304).

  51. 51.

    Ibid.

  52. 52.

    Cellucci (2009, 13). Tennant speaks of the ‘Triangle ABC’ as the “general triangle” (Tennant 1986, 304). Thus he links his view with Locke’s. For Locke claims that ‘Triangle ABC’ does not stand for a particular object but rather for the ‘general triangle’, that is, “the general idea of a triangle,” which “must be neither oblique, nor rectangle, neither equilateral, equicrural, nor scalenon; but all and none of these at once” (Locke 1975, 596). But the idea of general triangle is an impossible one, because it can be shown that “general objects cannot exist” (Cellucci 2009, 5).

  53. 53.

    Ibid., 14.

  54. 54.

    Klein (2004, 201).

  55. 55.

    Giaquinto (2007, 174).

  56. 56.

    Ibid.

  57. 57.

    Bråting and Pejlare (2008, 354).

  58. 58.

    Ibid.

  59. 59.

    Avigad et al. (2009, 758).

  60. 60.

    Ibid., 760.

  61. 61.

    Ibid., 764.

  62. 62.

    This is also clear from Mancosu (2008), a sort of manifesto for the attempt to fit not only diagrams, but all mathematical practice into the axiomatic method. For a criticism of this manifesto, see Cellucci (2013b), section 2.

  63. 63.

    Dieudonné (1969, ix).

References

  • Avigad, Jeremy, Edward Dean, and John Mumma. 2009. A formal system for Euclid’s Elements. The Review of Symbolic Logic 2: 700–768.

    Article  Google Scholar 

  • Bråting, Kajsa, and Johanna Pejlare. 2008. Visualization in mathematics. Erkenntnis 68: 345–358.

    Article  Google Scholar 

  • Cellucci, Carlo. 2008a. Perché ancora la filosofia. Rome: Laterza.

    Google Scholar 

  • Cellucci, Carlo. 2009. The universal generalization problem. Logique & Analyse 52: 3–20.

    Google Scholar 

  • Cellucci, Carlo. 2012b. Top-down and bottom-up philosophy of mathematics. Foundations of Science. doi:10.1007/s10699-012-9287-6.

  • Clagett, Marshall. 1999. Ancient Egyptian science. A source book, vol. 3: Ancient Egyptian mathematics. Philadelphia: American Mathematical Society.

    Google Scholar 

  • Dieudonné, Jean. 1969. Foundations of modern analysis. New York: Academic Press.

    Google Scholar 

  • Eilenberg, Samuel, and Saunders MacLane. 1945. General theory of natural equivalences. Transactions of the American Mathematical Society 58: 231–294.

    Google Scholar 

  • Frege, Gottlob. 1967. Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In From Frege to Gödel. A source book in mathematical logic, 1879–1931, ed. Jean van Heijenoort, 5–82. Cambridge: Harvard University Press.

    Google Scholar 

  • Frege, Gottlob. 1979. Posthumous writings, ed. Hans Hermes, Friedrich Kambartel, and Friedrich Kaulbach. Oxford: Blackwell.

    Google Scholar 

  • Frege, Gottlob. 1984. Collected papers on mathematics, logic, and philosophy, ed. Brian McGuinness. Oxford: Blackwell.

    Google Scholar 

  • Galilei, Galileo. 1968. Opere, ed. Antonio Favaro. Florence: Barbera.

    Google Scholar 

  • Giaquinto, Marcus. 2007. Visual thinking in mathematics. An epistemological study. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Graham, Ronald L., Donald E. Knuth, and Oren Patashnik. 1994. Concrete mathematics. Reading: Addison-Wesley.

    Google Scholar 

  • Grosholz, Emily Rolfe. 2007. Representation and productive ambiguity in mathematics and the sciences. Oxford: Oxford University Press.

    Google Scholar 

  • Hadamard, Jacques. 1954. The psychology of invention in the mathematical field. Mineola: Dover.

    Google Scholar 

  • Hilbert, David. 1962. Grundlagen der Geometrie. Stuttgart: Teubner.

    Google Scholar 

  • Hilbert, David. 1996b. Axiomatic thought. In From Kant to Hilbert: A source book in the foundations of mathemtics, vol. 2, ed. William Bragg Ewald, 1107–1115. Oxford: Oxford University Press.

    Google Scholar 

  • Hilbert, David. 2004a. Die Grundlagen der Geometrie. In From Kant to Hilbert: A source book in the foundations of mathematics, vol. 2, ed. William Bragg Ewald, 72–81. Oxford: Oxford University Press.

    Google Scholar 

  • Hilbert, David, and Paul Bernays. 1968–1970. Grundlagen der Mathematik. Berlin: Springer.

    Book  Google Scholar 

  • Kant, Immanuel. 1997a. Critique of pure reason, ed. Paul Guyer and Allen W. Wood. Cambridge: Cambridge University Press.

    Google Scholar 

  • Klein, Christian Felix. 2004. Elementary mathematics from an advanced standpoint: Geometry. Mineola: Dover.

    Google Scholar 

  • Levi, Carlo. 2005. Words are stones. Impressions of Sicily. London: Hesperus Press.

    Google Scholar 

  • Locke, John. 1975. An essay concerning human understanding, ed. Peter Harold Nidditch. Oxford: Oxford University Press.

    Google Scholar 

  • Mancosu, Paolo (ed.). 2008. The philosophy of mathematical practice. Oxford: Oxford University Press.

    Google Scholar 

  • Newton, Isaac. 1967–1981. The mathematical papers, ed. Derek Thomas Whiteside. Cambridge: Cambridge University Press.

    Google Scholar 

  • Newton, Isaac. 1972. Philosophiae Naturalis Principia Mathematica. Facsimile of third edition (1726) with variant readings, ed. Alexandre Koyré, I. Bernard Cohen, and Anne Whitman. Cambridge: Cambridge University Press.

    Google Scholar 

  • Pascal, Blaise. 1904–1914. Oeuvres, ed. Léon Brunschvicg and Pierre Boutroux. Paris: Hachette.

    Google Scholar 

  • Popper, Karl Raimund. 1945. The open society and its enemies. London: Routledge.

    Google Scholar 

  • Sartorius von Wantershausen, Willem. 1856. Gauss: zum Gedächtnis. Leipzig: Hirzel.

    Google Scholar 

  • Tennant, Neil. 1986. The withering away of formal semantics? Mind and Language 1: 302–318.

    Article  Google Scholar 

  • Whitehead, Alfred North, and Bertrand Russell. 1925–1927. Principia Mathematica. Cambridge: Cambridge University Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sapienza University of Rome, Rome, Italy

    Carlo Cellucci

Authors
  1. Carlo Cellucci
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Cellucci, C. (2013). Other Rules of Discovery. In: Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method. Logic, Argumentation & Reasoning, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6091-2_21

Download citation

Publish with us

Policies and ethics

Search

Navigation