Atomism and Infinite Divisibility

Chapter 5

Notes and References

  1. David J. Furley, Two Studies in the Greek Atomists, (Princeton: Princeton University Press, 1967), p. 7. text
  2. Furley, p. 7. text
  3. Furley, p. 8. text
  4. After one bisection the parts are half the size of the original and are both extended. If a part after N bisections is extended, then half that is still extended. The parts are extended after one bisection, and if the parts are extended after N bisections, then they are also extended after N+1 bisections. Mathematical induction concludes that the parts are extended after K bisections for all K. text
  5. Furley, p. 10. text
  6. Furley, p. 13. text
  7. Furley, p. 13. text
  8. Furley, p. 13. text
  9. Furley, p. 14. text
  10. Furley, p. 16. text
  11. Furley, p. 17. text
  12. Furley, p. 17. text
  13. Furley, p. 19. text
  14. Furley, pp. 17-18. text
  15. Furley, p. 18. text
  16. Furley, p. 18. text
  17. Furley, p. 22. text
  18. Furley, p. 25. text
  19. Furley, pp. 36-37. text
  20. Furley, p. 36. text
  21. Arnauld, Antonie, The Art of Thinking: Port-Royal Logic, trans. James Dickoff and Patricia James, (New York: Bobbs-Merrill, 1964), p. 299. text
  22. 22. Such a proof might go as follows. Suppose is rational. Then there exists numbers P and Q such that  = P/Q (1). We may suppose that the fraction P/Q is expressed in lowest terms. This would require that P and Q are relatively prime, that is, that they have no greatest common divisor (GCD) (2). Now then, squaring both sides gives us the equation 2 = P2/Q2 or that Q2 = 2·P2 (3). Since the right hand side of the equation is divisible by 2, so must the left hand side be. But for that to be possible, Q must itself be divisible by 2, and Q must be of the form 2·R (4). Consequently Q2 (= 2·P2) must equal (2·R)2 (5). This allows us to conclude that P must also be divisible by 2 (8,9) contradicting the assumption that could be expressed as a rational number in lowest terms.
    (1) = P/Q. Assume is rational.
    (2) GCD(P,Q)=1 P/Q is expressed in lowest terms
    (3) Q2 = 2·P2 Square both sides & multiply by Q2
    (4) Q = 2·R Both sides must be divisible by 2
    (5) (2·R)2 = 2·P2 Substituting
    (6) 4·R2 = 2·P2 Expanding
    (7) 2·R2 = P2 Divide both sides by 2.
    (8) P = 2·S Both sides must be divisible by 2.
    (9) GCD(P,Q)=2 Both P and Q divisible by 2.
    (10)   P/Q By reductio cannot be rational.

    text

  23. See note 8 of chapter VI and page 174. text
  24. Arnauld, 1964. p. 299. text
  25. N. Kretzmann, ed., Infinity and Continuity in Ancient and Medieval Thought, (Ithica, NY: Cornell University Press, 1982). text
  26. J. D. North, "Finite and otherwise: Aristotle And Some Seventeenth Century Views", in Nature Mathematized vol. 1., University of Western Ontario Series in Philosophy of Science, no. 20. William R. Shea, ed., (Dordrecht, Holland and Boston, U. S. A.: D. Reidel, 1983), 113-148. text
  27. Donald L. M. Baxter, "Hume on Infinite Divisibility", History of Philosophical Quarterly 5 (April 1988): 133. text
  28. Baxter, pp. 134-5. text
  29. D. E. Thomsen, "Atomic nuclei: Quarks in leaky bags", Science News Magazine Vol. 125, No. 18, May 5, 1984, p. 297. text
  30. Dietrick E. Thomsen, "Experimenting With 40 Trillion Electron-Volts", Science News Magazine Vol. 132, No. 20, November 14, 1987, p. 315. text