# 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感2 (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愛 (4). Consequently Q2 (= 2感2) must equal (2愛)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感2 Square both sides & multiply by Q2 (4) Q = 2愛 Both sides must be divisible by 2 (5) (2愛)2 = 2感2 Substituting (6) 4愛2 = 2感2 Expanding (7) 2愛2 = P2 Divide both sides by 2. (8) P = 2惹 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