Atomism and Infinite Divisibility

Chapter 3

Notes and References

1. Salmon, Wesley, Zeno's Paradoxes, (Indianapolis: Bobbs-Merrill, 1970), p. 12. text
2. Max Black, "Achilles and the Tortoise", in Zeno's Paradoxes, ed. Wesley Salmon, (Indianapolis: Bobbs-Merrill, 1970), p. 69. text
3. Black, p. 74. text
4. Adolf Grünbaum, "Zeno's Metrical Paradox of Extension", in Zeno's Paradoxes, (Indianapolis: Bobbs-Merrill, 1970), p. 189. text
5. It is a developmental enhancement to resolve one concept into two. When the need to do so arises, both concepts usually apply to previously considered cases, but the resolving case necessitates the distinction because one concept applies while the other does not. text
6. Ernest Nagle and James R. Newman, Gödel's Proof, (New York: New York University Press, 1958), p. 86. text
7. Salmon, p. 9. text
8. Suppose the general model is <L,I,A>, the limited model is <l,i,a>, and a A. Let z be a sequence of terms in l. Then i(z) is a subset of a. Let Z be a sequence of terms in L such that I(Z) = i(z). We would represent this as I-1(i(z)). Clearly Z is not all L. This would contradict the hypothesis that a  A. Since a  A, A-a is not Null. Let X be a sequence in I-1(A-a). If we combine the sequences Z and X, objects will be picked out which include the sequence i(z) as well as additional objects in A-a which cannot be picked out by any sequence in language l. Nothing in the foregoing precludes the possibility that l is a subset of L and i is just I restricted to l. text
9. Bertrand Russell, "The Problem of Infinity Considered Historically", in Zeno's Paradoxes, ed. Wesley Salmon, (Indianapolis: Bobbs-Merrill, 1970), p. 56. text
10. J. O. Wisdom, "Achilles on a Physical Racecourse", in Zeno's Paradoxes, ed. Wesley Salmon, (Indianapolis: Bobbs-Merrill, 1970), p. 87. text
11. James Thomson, "Tasks and Super-Tasks", in Zeno's Paradoxes, ed. Wesley Salmon, (Indianapolis: Bobbs-Merrill, 1970), p. 91. text
12. Keith J. Devlin, Fundamentals of Contemporary Set Theory, (New York: Springer-Verlag, 1979), p. 52. text
13. Russell, p. 56. text
14. Thomson, p. 100. text
15. G. E. L. Owen, "Zeno and the Mathematicians", in Zeno's Paradoxes, ed. Wesley Salmon, (Indianapolis: Bobbs-Merrill, 1970), p. 143. text
16. Herbert A. Pohl, Quantum Mechanics for Science and Engineering, (Englewood Cliffs, New Jersey: Prentice-Hall, 1967), pp. 11-12. text
17. Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics , (San Francisco: W. H. Freeman and Company, 1963), pp. 64-66. text
18. Taylor and Wheeler, p. 66. text
19. Pohl, pp. 50-56. text
20. While some radioactive atoms decay by exactly this process, others decay by a similar process, but by emitting other particles than protons. text
21. There are no known macroscopic models for such behavior -- passing a place without being there. text
22. The process is called "tunneling", it seems to me, as a direct result of the perceived absurdity of passing a point without being there and the strength of our commitment to the premiss that objects remain in existence. text
23. We have even invented a number of electronic devices that depend upon this theory to work -- (the tunnel diode is one). text
24. Taylor and Wheeler, pp. 50-51. text
25. Taylor and Wheeler, p. 66. text
26. William McGowen Priestly, Calculus: An Historical Approach, (New York: Springer-Verlag, 1979), p. 278. text
27. The factorial function, written N!, can serve as an example of a recursive definition. The factorial of a number, N, is the product of that number and all numbers less than it, down to and including 1. 0! is 1 by definition. The above definition can be expressed by a recursive definition as follows:

The factorial of N is defined recursively as follows: If N is zero then the value of N! is one. Otherwise, the value of N factorial is N times N-1 factorial.

`     0! = 1, N! = Nˇ(N-1)!.`

If we tried this definition for N = 4, we would get the following sequence:

```                          4! = 4ˇ3!
First reentry  (N=3)        3! = 3ˇ2!
Second reentry (N=2)               2! = 2ˇ1!
Third reentry  (N=1)                      1! = 1ˇ0!
Fourth reentry (N=0) (and return)                0! = 1
Return from Third                         1! = 1ˇ1 = 1
Return from Second                 2! = 2ˇ1 = 2
Return from First            3! = 3ˇ2! = 3ˇ2 = 6
4! = 4ˇ3! = 4ˇ6 = 24```

Because the number N is decreased at each "reentry", it is guaranteed to reach zero and terminate the sequence after a finite number of times. text

28. Robert A. Meyers, ed. Encyclopedia of Physical Science and Technology Vol. 5. (Orlando: Academic Press, 1987), s.v. "Elementary Particle Physics", by Timothy Barklow and Martin Perl, p. 16. text
29. Meyers, pp. 13-14. text
30. Adolf Grünbaum, "Modern Science and Refutation of the Paradoxes of Zeno", in Zeno's Paradoxes, ed. Wesley Salmon, (Indianapolis: Bobbs-Merrill, 1970), pp. 167-8. text
31. William H. Beyer, ed., CRC Standard Mathematical Tables, 25th ed., (West Palm Beach, Florida: CRC Press, 1978), p. 72. text
32. Thomson, pp. 101-2. text