|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
|
 |
|
|
|
|
|
|
|
|
|
|
|
|
Simon asked:
|
|
Hello, I'm 15 and there is an unanswerable question on which I would like a philosopher's view. If an
object were to fall to the ground, where the object started would be "1", and where the ground is
would be "0". Starting at "1", the object falls half of the way... then one fourth of the way... then one
eighth of the way... then one sixteenth of the way, and so on as the number doubles. The question
comes down to, when is the fraction so small to call "0"? if as learned in school there is an infinite
number of fractions, if this is true the object never becomes "0" and never hits the ground. Please
explain this.
|
|
============
|
|
Simon's (and, before him, Zeno's) puzzle falsely presupposes that the finite distance to be traversed
is a continuum of extensionless points. The potential to double the denominator of a fraction may be
infinite — you can double that number forever — but the number of places to be traversed by a
moving object is not infinite, and so the falling object does hit the ground. To put it another way, no
falling object can reach the ground if it must traverse an infinite number of places to get to its resting
place. It can't even leave the place it is supposed to be falling from. Fortunately, a moving object
traverses a finite number of discrete (separate) places, not an infinity of points on a continuum.
|
|
Tony Flood
|
|
Well this is a neato version of Zeno's paradox; look that up. There is no fraction so small to call "0",
that is the formulation of the paradox. As you can see, when you drop an object it can never reach the
ground. It's the same for reaching any destination, and that's why when you shoot an arrow or a bullet
at someone, you never can quite hit them.
|
|
Whoops... there's a problem here, isn't there.
|
|
So should I tell you the answer? No... that would spoil the fun. Just go look up Zeno's paradox, and
you'll find tonsof stuff written about this. But I'll say this; there are different types of series of fractions:
convergent and divergent. If you add 1/2 + 1/4 + 1/8... etc., you get, after an infinite number of
additions, the sum of 1. But if you add 1/2 + 1/3 + 1/4... you get, after an infinite number, the sum of
infinity. Something else for you to play with.
|
|
Steven Ravett Brown
|
|
17
|
|