Disclaimer for this Book
and commentary
Disclaimer:
Some statement in this book is false.
Commentary on the Disclaimer:
If some other statement in this book is indeed false, then the Disclaimer is true. If no other statement in this book is false, then the Disclaimer declares itself false; a paradox, hence a half-truth. Therefore the Disclaimer is true or at worst half-true. By saying that some statement in this book is false, the Disclaimer guarantees that it is, itself, not entirely false.
Zeno’s Arrow Fallacy
Consider Zeno’s Paradox of the Arrow: the arrow does not move within any instant during its flight; so when does it move?
To this I reply: we measure motion not in meters, but in meters per second. Within the instant the arrow moves zero meters, but this is in zero seconds; so the speed of the arrow is zero divided by zero, the indefinite ratio. Within the instant the arrow moves at an indefinite rate. It moves at any speed.
For proof of this, consider this experiment; we fire Zeno’s Arrow past another arrow, which we release the moment Zeno’s Arrow passes. At that instant we take our flash photo. At the instant of the flash, the two arrows are side-by-side; one is moving rapidly to the side and the other is slowing falling down; otherwise they are indistinguishable. In the instant, speed is indeterminate. You’d need another flash picture a millisecond later to tell their velocities.
Zeno argued that the arrow does not move in any instant, and so does not move. His mistake is confusing 0/1 = zero speed with 0/0 = indefinite speed. In each instant the arrow’s speed is indeterminate; we learn its speed when we compare instants. So the arrow’s speed isn’t zero at every instant of its flight; it’s unknown; a mystery that clears up when we compare two instants.
So Zeno’s Arrow Fallacy is the misleading equation:
0/0 = 0/1
Actually 0/0 equals every number equally. It’s as much 0 as it is 1 or 17 or 42 or 1/137. The 0/0 = 0 equation is misleadingly specific.
My students consistently make this mistake whenever I mention division by zero. They get 0/1 mixed with 1/0 and with 0/0. I have been driven to invent these nicknames:
0/1 = POOF ; for vanishing
1/0 = BOOM ; for exploding
0/0 = GOOP ; for vagueness
I will give Zeno this much: this resolution of the Arrow requires that some physical quantities are indefinite.
Epimenides Trilemma
Consider these three voters:
Moe says:
Some philosophers are Cretans;
All Cretans are liars;
Some philosophers are liars.
Larry says:
No philosophers are liars;
Some philosophers are Cretans;
Not all Cretans are liars.
Curly says:
All Cretans are liars;
No philosophers are liars;
No philosophers are Cretans.
If Moe, Larry and Curly vote, then each of these propositions pass by 2/3 majorities:
Some philosophers are Cretans;
All Cretans are liars;
No philosophers are liars.
That is the “Epimenides Trilemma”; a voter’s paradox. All three propositions pass by majority rule, but then cannot all three be true at once. If two are true, then the third must be false. So:
If
Some philosophers are Cretans
and
All Cretans are liars
then
Some philosophers are liars.
If
No philosophers are liars
and
Some philosophers are Cretans
then
Not all Cretans are liars.
If
All Cretans are liars
and
No philosophers are liars
then
No philosophers are Cretans.
The trilemma both defies syllogisms, and yields three of them!
