Oh boy, the comments continue to poor in on my post with regard to the Gene Callahan statement. Callahan, himself, has stepped in screaming:
Absolutely incredible, folks! I did not invent this example: it is absolutely standard probability theory. Go ask any math PhD. I think next I will post, "The square root of two is an irrational number," and then wait for the stampede of lemmings to arrive here declaring "Callahan has just shown his Keynesian irrationality."Which now forces me to prove that there is 0% chance for Callahan to have a logical thought. Let's go back to Callahan's original statement:
"There's a 0% chance that X will happen."
Most people interpret that to be equivalent to, "X will never happen."
But that isn't so. An example: If one were to toss an infinitely fine-pointed dart at the real number line and hit a spot between 0 and 1, for any particular number in that range, there is a 0% chance you will hit it. But you *will* have hit *some* number, so even though there was a zero percent probability that number would be hit, it was hit.Now, when most people hear that "There's a 0% chance that X will happen." in a dart throwing contest (using darts in this example as Callahan does), people are assuming we are discussing a dart that has some physical substance to it, to be thrown at some real target. Since at this point, Callahan has not added the non-common way of thinking about the situation in terms of an "infinite dart". Now, it is perfectly legitimate to construct any type of thought experiment you want, but this has zero to do with Callahan's original statement. As David Gordon points out Callahan's first sentence is ambiguous.
He is saying, "Ha, ha most people aren't as smart as I am because they think 0% chance means something will never happen". But "most people" are correct using every day non-infinite concepts. When Callahan introduces an "infinitely fine-pointed dart", he introduces a factor not considered in everyday use of the term probability. The people are right and Callahan is wrong and I can prove this by using Callahan's infinite concept and proving that he has 0% chance of coming up with a logical thought.
Let us examine Callahan's thinking process along a timeline divided up into an infinite number of points (slyly slipping in the concept of infinite as does Callahan in his dart argument). Now, we all know that a thought takes time to develop, so that at any point on Callahan's infinite timeline it is impossible for him to have a logical thought. The chance is 0%.
Now, this is nutty and has nothing to do with reality, since we generally do not consider thoughts at a point on an infinite timeline, but neither do we consider infinite darts. But on second thought, as I assemble a number of my infinite fractions of thought, I see something unique happens with Callahan logical thought, his logical thought here over time in this discussion does intersect with his infinite thoughts at 0%.