Surprisingly, my post on probability theory has generated quite a discussion thread. (And here I thought I saying Ron Paul won't win the GOP nomination was the key to lengthy threads!) In the course of the discussion, I thought of a way to refine my first example to clarify the issues a bit.

Instead of an infinitely fine-pointed dart, let's take an ordinary one, and instead of the real number line, we'll just throw it at the wall. Now, I ask you, "Does the exact middle of the head of the dart have some exact x, y coordinates on the grid of that wall, despite the fact that, of course, we could never measure that exact x or that exact y?"

If you answer "yes," then I will note that there was a 0% chance that the dart would land at those coordinates, and yet there it is.

If you answer "no," you have adopted a respectable philosophical position, denying the physical reality of the continuum. But you have changed subjects on us: you've dodged the mathematical problem by introducing a philosophical problem concerning the applicability of the mathematics. If this is the way you are inclined, I highly recommend

Instead of an infinitely fine-pointed dart, let's take an ordinary one, and instead of the real number line, we'll just throw it at the wall. Now, I ask you, "Does the exact middle of the head of the dart have some exact x, y coordinates on the grid of that wall, despite the fact that, of course, we could never measure that exact x or that exact y?"

If you answer "yes," then I will note that there was a 0% chance that the dart would land at those coordinates, and yet there it is.

If you answer "no," you have adopted a respectable philosophical position, denying the physical reality of the continuum. But you have changed subjects on us: you've dodged the mathematical problem by introducing a philosophical problem concerning the applicability of the mathematics. If this is the way you are inclined, I highly recommend

*Atomism and Its Critics*, by Andrew Pyle.
At first it seemed to me that you were trying to take out the key word "infinite". Though I realize yet that instead of an infinite fine dart, you now ask for infinite exact measurement.

ReplyDeleteMeasurement is always as exact as needed. If you ask for infinite 100% exact measurement, then you did not really alter the example... It still has "infinite" in it. For my point of view this example is exactly the same as the first one.

If you do not ask for infinite exact measurement, then there would be a measurable middle of the head, and a finite measurable and countable number of points at wall it may hit, and therefore not a Zero percent chance to hit one of those. But then it is not an example for the initial statement.

Correct me if I am wrong.

Yes, you are wrong, skylien, and I will correct you:

Delete"you now ask for infinite exact measurement"

No, I did not ask for any measurement at all. In fact, I very explicitly acknowledged that we could not get infinitely exact measurement: "despite the fact that, of course, we could never measure that exact x or that exact y?"

"then you did not really alter the example..."

Exactly correct. I just tried to take out some elements that were distracting people.

We are doing mathematics here, skylien, not engineering. When your teachers asked you, "Suppose we have a right triangle with sides of length 3 and 4..." did you stop them and say, "But wait! We could never measure that they were exactly 3 and 4!"

Of course not. But we are doing math, not surveying. We simply posit that we have sides of those exact lengths. The difficulty of exact measurement is a total red herring.

Look at my question again: "Does the exact middle of the head of the dart have some exact x, y coordinates on the grid?" NOT "Can we measure those exact locations?" but "Do you think those locations exist, despite not being able to measure exactly where they are?"

I guess I am still distracted. You started to speak of the inability to measure it “exactly”. If you do not refer to infinitely exact, I fail to see how you can say that we cannot measure them exactly with absolute certainty. It is only a matter of how exact do you want it to have.

DeleteIf you set the grid on the wall that it counts only as 1 point and you cannot shoot past the wall, it is a 100% chance to hit that point. If you set it at a 1000 points (and you have a perfect random distribution of throwing darts, while not throwing past the wall), you will have a chance of 1 per mill to hit that point. If you say the amount of points on the wall is infinite, then the chance is infinitely small to hit a definite point, but not zero. Only in this case I would have said ok it is approaching zero so let’s round down to zero. But then the seeming conflict that although the chance to hit some definite point on the wall is “zero” while you really must hit some point is resolved by acknowledging the fact that we did round down in the first place. I just don’t see any surprising conflict here...

Skylien, you are still distracted: you are still thinking of measurements and rounding. These are TOTALLY IRRELEVANT. We are not in physics lab: we are doing pure mathematics here, and only using the dart and the wall as thought tools. Once again, when you got an algebra problem saying, "A train leaves Rotterdam traveling at 60 miles an hour..." it is a symptom of confusion if you start asking "Measured to how many decimal places?"

DeleteAnd, no, skylien, we are not "rounding down to zero," the answer just is zero. This is standard probability theory.

ReplyDeleteProof:

ReplyDelete1) All probabilities are, by definition, between 0 and 1.

2) For any point on the wall, the probability of hitting it must either be 0, or greater than 0.

3) The probability of hitting any point on the wall is the sum of the probabilities of hitting each individual point.

4) But if the probability of hitting the individual points is greater than 0, the sum of an infinite number of such probabilities will be infinite. This contradicts 1)

5) QED, the probability of hitting any individual point is 0.

Statement 3) is false. If it were true, then "the probability of hitting any particular point is 0" would imply "the probability of hitting any point at all is 0."

DeleteThe point is that probability measures are sigma-additive, i.e., the probability of the union of

countablymany pairwise disjoint events is the sum of the probabilities of the individual events. This doesn't work for uncountably many events.Stements 4) could be replaced by "But if the probability of hitting the individual points is greater than 0, the probability of hitting a rational number will be infinite. This contradicts 1)"

This works since there are only countably many rational numbers.

Thanks for the correction, David.

DeleteTo me, the problem here is a semantic one: "wow, when probability theorists say '0% chance' they apparently don't always mean the same thing I would mean if I said that."

ReplyDeleteWell, I think you are right that they don't, but I don't see why you think that is a problem!

DeleteSimply that it creates possibilities for miscommunication and misunderstanding.

Delete