Suppose you have encountered Zeno’s Achilles paradoxfor the first time. Zeno offers a rigorous (looking) proof that, having once given the tortoise a head start, Achilles can never overtake it. Would you regard this as
# A startling new discovery in athletics;
# A demonstration of the transcendent capacity of the human spirit – although the laws of logic forbid it, Achilles does in fact catch and overtake the tortoise; or
# A warning about how not to take limits?
In this case, I assume nearly all readers will go for option 3. But things aren’t always so easy. The Einstein-Podolsky-Rosen Paradox was supposed to be a type-3 paradox demonstrating the incompleteness of quantum mechanics. But on most modern views, it is really a type-1 paradox, predicting various highly counter-intuitive consequences of quantum mechanics that nonetheless turn out to be be empirically valid.
Although I don’t accept that there are any good examples of type-2 paradoxes, plenty of others would offer this solution in relation to both Godel’s theorem and Schrodinger’s cat.
With these three possibilities in mind, how should we think about paradoxes involving probability measures over infinite sets that are finitely, but not countably additive? The two-envelopes problem we’ve been discussing here falls into this class and so, with a little bit of tweaking, do St Petersburg and related paradoxes. I’ll leave this question hanging and offer my own answer in a later post.
fn1. This is an expanded version of a point made by my friend and occasional co-author, Peter Wakker. effexors
5 thoughts on “The point of paradox”
One limit paradox I liked was constructing a staircase with n steps down the diagonal of the unit square. The amount of carpet you needed to cover the n steps was n*1/n +n*1/n = 2 units — the sum of the lengths of each side of the square. But in the limit n -> infinity so the staircase coincides with the diagonal, the length of carpet needed -> sqrt(2).
Seemed more physical and real than Zeno. And less atmospheric and abstract (for non-maths specialists) than those you mention.
harry, how about that the coastline of australia is actually infinite?
(ughhh…not again with the two envelopes being about infinity…it decidedly isnt! see my comments on the original post: http://www.johnquiggin.com/archives/001707.html#more )
john, clearly the extra .25 comes from incorrectly replacing x or 2x with a single variable. as my comment shows.
the two new envelopes (http://www.crookedtimber.org/archives/001873.html) is a problem with infinities…
both envelopes have expected sentences of infinity:
1 + 1 + 1 … -> infinity
.5 + .75 + .875 -> infinity
so if you can open the first envelope, and you see a finite number, you are meant to prefer this opened envelope.
the problem i feel is that although god finishes coin tossing in less than a second…he could be still ‘going’, stuck in that second, furiously throwing the coin faster and faster, waiting for a head…in which case you can relax and play some cards with the angel…
Another famous paradox involving measures is the Banach-Tarski Paradox: Take a solid ball, cut it into half a dozen pieces, rearrange the pieces and end up with two solid balls the same size as the original.
You may not want to try this at home however since the existence of some of the pieces relies on the Axiom of Choice which permits non-constructable functions.
There’s a good introduction to the theorem, its implications and “paradox” at Kuro5hin.org.
As a paradox, I think it falls into a more general category than Prof. Q’s category three regarding limits and that is “never mix intuition and infinity” (reminiscient of “never drink and derive”). I mean how intuitive is the fact that you can put the set of even numbers into a one-to-one correspondence with the all the integers?
What’s 2 minus 3?
Minus 1, you cry.
This is at the heart of why Achilles and the turtle is not a paradox at all.