- #1
yonat83
- 12
- 0
Hi,
I am currently working on probability densities and one simple question came to my mind:
Say we lived in an "2^aleph_0 world" which contains 2^aleph_0 mathematicans (dont screem already!). Now each mathematician chooses one real number between 0 and one in a bijective way.
Now we can pick some random number between 0 and one following say a uniform repartion on [0,1].
Obviously, the probability of getting one specific number is 0. But then, one mathematician is going to get his number chosen, wish apparently contradicts the fact the probability of him to be chosen was 0 a priori.
I can think of 3 ways to interpret this apparent contradiction but none of them satisfies me completely:
1 - the problem comes from the fact our world is finite and talking about aleph_1 mathematicians is absurd. Well, it doesn't convince me because first, I wouldn't have any problem with a discrete probability on the natural numbers N for exemple: then if some mathematician picked a number associated with a null probability to happen, well, then it will never be picked! That eliminates the paradox. So the contradiction comes from the type of infinity used in the experiment. Why would there be such a "phylosophical" gap between those 2 apparently equivalent ways of describing an imaginary world?
Moreover, I don't really see why our physical would be the limiting factor. We always use mathematical concepts involving infinite quantities and I can't see why I shouldn't be able to do the same here.
2 - the problem comes from the way we are physically going to choose a number between 0 and 1: Well assuming I can randomly pick a number from a finite set, say {0,1} in some amount of time, I would also be able to choose some random sequence of them forming the binary development of some random number between 0 and 1. One can argue then, that it would take an infinite amount of time to get my final number. While I agree with that, I could then imagine that each pick takes half the time of the one before. The resulting total time would then be finite as this geometric series converges. I m assuming here that the time taken for one to randomly pick from {0,1} in not bounded, but why not? maybe because, once again, our universe is finte? Every single event takes some minimal time to occur, I must agree. But isn't that due to the specific laws of our universe? I mean, I could imagine some universe where it wouldn't be the case at all!
Makes me think that maybe it comes from the fact that each "picking" takes some energy to be done in our world, so if the time of pick wasn't bounded, it would require us unbounded "power" and energy to do it wish once again contradicts the belief that our universe has limited supply of energy...
3 - The probability function is NOT a physical concept, and consequently, the apparent absurdity of the result comes from the fact we are naturally interpreting a probability as the chance of some event to happen. But maybe this interpretation is the problem: maybe in our "2^aleph_0" world , probabilities should be interpreted in a different way. Then what would be that meaning?
Anyways, I have the feeling the answer lies somewhere between those, and I m not even sure there is only one since my question is more about philosophy than mathematics.
Yonathan
I am currently working on probability densities and one simple question came to my mind:
Say we lived in an "2^aleph_0 world" which contains 2^aleph_0 mathematicans (dont screem already!). Now each mathematician chooses one real number between 0 and one in a bijective way.
Now we can pick some random number between 0 and one following say a uniform repartion on [0,1].
Obviously, the probability of getting one specific number is 0. But then, one mathematician is going to get his number chosen, wish apparently contradicts the fact the probability of him to be chosen was 0 a priori.
I can think of 3 ways to interpret this apparent contradiction but none of them satisfies me completely:
1 - the problem comes from the fact our world is finite and talking about aleph_1 mathematicians is absurd. Well, it doesn't convince me because first, I wouldn't have any problem with a discrete probability on the natural numbers N for exemple: then if some mathematician picked a number associated with a null probability to happen, well, then it will never be picked! That eliminates the paradox. So the contradiction comes from the type of infinity used in the experiment. Why would there be such a "phylosophical" gap between those 2 apparently equivalent ways of describing an imaginary world?
Moreover, I don't really see why our physical would be the limiting factor. We always use mathematical concepts involving infinite quantities and I can't see why I shouldn't be able to do the same here.
2 - the problem comes from the way we are physically going to choose a number between 0 and 1: Well assuming I can randomly pick a number from a finite set, say {0,1} in some amount of time, I would also be able to choose some random sequence of them forming the binary development of some random number between 0 and 1. One can argue then, that it would take an infinite amount of time to get my final number. While I agree with that, I could then imagine that each pick takes half the time of the one before. The resulting total time would then be finite as this geometric series converges. I m assuming here that the time taken for one to randomly pick from {0,1} in not bounded, but why not? maybe because, once again, our universe is finte? Every single event takes some minimal time to occur, I must agree. But isn't that due to the specific laws of our universe? I mean, I could imagine some universe where it wouldn't be the case at all!
Makes me think that maybe it comes from the fact that each "picking" takes some energy to be done in our world, so if the time of pick wasn't bounded, it would require us unbounded "power" and energy to do it wish once again contradicts the belief that our universe has limited supply of energy...
3 - The probability function is NOT a physical concept, and consequently, the apparent absurdity of the result comes from the fact we are naturally interpreting a probability as the chance of some event to happen. But maybe this interpretation is the problem: maybe in our "2^aleph_0" world , probabilities should be interpreted in a different way. Then what would be that meaning?
Anyways, I have the feeling the answer lies somewhere between those, and I m not even sure there is only one since my question is more about philosophy than mathematics.
Yonathan
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