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Virgil wrote:
In article <MPG.1e9dd8e096dfe76398abee@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
Han's original point is that calculus is very precise and works, whereas the
case where you have a uniform probability distribution over an infinite set
of
possibilities is not well handled by the classical notion that all individual
probabilities sum to 1, because the individual probabilities are considered
equal to zero. It is the opinion of both of us that this can be resolved,
among
other ways, by assigning infinitesimal nonzero probabilities to each
possibility, leaving intact the notion that the sum of the individual
probabilities sums to 1. I am not sure why this is roundly rejected. Can you
address that?
Because there is no model of the reals, either standard or nonstandard,
in which the sum of countably many equal values can equal 1.
You can repeat this a thousand times, but Tony and I don't get it. In my
not so humble opinion, you must also reject then the integral(0,1) dx ,
because it is derived from the Riemann sum n.1/n , which is exactly the
same as summing up (n) probabilities with 1/n chance for each. dx = 1/n.
Now take the limit for n->oo and you're done. What's the problem? (Well,
I _can_ understand that mainstream mathematics can't drop the whole wide
world of calculus because of this little issue)
Han de Bruijn
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