Re: A modest proposal (was Calculus XOR Probability)

Han de Bruijn a crit :
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

Then HdB clearly does not understand the definition of the Reimann
integral or the properties of the field of real numbers in which it is
defined.

The Reimann integeral exists if one can show that if the limit of the
set of upper Darboux sums and the set of lower Darboux sums for a fixed
function and fixed closed real interval converge to a common limit as
the size of the maximum partiion decrease towards zero, and then the
limit is the integral.

There is nothing is probability theory that says anything like that
limit process applied to finite spaces has to carry over in any way to
any infinite space.

.

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