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Randy Poe said:
>
> Tony Orlow wrote:
> > Randy Poe said:
> > >
> > > Han de Bruijn wrote:
> > > > Randy Poe wrote:
> > > > > Ah, so when you said "physicists use infinitesimals" what you
> > > > > really mean was "physicists use integrals".
> > > >
> > > > No. Physicists use "infinitely small element"s and "an infinite number
> > > > of elements".
> > >
> > > In the form of integrals.
> > >
> > > > How can you possibly deny? Can't you just READ the above?
> > >
> > > Yes, I've read a thousand such passages, which are always deriving
> > > an integral equation or differential equation.
> > >
> [snip]
> > >
> > > Your infinitesimals are only used under integral signs.
> > >
> > > - Randy
> > >
> > >
> >
> > I think that Han's whole point is that, while the final solution leads to an
> > integral,
>
> The final expression is an integral, yes.
>
> > the derivation of the integral itself rests on the very notion of
> > infinitesimals that he's trying to put forth, as the limit of 1/n as n->oo.
>
> No it doesn't. Physicists know they aren't being rigorous
> when they throw stuff like that around, they know they
> aren't following a rigorous derivation. But they know that the
> integral is valid, and it is that endpoint they are interested in.
>
> Nobody cares about the actual "infinitesimals". They care about
> the limit of the sum, i.e. the integral.
>
> > So,
> > if calculus itself is derived using this notion,
>
> It isn't.
>
> > the question becomes why this
> > notion of the infinitesimal cannot be discussed in its own right.
>
> It can. That's called NSA. And calculus can be developed within
> NSA. But it doesn't have to be, and Han's claim is that physicists
> are routinely doing something like NSA, or treating infinitesimals
> separately from their sums.
>
> - Randy
>
>
Well, that claim may or may not have ny real veracity. I am not familiar with
how physicists go about their business all the time, but as far as I know from
the physics I've taken, it IS generally dealt with in terms of calculus. So,
maybe you have a valid point with respect to that. Still, the calculus was
originally developed in a non-rigorous manner with the use of the notion of
infinitesimals. Do you disagree?
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?
If the probability is zero in standard math, but each possibility has a chance
of being true, then what you really have is a nonzero but nonreal probability
for each. I don't understand why this probability cannot be considered
infinitesimal.
--
Smiles,
Tony
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