[FOM] The irrelevance or its relevance

[Lasse Rempe]

Ben Crowell wrote:

>(1) The kinds of
>objects involved in the Banach-Tarski paradox are clearly not
>interesting physically, because they owe their existence to a
>historical accident. If Newton and Leibniz's ideas had been
>formalized using smooth infinitesimal analysis instead of sets
>of points in R^n, then such objects would never have come up.
>  
>

I do agree with your sentiment in (2) below, but I think one should be 
careful about (1). There are many limit processes (Brownian motion, 
percolation etc.) which are useful to physicists, yet yield highly 
non-smooth ("fractal") objects.

On the other hand, I am rather doubtful about non-measurable sets ever 
being of interest to physicists.

Lasse

>(2) The correspondence principle tells us that we should expect
>any physical theory to be superseded later on by a more general
>theory. Therefore, any treatment of infinities or infinitesimals
>is really describing a limiting process that has to stop when we
>get beyond the frontiers of the relevant theory. For example, when
>a physicist says there's a discontinuity in density at the surface
>of a lake, he's talking about a limiting process that he knows
>will stop when he reaches the scale of individual atoms. He expects
>this to happen with *all* limiting processes, even if he doesn't
>know the nature of the more general theory or exactly at what
>point the breakdown will happen.
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>  
>


-- 
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Dr. Lasse Rempe
Dept. of Math. Sciences, Univ. of Liverpool, Liverpool L69 7ZL
Office 505; tel. +44 (0)151 794 4058
http://pcwww.liv.ac.uk/~lrempe
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