FOM: Re: Berkeley and nonstandard analysis

[Robert Tragesser]

        It is worth considering that the logical error Berkeley points 
up is not one the logically hyper-acute Leibniz (could have) made.-- 
Leibniz sharply distinguished methods of demonstration [where logic is 
central] from methods of discovery [where the method leads to an 
obviousl;y correct formula but not necessarily to a logically sound 
demonstration of the correctness of the formula].  The latter (his 
"calculus" was a method of discovery,  not of demonstration) is happy to
come up with the obviously correct formula and is not gloomy about not 
being able to give a logically sound/rigorous demonstration.  Bos made 
the point that when looking at the geometric picture(s),  it is 
obviously good mathematical sense to let the dx vanish to get the right 
formula for (dx\2)/dx,  say;  never mind that the algebra/logic leaves 
much to de be desired -- that the algebra/logic is deficient doesn't get
in the way of seeing that the formula sans the tail dx is the right one,
 it just gets in the way of giving a logically coherent demonstration 
that it is. (By the time one moves from division to cancelling the dx,  
one has shifted frames from algebra to intuitive geometry. . .so there 
isn't any sort of logical error FORCING drawing a contradiction,  just a
switch of frame of consideration.)
        (By the way,  indeed,  Bos observes that for higher order 
_Eulerian_ infinitesimals,  the geometric picture breaks down;  here the
failure of the logic catches up with "the infinitesimal calculus" as a 
mathod of discovery.--In this sense one would expect that Berkeley could
make his strongest case with higher order fluxions.  But of course 
Berkeley still loses -- for the mathematicians' response to this dead 
end is to sharpen up the logic,  that is,  to find the right concepts.)

r.tragesser
west(running)brook connecticut usa