[FOM] Davis on Torricelli

Dean Buckner Dean.Buckner at btopenworld.com
Wed Oct 29 17:21:53 EST 2003

>The  reason that [Torricelli's] result (which today is a homework problem
in a freshman
>calculus class) was so shocking to contemporaries was because it violated
>classical ideas going back to Anaximander and codified by Aristotle about
>the unapproachability of the infinite.

That's a bit of a simplification, to say the least!  It was not shocking to
Leibniz, for example, who compared it to the fact that the series 1 + 1/2  +
1/4 . has a definite sum.  It was shocking to Hobbes, but only because of
his (indefensible) position that the infinite cannot be given in sense
experience, a position that arose from his extreme empiricism & antagonism
to scholastic (i.e. Aristotelian) philosophy.

And we need to distinguish two views on "potential infinity".  One is that
the parts of a body are not there until "actually" divided.  This was
Aristotle's and Aquinas' view.  The other is that they are actually there,
but that, however many times the body is divided, the result is always
finitely large bits.  So we can speak of "every part" of a body, but not
"all of the parts".  Joe Shipman characterised this view very well in a
recent posting ("Sharp mathematical distinction between potential and actual
infinity?" - Sep 28 2003).  This was Ockham's view.

Suppose for example I put a 3kg block on a scale, then a block that weighs
1/10kg, then 4/100kg, then 1/1000kg, so that the first 6 blocks weigh

 3.14159 kg

and then keep on piling on blocks whose weight corresponds to the decimal
expansion of pi.  Then suppose I have piled every block that it is possible
to pile, according to this rule (infinitely many blocks).  Then the weight
of the pile of blocks is exactly pi.  Moreover, remove any one of the
infinitely many blocks on the scale, and the weight will be finitely
different from pi.  Every block on the pile has a finite weight, in other
words.  That is the Ockhamist position.

Dean Buckner,    : If one person can see it as a paradise
London,              : of mathematicians, why should not
ENGLAND        : another see it as a joke?

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