[FOM] Davis on Torricelli

[William Tait]

On Oct 29, 2003, at 4:21 PM, Dean Buckner wrote:

> Davis:
>> 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.

Simplifications abound.

Leibniz also believed (at least sometimes) in the actual 
infinite---which was not a commonly held belief at the time.

I mentioned once before, in connection with Torricelli,  Richard 
Swineshead's proof (in the 14th century) that  1+1/2+2/4+   3/8+ ...= 
2. I mentioned this once before: the proof proceeds by assuming that a 
rod of length 1 has  intensity 1 of heat (or some other intensive 
magnitude) along the first 1/2 of the rod, of intensity 2 along the 
next 1/4, intensity 3 along the next 1/8, etc., and then redistributing 
the heat so that it is of intensity 2 along the whole. What is 
interesting is that he did not take for his proof instead the area 
obtained by summing the rectangles, for each n,  of height n along the 
nth  1/2^n part of the rod (i.e. line segment of length 1), and then 
cutting and pasting to obtain the rectangle of length 1 and height 2. 
For he would then have had a simpler example than Torricelli's of an 
unbounded figure with finite area.

I suggested that the reason Swineshead avoided this more natural 
example (which had to be behind his argument)  might have been that 
such a figure would contradict Aristotle's and the Church's dictum that 
space is bounded.

In any case, independently of the issue of how one is to understand the 
distinction between actual and potential infinite , the very 
possibility of these figures (Torricelli's and the one I wish 
Swineshead had constructed)  contradicts Aristotle, which would seem to 
support Martin Davis's statement.

> 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.

This is a strange observation: it was Aristotle's view and the almost 
unanimous view of the medieval philosophers (after the transmission of 
Aristotle's works) that there is no actual infinite and, in particular, 
that the infinite is not given in sense experience---in the sense that 
there are no infinities in the natural world at any particular time. 
(The qualification is needed because the infinity of days and nights up 
to any given time was a problem for them.) So Hobbes is in agreement 
with them on this.

Bill Tait