[FOM] a correction to "on bill tait's answers to my questions (V)"
Gabriel Stolzenberg
gstolzen at math.bu.edu
Fri Mar 24 11:38:05 EST 2006
Here I want to make a correction to something I said in "on bill
tait's answers to my questions (V)." But I will begin by posing two
more questions for Bill.
How did you get into this stuff?
How did I get into this stuff?
The correction is to my reply to one sentence in Bill's remarks
about proofs of the Intermediate Value Theorem (IVT) by Bolzano and
Cauchy.
> > Of course, [Bolzano's] proof [of the intermediate value theorem]
> > was not well-known, but Cauchy's proof some four years later seems
> > to have been widely known.
> If I remember correctly, [in Cauchy's Cour d'Analyse] the IVT is
> "proved" by an appeal to intuition like the one in "If a man goes
> up a mountain on one day and comes down the next, there is a time
> of day at which he was at the same position on both days. Proof.
> Suppose it is two men on one day, one going up, the other going
down. Then they will meet."
> The error here is to mistake an intuition of a position at which
> the men will meet for one of a time at which they are both at that
> position.
On second thought, I don't wish to defend my claim that Cauchy
"proved" IVT by an appeal to an intuition *like* the one for "Two
men on a mountain." The relationship now seems less obvious than I
implied. So, unless my intuitions about these intuitions continue
to be unreliable, it would have been better to have likened Cauchy's
"argument" (as I recall it) to an implicit appeal to the intuition
of running your finger along the graph from one end to the other
(in your mind's eye!) and "seeing" that, at some point, it must
touch the x-axis.
Here the problem is that, because your finger is thick, on close
inspection, this can be seen to support only the weaker version of
IVT that Bill mentions in the remark from which I took the quote at
the top of this message.
Gabriel Stolzenberg
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