From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: TIME HAS INERTIA
Keywords: CORRECTION        FUNDAMENTAL THM. OF ALGEBRA
Organization: Computer Science Department,  Stanford University.
Date: Sun, 22 Nov 1992 23:20:21 GMT

Let me shortcircuit some of these objections to Abian's proof by
pointing out that it is a variant of a one-paragraph proof of FTA that
appears for example in Mac Lane, "Mathematics: Form and Function".

After proving Liouville's theorem, that the only bounded complex
functions with a derivative everywhere are the constant functions,
Mac Lane continues:

"From Liouville's theorem one readily derives a proof of the
fundamental theory of algebra.  For let

	g(z) = b_0 + b_1 z + ... + b_n z^n, b_n != 0,

be a polynomial of degree n > 0 with complex coefficients b_i  If g has
no complex zeroes at all, then f(z) = 1/g(z) is defined and holomorphic
in the whole complex plane.  One readily shows also that f is bounded
there (because, for large z, the term |b_n z^n| dominates |g(z)|.
Hence, by Liouville, f(z) is constant --- nonsense."

Liouville's theorem plays the same role for this simple proof as
Abian's algorithm for producing expansions valid in some region plays
for his proof.


In article <1992Nov22.125437.16062@prl.philips.nl> dacosta@prl.philips.nl (Paulo da Costa 42147) writes:
>In article <1992Nov20.230233.18271@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>[...] Whether or not it can be, you are
>>forgetting that any reductio ad absurdum *must* contain some absurdity
>>in order to succeed.  Pointing out a falsehood during a reductio ad
>>absurdum does not demonstrate a flaw in the proof. [...]
>
>Yes, it does. The whole point of a proof by contradiction is the
>following:
>
>(1) You assume a proposition you want to disprove.
>
>(2) By a sequence of _logical_ steps, and making only _valid_
>    assumptions, you arrive at a contradiction.
>
>(3) The fact that you arrived at a contradiction tells you that one
>    of your assumptions is invalid, or one of the steps in the proof is
>    illogical. SINCE ALL INTERMEDIATE ASSUMPTIONS AND STEPS ARE VALID,
>    you can conclude that your original assumption was false, QED.
>
>If you introduce a falsehood "during" a reductio ad absurdum -- i.e.,
>during step (2) above -- you're bound to arrive at a contradiction. 
>So what? You'll never know whether the contradiction was caused by
>your original assumption or by the falsehood you introduced, and
>so step (3) fails and your "proof" proves nothing.

If by "introduce" you mean simply "appeal to" then you are right.  But
if you mean "infer" (from the hypotheses you are working on showing
inconsistent) then I disagree.

Your objection was that a certain statement occurring in his proof was
false.  Had he simply appealed to that statement as a mathematical fact
then your point would be well taken.  But if he *infers* that statement
from the hypotheses to be shown inconsistent then you cannot object to
the argument merely because in so doing he happens to be inferring a
false statement.

Abian obtains the "falsehood" legitimately (he claims), by applying a
procedure he claims delivers a valid expansion and observing that it
delivered an invalid one.

You can't object to the legitimacy of his claim to sound inference
merely by pointing to the falsehood of the result.

>Just for fun, let's point out the hole in the original "proof":
>
>>For the record here's Abian's proof (that 1+z+z^2 has a zero) again.
>[...]
>>>    1/ (1+z+z^2)  =   1 - z + z^3 - z^4 + z^6 - ...
>>> 
>>>    1/ (z^2+z+1)  =   1/(z^2) - 1/(z^3) + 1/(z^5) - 1/(z^6) + ... 
>>>   
>>>The two expressions on the right side of =  must be identical. [...]
>
>No. Why should they? The series on the first line converges only for
>|z|<=1; the series on the second line converges only for |z|>=1. You
>can use either series in its own region of convergence, but you can
>only equate them at the intersection of those regions (i.e., |z|=1),
>where they do indeed converge to the same value (though not absolutely,
>so you can't even change the order of the terms in either side of the
>equation!).

You're continuing to argue the facts locally, ignoring how Abian
arrived at them.  If he obtained each expansion by a method guaranteed
to deliver a sequence known to be valid in the whole plane, and if the
two expansions differ in any coefficient, we have a contradiction
already without even having to argue that one of the series is
singular.
-- 
Vaughan Pratt	  		A fallacy is worth a thousand steps.
Newsgroups: sci.physics