From: Harvey Newstrom (mail@HarveyNewstrom.com)
Date: Fri Jun 14 2002 - 10:24:44 MDT
On Friday, June 14, 2002, at 07:52 am, Lee Corbin wrote:
> Assume a = b
> then a^2 = ab ;multiplying both sides by a
> also a^2 - b^2 = ab - b^2 ;subtracting b^2
> (a-b)*(a+b) = b(a-b) ;factoring
> a + b = b ;dividing through by a-b
If a=b, then a-b=0. You are dividing by zero. This is bad math at this
point.
> b + b = b ;since a=b
This is only true if b=0. For this and any other values, it should be
b+b=2b.
> 1 + 1 = 1 ;dividing through by b
Since the previous statement establishes that b=0, you are dividing by
zero again here.
> 2 = 1
Only if you break the rules of algebra and divide by zero twice. Any
number divided by zero is "undefined". You are supplying whatever
answer you desire after dividing by zero. This "proof" can be rewritten
to show any number equals any other number, since there are two stages
where the answer is really undefined and an arbitrary number is inserted
to continue.
> There are many other faulty proofs in mathematics and
> logic whose flaw isn't so apparent, yet it's hardly
> intelligent to presume that an argument is air tight
> just because you haven't yet found a flaw in it.
This "proof" is instantly apparent to anybody who knows basic algebra.
Anybody who has difficulty in seeing the flaws in this proof simply are
not skilled enough in math to do basic algebra reliably.
> I don't know of any rigorous guidelines to suggest to
> one that an apparently completely rational argument
> may have a hole in it. But one is that if the conclusion
> of an argument goes against long deeply held beliefs that
> you have, then it's wise to reserve judgment for an
> extended period.
The rigorous guidelines are called math, logic, and science. They
provide clear knowledge of what is proven, what is theorized, what is
unproven, and what is proven false.
> I suppose that the proper course is to acknowledge the
> seemingly conclusive nature of a new argument or rational
> deduction, and to admit that one is not in possession of
> any demonstration of the flaw in said argument, but to
> maintain that the conclusion flies so strongly against
> other beliefs, or appears to, that a suspension of
> judgment is called for.
This is only a prudent course of action if one is well-versed in math,
logic and science. If one makes common errors, such as shown in the
proof above, one cannot rely on these faulty skills to evaluate new
ideas.
> Of course, it's then your intellectual duty---that is,
> if you are concerned about what is true in the end---
> to quietly pursue a rectification of all your beliefs.
> To do anything else, naturally, would be to simply bury
> your head in the sand.
Agreed. But first you need a reliable methodology. Pursuing any belief
system with a faulty evaluation mechanism won't produce reliable
conclusions.
-- Harvey Newstrom, CISSP <www.HarveyNewstrom.com> Principal Security Consultant <www.Newstaff.com>
This archive was generated by hypermail 2.1.5 : Sat Nov 02 2002 - 09:14:47 MST