From: rickert@mp.cs.niu.edu (Neil Rickert)
Subject: Re: Sets in mathematics
Organization: Northern Illinois University
Date: Mon, 13 Apr 1992 17:41:52 GMT

In article <sbahlINN219@agate.berkeley.edu> epfaith@twinkies.berkeley.edu (Edward Paul Faith) writes:
>In article <1992Apr12.051222.25154@mp.cs.niu.edu> rickert@mp.cs.niu.edu 
>(Neil Rickert) writes:
>...
>] Russell's paradox pointed out some mistakes which did require deep
>]reflection on the nature of mathematics.
>
>  Is this paradox the result of mistakes, or of inconsistent axioms,
>which is another thing entirely?

  I guess that is all a matter of definition.  But the point is this --
if you define the field of mathematics in such a way that the only
type of error possible amounts to the use of inconsistent axioms, you
should not find it at all surprising if every mistake turns out to be
due to inconsistent axioms.

>  Anyway, the paradox "required" deep reflection from thoughtful
>individuals.  But the solution to it required only a new set of axioms
>which avoided it.

  I'm not sure what point it is you are trying to make.  Are you saying
that requiring a new set of axioms is trivial in comparison to requiring
a new theory (as happens in the physical sciences)?  If that is your
point, I strongly disagree.

>]>Mathematics all the way back to Euclid is perfectly usable today. This
>]>is not true of alchemy, which is one percent gold, 99 percent lead.
>]
>] This is a rather naive statement.  It is true that a theorem, once
>]proved, stays proved.  But there have been many things that were thought
>]to have been proved, yet it later turned out that they were not.
>
>   Often I will think I have proved a thing, then realize I have not.
>This is essentially the same thing as what you say, only in your case
>more people were in error for a longer period of time.  In my
>experience, faulty proofs are invariably the result of skipping too
>many steps and vaguely defining the terms. There is an obvious cure
>for that.

  The "obvious cure" is often not so obvious.  When there "too many steps"
are skipped, and there is a vague defining of terms, it is usually because
of a lack of underlying understanding.  Just looking for the skipped steps
does not solve the problem, because much more knowledge may be required
before it is even recognized that those steps are missing.

>] Quite apart from that, a considerable amount of old mathematics is of
>]little use.  In Euclid's era geometry was used to prove the equivalent
>]of
>]	(a + b)^2  = a^2 + 2ab + b^2
>]
>]but it would be difficult to make a case that this approach to algebra
>]is useful today.
>
>  Ah, you are testing my memory.  I said usable, not useful.  A
>flintlock pistol is not useful to someone who owns automatic weapons,
>but it is usable.  It still works.  Most of alchemy is not usable.

  But most of alchemy was never usable.  The parts that were usable are
probably still usable, even if not useful.  The ptolemaic model of
the universe is still just as usable as ever for the kinds of things
for which it was originally usable.  Newton's laws are not only usable,
but commonly used, even though they have been surpassed by Einstein.

>] You only hear of the cases where the errors were discovered and corrected.
>]You don't hear of the errors which have never been discovered, because
>]the fact that they have never been discovered means that this information
>]is not readily available.
>
>  Do you suspect that there may be an error in the proof that rules
>for multiplication in the decimal system correspond to the definition
>as we now have it?  What might be wrong?

  My point was not to claim that mathematics is riddled with errors -- I
do not have any evidence to suggest that.  My point was that the source
of information about errors is biased toward only having information on
errors that have been discovered and corrected.  It is unwise to draw
conclusions from such biased evidence.

>  There are many possibilities.  All of them boil down to inconsistency,
>I think.  For example, we show by induction the rules work.  But then
>when we apply the rules we do something that induction does not
>anticipate.  An inconsistency exists then between the rules as applied
>and the rules as represented by induction.

  I think you are under the illusion that the rules of multiplication are
derived from the axioms of a field.  Surely that is wrong!  The rules of
multiplication come from experience with counting.  The axioms come later
essentially as a theory to formalize the important features of
mathematics.  The derivation of arithmetic from the axioms is important
mainly because it serves to validate that a correct set of axioms has been
found.

>  Please reply.  However, I am sorry but the obligations of life
>prevent me from disturbing you again for some time.  It was a pleasure

 By the way, the last line of you posting is being chopped off somewhere.
Your articles seem to end in mid sentence.

-- 
=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=*=
  Neil W. Rickert, Computer Science               <rickert@cs.niu.edu>
  Northern Illinois Univ.
  DeKalb, IL 60115                                   +1-815-753-6940
Xref: sparky sci.philosophy.tech:2080 comp.ai.philosophy:4231