From: epfaith@twinkies.berkeley.edu (Edward Paul Faith)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 13 Apr 1992 06:43:01 GMT
Organization: U.C. Berkeley Math. Department.
NNTP-Posting-Host: twinkies.berkeley.edu

In article <1992Apr12.051222.25154@mp.cs.niu.edu> rickert@mp.cs.niu.edu 
(Neil Rickert) writes:

]In article <s7s97INNmmc@agate.berkeley.edu>
]epfaith@twinkies.berkeley.edu (Edward Paul Faith) writes:
]
]>  I have trouble understanding what "intrinsic meaning" means for you
]>in this context.  "Meaning" alone I can understand. Mathematical
]>results have real consequences for mathematicians.
]
]  This example is not original with me, and may in fact be quite old.
]
]  Pick an integer, preferably a very large one.
]
]  Square it.
]
]  You now have a number which is a square.  If you chose a large enough
]number, the fact that it was a square was probably not known before.
]
]  Most mathematicians would dismiss this "discovery of a square" as having
]no real mathematical significance.  If you treat mathematics as a mere
]manipulation of meaningless symbols, it is hard to understand why they
]would react this way.  The result is surely as important as many
]published results, and its proof is no more trivial than the proof
]of many published theorems.  However it is easy to understand why
]this is dismissed if we assume mathematicians do associate intrinsic
]meaning to their mathematics.

  But is it both nontrivial enough and important enough to be published?

  Of what possible use to any mathematician is the result that
323713205764 is or is not a square?  (It is.)  Publishable theorems
are generally either interesting or useful.  Useful theorems derive
their usefulness eventually from interesting uses.  And what makes
a theorem interesting?  Not its meaning.  What makes a puzzle
interesting, or a game interesting?  Its meaning?  Are interesting
games somehow embedded in a meaningful field of games?  Not that I
am aware.  Meaning is not necessary to explain the interest of
publishable theorems, and the perfectly understandable boredom of
a multiplication (your above example).  If I were at a bar, and
someone counted from one billion to one billion and ten, a feat I
have never observed, I would not be amused.  But if someone did
something amusing (and it is hard to think of amusing examples, which
is part of the reason they are so amusing when they occur), then
I would be amused.  What has meaning got to do with this?

In article <1992Apr12.053648.29361@mp.cs.niu.edu>rickert@mp.cs.niu.edu
(Neil Rickert) writes:

] In article <s85diINNnet@agate.berkeley.edu>
]epfaith@twinkies.berkeley.edu (Edward Paul Faith) writes:
]
]>What is the astounding fact here?  The astounding fact seems to be
]>that even though mathematicians don't understand what they are doing,
]>as you put so diplomatically, nevertheless what they do works.
]
] I don't like to put words into David's mouth.  But I do think you are
]misstating his position.  I didn't find anything in his messages which
]implied that mathematicians do not understand what they are doing.  He
]does perhaps question whether they understand the philosophical
]underpinnings of what they are doing, but that is very different.

  My point was that mathematics survives the absence of deep
philosophical reflection, while the other sciences seem to require a
certain development of science.  Mathematicians with mystical ideas
about the significance of numbers nevertheless usually did not do
incorrect math.  Alchemists and medieval doctors with funny ideas, on
the other hand, had mostly incorrect theories about chemistry and
medicine.

]>Mathematicians make mistakes, but these are eventually recognized as
]>such without need for deep reflection as to the nature of mathematics.
]
] 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?

  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 am forgetful of Russell's paradox.  Is it the one
where a set is defined to contain all non self-containing sets?

]>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.

] 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.
Most of medieval medicine is not usable.  Bleeding a person hurts him,
it does not help him. Our modern algebra is a different system from
that which the Greeks used. We have replaced their flintlock algebra
with an automatic algebra that does not need any assistance from
geometry.  Are you saying that the above proof is incorrect?  Applied
to our algebra, perhaps it is.  But to theirs?  To them, multiplication
apparently is defined to be a certain operation on lengths and
volumes, and so our trivial proof through distributivity would not
apply.

] 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?

  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.

  In this case, where is the error?  The "error" is no error at all,
but only an inconsistency between two things. Inconsistencies become
errors when we decide that one thing we do is correct, and the
inconsistent thing is incorrect.  But when we decide such a thing, we
change our mathematics, adding new do's and don'ts to it.Your
statement that there may be hidden errors means to me that there may
be hidden inconsistencies. Perhaps there are.

  Please reply.  However, I am sorry but the obligations of life
prevent me from disturbing you again for some time.  It was a pleasure
Newsgroups: sci.philosophy.tech,comp.ai.philosophy