From: gudeman@cs.arizona.edu (David Gudeman)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 10 Apr 92 18:24:18 GMT
Organization: U of Arizona CS Dept, Tucson

In article  <1992Apr10.024357.15902@midway.uchicago.edu> Anil Trivedi writes:
]In article <15127@optima.cs.arizona.edu> gudeman@cs.arizona.edu (David Gudeman) writes:
]
]DG:
]>The term "foundational" in this sense does not refer to relatively
]>permanency but to relative primacy.  In other words, mathematics in
]>general is built upon a foundational theory, 
]
]Where did you get that? You should know that mathematics was built,
]not upon these "foundations", but long before anyone thought of them.

As a couple of people have pointed out, the present tense is important
here.  However mathematics developed historically, the purpose of
foundational theories is to justify (or even correct) the original
development.  I can justify my statement that "mathematics _is_ built
on a foundational theory" in any one of three ways, take your pick:

(1) current mathematics practice almost always defines arithmetic,
geometry, and other basic disciplines in terms of set theoretic
notions.

(2) the thinking processes that led to arithmetic and geometry
_historically_ is equivalent to some foundational approach (which
approach is still unknown), otherwise it would not be correct.  And in
fact any correct thinking process that leads to a correct mathematics
must rely upon this hypothetical foundational theory at some level
whether we can recognize it or not.  The purpose of foundational
theories is to _discover_ the fundamental reasoning behind
mathematics, not to replace it.

(3) the thinking processes that led to mathematics historically were
unjustified.  In other words, conclusions were drawn without
sufficient logical reason, and if they were right, they were lucky.
The purpose to foundational theories is to replace this incorrect
reasoning with something better.

]>The obvious implication is that such
]>studies are superfluous and have no fundamental importance.  But there
]>are some serious questions...
]
]You have a chip on your shoulder or what? :-) 

Not really.  I think the view that foundational studies are
superfluous is a legitimate one, worth arguing against.  If I thought
the view was totally off-the-wall, I would not bother answering.

]Such studies are neither superfluous nor unimportant, but they
]do not constitute "foundations" upon which mathematics is built.

Then what good are they?  If they don't do what they were designed to
do then I say that they are worthless.

]>A more difficult and fundamental questions is: Is mathematical
]>reasoning purely logical, or do we somehow just "know" things about
]>mathematical objects so that we can make leaps of intuition? 
] 
]A reliable answer to such questions will come from actually understanding
]how the brain works; I don't know how far we are from that kind of knowledge, 
]I wish I did.

I disagree.  The _brain_ is a physical object with physical
operations, and doesn't _know_ anything.  If mathematics is done
entirely in the brain then we can have no confidence whatsoever that
any of it is right.  (Darn.  Just for a moment there, Neil Rickert and
I actually ageed on something, but I guess that's passed :-).
-- 
					David Gudeman
gudeman@cs.arizona.edu
noao!arizona!gudeman