From: epfaith@twinkies.berkeley.edu (Edward Paul Faith)
Newsgroups: sci.philosophy.tech
Subject: Re: Sets in mathematics
Date: 11 Apr 92 23:21:11 GMT
Article-I.D.: agate.s7s97INNmmc
Organization: U.C. Berkeley Math. Department.
NNTP-Posting-Host: twinkies.berkeley.edu


  In article <1992Apr11.165604.13414@mp.cs.niu.edu>, rickert@mp.cs.niu.edu 
(Neil Rickert) replies to a comment:

>>I do not usually think
>>of math (when I do it) as simple symbol-manipulation.  However, I can do so,
>>even though it would make me a much less motivated mathematician.

  Neil Rickert replies:

>  Let me expand a little on my comment that this is perhaps the preponderant
>view.  I think many mathematicians would state this as their view if asked,
>although it might depend on who asked them, and how the question were
>phrased.  But it does not seem to be the view they use when they do their
>research.  They tend, instead, to behave as if they believe there is
>some intrinsic, perhaps undefineable, meaning in mathematics.

  Yes, they behave as if they believe there were some intrinsic meaning.
I would compare this to the suspension of disbelief of a book's readers.
If mathematicians say that they are merely using their imaginations
because for some reason it is useful to do so, then I don't see why we should
doubt their word.  Our minds are designed to deal with the real world,
and have trouble doing mathematics without imagining that they are
studying something real and "out there".

>  My personal view is that mathematics does have intrinsic meaning, value
>and purpose.

  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.  When someone
proves the unconstructibility of a figure on the plane, mathematicians
stop trying to construct it.  Mathematics the activity is affected by
mathematical results.  I think mathematics the activity would derail
completely if results were not consistently taken by mathematicians as
real-world discoveries about the activity of doing mathematics.  So
one can say that mathematical results have immediate purpose and value
for mathematicians, without needing to be applied to physics or
economics or whatever.  In this sense, mathematical results have
mathematical meaning, insofar as meaning is use.  But "intrinsic meaning"
seems to refer to something ineffable.

>However the view that mathematics is empty symbol manipulation
>is a valid (even if incorrect) interpretation.  It is important that this
>interpretation remain valid, for this is what makes mathematics developed
>in one context applicable in a completely different context.  In particular
>the validity of the proof of a theorem must be based on its correctness
>under the interpretation of mathematics as empty symbol manipulation.
>In contrast, the importance of the theorem is judged in terms of its
>relevance to the intrinsic meaning of the mathematics within some context.

  I see now the error of stressing the symbols of
mathematics. Mathematics is generally explained to fellow
mathematicians using ordinary language, and it's generally
done by imagining that there are real things somewhere with
real properties, and by referring in a normal linguistic way
to mathematics itself, or so I gather. Ultimately it is
always formalizable. Nevertherless, the symbols are in a way
merely the afterimage of the actual process of doing
mathematics, which is itself not completely formalizable all
at once as Godel showed.  Looking at it this way, I no
longer see what empty symbols have to do with the "essence"
of mathematics.  Symbols do not refer to anything, and
mathematics does at its root.

  Still, one must ask if there is any foundation for this
property of referring.  I suppose that mathematicians think
that the application of mathematical results to mathematical
activity is obvious and straightforward.  I personally will
never try to "trisect an angle using ruler and compass",
because it was proven to be impossible.  But what right do I
have to be so complacent about the impossibility? This is a
problem that I am interested in.
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