Re: [HM] the importance of notation
Robert Tragesser (RTragesser@compuserve.com)
Tue, 11 May 1999 09:00:00 -0400
The emergence and refining of logical notations adequate to
sustain finitary logical calculi seems to raise this question:
Are these centrally significant _mathematical_ notations?
[I do have in mind by notation something we directly employ,
work and write in; not as an abstract type.]
No one does or writes mathematics (anything like fully) in
these notations (except as illustrations); it is not just that there is
not world enough or time to do so, it is that there would be no point
to doing so (save that of the obsessive-compulsive hand washer,
whatever that might be). That logical notation is sometimes used as
shorthand counts for little, I think.
At the same time, the _very possibility_ (rather than actual
practice) of writing out mathematics -- calculations and proofs -- in
these notations has surely been having cumulatively profound effects on
mathematics. Such as: metamathematical solutions to, and explorations
of, mathematical problems; foundational explorations; the emergence
of new branches of mathematical science (mathematical logic, computer
science).
And it does seem that the very possibility has had a more
difficult to specify, and more unconscious, influence on the
confidence with which pure or abstract mathematics is practiced? [I
gathe that e.g. Saunders MacLane would make this point?]
I am wondering -- along these lines -- if other notations in
mathematics have played an analogous role? I suppose that one could
argue that a purist of Galois Theory would never write out a polynomial
(in the usual post-Viete or any notation), and to regard _applications_
of G.T. to what could be done in such notations as unseemly, or of
marginal interest. But this is not really quite so comparable since a
real mathematical life has lived through such notations; and I don't
think this was/is true of logical notations?
robert tragesser
west(running)brook, connecticut