FOM: examples of infinitesimals; f.o.m. testimonial

Stephen G Simpson simpson at
Mon Nov 24 14:31:13 EST 1997

(1) examples of infinitesimals

  Several people have said that it's easy to give examples of
  infinitesimals in nonstandard analysis.  But nobody seems to have
  commented on Harvey's highly relevant posting
     FOM: 7:Undefinability/Nonstandard Models
  on November 17.  The essential point there is that it's impossible
  to give an example of an infinitesimal under the Abraham Robinson
  setup, because it's consistent with ZFC that there is no definable
  nonprincipal ultrafilter on N, the natural numbers.  If e were a
  definable infinitesimal, then

     {X subset of N: [1/e] is an element of X*} 

  would be a definable nonprincipal ultrafilter on N.

(2) f.o.m. testimonial

  Lou and Dave have said that they personally find pure mathematics
  more interesting than foundations of mathematics.  I'd like to say
  that I personally find foundations of mathematics incomparably more
  interesting than pure mathematics.  (I would also go farther and say
  that foundations is *objectively* more interesting, not just more
  interesting to me personally.  But that is another story.)
  First let me comment that my credentials in pure mathematics are
  pretty good: (1) I went through the normal graduate training at MIT,
  taking beginning and advanced graduate courses in many areas of pure
  mathematics, including algebra, analysis, and topology/geometry.
  (2) I had some very inspiring teachers in these areas, including
  Guillemin and Quillen.  (3) I subsequently made some serious
  contributions to pure mathematics, perhaps most notably the
  Carlson/Simpson dual Ramsey theory, which was used by Furstenburg in
  his recent work in ergodic theory.  The point I am trying to make is
  that I had the option of becoming a pure mathematician, and I think
  I could have done well in that direction, had I been so inclined.
  Now to my main point.  I opted not to pursue pure mathematics
  because I found it too narrow, too confining, too technical (in the
  sense that pure mathematicians place too much emphasis on technique
  such as cohomology, and not enough on fundamental mathematical
  issues).  Instead I opted for foundations of mathematics.  And by
  the way, I always thought of the subject as "foundations of
  mathematics," never as "mathematical logic" or "model theory" or
  whatever.  What drew me to foundations was its broad intellectual
  appeal and scope, with its obvious philosophical significance and
  the possibility of contributing to major intellectual developments
  on a broad front.  The work of Hilbert and G"odel and Friedman was
  tremendously exciting.  This is what moved me.  Unfortunately,
  because of circumstances (including a thesis advisor who was
  explicitly hostile to foundations) it was quite a while before I was
  able to pursue my foundational interests full time.  But all along I
  knew that foundations of mathematics was what I wanted to do, and
  now I'm proud and happy to be doing it.

-- Steve

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