FOM: Formalisms

[Allen Hazen]

    A lot of very different points of view have gone under the name
"formalism."  Some of the cruder and more extreme 19th century ones were
criticized by Frege in his "Foundations of Mathematics."  For a reasonably
short, very readable, account of the history of these ideas, my favorite
book is Michael Resnik's "Frege and the Philosophy of Mathematics" (Cornell
U.P. 1980; I think out of print, alas).
   As for Robert Black's question about the name of the discipline tha
considers the non-Sigma-1 claims that can be made ABOUT formal systems...
the obvious one is PHILOSOPHY (though there is no evidence that people with
that word as part of their academic job title are any better at it than
those with the word "mathematics"), or maybe METAPHYSICS.  Matthew Frank's
reply suggests he'd say "metaphysics" (and that he would pronounce it in
the same condmnatory or dismissive tone that the logical positivists of the
1930s did!).
   --
   The Lady Welby who supposedly coined the word "significs" makes it into
the history of logic as a correspondent of C.S. Peirce.  As for Mannoury,
one of my teachers credited him with making a distinction (necessary in
thinking about non-classical logics) between "exclusion negation" (the
negation of any non-truth is true) and "choice negation" (only some
non-truths have true negations).  So he may have helped Brouwer and Heyting
clarify their thinking about logic.
Allen Hazen
Lecturer in Philosophy
  (interests: logic, philosophy of mathematics, &c)
University of Melbourne