FOM: Bourbaki

[Joseph Shoenfield]

     There are more interesting subjects on fom nowadays than I can find
time to write about.   Nevertheless, I cannot resist reminiscing a little
about Bourbaki, even though my remarks wil contribute little to recent
dicussions.
     Bourbaki's greatest days occured during my graduate career (circa
1950).   In those days he had great influene on mathematics, especially on
graduate education.   This was mainly through a series of outstanding
introductory texts.   For example, I believe that his work on multilinear
algebra revolutionized the way mathematicians looked at that subject.   In
addition, his pronouncements on mathematics often had great influence.
Some of them could still have significances for devotees of fom.   I
recently called attention to his belief that established terminology
should only be changed for compelling reasons.   Another pronouncement
relevant to some recent discussions on fom concerns applications, more
particularly, applications of one field of mathematics to another.   He
said that in the beginning of a new field of mathematics, such
applications were necessary to justify the field; but that after the field
became established, researchers were justified in devoting their time to
problems which arose within the field.   One could apply this to, for
example, Recursion Theory.   In its early days, it justified itself by
remarkable applications, such as the unsolvability of Hilbert's 10th
problem and the word problem for groups.   Nowadays researchers in RE
degree theory are justified in considering structural problems without
worrying to much about the lack of applications.
     As a digression, I have observed that very frequently, such
applications of the new field make comparatively little use of results in
the field.   For example, the two applications mentioned above use only
Church's Thesis and Kleene's definition of recursive functions via
primitive recursion and the mu operator.   Perhaps someone can cite an
example of an application of a new field which made use of deep results in
that field.
     My admiration for Bourbai did not conceal from me the fact the his
approach to some fields was distinctly below his generally high standards.
In particular, his approach to the language and axiom of mathemtics showed
in many places a complete lack of knowledge of the accomplishments of
logic and foundations in this century.   I give a few examples.
     (1) He elected to use the Hilbert epsilon symbol, which was designed
for proof-theoretic investigations, in his basic language.   On result was
that the Axiom of Choice was derivable from purely logical axioms.   He
may have thought this an advantage, since there were the still some who
questioned the inclusion of the Axiom of Choice.   He probably did not
know that he thereby eliminated from mathematics Godel's prood of the
consistency of the axiom and Cohen's proof of its independence, two
of the gems of 20th century logic.
     (2) His treatment of the propostional logic consisted of axioms and
many theorems; there is no attempt  to state, much less prove, the
completeness theorem for the propositional calculus.   This was also the
case in Principia, and I have often thought of this an an example of
Russell's failure to apply his ability to signficant questions becaus they
did not occur to him.   To repeat his error fifty years later is
imcomprehensible.
     (3) A small but revealing example concerns Kuratowsi's definition of
the ordered pair.   I recall some minutes of a Bourbaki in which Andre
Weil mentioned it; his comment was that it made him ill.   Of course it is
an artificial definition to enable discussion of ordered pairs in a
minimal language; but I can't see why anyone who approved of defining a
real number as an equivalence class of Cauchy filters should complain
about the this.   Bourbaki treated ordered pairs as an undefined notion,
but treated the ordered pair of two sets as a set.   Since only the basic
properties of ordered pairs werew assumed, this had strange results.   For
example, one could not prove the the sets of rank less than the first
inaccessible were a model, since a ordered pair of natural numbers could
have rank greater than that cardinal.
     I have not read Mathias's recent attack on Bourbaki's approach to
logic; but my first question for him would be why he though it was worth
attacking.   In fairness, I should point out that Bourbaki's treatment of
foundations improves somewhat in the later stages with his treatment of
structures of mathematics.   This topic has been rather neglected by
conventinal set theorists, but I think Bourbaki's treatment of it may have
contrubuted something (at least indirectly) to category theory as
foundations.   In any case, his contribution to my education was as least
as great as that of any teacher whom I actually met.