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http://www.tau.ac.il/~corry/publications/articles/pdf/bourbaki-structures.pdf

As already stated,
members of Bourbaki consistently declared themselves first and foremost to
be “working mathematicians”, and their actual views concerning philosophi-
cal or foundational issues is perhaps most frankly expressed in the following
quotation of Dieudonné:
On foundations we believe in the reality of mathematics, but of course when phi-
losophers attack us with their paradoxes, we run to hide behind formalism and
say: “Mathematics is just a combination of meaningless symbols” and then we
bring out chapters 1 and 2 [of the Eléments] on Set Theory. Finally we are left in
peace to go back to our mathematics and do it as we have always done, working
in something real. (Dieudonné 1970, 145)
This position of “Platonism on weekdays and formalism on Sundays”,
which is so widespread among working mathematicians, becomes especially
worthy of attention in the case of Bourbaki. It has been claimed elsewhere that
such a position is untenable as a consistent philosophical account of mathe-
matics, since it involves both logical inconsistency and a distorted description
of the actual doings of the mathematician.
56
Nevertheless, it is an accepted
image of mathematics, that has at least helped many a twentieth-century math-
ematician confer some meaning to his own scientific work. This seems to be
the case as well for Bourbaki.

It may come as a surprise that, while raising the banner of rigor and parsi-
mony in mathematics, Bourbaki was willing to adopt the above-mentioned
philosophical position without any reluctance. It is not a criticism of Bour-
baki’s philosophical sophistication, or lack of it, which concerns us here but
rather the question, how is the elaboration of the theory of structures con-
nected with Bourbaki’s images of mathematics. The above-described mixture
of a declared formalist philosophy with a heavy dose of actual Platonic belief
is illuminating in this regard. The formalist imperative, derived from that
ambiguous position, provides the necessary background against which Bour-
baki’s drive to define the formal concept of structure and to develop some
immediate results connected with it can be conceived. The Platonic stand, on
the other hand, which reflects Bourbaki’s true working habits and beliefs, has
led the very members of the group to consider this kind of conventional, for-
mal effort as superfluous. Indeed, of all the apparatus developed in the first
book of the treatise following that formalist imperative, only feeble echoes
appear in the other volumes, where Bourbaki’s real fields of interest are devel-
oped.