FOM: Brouwer, Constructivist/Intuitionist?
Robert Tragesser
RTragesser at compuserve.com
Fri Jul 28 13:44:16 EDT 2000
The recent disucssion of constructivism inspires me to query
whether it is correct to say that Brouwer was a constructivist. One has
the impression that for many 'Brouwer' and 'constructivist' are linked like
the bell ding and the drool are linked in Pavlov's dog. Reading Dirk van
Dalen's wonderful biography of Brouwer, I can't decide what he would say if
queried, but I think that there is space in his account for saying that
Brouwer was an intuitionist first, and that the constructive mathematics he
deployed was a means to an end.
Here's what I suspect.--
First, there is the intuition of (temporal) continuity, the
flowing/fluid (Weyl) and definitely not intuitively atomistic (like the
Dedekind real number continuum the Cantorian infinitary set-theoretical
representations of continua are atomistic).
Second, there is the problem of achieving a good mathematical
representation of such continua, the flowingness somehow intuitively
suggesting inseparability of continua and certainly suggesting that they
are not atomistic.
I sense that Brouwer was motivated to "return" to these intuitive
continua because the counter-intuitive atomistic, infinitary continua being
articulated through the concept of set emerging from Bolzano, Dedekind,
Cantor, et al. had led to logical and, worse, mathematical stumbling blocks
if not outright logical and, worse, mathematical dead-ends, cul-de-sacs.
Because this was what was inspiring Brouwer to return to the intuitive
fluidic continua, he would of course be very cautious about deploying
infinitary, set-theoretic reasoning. The natural thing for Brouwer to do
is to go back to a core mathematics which is not so radically problematic
from either a logical or mathematical point of view, and to go as far as he
can with that core mathematics toward a mathematicfally cogent and
something like full representation of fluidic continua. Brouer would
realize that at some point he would need to add a new idea that would be
alternative to the infinitary, set-theoretical reasoning of Cantor etc. And
that new idea would be the full idea of choice sequence leading to the all
functions are continuous theorem, entailing the inseparability of the
representable continua (representable through or in the mathematics of
choice sequences), thus assuring that a fundamental intuitively given train
of flowing (rather than atomistic) continua is captured in the mathematics.
What might look like a retreat to finitist cum constructivist
mathematics was more a regrouping there with a view to discovering and then
pursuing from out of there a wholly other mathematical strategy {"wholly
other" than classical set-theoretic reasoning), which turned out to be a
mathematics of continua based on the mathematical representation by means
of the novel choice sequences of fluidic continua. But Brouwer was not
substantively committed to the mathematics of the latter representation
being constructive mathematics.
[Of course one can wonder than if it were to have followed Brouwer,
mathematics would have found itself in more difficulties, encountered more
dead-ends or at least stumbling blocks than Brouwer had perceived occurring
in the in his day emerging classical set-theoretically reasoned
mathematics.]
robert tragesser
westbrook connecticut
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