[FOM] foundational philosophies and core mathematics
Nik Weaver
nweaver at math.wustl.edu
Mon Feb 13 03:32:46 EST 2006
A little while ago Curtis Franks posted a message containing the
assertion that "from the restrictive standpoint of Predicativism
lots of ordinary mathematics appears unjustified and even
paradoxical", which he attributed to Hilbert. I think that was
the consensus view in Hilbert's time and this was surely the main
reason why predicativism initially attracted so little support.
However, it is now generally understood to be false. I want to
make the point that the predicative conception of mathematical
reality is in fact in remarkably exact accord with normal
mathematical practice, much better than for any alternative
foundational stance of which I am aware.
I'll explain what I mean by the first part of that assertion
in a subsequent message. Here I want to show why I think other
foundational stances fit rather poorly with normal mathematics.
The main complaint against Cantorian set theory is, as Richard
Feynman said in a different context, that "the stage is too big
for the drama." Never mind large cardinals, even in plain ZFC
we have a cumulative hierarchy (V_alpha) which extends up to
alpha = aleph_1, aleph_omega, aleph_{aleph_omega} ... yet
mainstream mathematicians generally study only objects which
appear in the lowest levels of this hierarchy. Even in fields
like algebra or functional analysis where there is no a priori
cardinality restriction on the fundamental objects of study,
interest has settled on the countable/separable case and the
nonseparable setting is generally seen as a zoo of pathologies.
This can also be seen in the disconnection between core
mathematics and set theory. I gather that some of the more
emotional responses I've gotten to earlier expressions of my
views came from set theorists who were outraged by the suggestion
that most mathematicians would not see their work as central, but
I think this is clearly the case. A lot of set theory has to do
with uncountable cardinals that just don't appear, or are at best
extremely peripheral, in core mathematics.
The case that other foundational philosophies don't fit normal
mathematics very well is probably less controversial. I will
say that intuitionists and constructivists have shown how to
accomodate versions of an impressive amount of core mathematics
in their conceptual frameworks. However, here I think it is
clear that those basic conceptual frameworks are quite alien to
most practicing mathematicians. The idea that the twin primes
conjecture may be neither true nor false, the nonexistence of
discontinuous functions from R to R --- these suggestions would
be very hard for most mathematicians to accept. Also I do not
want to downplay the amount of rewriting that would be necessary
to translate standard mathematics texts into, say, intuitionistic
language.
It has been said that most mathematicians are more comfortable
being platonists with regard to number theory than they are with
regard to set theory. This basic intellectual orientation is
mirrored exactly in predicativism. Moreover, virtually all of
core mathematics can be made predicative with only minor
modifications, whereas vast regions of set-theoretic pathology
simply disappear. I'll say more about this in my next message.
Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu
http://www.math.wustl.edu/~nweaver
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