[FOM] Bishop Berkeley and ZFC

Nik Weaver nweaver at math.wustl.edu
Sun Feb 19 04:00:12 EST 2006

Martin Davis ("The lure of the infinite") mentioned Bishop
Berkeley's criticism of calculus and made the point that even
though Berkeley was correct, mathematicians rightly ignored him
and were eventually vindicated by the rigorous development of
calculus by Weierstrass and others.

Mark Bridger ("The ghost of Berkeley") comments that a modern
day Berkeley might mount a similar protest against Platonism
in foundations.  I'd like to follow up on that; I think I have
one or two interesting points to add.

In both cases (naive infinitesimals and platonic sets) the
objection is to the invocation of ill-defined metaphysical
entities in what is supposed to be precise mathematical
reasoning.  Davis believes that just as with infinitesimals,
set-theoretic reasoning will eventually be firmly justified.
He won't be surprised to hear me note that as far as core
mathematics is concerned, this has already happened.  Of
course I'm referring to the fact that essentially all core
mathematics can be interpreted predicatively, a development
due to many people but probably more to Weyl and Feferman
than anyone else.

The predicative development of core mathematics is highly
satisfying in the same way that the epsilon-delta method is.
My preferred approach is to carry out this development in J_2,
the second level of Jensen's constructible hierarchy.  (See my
paper "Analysis in J_2", available on my website.)  When one
does this one has a feeling of concreteness, a clear grasp of
structure, that is similar to what one gets from epsilon-delta
proofs.  Looking back either at naive infinitesimals or at
Cantorian reasoning one has a sense that a dark and murky
world has been replaced with something bright and clear.

There's a specific analogy between naive infinitesimals and
Cantorian objects like P(N) or aleph_1.  Predicatively the
latter are conceived of essentially as proper classes.  One
can generate lots of sets of numbers or countable ordinals but
it is always possible to go further and generate more.  In the
Cantorian picture we have a vague sense of "passing to the limit"
and actually grasping P(N) and aleph_1 as finished entities.  This
is rather like passing from a decreasing positive real variable to
an infinitesimal quantity.  Weierstrass showed that this obscure
transition is unnecessary: rather than working with actual
infinitesimals, we can use arbitrarily small positive quantities.
That's analogous to the predicativist approach to things like P(N)
and aleph_1.

I haven't yet mentioned nonstandard analysis.  Here we have a
precise setting in which the somewhat confused naive ideas can
be rigorously interpreted.  I think this is quite analogous to
ZFC.  In fact I predict that most future mathematicians will
think about ZFC in roughly the same way that analysts now think
about nonstandard analysis: as an alternative approach that allows
one to use somewhat confused naive ideas about sets in a precise
way, that is in some ways very elegant, and in other ways quite
pathological.  I think most analysts currently take the view
(fairly or not) that nonstandard analysis is rather exotic and
not worth learning since anything important that you can get using
it can be obtained in a more straightforward way without it.  On
the other hand, some people hold the opposite view.  So it's partly
a matter of taste.

My feeling is that in both nonstandard analysis and in ZFC we
are dealing with systems that have no straightforward transparent
model.  That's the source of the pathological aspect.  But at a
higher level, in both cases the systems themselves are chosen to
have very nice properties; that's the source of the elegant aspect.

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu

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