FOM: anti-foundationalism

[Stephen G Simpson]

Stewart Shapiro 26 Mar 1999 10:18:38 writes:

 > I will try to write something about foundationalism soon ..., but I
 > am not all that wedded to those parts of the book anymore (as
 > indicated in the PM article).

I now have your PM article (Philosophia Mathematica vol. 7, 1999,
42-64).  Thanks for sending it to me.

Unfortunately, looking at your PM article, I don't see that you have
retreated at all from the vehement anti-foundational stance which
seems to be the main point of your book.

The title of your book is `Foundations Without Foundationalism: the
Case for Second-Order Logic'.  In it you define foundationalism as
`the view that it is possible and desirable to reconstruct mathematics
on a completely secure basis, one maximally immune to rational doubt'
(page v).  You say `one of the main themes of this book is a thorough
anti-foundationalism' (page 220).  You say that `foundationalism
... has few proponents today, and for good reason' (page vi) and that
mathematics is `a house built on sand' (page 26).  You briefly mention
two foundationalist programs, logicism and Hilbert's program, only to
remark that `both these programmes failed to achieve the
foundationalist goal and, for various reasons, few people seriously
hold out hope for repairs' (page 29).  You cite venerable authorities
such as Weyl (page 25) and Quine (page 197) in support of your
anti-foundationalist stance.  You say `I argue that the assumptions
behind the foundational conception of logic [i.e., the conception of
logic in which deduction is central] are untenable' (page x) and `I
reject the foundational conception of logic altogether' (page 45).

So far as I can see, none of this anti-foundationalism is retracted in
your PM article.  The only comment on foundationalism that I can find
in your PM article concerns the rather arcane issue of whether
`foundationalism provides the *only* motivation for first-order logic'
(page 50).  I don't see anything in your PM article on the issue of
whether foundationalism and/or `the foundational conception of logic'
are true.

Could you please explain why you yourself (not Quine or Weyl or other
authorities) are so vehemently opposed to foundationalism?

In particular, could you please explain how you square your jaundiced
view of Hilbert's program with modern progress in that direction?

-- Steve