FOM: anti-foundationalism

[Stewart Shapiro]

This is very quick, and so will open me up to vigorous refutation, due to
lack of care on my part.  Well, that is what discussion is for.  (I will be
leaving town for about 10 days soon.)

>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.
>

Right.  What I have backed off from is the claim in the book that
foundationalism provides the only or the main motivation for first-order
logic.  Some of the reviews and critical studies of the book took me to
task on that.  As far as I know, the FOM discussion is the first time I
took some heat for the anti-foundationalist stance itself.  In his review
in the JSL, John Burgess said that Chapter 2 (the section on
foundationalism) is the weakest section of the book.  I don't think he
argued that I was wrong there, but that it does not provide a good support
for second-order logic.

>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).

Well, this particular quip is out of contect.  It was to pursue a metaphor
that Weyl made.  My claim was that mathematics does not need a solid
foundation in order to be set right.  Weyl argued that because classical
mathematics is less than absolutely certain, we have to retreat to either
predicative mathematics or intuitionism (depending on the period of Weyl's
career).

>  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).

Incidentally, one serious exception is Mic Detlefsen's excellent book on
the Hilbert Programme.

>You cite venerable authorities
>such as Weyl (page 25) and Quine (page 197) in support of your
>anti-foundationalist stance.

Not Weyl.  He went for an absolutely secure foundation and was willing to
cripple mathematics in order to achieve it.

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

I remain in admiration of contemporary proof theory, and do not mean to
denegrate it.  But do people think that the modern progress contributes to
the original *epistemological* goals of the Hilbert program?  Do any of the
results make mainstream mathematics more secure than it is already?  I am
not a scholar of the relevant history (and will be corrected), but I took
Hilbert's goal to be to remove doubt from mathematics (in light of the
antinomies).  Recall his quip that once the program is achieved, no one
will drive us out of Cantor's Paradise.

A more modest way to put the thesis of this aspect of my book is that there
is value to foundational studies independent of the epistemological goals
of the Hilbert program.

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

Again, I have to be quick.  Foundationalism goes back to the rationalism
of, e.g., Descartes, and is a quest to put science (or in this case
mathematics) on an absolutely secure foundation (or, failing that, as
secure a foundation as is humanly possible).  I guess I was not so much out
to refute foundationalism, but to point toward alternate orientations to
foundational studies.  I did not mean to be "vehement", as Steve suggested.