FOM: second-order logic is a myth

[Stephen G Simpson]

Today I spent a little more time with Shapiro's book `Foundations
Without Foundationalism: A Case for Second-order Logic'.  

Although I still think this book is completely misguided, I have to
give Shapiro credit for presenting a thoughtful account.  Contrary to
my earlier impression, I'll now say that this book is *much* better
than Hersh's.  I now regard Shapiro's `anti-foundationalism' as little
more than an attempt to be trendy.  I don't assign any serious weight
to it.

 From the viewpoint of mathematical logic, the defects of second-order
logic are well known and I don't want to go into them here and now.
Suffice it to say that second-order logic leads immediately into the
quagmire of set theory and never leads out of it.  `Second-order logic
is set theory in sheep's clothing.'  This is from Quine, quoted by
Shapiro.  Although Shapiro disagrees with Quine, he presents enough
technical results about second-order logic to make it obvious that
Quine's point is true.  [By the way, Shapiro on page 107 assumes the
consistency of AD_R, but it seems to me that this should be avoidable
by means of a forcing argument.]

Let me concentrate instead on Shapiro's philosophical case.

Boiled down to essentials, Shapiro's case for second-order logic seems
to be as follows:

  1. There is no sharp boundary between mathematics and logic.
  
  2. In the present historical era, mathematicians standardly assume
  set-theoretic realism, including the existence of actual infinities
  and an absolute powerset operation applying to them.
  
  3. Therefore, logicians ought to also assume these things.

An argument similar this appears in Church's logic textbook, page 326
(quoted on page 207 of Shapiro's book).

I agree with 2 as a historical fact, reserving judgement on whether
the mathematicians' assumptions are correct.  I sharply disagree with
1 and 3.

Against 1, the correct view of the matter (going back to Aristotle) is
that logic is a method or common background shared by all scientific
subjects, not only mathematics.  This key scientific/philosophical
distinction is reflected in the usual predicate calculus distinction
between logical and `non-logical' axioms.  The logical axioms are
common to all subjects (i.e. theories), while the `non-logical' ones
are subject-matter specific.  A major defect of second-order logic is
that it blurs this key scientific/philosophical distinction by
pretending that axioms regarding sets (a specific subject matter) are
part of the underlying logic.  For instance, Shapiro presents two very
different semantics for second-order logic and ambiguates on them
throughout the book.

Against 3, I reject the notion that f.o.m. professionals ought to
slavishly follow the current practice of `working mathematicians'.
F.o.m. should take and does take a higher, broader, more universal
perspective.  As I have said many times here on the FOM list,
f.o.m. addresses the place of mathematics in the totality of human
knowledge.  The Shapiro case leaves no way to do that.  For instance,
Shapiro says nothing about the obvious disconnect between Platonist
realism and applied mathematics.

To his credit, Shapiro does consider some views contrary to his own,
notably those of Skolem, von Neumann, and G"odel.  However, so far as
I have been able to discover by a cursory reading of Shapiro's book,
he doesn't address the objections that I have made above.

-- Steve