FOM: Feferman's first thesis

[Torkel Franzen]

  John Steel says, commenting on "Thesis 1" below:

   >1. Mathematics consists in reasoning about more or less clearly and
   >coherently conceived groups of objects which exist only in our
   >imagination.

   >If you drop "which exist only in our imagination", there isn't much to
   >disagree with there. What is the meaning of "exist only in our
   >imagination"?

  Chiefly that there is no problem of how we know that the real
numbers exist (or that a complete ordered field exists). It doesn't
matter if they exist or not (however these alternatives might be
interpreted) - let's just imagine that they do, and go
on from there, telling convincing stories (i.e. formulating axioms) about
them, making deductions from these stories, using them in our models
of physical events, and so on.

  Now, if somebody claims that on the contrary, the real numbers do exist,
the question arises how we know this, and what difference it makes. To
this question various answers might be proposed. The matter is problematic.

  Naturally there are also problems with the view that the objects of
mathematics are figments of our imagination. In particular, when "Thesis
1" is combined with "Thesis 6":

    6. While reasoning is our only known path to secure mathematical truth,
    there are objective questions of truth and falsity.

it becomes problematic how there can be objective questions of truth and
falsity about figments of our imagination.


Torkel Franzen
http://www.sm.luth.se/~torkel/