On Nov 18, 2011, at 4:51 AM, Irving wrote:
> ...
> All of this having been said, the best answer I can give is that, the
> "points, lines, and planes" and "tables, chairs, and beer mugs" remark
> aside, Hilbert would give different axiomatizations for different parts
> of mathematics. That is to say, there is one set of axioms and
> primitives suitable to develop, say, projective geometry, and another
> for algebraic numbers; there is one suitable for Euclidean geometry and
> another for metageometry. In the case of the latter, for example, one
> needs to devise an axiom set that is powerful enough to develop all of
> the theorems required for the articulation not only required for
> Euclidean geometry, but also for hyperbolic geometry and elliptical
> geometry, but which do not also generate superfluous theorems of other
> theories. Hilbert's axiom system for geometry, then, is not the same
> as that which he erected for physics.
>
> What I think is the correct understanding of Hilbert's "off-the-cuff"
> remark about points, lines and planes and tables, vs. chairs, and beer
> mugs, is the more profound -- or perhaps more mundane -- idea that
> axiom systems are sets of signs which are meaningless unless and until
> they are interpreted, and by themselves, the only mathematically
> legitimate characteristic of axiom systems is that they be
> proof-theoretically sound, that is, the completeness, consistency, and
> independence of the axiom system, and capable of allowing valid
> derivation of all, and only those, theorems, required for the piece of
> mathematics being investigated.
Dear Irving,
My own interpretation may be substantively different, it may not.
I take Hilbert's position to be that the formalism is independent of the
subject matter. That is, I take his view of formal interpretation to be
mechanistic, specifying valid transformations of the structure under
consideration, be it logical, geometric or physical. I am confused because you
use "signs" instead of "marks" here. In addition, since the formalism is
independent of the subject - as suggested by his appeal to Berkeley - a theorem
of the formalism remains a theorem of the formalism despite the subject.
In this view, how one selects an appropriate formalism for a given subject - if
there is a fitness ("suitability") requirement as you suggest for the
"different parts of mathematics" - appears to be a mystery, unless you think
empiricism is required at this point.
With respect,
Steven
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