[FOM] Numbers vs writhmetic. was: n-th order ZFC
Roger Bishop Jones
rbj at rbjones.com
Fri Jul 15 16:15:06 EDT 2011
On Wednesday 13 Jul 2011 07:26,
W.Taylor at math.canterbury.ac.nz wrote:
> Quoting Roger Bishop Jones <rbj at rbjones.com>:
> > ... one can believe in the objective truth of
> > arithmetic without also believing in the existence of
> > numbers.
>
> I would like to hear more about this, if possible, as it
> touches on a dichotomy that I have been hearing a lot
> about in the last few years.
>
> If one (a) DOES accept the objective truth of arithmetic,
> but (b) does NOT accept the existence of numbers,
> then wherein resides the objectivity of the arithmetic?
>
> If one has no semantics for arithmetic (which seems to be
> what (b) says), then on what grounds is truth to be
> defined for arithmetic?
The idea is that one can do semantics without doing
metaphysics, not that one can have objective truth
conditions without a semantics, for the latter is a
semantics.
Note that the "objectivity of truth" should be understood by
contrast with subjectivity. The idea is that the truth
conditions are unambiguously determined. If you are
reasoning about abstract entities objectivity arises when
the truth value of a sentence is entailed by the defining
characteristics of the relevant abstract entities and is
therefore a purely logical matter.
Two philosophers in whose philosophy mathematical truth is
divorced from metaphysics are Hilbert and Carnap.
In Hilbert we have the idea that mathematical concepts
should be defined exclusively by a formal axiomatisation, and
that the only requirement for the existence of entities thus
defined is the logical consistency of the definitions.
This is a kind of denial of metaphysics, and mathematical
declaration of independence from metaphysics (though Hilbert
is arguing in general against any considerations beyond the
axioms, not just metaphysical considerations).
Mathematicians are free to reason about any kind of entity
which is logically coherent.
In Carnap we find the starkest contrast with the idea that
semantics depends on metaphysics, for he was the staunchest
opponent of metaphysics (in the sense in which he used the
term) and a major part of his philosophical energy was
devoted to furthering the field of semantics.
His manner of reconciliation is most concisely presented in
his paper "Empiricism Semantics and Ontology".
Roger Jones
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