[FOM] Numbers vs writhmetic. was: n-th order ZFC

[Roger Bishop Jones]

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