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

Rob Arthan rda at lemma-one.com
Wed Jul 13 17:57:25 EDT 2011

On 13 Jul 2011, at 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?
> (I do not intend to be combatively rhetorical, I would just like
> to understand this combination position better.)

I have no idea what it means to say that the number 7 exists. But I believe I know exactly what 7-ness means of any collection of entities that I can conceive and for which I know how to decide equality of entities. The equivalence of (7+7)-ness and 14-ness is therefore true for me irrespective of the (to me) meaningless and irrelevant question of whether the numbers 7 or 14 exist.

May I throw your question back to you and ask what you think it means for a mathematical abstraction such as the number 7 exists?



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