[FOM] Numbers vs writhmetic.

[Roger Bishop Jones]

[with apologies for slow response]

On Saturday 16 Jul 2011 19:11, Daniel Mehkeri wrote:
[RBJ]
>  > 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.
> 
> To apply this to the natural numbers, obviously there
> will arise the question of what logical consistency
> means. It was not my understanding that Hilbert intended
> merely the absence of a "practically realisable"
> contradiction. Rather he took at least Pi_1 sentences at
> face value, didn't he?

I am not expert in Hilbert, but the ideas I spoke of 
predated Goedel's incompleteness results and possibly at 
that time Hilbert had not seriously considered the 
possibility that a contradiction might not be "practically 
realisable" (and had perhaps not considered the possibility 
that Pi_1 sentences might have any other than their face 
value).

The main point, however, is that the criteria for existence 
of mathematical entities should be logical (in the broad 
sense of that term which prevailed before the ascendency of 
first order logic) rather than metaphysical, and this point 
of view remains tenable, incompleteness notwithstanding.
The relevant distinction is probably that between necessity 
de dicto and de re.

Roger Jones