[FOM] Numbers vs writhmetic.
Roger Bishop Jones
rbj at rbjones.com
Tue Aug 9 15:23:48 EDT 2011
[with apologies for slow response]
On Saturday 16 Jul 2011 19:11, Daniel Mehkeri wrote:
> > 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
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.
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