[FOM] Falsify Platonism?
Roger Bishop Jones
rbj at rbjones.com
Mon Apr 26 05:43:29 EDT 2010
On Monday 26 Apr 2010 03:14, Timothy Y. Chow wrote:
> This is precisely the kind of proposal that philosophers
> might find satisfying but that is not likely to satisfy
> the typical mathematician. If everything hinges on
> drawing a sharp distinction between the natural numbers
> and our concept of the natural numbers, then the
> mathematician is going to be perplexed, since this is
> not the kind of sharp distinction that is customarily
> demanded in mathematical discourse.
It seems to me that a similar point can be made in a way
which may seem less mysterious to mathematicians if instead
of talking of the "concept" we speak of the definition.
Mathematicians do understand that a purported definition must
be shown to be consistent before it can properly feature in
a sound proof. i.e. if you wish to say in a proof "let a be
some X such that P" then you must show that there is an X
such that P, otherwise your proof will be unsound.
The proposal is then that one should distinguish between the
natural numbers and some purported definition of the natural
numbers. PA is then to be understood as an "implicit
definition" of the natural numbers (or of the concept of
natural number). But an implicit definition defines only if
it is consistent, and a demonstration that PA is
inconsistent shows only that PA fails to define the natural
numbers (or the concept of natural number, or anything
else). It does not show that the naive idea of the natural
numbers as those which we obtain by counting up from zero is
incoherent, or that ontological claims about abstract
entities lack an objective truth value.
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