Re: what makes it true?

Torkel Franzen wrote:
> Timothy Little <tim-usenet@xxxxxxxxxxxxxxxxxx> writes:
>
>> The problem is that I don't think that there is any such thing as
>> "the" natural numbers as distinct from any formal theory. "The"
>> natural numbers are partially refined by the properties that are
>> proved of them.
>
> This philosophical doctrine is not easy to grasp. The "as distinct
> from any formal theory" is particularly puzzling. Just what do you
> take to be the relation between formal systems and the natural
> numbers?

To me, formal systems are the basic concepts underlying communication
of reasoning about concepts between mathematicians. Actual
communication is generally done much more informally. The general
understanding is that in case of disagreement it could in principle be
resolved by translating the questionable steps into a suitably agreed
formal system and checking whether or not they are valid.

Hence: if I make a claim that some property of the natural numbers is
true, I'm expected to back it up with at the least, an informal proof.
If the readers agree that it can in principle be formalized in a
suitable system, then they accept it as a true property.


My philosophical position is that this is all there is to mathematical
truth. There is no such thing as "the" natural numbers (or indeed any
other mathematical entity) without considering which system acceptable
to most mathematicians is used to define their properties.


- Tim
.

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