[FOM] A Defence of Set Theory as Foundations

[Roger Bishop Jones]

On Thursday 13 October 2005 06:25, Robert Lindauer wrote:

> We are left, as Mr. Jones rightly puts it, with the option of
> 'semanticizing' our mathematics - making it a kind of
> word-play.

I don't agree with that gloss on "semanticizing".
If I had suggested a syntactic view similar to that
of the early Carnap or of radical formalists then talk
of word-play would be appropriate.
But here, we are assigning meanings to set theory
in domains which are not syntactic.

> This means giving a new kind of meaning to "true" 
> where the objects in question don't exist and don't "really"
> have the properties they are asserted to have.

A semantic view does not (in my conception) involve a
change in the meaning of "true".  It entails being sufficiently 
definite about truth conditions, but the meanings which are 
being assigned are, in this case, to the language of set theory.

> We therefore 
> have to separate the concept "true in reality" from "true in
> the story" or "true in the system".  We are left, however,
> with something rather unsatisfactory for most mathematicians -
> a true that means "not really true" and perhaps "not possibly
> true since originally fictional".

I would like to distinguish my semanticism from "fictionalism".
In my suggestion sentences of set theory are to be understood as 
making no metaphysical claims about the existence of sets, 
making claims only about what sets exist in, or are entailed by, 
the iterative conception. It is not intended that the claims of 
set theory be regarded as fictional.  To regard them as 
fictional is to accept that they are false.  Under the semantic 
view they really are true (some of them, including all the 
theorems).
Nor need a semanticist regard the metaphysical claim that the 
sets in the iterative conception "really" exist as false or 
fictional.  The truth or falsity of the metaphysical claim is 
immaterial to the mathematical discipline of set theory (though 
its consistency is not).

> Otherwise, we need to give an account of sets as "ideas" and
> then give an account of "ideas" as either soul-elements or
> brain-functional-systems.  Neither, I think, will serve as an
> adequate foundation for set theory if one wants to take it
> beyond psychologism.

Under the semantic view sets are not ideas.
They are pure well-founded extensional collections.

> Slightly more importantly, neither will serve as an
> explanatory system for the "basic physical facts" of a world
> in which "when you have two apples and give one to a friend
> and nothing destroys either apple and no new apples are given
> to you, you are left with only ___" has a definite answer. 

Which is OK by me, since this is not the purpose of set theory or 
of the foundational stance which I have suggested.

Roger Jones