[FOM] A Defence of Set Theory as Foundations

[Robert Lindauer]

On Oct 12, 2005, at 4:11 AM, Roger Bishop Jones wrote:
>
> Weaver seems to think that the iterative conception
> can be construed in only two ways, platonistically
> or constructively. (or does he to think it
> inconsistent because it can only be construed in
> both ways?)
>
> It seems to me that it can be construed "semantically",
> as a description of a collection of sets which is to
> be considered the domain of discourse of set theory
> but without any claim or presumption that the sets
> or the domain of sets "really" exist or that they
> can be "constructed".  Both of these elements are
> undesirable because they introduce considerations
> which are irrelevant to the mathematics for which
> a foundation is sought.

Using a "classical" definition of truth - correspondence - we have only 
two ideas of what sets could be - they could be objects in some 
invisible world, a world so unlike our world that the laws of it allow 
it to be "incomplete" in the sense that there is no such thing as "all 
the things in that world".  Thus platonism with regard to them, and 
therefore a classical "correspondence" theory of truth is not 
applicable.  One MUST if one is to talk about set-theoretical theorems 
being "true" make a new definition of true.  This has yet to be done 
explicitly.

The notion of a "constructed" infinite group is, I think, incoherent.   
The operational nature of constructing things leaves us a limited 
amount of time in which to do it, leaving us a limited number of things 
constructed.  Again, the iterative set-conception makes it impossible 
to think of these constructed things as the corresponding "real" 
objects of set theory.

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

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.

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.  That is, if mathematics has a foundation, it is to be 
found in very generalized and -too obvious- physical law - where we 
generalize from our physics some very, very general features of our 
universe - and, surprisingly, it may turn out to be that 1+1 = 4 - that 
even PRA is not a good representation of our world a the most general 
levels.  But this shouldn't be regarded as any more surprising than 
finding out that space is curved, if it ever comes up...

Personally, I prefer a combined approach.  I believe that mathematics 
is mostly to be explained with regard to the social and psychological 
processes that produce it and that MANY of those social and 
psychological processes are to be explained with regard to the actual 
experiences of the people that produce it.  Barring including 
transcendental experiences with third-world objects as a primary 
explanation for human behavior (despite its occasional claimants), many 
of these explanations come from two sources:

1)   The basic physical features of ordinary medium-sized objects in 
our perceptual world.
2)   The social pressure to produce new mathematical products - books, 
lectures, theorems, etc.

Best Wishes,

Robbie Lindauer