[FOM] A Defence of Set Theory as Foundations
Robert Lindauer
rlindauer at gmail.com
Thu Oct 13 01:25:24 EDT 2005
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
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