[FOM] A Defence of Set Theory as Foundations

[Robert Lindauer]

On Oct 17, 2005, at 11:43 AM, Roger Bishop Jones wrote:

> 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).

I guess here I'd like to give you a hard choice.

Either:

a)  Set theoretical statements are true, and therefore true about 
something and therefore, they carry some metaphysical claim (as I take 
y our "exist-in" language to be an extension of the more simple 
"exists" and hence have the "metaphysical import" about which you seem 
to be vague).

OR

b)  Set theoretical statements are not true, and therefore not true 
about anything, and therefore carry no metaphysical import (which is 
the other way of taking your "exists-in" predicate, like "exists in the 
Alice in Wonderland story", etc.)

The view I'm rejecting as absurd is probably the one you'd prefer - 
that the statements are true, but not true about anything.  I take this 
to be a straightforward example of nonsense on the order of "This 
sentence is does not mention my hand."

> The truth or falsity of the metaphysical claim is
> immaterial to the mathematical discipline of set theory (though
> its consistency is not).

I don't agree with this statement as a statement about mathematics -in 
general-.  In general, it is important that our mathematical statements 
are true - planes crash if our numbers are off, etc.  On the other 
hand, it is not important that ALL of our statements that appear 
mathematical (e.g. have the social aspects of mathematics without the 
-important and useful- factors) be true in the same sense that not all 
of our musings about possible interpretations of apocalyptic literature 
need be "true" in any serious sense.  Perhaps it's enough that we're 
directing our attention that way for the purpose, whatever it may be.

On the other hand, if someone is in fact interested in TRUTH for its 
own sake - e.g. in finding all and only true sentences - then the 
question of the literal truth of mathematical and therefore 
set-theoretical statements is the only relevant question.  This 
wouldn't -prevent- anyone from studying set theory as though it -could- 
be true absent a proof that it couldn't be true.  But it would make any 
such study a kind of futile game until the question of it's truth was 
at least put forward.  Someone surely should come up with an x for "if 
set theory is true, then x exists."  If it is the semantic-word-play 
thing that I imagine, then language exists, but set theory isn't true 
in any serious sense.  If it is the platonic other-worldly version then 
set-theory is (possibly) true but inaccessible to our sublunar 
intelligences.

>> 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.

Do they have a street address so we may go visit them and examine them 
ourselves?

To be explicit.  Let's imagine two people discussing the relative 
merits of PRA and ZFC.  One says "look, PRA is equivalent to ZFC for 
the integral cases."  The other says "but ZFC adds on all these 
transcendental sets."  A third walks up and says "What would decide the 
matter between you?"  Since none can point at the thing they're talking 
about except to attempt to express some ideas, I suggest that they 
really must be (psychological) ideas or at best Platonic Ideals since 
no one can devise a decisive test that will settle the matter.  I think 
this -in principle- because the rules of engagement on both sides of 
the argument are entrenched all the way back to what they think 
fundamental LAWS OF THOUGHT are.  Once one disagrees about how words 
must refer in order to be true, there is no longer room for rational 
discourse.

>> 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.


I regard mathematics as interesting because it is an essential part of 
a systematic knowledge of the whole of the world - without being able 
to say "there are two elephants in the cage" we wouldn't be able to 
understand meiosis.  Foundational mathematical questions like "what is 
the number two" as the obvious extension of the systematic knowledge of 
the world.  If we say there are two of something, we should know what 
is being said of those things - what makes them two.  My hope for 
"foundational mathematics" is that it will cohere nicely with my 
(admittedly naive) desire to understand what is asserted when it is 
said there are two of something along with my (again admittedly naive) 
desire to understand why 2 - 2 = 0 with something better than "because 
that's the way we subtract".  If nothing more is available than "those 
are the rules of the game" - I'd like to know that.  If something more 
is available, I'd like to know that too.

So I guess I have different questions in mind for "foundations" than 
you're interested in.  That can't be regarded as surprising.

Aloha,


Robbie Lindauer