[FOM] Permanent value revisited

[Neil Tennant]

On Mon, 17 May 2004, Robbie Lindauer wrote:

> ... the commitment to logicism (when it 
> arises) might be some kind of will to absolute knowledge explicable 
> primarily by religious psychology or something.

Should logicists, then, seek out an expert in "religious psychology"
(whatever that is) in order to find out what makes logicists tick?

> > ... what would those "rigorous proofs" begin with? Non-axioms?
> 
> How about transcendental arguments?  How about linguistic or 
> psychological arguments?  How about physical arguments?  There are lots 
> of possibilities for arguments to the truth or falsity of the axioms.  

You are now substituting "argument" for "rigorous proof". That is not a 
permissible move in the current context. (And indeed, some philosophers
would still insist that any good *argument*---let alone rigorous
proof---must appeal to certain principles which, in the context of the
argument, receive no further justification. These principles would then
be "axiomatic" in the context.)

> In the end it seems to me we have two possibilities:
> 
> 	A) Either there are sound proofs establishing the truth of every axiom 
> from which we derive our proofs.
> 		OR
> 	B)  There is no adequate accounting of our acceptance of those 
> theorems which depend on unproved axioms for their proofs.

(A) carries the danger of infinite regress. For what about
justifications for the axioms used in one's proofs? The regress can be
stopped only by eventually acquiescing in the truth of the ultimate axioms
one has chosen. One could add further commentary about their
self-evidence, or analyticity, or elegance, or systematicity,
or usefulness; but there would be no further possibility of "rigorous
proof" of those axioms. Even those---like logicists---who seek "deeper"
justifications for the conventional axioms of branches of mathematics such
as arithmetic, are fully aware that they will still be resting their
"deeper" derivations of the *old* axioms from *new* ones (such as
higher-order abstractive principles).

As for (B), the further commentary just mentioned could very well provide
an "adequate accounting of our acceptance" of the axioms, and therefore of
the theorems derivable from them. But the further commentary will not
constitute a "rigorous proof" of those axioms, on pain of having to *be* a
proof, in which, of course, yet "deeper" axioms will be
identifiable---contrary to our assumption that the regress of rigorous
justification has been terminated.

> > Cantor's proof does not require the powerset axiom. Please see
> >
> > http://www.cs.nyu.edu/pipermail/fom/2003-February/006237.html
> 
> What does the term "every subset Y of X" refer to except the powerset 
> of X?  What guarantees the existence, in every case, of the set of 
> things "every subset Y of X"?  I know there was recently on FOM a 
> thread about "what do we get if we lose powerset?"   There's no need to 
> rehash the specific criticisms here, that was not the point.

No, it is very much to the point. The point is that one ought to
understand the logical structure of Cantor's proof, before adducing it in 
support of a philosophical or epistemological claim. In reply to your
rhetorical question

> What does the term "every subset Y of X" refer to except the powerset  
> of X?

I obviously need to reiterate that the use of the quantifying phrase
"every subset Y of X" does NOT commit one to the existence of the set of
all subsets of X. In similar vein, an intuitionist (say) could make
sensible use of the quantifying phrase "every natural number" without
being committed to the existence of the set of all natural numbers (as a
completed totality). 

Cantor's proof, construed in this minimalist fashion, makes one realize
that IF one postulates that the set of all natural numbers exists AND
postulates that every set has a powerset, THEN one is committed to an
infinite sequence of infinite sets none of which stands in 1-1
correspondence with any of its predecessors.

Neil Tennant