🚀 go-pugleaf

[Some of you might have seen a forgery come by recently. Onwards.]

In article <10034@mimsy.UUCP>, flink@mimsy (Paul V Torek) writes:
>Matthew P. Wiener writes:
>w>This could only show one particular infinite regression (or family of
>w>such) was invalid.
>
>True, but I think it establishes a burden of argument on someone who claims
>that some other type of infinite regression is justifying.

I'll agree that it establishes the burden on someone who claims that
it's "intuitive".  I've seen enough strange proofs in my time that I
don't think "counterintuitive" has much meaning left in me, unless I
kick myself, or get sucked in by somebody's mathematical propaganda.

Now why should I think there's some plausibility behind any infinite re-
gression arguments?  Well, thinly disguised versions of these are ban-
died about quite freely in modern set theory--or at least at one time
they used to be--to "justify" large cardinal axioms (the higher infin-
ities).  This activity makes no sense to formalists, but to a Platonist
like myself it seems a necessary phase one goes through.

And not just an infinite regression, but transfinite regressions.  I've
seen people say things to the effect that, "0# is harmless, and why not,
0## will be too (which justifies 0# by the by), and hey, so will 0###,
(which justifies 0##) and so on by transfinite induction of thin-air
assumptions.  But this Pollock-like reasoning, while plausible--to
some people at least--is generally considered too suspicious or em-
barrassing even, so one rethinks the issue and discovers a single
higher principle that covers all the 0##...##..##...s.

The details of this example don't matter, my point is that it HAS been
a form of reasoning that has been taken seriously by a good number of
mathematicians.  Whether the future will come to a consensus is unknown,
but I am reminded of the groping infinitary arguments that went on to
"justify" the axiom of choice at the turn of the century, contrasted
with an almost universal acceptance of its "intuitiveness" today.  (And
the intuitive introductions one sees today, of course, are still as in-
finitary as ever.  The explicators are just not abashed anymore, or
even worried that anyone might object.)

>	        Anyway, the self-reference of your sentence above doesn't
>require infinite regress to verify, so it seems importantly different
>from, say, the English sentence "this sentence is true".  The latter
>sentence *does* require infinite regress to verify, and it seems
>objectionable (to me at least) for that reason.

Ah, but the mathematical sentence "this sentence is provable" does not
require an infinite regress to verify!  That's the surprising point of
Loeb's theorem.

>w>[The Pollock example] reminds me of the infinite regression inside Lewis
>w>Carroll's Achilles and the tortoise story
>w>....
>w>And so, by your reasoning, we should reject modus ponens.
>
>But wait, there's less!  Since the first set of premises is *transparently*
>implied by the Nth, anyone who accepts the Nth ought to accept the
>first, without need to deduce that set of premises from anything else.

But the tortoise snickers, "Uh uh uh, Mr T, you've just made another one
of those darned assumptions! Better put it down in your little notebook!"

>Hence, there's no need for the regress.

In formal logic, this is correct, since one is studying certain mathem-
atical sequences, and the very first step of the regression is blocked:
one's reasoning here is not one of those sequences.  It is translatable
into one of those sequences; that is how the Goedelian and Loebian proofs
work.

But as for ordinary language, ordinary beliefs?  I don't think anyone can
claim to know.

>					  By contrast, in the Pollock example,
>at any finite stage of the argument, there could be a point to appealing to
>the next set of premises.  E.g., someone might actually be convinced that q1
>and that q1->p, if it is suggested to him that q2 and q2->q1 and q2->(q1->p).

Right.  And in the Carroll example, there's also a point--albeit a twisted
point--in appealing to the higher premise!  Someone might not yet be con-
vinced that q, yet be convinced of p and p->q, because he has intuitionis-
tic style doubts about modus ponens.  And one can then up argument this ad
infinitum.

>One can consistently maintain that modus ponens is legitimate (i.e., that
>q is justified by p & p->q, provided that p is already justified and so
>is p->q) while maintaining that the above infinite regress fails to
>justify.

If I read your references correctly, you are saying one can be convinced
that the Carroll example is an unnecessary attempt to appeal to the re-
gression while the Pollock example isn't.  I agree: a pure formalist is
just such a person.

ucbvax!brahms!weemba	Matthew P Wiener/Brahms Gang/Berkeley CA 94720
A mathematician's wife overhears her husband muttering the name 'Nancy'.
She wonders whether Nancy, the thing to which her husband referred, is
a woman or a Lie group.			--Saul Kripke