[FOM] infinity and the noble lie
Mark Lance
lancem at georgetown.edu
Mon Jan 9 12:39:10 EST 2006
I'm finding this whole discussion confusing, and I think that is not
entirely my fault. In particular, I think there are many formal and
nonformal issues which are much more subtle than this discussion is
letting on. Consider the following from Joe Shipman:
"A theorem cannot be MORE certain than the axioms it is derived
from. Therefore, if you won't call the set-theoretical axiom of
Infinity "true", you had better explain whether you are willing to
call the Paris-Harrington Theorem "true". If you are so willing, you
should be able to identify "true" axioms it can be proved from."
Well of course a theorem can be more certain than the axioms it is
derived from. Let P&Q be an axiom and P a theorem. What is
generally true is that a theorem can't be LESS certain than the
axioms it is derived from. So what is going on here? If it was all
turned around in the way suggested by the generally true claim above,
I would understand the discussion. That is, if Prof. Shipman said
that some theorem someone thinks true entails the axiom of infinity
-- or entails it given some context -- then it would follow that they
had to think the axiom of infinity true also. That's obvious.
(Actually there are all sorts of fussy worries about even that, but I
don't think they are relevant in a mathematical context.) But that
doesn't seem to be what is bothering Prof. Shipman and others.
So maybe what is meant is something about the axiom being the only
way to derive the theorem. But that can't literally be what's at
stake, because one could just have the theorem itself as an axiom.
So the claim must be that there is a problem if the only non-ad hoc
axiom -- natural axiom? general, natural, non ad-hoc axiom? -- that
allows us to prove P is the axiom of intinity? There are two
points: first, I don't know what possibly true thing is being
claimed, and second, if it does go in something like this way, then
nothing is anywhere near as straightforward as many in this thread
are letting on, simply because nothing having to do with ad-hocness
and the like is straightforward. Third, even if something like this
is right, I don't see why the only non-ad-hoc axiom-like principle
that allows us to prove P couldn't be a good deal less certain than
P, simply because it will be more general. Finally, there seems to
be a general background assumption that our epistemic confidence in a
mathematical claim -- or is it our philosophical view that the claim
is either objectively true or objectively false, something that is
quite different? -- can only be based on our confidence in a set of
axioms that are used to to prove it. That is hardly uncontroversial.
Another point is with the rather dogmatic pronouncements about truth
that various folks are making. COnsider this from Prof. Mycielski:
"we cannot talk honestly about the truth (in the usual sense of the
word true) of any statement unless this statement refers in a clear
way to a real object or process"
Aside from not knowing what "real object or process" means here, and
leaving aside the colorful tactic of putting the point in terms of
"honesty," I think that "the ordinary sense of the word true" is
given by the anaphoric theory of truth-talk. And on this, asserting
that P is true is nothing beyond asserting that P. Many in this
thread seem to me to be assuming some sort of correspondence notion,
or even some sort of empirical verificationist notion of truth. The
point isn't that these theories are false -- I think they are, but
that's not the point -- but that people are writing in as if they are
JUST OBVIOUS, or even that others are dishonest if they don't believe
them.
That contributes to my not understanding the debate.
Mark Lance
Philosophy
Georgetown University
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