[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  

That contributes to my not understanding the debate.

Mark Lance
Georgetown University

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