[FOM] Infinity and the "Noble Lie"

[Timothy Y. Chow]

Joe Shipman wrote:
> My concern is with those who insist that key statements about infinite
> sets are "meaningless" or "fictitious" and refuse to grant axioms like
> the Axiom of Infinity the status of "True", while continuing to conduct
> or refer to ordinary mathematical research which depends in an
> essential way on such an axiom.

Most respondents seem to have slid into familiar grooves of argumentation 
and have missed the point of Joe Shipman's question, which is about 
ethics, not about mathematical truth.

To clarify the question further, perhaps it will help to push the 
religious analogy.  Suppose I am the pastor of a church, and suppose I 
started down this professional path with naive enthusiasm, but in the 
course of studying higher criticism of the Bible, skeptics' writings, 
comparative religion, etc., have come to believe that the basic statements 
of faith of my church are false when taken at face value (which is how 90% 
of my congratation takes them).  On the other hand, I haven't totally lost 
my faith, and I am comfortable with the religious language of my church, 
*provided* that that language is "suitably reinterpreted."  If I use 
conventional religious language in the pulpit, and operate among my 
congregation in a conventional manner, without ever divulging my secret 
reinterpretations, am I a hypocrite?

Whether you answer yes or no to this question, you probably *do* perceive 
that there *is* an ethical question here.  I certainly do.  So if the 
analogy holds, then my answer to Joe Shipman's original question as stated 
("Does anyone perceive an ethical issue here?") is yes.

The analogy might not be quite convincing, so maybe not everyone will 
answer yes, but I expect most people will be willing to grant a "yes" 
answer, if only for the sake of argument.  The more interesting question 
then becomes, given that there *exists* an ethical issue, what is the 
ethically correct path?

At this point let me drop the religious analogy and speak just to the 
mathematical case.  I think the answer depends on just how radical your 
disagreement with the "establishment view" is.  If you believe that their 
activity is pernicious, then certainly you should have the courage not to 
aid and abet it.  But if you believe that their activity is, on the whole, 
good and valuable, and the only problem is that they have a somewhat 
misguided philosophical view of their own activity, then I see nothing 
wrong with participating in and promoting that activity.  Even adopting 
their misguided language so as to keep things running smoothly without 
getting bogged down in interminable and unresolvable philosophical debates 
seems acceptable to me up to a point, provided you don't mislead people as 
to your true beliefs and provided you don't, for the sake of being a 
likable guy, cross the line into activity that your philosophy deems bad.

Tim