[FOM] Infinity and the "Noble Lie"

[William Tait]

One property of lies, noble or otherwise, is that they are  
*falsehoods*; and this presupposes an antecedent notion of truth. One  
can question whether there is an antecedent  notion of truth in  
mathematics,  which serves as a common ground upon which to resolve  
the debate about whether there are infinite sets. It seems to me to  
be very analogous to the debate between constructivists and non- 
constructivists, where, also, one has to ask: On what none question- 
begging grounds could one possibly resolve the issue. (I am also  
reminded of groundless and therefore endless debates over the  
centuries about religion---although the debates in math have been  
less bloody.)

There may be a sense in which one would sometimes say that  the axiom  
is antecedently true: Namely, it is true to a certain conception that  
we have of an ideal structure that the axioms are intended to make  
precise. (This is not really the semantical notion of truth.) But one  
can expect agreement about truth in this sense only if there is  
agreement about what one wants to axiomatize. When what is being  
axiomatized is the theory of transfinite numbers, say, then those who  
reject infinite objects from the outset will not be arguing about the  
truth of the axioms in this sense. Rather they will be rejecting the  
whole enterprise---the very grounds upon which any meaningful  
discussion of truth or falsity can take place.

Bill Tait