[FOM] Infinity and the "Noble Lie"

[William Tait]

On Dec 11, 2005, at 11:09 AM, Joe Shipman wrote:
> 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. ...
>
> My reply to Tait:
>
> My point is that there does appear to be an antecedent notion of truth
> in mathematics as far as "ordinary mathematics" is concerned --  for
> example, there is a fair consensus that the Riemann Hypothesis is
> either true or false but we don't know which. Those who deny that an
> antecedent notion of truth exists which settles the Axiom of Infinity
> had better explain what to make of statements which do not mention
> infinite sets but all known proofs of which require the axiom of
> infinity. (For example, Kruskal's theorem in Friedman's finite  
> form, or
> various other specializations of the Graph Minor theorem.)

As to the first point, it seems to assume that the notion of  
mathematical truth (as opposed to that of truth in a model)  is  
employed *within* "ordinary mathematics". it surely is used  
informally, but, I would argue, not essentially.To use your example,  
the statement that RH is true is no more than the statement that RH:  
certainly the two would be proved in the same way. So the assertion  
that RH is true or false is nothing more than the law of excluded  
middle, RH v  not-RH, which indeed is assumed in ordinary  
mathematics. For example, if we can prove P both from RH and from not- 
RH, we would conclude that P.

As for the example of theorems not referring to infinite objects  
requiring the introduction of infinite objects for their proof, I  
take it as a very strong motivation for accepting infinite objects.

I'm curious about how you feel about another case: The proof of  
projective determinacy (which certainly is formulated without  
reference to large cardinals) can be proved only using the hypothesis  
of infinitely many Wooden cardinals. Does this fact demonstrate the  
truth of the hypothesis? This is meant as a genuine question, not a  
challenge. If one thinks the cases are different, where is the line  
to be drawn?

John Corcoran wrote on Dec 11, fairly clearly in response to my  
statement that one property of a lie is that it is  false:

> Some of our esteemed colleagues whose mathematics is impeccable do not
> seem as well-versed in speech ethics. They might have misunderstood  
> the
> nature of the lie. A lie is a speech-act, not merely a sentence or a
> proposition. A lie is a statement of a proposition that is not a  
> belief
> of the speaker.

You are right, John. But I wasn't lying, since I believed it when I  
wrote it.  I should have restricted myself to noble lies:  It surely  
wouldn't be noble to tell the truth by accident.

Bill Tait