[FOM] Infinity and the "Noble Lie"

[A. Mani]

On Thursday 08 December 2005 23:26, joeshipman at aol.com wrote:
> Mani says axioms are not "lies" -- but my point is, are they "true"?
My point was that axioms may or may not be called 'admissible'. I did not say 
anything about the truth. Mathematically, I think the axiom of infinity is 
perfectly true. The concept of truth of axioms must be seen in a hierarchy 
defined by the abstraction levels associated. Interpreting mathematics in a 
particular 'semantic domain' involves many assumptions on the mechanism of 
interpretation. 

Actually mathematics starts from reality and from the truth corresspondences 
therein we eventually abstract these axioms. So axioms must be true by 
definition. Particular abstractions may be favoured over others. It is also 
not necessary to find a fault with a version to investigate another.
'Absolute truth of ZFC axioms' is all right for mathematicians not working on 
the foundations. 


>
> If you are unwilling to say the Axiom of Infinity is "True", does that
> mean you would have been unwilling to say that the Prime Number theorem
> was "True" in 1945 when only analytical proofs were known, but willing
> to say that the Prime Number Theorem was "True" in 1950 after the work
> of Erdos and Selberg?
>
> You can't get away with being indifferent to whether an axiom is "True"
> unless you are willing to be just as indifferent to whether any therems
> proved using that axiom are "True". Where I see a potential "ethical
> issue" is when mathematicians talk quite freely about whether
> statements like the prime number theorem are "true" but are unwilling
> to grant the same status to axioms used to prove those theorems.
>
Generally axioms are taken to be true by definition.
There are people who do mathematics admitting contradictory statements 
(inconsistent mathematics). But even they do not dare that... they write 
their models in the usual sense only.  

The ethical issue can and does arise for some 'scientists', who will not 
accept the material basis of reality and then truth has to arise arbitrarily 
and sometimes conditionally.  


A. Mani
Member, Cal. Math. Soc