[FOM] Infinity and the "Noble Lie"
A. Mani
a_mani_sc_gs at yahoo.co.in
Fri Dec 9 18:13:26 EST 2005
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
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