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On 23 Apr 2006 01:04:53 -0700, Starbles@xxxxxxxxxxxxx wrote:
>What is math's relation to the real world? Is math some higher ideal
>that exists on a different plane from reality, such as in platonism, or
>is there some fundamental connection between mathematics and reality?
>
For me, mathematics - as I know it ! - is not happening in some
platonic higher world, because this would mean that truth in math.
would depend on the contents of that world - which we humans
might not be able to know for sure (this is especially important
because of controversial math. statements like the axiom of choice
and the [generalized] continuum hypothesis). But it remains clear to
me that in some sense math. truths are not invented, but discovered.
Mainly because we may consider many axioms as implicit definitions
- even in logic and set theory. But the controversial statements
make problems here ... I see that mathematicians have preferences:
normally the axiom of choice is accepted (it is very 'intuitively
true' - so much that I suspect it is often used without realizing -
and apparently Cohen proved its consistency under the hyp.
that ZF without it is consistent), but continuum hypothesis is for
the moment a statement left open (i.e. mathematicians don't
assume it and its opposite isn't assumed either) despite what
some (a small minority) say.
For this I hold with Cohen himself (who has shown that CH is
independent from the axioms of ZF+Choice), who wrote that
may-be one day we'll find some new 'obvious' axiom, which
added to ZF+Ch. enables to prove or disprove CH - adding
that his feeling is that if this happens, it is more probable that
the latter will be the case (CH false). This corresponds to my idea
that mathematicians will always prefer a more 'inclusive'
math. 'world' - this being the main reason for accepting (for
instance) following axioms: axiom of infinity, axiom of the power set
of a set. In the past, this led to the acceptance of negative,
of irrational, of 'imaginary' (complex non-'real') numbers ...
To say things shortly, the 'math. world' we want is nothing
depending on physical reality, it resembles somehow to
a platonic world of ideas, but in a more subtle way than
usually understood. Because we want the freedom of
collective decision over it, based on (advanced) intuition ...
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