Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?

tchow_at_lsa.umich.edu
Date: 01/29/05 

Date: 29 Jan 2005 19:04:33 GMT

In article <ctgl6b$bck$1@wisteria.csv.warwick.ac.uk>,
 <mareg@mimosa.csv.warwick.ac.uk> wrote:
>Hmmm! Well I think of myself as a platonist, but I don't consider that
>it is meaningful to say that the axiom of choice, which is a meaningful
>statement about sets, is either true or false in any absolute sense.
>So perhaps I am being inconsistent?

I did say that a platonist was "likely" to believe in bivalence. I
don't think it's necessarily inconsistent to reject bivalence and
embrace platonism. Platonism is what I would call "mathematical realism"
and there many variations of realism.

>I find it strange that there are
>some mathematicians who do claim to believe that ACC is true or false,
>although they do not generally expect ever to find out which!

Well, "there is an earthlike planet in a distant galaxy with intelligent
life" is surely either true or false (after some of the vaguer words in
that sentence are made more precise), but many people do not expect to
find out which.

I think what makes the mathematical situation seem strange to many people
is that they find it hard to envision under what circumstances, even ideal
circumstances, they would come to believe that (say) the continuum
hypothesis is true. At least in the case of the distant planet, or even
the consistency of various axiomatic systems, they can envisage an idealized
situation under which they would have enough evidence to embrace the claim.
But somehow the "ideal situation" in which one has transfinite powers that
allow searching through *all* bijections for a counterexample to the
continuum hypothesis seems *too* idealized for comfort. 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences

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