FOM: mathematical certainty
pollack at dcs.gla.ac.uk
Fri Nov 20 11:09:39 EST 1998
>From Anatoly Vorobey's recent reply to Shipman:
> But our trust in that [that the computer "works right"] isn't at all
> like our trust in a mathematical truth, and is more like our trust
> in that the sun will rise tomorrow. It has more to do with
> philosophical induction than with whatever we percieve as
> mathematical certainty.
There clearly _is_ a difference; but what is the difference? What is
this *mathematical certainty*?
Whether Vorobey is talking crass platonism here (the natural numbers
in the sky) or something else (formal system ....) the question is not
just whether some sentence is (mathematically) true, but how we know
it is true.
Of course, it helps a lot if (for example) Friedman or Simpson tell me
a sentence is mathematically true, but I guess if I were to watch
Friedman or Simpson long enough I'd see them make a mistake. Even
without watching them very long, I'd probably see that they are
physical mechanisms, so prone to occasional erratic behavior for all
kinds of reasons.
Maybe everyone on the FOM list agrees that the hypothetical sentence
is mathematically true; that would be very convincing. But I might
run the four-color theorem program on as many different computers as
there are FOM readers. I believe there are now several independent
four-color theorem programs; and I can run them on different models of
computer with different processors and different operating systems,
.... I guess this will still not convince many FOMers that the
four-color theorem is mathematically certain.
Further, I suppose many FOM readers feel Fermat's Last Theorem is
mathematically certain, while the four-color theorem is not, although
most of us have not directly checked either proof.
So what is the meaning of *mathematical certainty*?
Randy Pollack <http://www.dcs.gla.ac.uk/~pollack/>
Computing Science Dept. <pollack at dcs.gla.ac.uk>
University of Glasgow, G12 8QQ, SCOTLAND Tel: +44 141 330-6055
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