Bruno Marchal wrote:
> Le 18-oct.-06, à 16:27, 1Z a écrit :
> >> Bruno: In computer science, a
> >> fixed universal machine plays the role of a coordinate system in
> >> geometry. That's all. With Church Thesis, we don't even have to name
> >> the particular universal machine, it could be a universal cellular
> >> automaton (like the game of life), or Python, Robinson Aritmetic,
> >> Matiyasevich Diophantine universal polynomial, Java, ... rational
> >> complex unitary matrices, universal recursive group or ring, billiard
> >> ball, whatever.
> > Peter: Ye-e-es. But if all this is taking place in Platonia, the
> > only thing it *can* be is a number. But *that* number can't
> > be associated with a computaiton by *another* machine,
> > or you get infinite regress.
> I don't think so. Remember that I assume arithmetical realism. This
> means that I take the arithmetical truth as being true independently of
> me. This means not only that the propositions saying that such numbers
> exists or not are true independently of me, but also that propositions
> asserting that such or such relations between numbers exist are true
> independently of me (you, ...).
As usual, the truth of a mathematical existence-claim does not
I have no problem with the idea that algorithms can be identified
with abstract structures consisting of relations. (As opposed, for
to Stathis's identification of algorithms with inteprretations by
That still means algorithms cannot be identified with numbers.
They might as well be regarded as abstract objects in their
own right. The anti-Platonist will not regard such objects as actually
existing, but thgen he will not regard numbers as actually existing
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