Re: Paradoxes and Platonism

On 2008-02-20, in sci.logic, Gc wrote:
I am saying that the status of those paradoxes depend which
philosophical stance you choose. Many of those "paradoxes" consider
axiom of choice, which is actually false by intuitionistic point of
view, which starts by asking the nature of mathematical objects.

I'm not sure what paradoxes you have in mind. The Russell paradox and
other familiar set theoretic paradoxes have nothing to do with the
axiom of choice. Whether choice is intuitionistically justified is a
delicate matter, depending on just one formulates it. In certain
formulations it is an evident intuitionistic truth, in others it
implies excluded middle.

These considerations are not really relevant here, though. In saying
that the paradoxes do not hinge on any specific philosophy of
mathematics, be it Platonistic, fictionalist, formalist, what have
you, I have in mind nothing more than that e.g. the Russell paradox is
just as pressing or irrelevant to the Platonist as to the
fictionalist, say.

What, then, of intuitionism? It is, as you say, a philosophical
stance, concerning the meaning of mathematical statements. But, unlike
Platonism, fictionalism, formalism etc -- which in any case can be
mutatis mutandis used to explain intuitionistic mathematics just as
well as classical mathematics, e.g. in the form that there are
objective facts about the "idealised mathematician" or that the
"idealised mathematician" and his antics are just a fairy tale with
peculiar appeal to the intellect -- it is also more or less a
mathematical conception, in that we clearly explain what the meaning
of mathematical statements is, in such a way that we can with
reasonable clarity decide which mathematical and logical principles
are and are not acceptable. Platonism, formalism, fictionalism and so
on, on the other hand, are entirely silent on such mathematical
questions, and indeed compatible with pretty much any mathematical
principles, such as those of strict finitism or those of higher
flights of abstract set theory.

We may also note that most paradoxes rely on nothing but
intuitionistic logic, and that in this sense they're just as
paradoxical for the intuitionist as for the classical mathematician,
even if, as it may be, their "solutions" are peculiarly
intuitionistic.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

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