FOM: 'constructivism' as 'minimalistic platonism'
V. Sazonov
V.Sazonov at doc.mmu.ac.uk
Mon Jun 5 14:56:49 EDT 2000
Fred Richman wrote:
> Stephen G Simpson wrote:
>
> > From the realist point of view, there is nothing wrong with tertium
> > non datur. If P states something unambiguous about something real,
> > then necessarily P is either so or not so, i.e., we can confidently
> > assert "P or not P", even if we don't know which of the two is the
> > case.
>
> This is an issue that has puzzled me for some time. I take it that the
> last sentence in the quote is some sort of argument that deduces the
> (confident) law of excluded middle from the realist point of view.
> I've heard this stated a number of times before, but it is normally
> just stated, as here. Is there some more convincing argument, with the
> details filled in?
>
> My suspicion is that the law of excluded middle is simply being
> assumed as part of the realist point of view, in which case, of
> course, no derivation is necessary. But why is the law of excluded
> middle necessarily part of a realist point of view? If I believe in
> the objective existence of the natural number series, am I committed
> to the proposition "either there exists an odd perfect number, or all
> perfect numbers are even"?
And what does it mean at all "to believe in the
objective existence of the natural number series"
if it is not just a play with words? By the way,
which one series? The shorter and vague one
ultrafinitistic feasible number series (which seems
is the only one which could pretend to be real) or
a longer (and also vague) one? How long? Should it
be closed only under successor or under primitive
recursive functions or may be also under
epsilon-0-recursive functions or even more closed?
How more?
(Another reading of "to believe in the existence
of the natural number series" [without unnecessary
word "objective"] which I understand consists in:
(i) taking some formal system of arithmetic, say,
PA, (ii) switching on corresponding intuition based
on the underlying [here classical] logic and (iii)
believing after some reflection and experience that
it is formally consistent.)
And also, why such unrealistic way of thinking
to believe in the *objective* existence of evidently
*unrealistic* things is called "realism"?
I think that the law of excluded middle does not
characterize any realist point of view on (classical)
mathematics using this law. There is only a small
analogy with some very simple situations of using
this law in the real world (real in the real sense of
this word - I am sorry, but the reason for such
comments is a very strange philosophical term
"realism").
Vladimir Sazonov
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