[FOM] To Vladimir Sazonov and others doubting the unambiguity of N
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Mon Jun 16 02:15:15 EDT 2003
Lucas Wiman wrote:
> You'll (no doubt) say that I'm "really" working in PA above, though
> this, too, denies mathematical reality. (I know, for example that
> addition is commutative and associative, but I don't know the proof of
> it in PA.) If you explain the proof of Godel's first incompleteness
> theorem to most mathematicians, they'll say something like "yes, but
> then the Godel sentence has to be true." What this tells you is not
> that mathematicians are working in second order number theory (because
> one can obtain a Godel sentence about that, and a mathematician will
> tell you the same thing), but that mathematicians aren't working in a
> formalized system at all.
If this is supposed to be a substantial claim (i.e. not simply the claim
that mathematics are not prima facie working in any formalised theory),
it needs *much* more arguments to support it. It does not follow from
Gödel's theorem that there can't be a formal system F, s.t.
<>(mathematicians prove \phi) <--> F |- \phi (if we don't use the
diamond operator we get a trivial truth, since presumedly human
mathematics will have a finite life span and any finite set of sentences
is recursive).
> If one thinks of logic as the "projection" of
> mathematical arguments, it's interesting to study, but it is objectively
> different from real mathematical arguments. Gauss never needed PA to
> understand arithmetic.
>
> It's interesting to note that Vladimir has had precisely the opposite
> reaction of some notable mathematicians to Godel's theorems. Alain
> Connes (a Fields medalist), for example, said in a recent book that
> because of the ability to see that Godel's statement is true,
> mathematicians must in fact be accessing some kind of reality separate
> from formal systems.
The Fields medalist Connes is wrong, at least as regards to the argument
you attribute to him. As Gödel himself realised, there is simply no way
to proceed from his incompleteness theorems to a refutation of mechanism
(even when restricted to mathematics) without substantial additional
assumptions.
It's precisely because Gödel's incompleteness theorem are
*constructive*, i.e. they provide a mechanistic way to diagonalise the
non-provability predicate (a way of calculating a fixed point, if you
wish), that there can't be any refutation of mechanism forthcoming from
Gödel's theorems. Whatever ingenious ways you have to construct
undecidable truths can be imitated by a machine. Such a machine can even
produce truths it *itself* can't decide to be true, as can humans. So
the Lucasian comptetition between himself and a machine does not have a
winner.
It's very easy to diagonalise a turth-like predicate (the details
needn't bother us here) to produce truths that no humans can ever prove
or know to be true or truths that a particular human beign cannot know.
For example, suppose I ask you whether you will answer this question to
the negative? Whatever answer you will give will be false. Or consider
the statement "Lucas Wiman can never know this sentence to be true". The
sentence is obviously true, but nevertheless you can never know it to be
true. Consider now the function f(x) = "x can never know this sentence
to be true". We've met f(Lucas Wiman), but how about f(Aatu Koskensilta)?
> No formal system can totally capture all the
> ability of mathematicians have to reason about mathematics.
This is a claim that has not been substantiated.
--
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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