FOM: Goedel: truth and misinterpretations
torkel at sm.luth.se
Fri Oct 27 03:19:39 EDT 2000
Vladimir Sazonov says:
>Yes, I explained this in terms of their illusions.
>There is (practically) nothing dangerous in this way
>of thinking of working mathematicians because all they
>do is under a strongest control of formal systems in
>which they are working.
They don't usually work in any formal systems at all. We learn, explain,
talk about and apply mathematical concepts informally in innumerable
contexts. I'm sure I haven't grasped what you regard as the proper way
of thinking about mathematics, but it seems to me extremely esoteric.
It's not clear what relevance it has to mathematics as it actually
exists and as it is pursued (or can be pursued) by human beings.
>But returning again to yours
>> Why, in the ordinary mathematical sense, of course.
>I recall that this was said concerning
> >from some
> >philosophical point of view ...
"In the ordinary mathematical sense, of course" was in response to
your parenthetical "(IN WHICH SENSE, PLEASE?)".
The "philosophical point of view" of the observation that Goldbach's
conjecture, even if true, need not be provable in PA, is that of
ordinary informal mathematics. This point of view has indeed been
criticized by philosophers (I think most forcefully by Wittgenstein)
as resting on an illusion or misconception ("false picture") of
arithmetic as dealing with arithmetical facts analogous to physical
facts ("the mineralogy of numbers", in Wittgenstein's phrase).
As in other similar cases (such as Hume's criticism of everyday
assumptions) it is a weakness of the criticism that it doesn't really
offer any workable alternative to our ordinary ways of thinking.
It's more in the nature of a general metaphysical conviction, as
when people are convinced that "in principle" everything that humans
do must be explainable in terms of chemical and physical processes.
Torkel Franzen, Luleå university
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