Re: all the incompleteness proofs are worthless untill...

On 2008-03-21, in sci.logic, Marshall wrote:
Certainly there is a style of interaction where the senior
person does nothing more than to negate what the junior
person is doing or thinking.

Man: No it isn't, an argument is an intellectual process...
contradiction is just the automatic gainsaying of
anything the other person says.

Mr Vibrating: No it isn't.

If I come off as merely negating everything my interlocutors, or you
in particular, think or do then my attempts at communication have
failed. Again, this is entirely possible.

It is not enough to say simply "not like that." There needs also to
be something that says "more like this."

Quite so. But in a Zen-like fashion sometimes the "more like this"
consists in precisely the realisation that something perceived as
problematic is not a problem at all, that this or that doubt or worry
is not grounded in anything in the subject matter but is just an
expression of empty abstract skepticism.

For example, it is a recurring idea, an idea that naturally suggests
itself to students of logic, that mathematics must be understood in
terms of formal theories, and that mathematical results about formal
theories and related stuff take place on some special "foundational
plane". They then are, naturally enough, puzzled about how we can use
set theoretic language and principles when studying formal theories of
sets and so on. The solution is not any elaborate philosophical system
explaining this apparent circularity but the disappointing observation
that mathematical logic does not provide "foundations" in any
epistemologically relevant sense but is just a branch of mathematics
itself, and that if we wish to draw conclusions about our mathematical
reasoning and practice we must carefully examine just how well a
mathematical model, say in terms of formal theories, captures the
relevant aspects of this reasoning and practice, the same way we must
use our good sense to judge whether a given mathematical model is
applicable to this or that physical situation. That is, what seems
puzzling at first is almost entirely due to philosophical assumptions
people almost unwittingly make. Getting them to unmake such
assumptions is what pithy questions such as I'm wont to present are
designed to achieve. As noted, whether I succeed or not is another
matter.

But I feel that if we just leave it at that, we have left something
important out. It seems to me that getting to the "point of boredom"
as it were, is much the same thing as getting to where there is a
formal system. So even if most of what happens is the journey, the
journey is still in some very important sense *about* the
destination. This is true even if we liken the steps of our journey
to the natural numbers, such that there is no destination.

Well, I would say for example that a sober and clear understanding of
the epistemological significance of the incompleteness theorems does
not at all consist in any formal theory. Rather, it is an informal
understanding of how results about formal theories incorporating
formalisations of all kinds of mathematical principles relates to
these and those aspects of our actual mathematical reasoning and
practice. Of the sort of "informal rigour" that goes into such an
understanding I've written on in my post on formalisation, archived at

http://groups.google.com/group/sci.logic/msg/1cf3026be617d644

which you might find of some interest.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

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

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