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From: Nam Nguyen on 24 Sep 2009 10:57
Frederick Williams wrote:
> Newberry wrote:
>
>> Jargon and group think will not help us to solve the outstanding
>> problems of the foundations of mathematics.
>
> Those problems being what?
>

For one example, being our failure to recognize that the encoding of
Godel sentence G could be an absolute undecidable, (in the sense of
being independent in any extension of Q), as GC could be.
From: Aatu Koskensilta on 24 Sep 2009 11:00
Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> The pitfall in self- learning to rely on one's own knowledge, when it
> turns out to be bad, is as equally grave as when relying on,
> respectfully speaking, a professional opinion in the field. In
> mathematics and reasoning, no one is above the possibility of being
> (inadvertently) wrong, especially when it comes to the issues of
> foundation.

I didn't say anything about the danger of being wrong. The danger in
being an autodidact I mentioned is that without contact with people
working in a given field it is difficult to develop a sense of what sort
of considerations, arguments, reflections, problems, notions, techniques
are considered significant or of interest, of what the whole thing is
about. This is not a matter of accepting this and that, or of being
right or wrong, but of awareness and understanding.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 24 Sep 2009 11:02
Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> For one example, being our failure to recognize that the encoding of
> Godel sentence G could be an absolute undecidable, (in the sense of
> being independent in any extension of Q), as GC could be.

There is no sentence undecidable in all (axiomatisable) extensions of
Robinson arithmetic. It's obscure what you have in mind.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 24 Sep 2009 11:39
Aatu Koskensilta wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> For one example, being our failure to recognize that the encoding of
>> Godel sentence G could be an absolute undecidable, (in the sense of
>> being independent in any extension of Q), as GC could be.
>
> There is no sentence undecidable in all (axiomatisable) extensions of
> Robinson arithmetic. It's obscure what you have in mind.
>

My wording for the sense of "absolute undecidable" was bad. What I had
in mind is that _assuming_ the consistency of an extension of Q in question,
it could be *impossible to decide* (hence a sense of "absolute undecidable")
which one - the formula or its negation - would syntactically contradict
the assumed consistency.

For instance, T = PA + GC would prove GC for sure but in that case it
could be impossible to know if we still could assume T be consistent.
From: Nam Nguyen on 24 Sep 2009 11:49
Aatu Koskensilta wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> The pitfall in self- learning to rely on one's own knowledge, when it
>> turns out to be bad, is as equally grave as when relying on,
>> respectfully speaking, a professional opinion in the field. In
>> mathematics and reasoning, no one is above the possibility of being
>> (inadvertently) wrong, especially when it comes to the issues of
>> foundation.
>
> I didn't say anything about the danger of being wrong. The danger in
> being an autodidact I mentioned is that without contact with people
> working in a given field it is difficult to develop a sense of what sort
> of considerations, arguments, reflections, problems, notions, techniques
> are considered significant or of interest, of what the whole thing is
> about. This is not a matter of accepting this and that, or of being
> right or wrong, but of awareness and understanding.
>

In these days self-learning (say as an autodidact) it's virtually guaranteed
that one would mostly learn _from others_: from "considerations, arguments,
reflections, ...notions, techniques,..." of others who might have already
presented, wrote what they - people in the field - worked, opined, etc...

Self-learning is like a detective work. We'd *use* others' theories and
field collected data. But in rendering logical arguments/assertions of
the case, we got to be mindful to the fact professional theories and
data "interpretations" could be quite wrong or illogical.
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