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From: Aatu Koskensilta on 7 Jul 2010 18:37
"|-|ercules" <radgray123(a)yahoo.com> writes:

> A counter example?

Take the Heine-Borel theorem. It is not in any apparent sense "about
words and their definitions" nor can it be with any accuracy described
as "study of formal language and what can be said using formal
language". The study of formal languages and their expressiveness is a
very marginal part of mathematics.

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

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Curt Welch on 7 Jul 2010 19:55
Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote:
> "|-|ercules" <radgray123(a)yahoo.com> writes:
>
> > A counter example?
>
> Take the Heine-Borel theorem. It is not in any apparent sense "about
> words and their definitions" nor can it be with any accuracy described
> as "study of formal language and what can be said using formal
> language". The study of formal languages and their expressiveness is a
> very marginal part of mathematics.

Well, the term "formal language" might be something very specific to you.
I was using it very loosely just to mean a langauge made up of generally
absolute concepts and definitions, instead of our normal natural language
where almost nothing is defined absolutely. That is, a language full of
absolute truths. So maybe you didn't agree with my position becuase your
view of what an "formal language" was didn't match my idea. Or maybe you
just don't see my point at all.

I've never heard of the Heine-Borel theorem and reading the Wikipedia entry
doesn't help me understand what it's really about, however, let me just
point out something obvious (to me).

Here is the beginning of the Wikipedia entry:

In the topology of metric spaces the Heine�Borel theorem, named after
Eduard Heine and �mile Borel, states:

For a subset S of Euclidean space Rn, the following two statements are
equivalent:

* S is closed and bounded
* every open cover of S has a finite subcover, that is, S is compact.

Those are all words. They make reference to other words. The "Heine�Borel
theorem" (like all of mathematics) has no existence other than in the
words that define it (and in the machine hardware that is able to
correctly produce those words - aka humans).

Even when the words have a loose connection to physical reality though
concepts such as space and topology (aka arrangement of objects in space)
they normally fail to be accurate descriptions of reality in that they
assume the existence of absolute truth which as far as I can tell, doesn't
exist in our universe.

The concept of absolute truth is a core foundation of the langauge of
mathematics, but yet, it seems to be lie. That is, it's something that
doesn't exist in the universe. It's highly useful to us to PRETEND that it
does, so it's a very valuable and useful lie, but it's a lie none the less.
There are no absolute facts about the universe - only facts that have such
a high probability of being true, that there is no _practical_ point in
trying to deal with the probability of it being wrong. It's far easier,
from a practical sense, to just assume they are absolute facts.

Mathematics is all about understanding what we can define and do, with a
language of absolute facts, even though such absolutes don't exist anywhere
except as concepts in that language. As such, it's a study of what we can
do with the _language_ and not a study of what we do with physical stuff.
It's highly useful, simply becuase it does allow us to make highly accurate
predictions about some aspects of the future, by pretending something is
absolute, even when it's not.

--
Curt Welch http://CurtWelch.Com/
curt(a)kcwc.com http://NewsReader.Com/
From: |-|ercules on 7 Jul 2010 20:26
"Aatu Koskensilta" <aatu.koskensilta(a)uta.fi> wrote ...
> "|-|ercules" <radgray123(a)yahoo.com> writes:
>
>> A counter example?
>
> Take the Heine-Borel theorem. It is not in any apparent sense "about
> words and their definitions" nor can it be with any accuracy described
> as "study of formal language and what can be said using formal
> language". The study of formal languages and their expressiveness is a
> very marginal part of mathematics.
>

I can only decipher your 'argument' as having 2 forms.

1/ Heine-Borel theorem IS mathematics.
2/ Heine-Borel theorem is not the output of any formal system.

(1) is what you literally asserted given the question, but obviously nonsense.
(2) contradicts your famous informal theorem that all informal proofs can be formalized.

Herc
From: |-|ercules on 7 Jul 2010 20:33
"Curt Welch" <curt(a)kcwc.com> wrote ...
> Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote:
>> "|-|ercules" <radgray123(a)yahoo.com> writes:
>>
>> > A counter example?
>>
>> Take the Heine-Borel theorem. It is not in any apparent sense "about
>> words and their definitions" nor can it be with any accuracy described
>> as "study of formal language and what can be said using formal
>> language". The study of formal languages and their expressiveness is a
>> very marginal part of mathematics.
>
> Well, the term "formal language" might be something very specific to you.
> I was using it very loosely just to mean a langauge made up of generally
> absolute concepts and definitions, instead of our normal natural language


Your usage is fine. As long as all statements in the system can be enumerated
by some grammar. The fact you include 'study of words' makes your definition
of mathematics very apt. That was the problem with systems like Godel's proof
which was purely numerical and didn't consider WHO verifies the proof.

Aatu still believes formal systems will always be incomplete because
"computer123 cannot prove this statement" is true, but not provable by computer123!

Even though Aatu cannot verify yes or no whether "Aatu cannot prove this statement"
is provably true.

Herc

PS
Isn't the fact you perceive something an absolute truth?
From: |-|ercules on 7 Jul 2010 21:16
"|-|ercules" <radgray123(a)yahoo.com> wrote ...
>>The study of formal languages and their expressiveness is a
>> very marginal part of mathematics.

And the study of the mechanics of the physical universe is a
very marginal part of mathematics.

Herc
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Next: NP+complete-problem navigation, search In computational complexity theory, the complexity class NP-complete (abbreviated NP-C or NPC), is a class of problems having two properties: * It is in the set of NP (nondeterministic polynomial time) pr