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From: Wolf K on 15 Jul 2010 18:58
On 15/07/2010 14:30, Vesa Monisto wrote:
[...]
> Dedekind (?): 'The numbers are free creations of man's mind,
> they serve as a means of apprehending the difference of things
> more easily and more sharply.'
> ("Die Zahlen sind freie Schoepfungen des menschlichen Geistes;
> sie dienen als ein Mittel, um die Verschiedenheit der Dinge leichter
> und schaerfer aufzufassen." )

[Snip interesting anthology]

Actually, it's in our genes. Language merely allows us to extend the
limited number sense we're born with (and which we share with other
animals) to, well, to infinity.

wolf k.
From: Vesa Monisto on 15 Jul 2010 21:54
Wolf k wrote:
> Vesa Monisto wrote:
>> [...]
>> Dedekind (?): 'The numbers are free creations of man's mind,
>> they serve as a means of apprehending the difference of things
>> more easily and more sharply.'
>> ("Die Zahlen sind freie Schoepfungen des menschlichen Geistes;
>> sie dienen als ein Mittel, um die Verschiedenheit der Dinge leichter
>> und schaerfer aufzufassen." )
>
>[Snip interesting anthology]
>
> Actually, it's in our genes. Language merely allows us to extend the
> limited number sense we're born with (and which we share with other
> animals) to, well, to infinity.

But (there are always buts for the existence of But) it might be
a product of mental/behavioral *economy*, too. Genetic IOR 'fenetic'.
From numbers to variables to pointers, and then we have the modern
metastructure P(A(D)) = Pointers(Addresses(Data)), which could
be expressed by 1 to many "<)" relations "P<)A<)D", 1-1-1 at time.
-- MST is for economy, for saving energy/resources, e.g., by P(A(D)).

BTW, I translated "Geist" by "mind" but could have used a word
"spirit" or "genius" (all mere alias-pointers to excellent inner behavior).

V.M.
(What were it like to be a But? -- I don't know, just corrected my grammar
;)


From: R. Srinivasan on 19 Jul 2010 07:20
On Jul 17, 8:09 am, c...(a)kcwc.com (Curt Welch) wrote:
> "K_h" <KHol...(a)SX729.com> wrote:
> > "Curt Welch" <c...(a)kcwc.com> wrote in message
> >news:20100715013832.531$Gi(a)newsreader.com...
>
[...]
>
> > Well, this is simply wrong. The truth underlying "1+1=2" is an absolute
> > truth. [...]
>
[...]
>
> Gee, how many times do I have to tell you that absolute truth doesn't exist [...]
>
>
I wouldn't go so far as to say that absolute truth does not exist.
However, I would definitely claim that absolute truth is not something
that human beings can discuss intelligently. And, to quote
Wittgenstein in a different context: "That which we cannot talk about,
we must pass by in silence".

I claim that human beings can only discuss *formal* truth
intelligently when it is characterized as provability. Further,
provability is always with respect to (axiomatic) theories, and
therefore truth (for formal propositions) must be with respect to such
theories. When human beings formulate a theory, they are basically
asserting a number of truths *axiomatically*, via provability of
various propositions in such theories. When this line of thinking is
taken to its logical conclusion, one gets the logic NAFL and one can
show that infinite sets are not permitted in such a logic. Again, it
is not the claim of NAFL that "absolute truths" do not exist; it is
just that formal logic is not about such absolute truths, even if they
exist.

Let us look at the claim made by K_h that '"the truth underlying
'1+1=2' is an absolute truth". From other posts of K_h, it is clear
he claims that such a truth is existent in *our* universe, rather than
in some ideal Platonic world. Let us see if we can falsify this claim.

There is a theory of cosmology to the effect that our universe has
zero total energy and in fact originated from absolutely nothing:

http://161.58.115.79/pubs/mercury/31_02/nothing.html

So as per this theory (which I do not necessarily agree with) there
was a time when there was absolutely "nothing" in our universe. To
quote selectively from the above reference:

"....the universe is the ultimate free lunch! It came from nothing,
and its total energy is zero. ...The meaning of "nothing" is
somewhat ambiguous here. It might be the vacuum in some pre-existing
space and time, or it could be nothing at all – that is, all concepts
of space and time were created with the universe itself."

In fact I agree with the latter part of the above quote. If there is
nothing in the universe, then we do not have a length scale, and we
have nothing to characterize time or mass either. So the concepts of
space, time and matter become meaningless.

Let us go back to the time when there was (supposedly) absolutely
nothing in the universe. In fact, there is no proof that the universe
will not relapse to such a state in the future (presumably via the
"Big Crunch" predicted by general relativity, when all matter in the
universe collapses).

I claim that in such a universe, there is no evidence whatsoever for
"1+1=2" (leave alone evidence for infinite sets). In fact, one can
reasonably conclude only the existence of the number zero, or the
empty set, which would characterize the universe itself.

I am sure K_h is now going to invoke stuff like "quantum fluctuations"
and "virtual particles", and so on, which must characterize the vacuum
as per quantum field theory. However, the existence of virtual
particles is pure theory and not observable. See the following
website:

http://www2.slac.stanford.edu/vvc/theory/virtual.html

Quote:

"It is meaningless to argue whether they[virtual particles] are or are
not there, as they cannot be observed. Any attempt to observe them
changes the outcome of the process".

Clearly if one wishes to demonstrate "1+1=2" with physical objects,
then these must be concrete physical objects. E.g. "If we put one
apple into an empty basket, and then put one more, we will find two
apples in the basket". You can't do that with virtual particles. Even
if you could put one into the (hypothetical) basket, it will disappear
in next to no time.

In conclusion, the truths that K_h is talking about can be
characterized as "absolute truths" only when one postulates an ideal
Platonic world, independent of our universe. In such a Platonic world,
K_h can posit the existence of numbers, infinite sets and the like as
concrete physical objects. I assert that classical infinitary
reasoning is sustainable if and only if one subscribes to Platonism in
this form. The logic NAFL which I have proposed rejects Platonism, and
hence rejects classical infinitary reasoning.

RS
From: Vesa Monisto on 19 Jul 2010 09:27

"R. Srinivasan" <sradhakr(a)in.ibm.com> wrote in message
news:b9a7f959-baa5-45e2-b9f8-beaa3d9c9984(a)g35g2000yqa.googlegroups.com...
> ...
> I claim that human beings can only discuss *formal* truth
> intelligently when it is characterized as provability.
> ...

Well, OTOH, provers are like salesmen
traveling in late night trains, IMHO.
A picture of Russell's and Whitehead's proof that 1 + 1 = 2:

http://www.cut-the-knot.org/selfreference/russell.shtml

V.M.
[I've got it: *Bugs for Bats, Bags for Buts!* For programmers
sets {a,b,c, ...}, flats [a,a,a, ...], bags (a,a,b, ...), and relations
1-1, 1-n, n-n. For all others cardinals of any-to-any mappings
of max entropy. -- Patterns a la "I've Got Rhythm" (Gerschwin).]


From: K_h on 22 Jul 2010 17:40

"R. Srinivasan" <sradhakr(a)in.ibm.com> wrote in message
news:b9a7f959-baa5-45e2-b9f8-beaa3d9c9984(a)g35g2000yqa.googlegroups.com...
On Jul 17, 8:09 am, c...(a)kcwc.com (Curt Welch) wrote:
> "K_h" <KHol...(a)SX729.com> wrote:
> > "Curt Welch" <c...(a)kcwc.com> wrote in message
> >news:20100715013832.531$Gi(a)newsreader.com...
>
> I claim that human beings can only discuss *formal* truth
> intelligently when it is characterized as provability. Further,
> provability is always with respect to (axiomatic) theories, and
> therefore truth (for formal propositions) must be with respect to such
> theories. When human beings formulate a theory, they are basically
> asserting a number of truths *axiomatically*, via provability of
> various propositions in such theories. When this line of thinking is
> taken to its logical conclusion, one gets the logic NAFL and one can
> show that infinite sets are not permitted in such a logic. Again, it
> is not the claim of NAFL that "absolute truths" do not exist; it is
> just that formal logic is not about such absolute truths, even if they
> exist.

Accepted axioms are assertions that are observed to be true and cannot be reduced
to anything simpler. Since most mathematicians, especially set theorists, accept
the axiom of infinity, they all disagree with you on a few of these points --
especially double negation and mathematical realism (aka mathematical Platonism).

> Let us look at the claim made by K_h that '"the truth underlying
> '1+1=2' is an absolute truth". From other posts of K_h, it is clear
> he claims that such a truth is existent in *our* universe, rather than
> in some ideal Platonic world. Let us see if we can falsify this claim.

The absolute truth underlying '1+1=2' exists everywhere: in "our" universe and in
any ideal platonic world.

> There is a theory of cosmology to the effect that our universe has
> zero total energy and in fact originated from absolutely nothing:

By definition, absolute nothing does not exist (absolute nothing being the
absence of matter, energy, quantum fields, space, time, etc). Therefore it
cannot do, nor can it produce, anything. One has to be careful with popular
expositions of cosmology that bring in philosophical impossibilities; there is
lots of misinformation out there.

> http://161.58.115.79/pubs/mercury/31_02/nothing.html
>
> So as per this theory (which I do not necessarily agree with) there
> was a time when there was absolutely "nothing" in our universe. To
> quote selectively from the above reference:
>
> "....the universe is the ultimate free lunch! It came from nothing,
> and its total energy is zero. ...The meaning of "nothing" is
> somewhat ambiguous here. It might be the vacuum in some pre-existing
> space and time, or it could be nothing at all - that is, all concepts
> of space and time were created with the universe itself."

As I mentioned above, it cannot be nothing at all. The field equations of
general relativity permit one to set the zero point at 0 but that is just a
convention; it could just as easily be set at 10^600 Kg*c^2. In the theories you
refer to the gravitational energy becomes vacuum and expansion energy very early
on.

> Let us go back to the time when there was (supposedly) absolutely
> nothing in the universe. In fact, there is no proof that the universe
> will not relapse to such a state in the future (presumably via the
> "Big Crunch" predicted by general relativity, when all matter in the
> universe collapses).
>
> I claim that in such a universe, there is no evidence whatsoever for
> "1+1=2" (leave alone evidence for infinite sets). In fact, one can
> reasonably conclude only the existence of the number zero, or the
> empty set, which would characterize the universe itself.

The truth of "1+1=2" exists even if it is not physically instantiated in the
universe.

_


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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