A Proposed New Principle of Mathematics

From: namducnguyen (namducnguyen_at_shaw.ca)
Date: 09/14/04 

Date: Tue, 14 Sep 2004 04:55:55 GMT

In the thread "A question on GIT.",

George Greene wrote:

>"peter_douglass" <baisly@gis.net> wrote in message
news:<vAE%c.401027$%_6.342704@attbi_s01>...
>
>>"Herman Jurjus" wrote in message
>>news:2q8dv6Fs5hohU1@uni-berlin.de...
>>
...
>to finitely many premises. That the resulting logic then
>turns out to be INadequate to define "finite" is either
>a) the greatest intellectual embarrassment of all time, or
>b) yet one more concrete example of a very Deep Principle
>that knowers can't know "Everything" about themSELVES.
>
...

I propose today a new arithmetic principle, say PId [Principle of Identity],
that GG has alluded to in "b)" above. PId would state:

(1) There exists an arithmetic number that we don't know if it's even or
odd.

The motivation for Pid is actually is straightforward:

To know whether a given number is even or odd, we have to *know the
number enough*
so that we could determine if it's in the form/category of "2*n". for
instance,
we have to know all of its decimal digits so that we could know if the
ending
one is 0, 2, 4, 6, 8, or anything else. *But we can't not know in such
_individual_
manner for infinitely many numbers*. In Brief, we should agree this meta
statement
as FALSE:

(2) We know the even-odd statuses, individually, of all the arithmetic
numbers.
      
And so (1), which is essentially the negation of (2), should be accepted
as TRUE.

Now, by the look of it, (1) is in no way an expression of FOL framework:
FOL doesn't
have anything like "don't know", as appearing in (1). FOL framework's is
built
principally on human knowing: as GG has alluded to, we *know what
_finite_ is*,
we know a wff is a theorem because we know its finite proof, etc...

The point is (1) is an expression in a meta level; and making use of its
*merit*
in the 1st order level of FOL framework might not be an easy task. In
fact, certain
core concepts such as the uniqueness of the standard model of arithmetic
might have
to be re-visited. And in fact certain fundamental assumptions in *human
reasoning*
might have to be changed. [One of such assumption is that basic
mathematical concepts
would be "absolute" w.r.t. human cognition: every human would
know/understand the
"same" basic concept. PId seems to reverse human mathematical knowledge
to a
"relative" basis: what might be an even number to one human's knowledge,
would
be an odd number to another human's one.]

It'd probably require _more work_ but initial intuition seems to suggest we
could arrive at Incompleteness Theorem via an entirely different route than
Godel's numerization for G, should we adopt (1) as a principle. Also the
fact
that we wouldn't know the even-odd status of an arithmetic number somehow
seems implicate GC (Goldbach Conjecture) in some way. [I'm not quite
sure though].

In any rate, (1) is something I've been wondering for a while. I believe
it should
be adopted based on its simple truth that, as human being with finite
knowledge,
we simply can't know everything there might be about infinity. [My very
limited
understanding in formalism also seems to convince me that perhaps this
simple truth
is reflected somewhat by LCT, as well.]

I do thank in advance for your suggestion, correction, insights
regarding to my
proposed (1).

Best Regards.

---Nam

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