Registered: Feb 2004
Posts: 172

I was pondering the logical statement: "x = x"
( 0 = 0, 5 = 5, etc.)

These kind of statements are so basic. But, I am wondering if there is a bit more complexity than what first appears. Hopefully, I can explain clear enough so that people can point out my mistake-- since, I am most likely making one here!

Okay,
I think that "x=x" is coupled to (or encodes) a meta-mathematical version of substitution. Meta-mathematically speaking, you can distinguish the left 'x' from the right 'x.' Mathematically, speaking, you can not. Consider this "meta-mathematical" dialogue:

a:"hey look at that left 'x'!"
b:"Oh that is nice! I wonder what it is?"
a:"well, lets take a closer it..."
a:"hmmm, it appears to be the right 'x'!"
b:"well isn't that interesting?"

So, within the conversation, a substition was applied was it not? Also, the conversation could have gone back and forth for infinity because you have to look at the right side in terms of the left side also. I am calling this an "encapsulated meta-mathematically inherit recursion."

So, my thought experiment makes me think that the equals constant encapsulates some kind of meta substitution and some kind of meta recursion....


Such meta mathematical substitution will always have to be used within a computation model (i cant yet find words to explain why i think this..maybe, to start: you can not represent that statement within computation models and not have to use a substitution for analysis!?) Hence I smell seeds of computation being meta-mathematical.....oops: I should not have said that!

And now, I apologize for my rookie mistakes (whatever they are!).
:)

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Last edited by Philip Ronald Dutton on 03-16-2004 at 05:25 PM

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Registered: Mar 2004
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This is interesting.

It made me think of this statement: X + people = x + people.

On the surface it states that an object «X» placed in a group of people is the same as that same objet in a group of people. The term «people» is generic and may include anywhere from 2 to as many as you want. It maybe safe to assume that although the numbers of people maybe the same the individuals making up the two groups are different.

Would the equation be both meta-mathematically true as well as mathematically true?

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My original post required the comfort of well behaved variables... but, also the use of constants:

3.5=3.5
007=007

In the case of variables:

u = u ,

I am talking about 'u' representing the same "thing" (whatever that is: equation, number, sentence, james bond figure) on both sides.



(
note:
I am trying to understand how to get to a point where I can
consider if computation is a meta mathematical phenomenon.
)


(
Along similar lines: I am wondering if the pure entity which we call, a paradox, is something that is only alive in the meta-mathematical realm.... please excuse my loose poetic terminology! Mathematics can arrive at two different answers in certain experiments: I say big deal! Mathematics is just doing it's thing! When we try to look at the two endings together then we are executing meta-mathematics, within which, we end up with a paradox.... so how can we use a paradox inside a mathematical proof when it does not belong in mathematics anyway? Again, I am trying to keep the discussion related to NKS... maybe this paradox stuff should be on a Philosophy forum.
)

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Last edited by Philip Ronald Dutton on 03-16-2004 at 09:55 PM

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I am trying to understand how to get to a point where I can consider if computation is a meta mathematical phenomenon.

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...that the world is actively and continually computing its next state through "local" atomic computations. ...


CAs are capable of being "productive" while using only simple "local" atomic computations.

Is there a particular type of Logic "world" which restricts itself to "local" resources?

I ask this because it seems as though, generally, human logical thought has a whole lot of global resource, which perhaps makes your "current state" (within a logical sequence) difficult to interpret (due to complex history- an effect of the global aspect).

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Months ago, driving down a leafy road with the city skyline ahead in the distance, I suddenly appreciated just how much information there is to be taken into account within each human moment, and that such an arrangement is essential to our maintenance of our own complexity.

it seems as though, generally, human logical thought has a whole lot of global resource

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