[equals constant] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
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equals constant
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Posted by: Philip Ronald Dutton
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!).
:)
Posted by: Gilles Robitaille
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?
Posted by: Philip Ronald Dutton
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.
)
Posted by: Tony Smith
I am trying to understand how to get to a point where I can consider if computation is a meta mathematical phenomenon.
Practitioners in the field can slide a bit too easily between mathematics and computation without bothering to note the change of context. Common use of the mathematical equals symbol as a computational assignment operator only compounds this problem.
The question "What is a computation?" is generally assumed to not require an answer, yet it may be that differences in our understanding of the atomic nature of computation may be at the heart of difficulties comprehending how many of the facts revealed in NKS map onto the world we find ourselves in.
The most that the PCE would appear to be claiming is that a whole bunch of atomic computations (aka simple programs) interacting in parallel can emulate a general purpose Turing machine, and thus, ultimately, any other digital computer.
Implicitly, it is also claiming that those atomic computations just happen because that is the nature of things, that the world is actively and continually computing its next state through "local" atomic computations.
Mathematical equality always says as much about the measure as it does about the items being noted as equal. Only x = x escapes this concern about measurement because it is more a tautological assertion about "=" than an assertion about x.
However in the world of simple programs, x = x is implicitly an assertion that x(t+1) = x(t), a very common and necessary state of affairs in a comprehensible world.
I am currently exploring a simple program for generating a next graph from a current graph where there is no atomic x = x, yet where X = X emerges quickly for even small configurations, as well as enough more interesting phenomena to keep my attention.
But even that simple model begs the question as to the true atomic nature of computational assignment. It still takes a few lines of general purpose computer code to emulate my particular atomic computations, so it becomes hard to escape the habit of looking for some structural dynamic which might actually facilitate such atomic computations, rather than accepting, as eventually we must, that at the truly atomic level that is just what the computational atoms do.
Posted by: Philip Ronald Dutton
...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).
Posted by: Tony Smith
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
In every hundredth of a second, an interval too brief to differentiate consciously, we can access current information from a "light cone" of radius 3000 Km.due to complex history- an effect of the global aspect
Yet it is our brain's (and our DNA's) ability to preserve representations across time of viable responses to perceivable patterns that underpin our complex potential. Via optic lenses only millimetres across we see the world around us through a process that only makes sense when you appreciate just how much smaller the atomic scale is.
Things might only be considered truly "local" at the Planck scale, and it is in those units that our slowness becomes much more apparent. If our time measuring scales were compressed so that a human life time was one second, then the age of our cosmos would only be measured in years. Effectively we take almost forever to do anything, at least relative to the tiny fraction of the space of our cosmos we are able to act within. We can know of but cannot in any way influence the hundreds of billions of galaxies, each with hundreds of billions of stars, as even the nearest other star is light years away.
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