[The Integers; Algorithm Axioms] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
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The Integers; Algorithm Axioms
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Posted by: Philip Ronald Dutton
The integers as we normally view them are in the typical listed representation:
...,3, 4, 5, 6, 7, 8, ... , N, ...
Are such numbers equivalent to instances of terminating algorithms? Whichever direction you progress through the list it appears that each position is an instance of a terminating algorithm (when viewed in the light of, say, the Peano axioms).
At this point we must state our core assumption: Is each position in the list a unique terminating algorithm or is it a unique termination instance of the same algorithm at each position?
They appear to be both. Intuition wishes them to be one or the other.
Each position (or algorithm in the list) is related to the neighboring algorithm. My slightly undeveloped thoughts limit my description of this relation to the following statement:
The algorithm in one position can not terminate unless the neighboor terminates (in relation to the direction you are working).
I think the integers may be just a list of simple algorithms which have "neighbor" relationships.
A sea of simple algorithms is fun to explore but soon enough we must understand how they can affect each other (if they can). How do simple algorithms (rules) interact/relate?
Someone needs to build a set of axioms for algorithm manipulation/exploration instead of manipulation/exploration of numbers (basic arithmetic).
Why can't we say that numbers are just terminated algorithms. Then go back and recast the axioms of arithmetic to handle algorithms? Then maybe we can start doing things with algorithms?
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