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Calculus of Rules

Set of All Possible Numbers

We are taught in school that the number line is continuous. I believe that all numbers are infinitely discontinuous by definition. Every number is unique, that implies a discontinuity between it and every other number. Extracting a number from the number line is done with a ‘cut’. A cut is just another name for applying a discontinuity. We see we cannot get an individual number without making it discontinuous from whatever line it is on.

The number line is a poor base to understand numbers from. The issue is that the number line is produced from rules. Rules apply continuity to numbers. The true set of all numbers is the set such that there are no rules applied. Totally abstract numbers are possible only with no rules applied. A total abstraction of numbers is also meaningless.

Numbers are better abstracted as a cloud of points. Think of viewing all the stars in the night sky. Each number is distinct from all other numbers. A way this can occur is if each number is its own dimension. Every number is orthogonal to every other number. Applying a rule to a set of numbers acts like a magnet to align the numbers according to the rule and makes the resulting shape continuous.

The number line is really the set of all numbers with the rule of linear continuity applied to them.

A circle is the set of all numbers with the rule of circularity applied to them.

 

Is there a Calculus of Rules?

I asked this question on reddit and got several responses. I am not sure where I am going with this idea. The closest thing to it is lambda calculus introduced by Alonzo Church in the 1930’s. Worth reading up on. https://en.wikipedia.org/wiki/Lambda_calculus

I was thinking about the definition of a number, and the definition of arithmetic operators like add, and multiply. There are proofs of the operation of addition and so forth. More complex functions are built out of the primary operators. The proofs can be written down. There is a symbolic representation to it.

I was thinking also of computer graphics operations like and, or, nand, and nor, which give different sets of colors depending on the operation.

In my computer, the operation can be represented by a variable, for instance called ‘op’

‘op’ could be one from a list of operations.

To draw something the rule A = B or C could be used. A variable ‘op’ can substitute for the rule, so the rule becomes A = B op C. A = B op C is a more general rule than A = B or C. It is a union of rules for graphic operations.

The union of a list is somewhat like an integral. If one was to “integrate” the list ‘and, or, nand, nor’ a new rule could be defined.

 

A calculus of rules would be a stepwise definition of rules. Rule N+1 would be one step away from Rule N. Given a stepwise definition potentially allows the ideas from calculus to be applied.

In calculus the area under a curve can be determined by integration. Integration is just a repeated stepwise addition. A rule of multiplication would be a specialized rule of integration.

Is there a minimum rule step that can be used to build higher order rules?

Looking at the definition of a number and taking into consideration that numbers are infinitely discontinuous, they must all be ‘at right angles’ to each other.