© 2025
Robert Finch
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.
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.