what if/then what

Considering the connection between algorithm and topology, I am drawn to the idea raised in our last discussion regarding the fundamental nature of the if/then operation. In reading an essay about topology, I came across the Jordan Curve Theorem, which I think has an interesting correlation with the if/then algorithmic operation in both its function and simplicity.

A simple closed curve (one that does not intersect itself) is drawn in a plane. This curve "C" divides the domain into two domains, an inside "A" and an outside "B." Even if the plane and the curve are deformed, these two classes persist and force any curve traversing A to B to cross C, regardless of the deformations.

Within the topological theorem, there are discrete operations concerning discrete elements (in this case, geometrical forms). While a curve may be reducible to a set of points, this operation is contingent on this curve being irreducible. It is essential to this particular function, and therefore it is discrete. While I do not intend to deny the divisibility test for discreteness, I do think this speaks to the nature of discrete in terms of algorithm.

For instance, the if/then statement, as part of a rule set, functions in a similar fashion as the curve in the Jordan Curve Theorem. It acts as a dividing line that creates, or forces the emergence of, two distinct states that are both contingent on the single statement.

Furthermore, the sequence of rules (rule set) can be discrete when the the operation of another algorithm is contingent upon the entirety of this rule set. Von Neumann speaks to this in his discussion of self-reproduction in the General and Logical Theory of Automata.

If we look at the nature of the algorithm through the lens of "topology," does the algorithm change if the order of rules are re-organized to yield different results? Is the Turing machine "topologically" consistent even if the order of operations are varied based on inputs?

While I think the idea of an infinite rule set is logically sound, I think it should be appended to reflect the necessity for recursion in this infinite string. This would suggest genetic instructions akin to evolution, rather than a sprawling sequence that simply doesn't end.