Complexity Beyond Computation

Stephen Wolfram's Principle of Computational Equivalence is a landmark in complexity study. Even though it's wrong at large, it did show how seemingly simple programs can exhibit extremely complex behavior. The fallacy results from the desire to generalize without proper scrutiny. For example, it claims that complexities in different dimensions are similar. But, topology and geometry make very striking distinctions between dimensions. In 2D, we have Poincare-Bendixson theorem. In 3D, we have knots. All are specific to dimensionality. Unless we are going to say these constructs do not reflect complexity, we are forced to accept that complexity is much more than computation and things like geometry should be a subject of study.

The field is still very young. A theory to rank knots is a good beginning. There is good reason to speculate that final results will involve vast new areas like statistics, and will provide insights into deep philosophical questions like rationalism versus empiricism.