On the World
Principle of Computational Equivalence Fallacy
NEVATHIR
October 1, 2017
A core thesis in modern complexity study according to Stephen Wolfram's NKS is the Principle of Computational Equivalence. Statement: There are various ways to state the Principle of Computational Equivalence, but probably the most general is just to say that almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication.
PCE is the culmination of a tradition to understand complexities with Turing machines. Because Turing machines were used to perform calculations in many sciences with high degree of efficiency, people began to speculate computational complexity is the answer to measuring complexity.
It's easy to understand people's aspiration for computational complexity, as well as its fallacies. If complexity is computation, then universal Turing machines elegantly close our quest for a proper mechanism to generate all complexities. However, the fallacy is glaringly obvious. Life is complex, but a Turing machine automatically recording symbols on a infinite tape is no life at all.
The fallacy of PCE opens up a wide variety of complexity measures. Here we briefly explore the notion of complexity as ability and its relationship with PCE.
As argued before, Eurozone crisis can be understood as inability to apply Keynesian policies with serious consequences. Certainly, Greeks can perform computations on Macs as well as Germans on iPads. The difference between Germany and Greece regarding Eurozone issues is not ability to perform computation. Or hiring German computer scientists will end Eurozone crisis for Greece.
Computation is a kind of ability but not all abilities are computations. Complexity as ability thus provides a wider view than Turing machines.
Ability to earn a life, ability to maintain a good economy, ability to form a good democracy, ability to build valuable sciences, ability to make critical engineering decisions, ability to produce great artworks, etc. are all very complex.
Abilities can often be represented as a partial order system due to the existence of high and low abilities. With a degree of justification, higher abilities may be regarded as more complex than lower abilities. This is not to say Lagrange is more complex than Michael Schumacher, but the former can produce more complex mathematics, while the latter can drive for more complex automobile races. Understanding abilities is essential if Lagrange or Michael Schumacher is to be understood properly.
Our ability to perform rational deduction and empirical exploration is required for scientific study as well as a reformed foundation for modern empirical sciences, which is a topic we will elaborate with many philosophical and engineering applications.
We may conclude with Turing machines as a special subclass of complexity measures for ability both concrete and abstract as a broader though not yet comprehensive complexity class. Serious consequences are abundant that will require many publications to elucidate the precise connection between complexity study and various sciences.