Stephan Wolfram

Stephen Wolfram

A New Kind of Science

“…nature samples a broader swath of the computational universe than mathematics and engineering. But it too is limited, whether because natural selection tends to favor incremental change, or because some physical process just follows one particular rule.”

“Principle of Computational Equivalence which implies that even when the underlying rules or components of a system are simple, the behavior of the system can correspond to a computation that is essentially as sophisticated as anything.”

One of the great surprises of my work has been just how easy it is to find universal computation. If one systematically explores the abstract universe of possible computational systems one does not have to go far. One needs nothing like the billion transistors of a modern electronic computer, or even the elaborate axioms of logic and arithmetic. Simple rules that can be stated in a short sentence—or summarized in a three-digit number—are enough.”

“Godel’s original work was quite abstruse. He took the axioms of logic and arithmetic, and asked a seemingly paradoxical question: can one prove the statement “this statement is unprovable”? … So here, then, is a statement within mathematics that is unprovable by mathematics: an “undecidable statement”. And its existence immediately shows that there is a certain incompleteness to mathematics: there are mathematical statements that mathematical methods cannot reach.”

Stephan Wolfram

http://forum.wolframscience.com/showthread.php?s=&threadid=1063