The thesis of A New Kind of Science (NKS) is twofold: that the nature of computation must be explored experimentally, and that the results of these experiments have great relevance to understanding the physical world.
[2] The basic subject of Wolfram's "new kind of science" is the study of simple abstract rules—essentially, elementary computer programs.
Simply enumerating all possible variations of almost any class of programs quickly leads one to examples that do unexpected and interesting things.
In order to study simple rules and their often-complex behavior, Wolfram argues that it is necessary to systematically explore all these computational systems and document what they do.
An extension of this idea is that the human mind is itself a computational system, and hence providing it with raw data in as effective a way as possible is crucial to research.
While Wolfram advocates simple programs as a scientific discipline, he also argues that its methodology will revolutionize other fields of science.
The basis of his argument is that the study of simple programs is the minimal possible form of science, grounded equally in both abstraction and empirical experimentation.
Every aspect of the methodology NKS advocates is optimized to make experimentation as direct, easy, and meaningful as possible while maximizing the chances that the experiment will do something unexpected.
Just as this methodology allows computational mechanisms to be studied in their simplest forms, Wolfram argues that the process of doing so engages with the mathematical basis of the physical world, and therefore has much to offer the sciences.
Wolfram argues that science is far too ad hoc, in part because the models used are too complicated and unnecessarily organized around the limited primitives of traditional mathematics.
Wolfram argues that one of his achievements is in providing a coherent system of ideas that justifies computation as an organizing principle of science.
Some examples include the first primitive recursive function that results in complexity, the smallest universal Turing machine, and the shortest axiom for propositional calculus.
In a similar vein, Wolfram also demonstrates many simple programs that exhibit phenomena like phase transitions, conserved quantities, continuum behavior, and thermodynamics that are familiar from traditional science.
Simple computational models of natural systems like shell growth, fluid turbulence, and phyllotaxis are a final category of applications that fall in this theme.
Wolfram suggests that the theory of computational irreducibility may explain how free will is possible in a nominally deterministic universe.
In 2007, as part of commemorating the book's fifth anniversary, Wolfram's company offered a $25,000 prize for proof that this Turing machine is universal.
[3] Alex Smith, a computer science student from Birmingham, UK, won the prize later that year by proving Wolfram's conjecture.
[4][5] Periodicals gave A New Kind of Science coverage, including articles in The New York Times,[6] Newsweek,[7] Wired,[8] and The Economist.
For instance, NKS does not establish rigorous mathematical definitions,[15] nor does it attempt to prove theorems; and most formulas and equations are written in Mathematica rather than standard notation.
[16] Along these lines, NKS has also been criticized for being heavily visual, with much information conveyed by pictures that lack formal meaning.
"[18] The principle of computational equivalence (PCE) has been criticized for being vague, unmathematical, and not making directly verifiable predictions.
[22] Jürgen Schmidhuber has also charged that his work on Turing machine-computable physics was stolen without attribution, namely his idea on enumerating possible Turing-computable universes.
He suggests that space consists of a set of isolated points, like cells in a cellular automaton, and that even time flows in discrete steps.