# A New Kind of Science

## A New Kind of Science

A New Kind of Science is a book by Stephen Wolfram, published in 2002. It contains an empirical and systematic study of computational systems such as cellular automata. Wolfram calls these systems simple programs and argues that the scientific philosophy and methods appropriate for the study of simple programs are relevant to other fields of science.

Contents [hide]

1 Computation and its implications

2 Simple programs

3 Mapping and mining the computational universe

4 Systematic abstract science

5 Philosophical underpinnings

5.1 Principle of computational equivalence

6 Applications and results

7 NKS Summer School

8 NKS Conferences

9 Reception

9.1 Scientific philosophy

9.2 Methodology

9.3 Utility

9.4 Principle of computational equivalence

9.5 The fundamental theory (NKS Chapter 9)

9.6 Natural selection

9.7 Originality and self-image

10 See also

11 References

12 External links

[edit]Computation and its implications

The thesis of A New Kind of Science is twofold: that the nature of computation must be explored experimentally, and that the results of these experiments have great relevance to understanding the natural world. Since its crystallization in the 1930s, computation has been primarily approached from two traditions: engineering, which seeks to build practical systems using computations; and mathematics, which seeks to prove theorems about computation (albeit already in the 1970s computing as a discipline was described as being at the intersection of mathematical, engineering, and empirical/scientific traditions[1][2]).

Wolfram describes himself as introducing a third major tradition, which is the systematic, empirical investigation of computational systems for their own sake. This is where the "New" and "Science" parts of the book's title originate. However, in proceeding with a scientific investigation of computational systems, Wolfram eventually came to the conclusion that an entirely new method is needed. Traditional mathematics was failing to describe the complexity seen in these systems meaningfully. Through a combination of experiment and theoretical positioning, the book introduces a new method that Wolfram argues is the most realistic way to make scientific progress with computational systems. This difference in method casts A New Kind of Science as a "kind" of science, and allows its principles to be potentially applicable in a wide range of fields.

[edit]Simple programs

The basic subject of Wolfram's "new kind of science" is the study of simple abstract rules — essentially, elementary computer programs. In almost any class of computational system, one very quickly finds instances of great complexity among its simplest cases. This seems to be true regardless of the components of the system and the details of its setup. Systems explored in the book include cellular automata in one, two, and three dimensions; mobile automata; Turing machines in 1 and 2 dimensions; several varieties of substitution and network systems; primitive recursive functions; nested recursive functions; combinators; tag systems; register machines; reversal-addition; and a number of other systems. For a program to qualify as simple, there are several benchmarks:

Its operation can be completely explained by a simple graphical illustration.

It can be completely explained in a few sentences of human language.

It can be implemented in a computer language using just a few lines of code.

The number of its possible variations is small enough so that all of them can be computed.

Generally, simple programs tend to have a very simple abstract framework. Simple cellular automata, Turing machines, and combinators are examples of such frameworks, while more complex cellular automata do not necessarily qualify as simple programs. It is also possible to invent new frameworks, particularly to capture the operation of natural systems. The remarkable feature of simple programs is that a significant percentage of them are capable of producing great complexity. Simply enumerating all possible variations of almost any class of programs quickly leads one to examples that do unexpected and interesting things. This leads to the question: if the program is so simple, where does the complexity come from? In a sense, there is not enough room in the program's definition to directly encode all the things the program can do. Therefore, simple programs can be seen as a minimal example of emergence. A logical deduction from this phenomenon is that if the details of the program's rules have little direct relationship to its behavior, then it is very difficult to directly engineer a simple program to perform a specific behavior. An alternative approach is to try to engineer a simple overall computational framework, and then do a brute-force search through all of the possible components for the best match.

Simple programs are capable of a remarkable range of behavior. Some have been proven to be universal computers. Others exhibit properties familiar from traditional science, such as thermodynamic behavior, continuum behavior, conserved quantities, percolation, sensitive dependence on initial conditions, and others. They have been used as models of traffic, material fracture, crystal growth, biological growth, and various sociological, geological, and ecological phenomena. Another feature of simple programs is that making them more complicated seems to have little effect on their overall complexity. A New Kind of Science argues that this is evidence that simple programs are enough to capture the essence of almost any complex system.

[edit]Mapping and mining the computational universe

In order to study simple rules and their often complex behaviour, Wolfram believes it is necessary to systematically explore all of these computational systems and document what they do. He believes this study should become a new branch of science, like physics or chemistry. The basic goal of this field is to understand and characterize the computational universe using experimental methods.

The proposed new branch of scientific exploration admits many different forms of scientific production. For instance, qualitative classifications like those found in biology are often the results of initial forays into the computational jungle. On the other hand, explicit proofs that certain systems compute this or that function are also admissible. There are also some forms of production that are in some ways unique to this field of study. For instance, the discovery of computational mechanisms that emerge in different systems but in bizarrely different forms.

Another kind of production involves the creation of programs for the analysis of computational systems — for in the NKS framework, these themselves should be simple programs, and subject to the same goals and methodology. 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. Wolfram believes that programs and their analysis should be visualized as directly as possible, and exhaustively examined by the thousands or more. Since this new field concerns abstract rules, it can in principle address issues relevant to other fields of science. However, in general Wolfram's idea is that novel ideas and mechanisms can be discovered in the computational universe — where they can be witnessed in their clearest forms — and then other fields can pick and choose among these discoveries for those they find relevant.

[edit]Systematic abstract science

While Wolfram promotes simple programs as a scientific discipline, he also insists that its methodology will revolutionize essentially every field of science. The basis for his claim is that the study of simple programs is the most minimal possible form of science, which is equally grounded in both abstraction and empirical experimentation. Every aspect of the methodology advocated in NKS 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 NKS allows computational mechanisms to be studied in their cleanest forms, Wolfram believes the process of doing NKS captures the essence of the process of doing science — and allows that process's strengths and shortcomings to be directly revealed.

Wolfram believes that the computational realities of the universe make science hard for fundamental reasons. But he also argues that by understanding the importance of these realities, we can learn to leverage them in our favor. For instance, instead of reverse engineering our theories from observation, we can simply enumerate systems and then try to match them to the behaviors we observe. A major theme of NKS style research is investigating the structure of the possibility space. Wolfram feels that science is far too ad hoc, in part because the models used are too complicated and/or unnecessarily organized around the limited primitives of traditional mathematics. Wolfram advocates using models whose variations are enumerable and whose consequences are straightforward to compute and analyze.

[edit]Philosophical underpinnings

Wolfram believes that one of his achievements is not just exclaiming, "computation is important!", but in providing a coherent system of ideas that justifies computation as an organizing principle of science. For instance, Wolfram's concept of computational irreducibility — that some complex computations cannot be short-cutted or "reduced" (cf. NP-hard) , is ultimately the reason why computational models of nature must be considered, in addition to traditional mathematical models. Likewise, his idea of intrinsic randomness generation — that natural systems can generate their own randomness, rather than using chaos theory or stochastic perturbations — implies that explicit computational models may in some cases provide more accurate and rich models of random-looking systems.

Based on his experimental results, Wolfram has developed the Principle of Computational Equivalence (see below), which asserts that almost all processes that are not obviously simple are of equivalent sophistication. From this seemingly vague single principle Wolfram draws a broad array of concrete deductions that reinforce many aspects of his theory. Possibly the most important among these is an explanation as to why we experience randomness and complexity: often, the systems we analyze are just as sophisticated as we are. Thus, complexity is not a special quality of systems, like for instance the concept of "heat", but simply a label for all systems whose computations are sophisticated. Understanding this makes the "normal science" of the NKS paradigm possible.

At the deepest level, Wolfram believes that like many of the most important scientific ideas, the Principle allows science to be more general by pointing out new ways in which humans are not special. In recent times, it has been thought that the complexity of human intelligence makes us special — but the Principle asserts otherwise. In a sense, many of Wolfram's ideas are based on understanding the scientific process — including the human mind — as operating within the same universe it studies, rather than somehow being outside it.

[edit]Principle of computational equivalence

The principle states that systems found in the natural world can perform computations up to a maximal ("universal") level of computational power. Most systems can attain this level. Systems, in principle, compute the same things as a computer. Computation is therefore simply a question of translating input and outputs from one system to another. Consequently, most systems are computationally equivalent. Proposed examples of such systems are the workings of the human brain and the evolution of weather systems.

[edit]Applications and results

There are a vast number of specific results and ideas in the NKS book, and they can be organized into several themes. One common theme of examples and applications is demonstrating how little it takes to achieve interesting behavior, and how the proper methodology can discover these cases.

First, there are perhaps several dozen cases where the NKS book introduces the simplest known system in some class that has a particular characteristic. 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 a large number of minimal examples of how simple programs exhibit phenomena like phase transitions, conserved quantities and 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.

Another common theme is taking facts about the computational universe as a whole and using them to reason about fields in a holistic way. For instance, Wolfram discusses how facts about the computational universe inform evolutionary theory, SETI, free will, computational complexity theory, and philosophical fields like ontology, epistemology, and even postmodernism.

Wolfram suggests that the theory of computational irreducibility may provide a resolution to the existence of free will in a nominally deterministic universe. He posits that the computational process in the brain of the being with free will is actually complex enough so that it cannot be captured in a simpler computation, due to the principle of computational irreducibility. Thus while the process is indeed deterministic, there is no better way to determine the being's will than to essentially run the experiment and let the being exercise it.

The book also contains a vast number of individual results — both experimental and analytic — about what a particular automaton computes, or what its characteristics are, using some methods of analysis.

[edit]NKS Summer School

Every year, Wolfram and his group of young top class instructors[3] organizes a summer school.[4] The first four summer schools from 2003 to 2006 were held at Brown university. Later the summer school was hosted by the university of Vermont at Burlington with the exception of the year 2009 that was held at the Istituto di Scienza e Tecnologie dell’Informazione of the CNR in Pisa, Italy. After seven consecutive summer schools more than 200 people have participated, some of which continued developing their 3-week research projects as their Master's or Ph.D thesis.[5] Some of the research done in the summer school has yielded important published results.[6][7]

[edit]NKS Conferences

Wolfram Research has organized three Wolfram Science conferences in Boston, MA, Washington, D.C. and Burlington, VT in the United States in the years 2003, 2006 and 2007 respectively. Two other independent NKS Midwest conferences have been organized at the Indiana University, Bloomington in 2005 and 2008. Other independent workshops related to NKS research have been also organized overseas, such as JOUAL (Just One Universal Algorithm) at the CNR in Pisa, Italy in 2009.

[edit]Reception

A New Kind of Science received extensive media publicity for a scientific book, generating scores of articles in such publications as The New York Times[8], Newsweek[9], Wired[10], and The Economist[11]. It was a best-seller and won numerous awards. NKS was reviewed in a large range of scientific journals, and several themes emerged in these reviews. Many reviewers enjoyed the quality of the book's production and the clear way Wolfram presented many ideas[12]. Even those reviewers who engaged in other criticisms found aspects of the book to be interesting and thought-provoking. On the other hand, many reviewers criticized Wolfram for his lack of modesty, poor editing, lack of mathematical rigor, and the lack of immediate utility of his ideas. Concerning the ultimate importance of the book, a common attitude was that of either skepticism or "wait and see". (Further detail of criticisms provided below.) Many reviewers and the media focused on the use of simple programs (cellular automata in particular) to model nature, rather than the more fundamental idea of systematically exploring the universe of simple programs.

[edit]Scientific philosophy

A key tenet of NKS is that the simpler the system, the more likely a version of it will recur in a wide variety of more complicated contexts. Therefore, NKS argues that systematically exploring the space of simple programs will lead to a base of reusable knowledge. However, many scientists believe that of all possible parameters, only some actually occur in the universe; that, for instance, of all possible variations of an equation, most will be essentially meaningless. NKS has also been criticized for asserting that the behavior of simple systems is somehow representative of all systems.

[edit]Methodology

A common criticism of NKS is that it does not follow established scientific methodology[who?]. NKS does not establish rigorous mathematical definitions[13], nor does it attempt to prove theorems.[14] Along these lines, NKS has also been criticized for being heavily visual, with much information conveyed by pictures that do not have formal meaning. It has also been criticized for not using modern research in the field of complexity, particularly the works that have studied complexity from a rigorous mathematical perspective.

Critics also note that none of the book's contents were published in peer-reviewed journals, the standard method for distributing new results, and complain it insufficiently credited other scientists whose work it is built on. Wolfram relegates all discussion of other people to his lengthy endnotes and thus no one is directly credited in the text. His critics argue that even the endnotes are misleading, glossing over many relevant discoveries and thus making Wolfram's work seem more novel.

[edit]Utility

NKS has been criticized for not providing specific results that would be immediately applicable to ongoing scientific research. There has also been criticism, implicit and explicit, that the study of simple programs has little connection to the physical universe, and hence is of limited value. Steven Weinberg has pointed out that no real world system has been explained using Wolfram's methods in a satisfactory fashion.[15]

[edit]Principle of computational equivalence

The PCE has been criticized for being vague, unmathematical, and for not making directly verifiable predictions; however, Wolfram's group has described the principle as such, not a law, theorem or formula. It has also been criticized for being contrary to the spirit of research in mathematical logic and computational complexity theory, which seek to make fine-grained distinctions between levels of computational sophistication. Others suggest it is little more than a rechristening of the Church-Turing thesis. However, the Church-Turing thesis imposes an upper limit while Wolfram's PCE suggests the nonexistence of intermediate degrees of computation sending a computational system either to the upper level (universal) or to the lowest degree, explained by Klaus Sutner[16] in terms of physics-like computation as a zero-one law claiming that, in practice, constructing actual computers with intermediate degrees is highly artificial and hasn't ever been done, hence endorsing Wolfram's intuition captured in his PCE.

[edit]The fundamental theory (NKS Chapter 9)

Wolfram's speculations of a direction towards a fundamental theory of physics have been criticized as vague and obsolete. Scott Aaronson, Assistant Professor of Electrical Engineering and Computer Science at MIT, also claims that Wolfram's methods cannot be compatible with both special relativity and Bell's theorem violations, which conflicts with the observed results of Bell test experiments.[17]

In a 2002 review of NKS, the Nobel laureate and elementary particle physicist Steven Weinberg wrote, "Wolfram himself is a lapsed elementary particle physicist, and I suppose he can't resist trying to apply his experience with digital computer programs to the laws of nature. This has led him to the view (also considered in a 1981 paper by Richard Feynman) that nature is discrete rather than continuous. 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. Following an idea of Edward Fredkin, he concludes that the universe itself would then be an automaton, like a giant computer. It's possible, but I can't see any motivation for these speculations, except that this is the sort of system that Wolfram and others have become used to in their work on computers. So might a carpenter, looking at the moon, suppose that it is made of wood."[18]

According to NKS Chapter 9, special relativity theory and quantum field theory are merely approximations to a digital network with inaccessible signal propagation below the Planck scale. NKS Chapter 9 and M-theory both attempt to unify general relativity theory and quantum field theory. M-theory postulates that there is a minimum physical wavelength and that vibrating string-like entities can model all of physics. NKS Chapter 9 postulates that there is a finite automaton that builds time, space, and energy from informational substrate below the Planck scale. According to Wolfram, infinities and infinitesimals do not occur in nature, except perhaps for time as a potential infinity. In particular, there is a maximum physical wavelength in addition to the minimum physical wavelength postulated by M-theory. Neither of these scales are accessible to current physics, however, and hence this is untestable speculation until experiments reach those scales.

In the NKS theory, the basic physical realities of time, space, and energy are merely approximations that arise from a few simple rules that operate with hidden determinism below the Planck scale. According to Wolfram, "building on the discovery that even simple programs can yield highly complex behavior, A New Kind of Science shows that with appropriate kinds of rules, simple programs can give rise to behavior that reproduces a remarkable range of known features of our universe — leading to the bold assertion that there could be a simple short program that represents a truly fundamental model of the universe, and which if run for long enough would reproduce the behavior of our world in every detail.”[19]

[edit]Natural selection

Wolfram's claim that natural selection is not the fundamental cause of complexity in biology has led some to state that Wolfram does not understand the theory of evolution[20]. However, some experts have acknowledged that natural selection leaves many unanswered questions, which information theory might be able to explain[21][22]. In this context, Wolfram's work is similar to that of D'Arcy Thompson.

[edit]Originality and self-image

NKS has been heavily criticized as not being original or important enough to justify its title and claims, mostly by people who argue that the book is about simple systems generating complex behavior. However, even though the fact that simple systems are capable of complicated behavior is an important part of the book, the main contribution is the new methodology of mining the computational universe. Edward Fredkin and Konrad Zuse pioneered the idea of a computable universe, the former by writing a line in his book on how the world might be like a cellular automaton, and later further developed by Fredkin using a toy model called Salt[23]. It has been claimed that NKS tries to take these ideas as its own. This has been mainly suggested by people thinking that Wolfram's main thesis is that the universe is a cellular automaton in spite of the fact that Wolfram's proposal as a discrete model of the universe is a trivalent network. Wolfram himself considers that a cellular automaton model is unsuitable to describe quantum and relativistic properties of nature, as explained in his NKS book.

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. However Schmidhuber did his original suggestion in purely algorithmic-theoretic terms, while Wolfram's proposal has been to test every candidate to find the right one in physical terms, in which the known laws of physics actually hold, rather than a theoretical description of what the candidates should algorithmically be.

Additionally, the core idea that very simple rules often generate great complexity is already an established idea in science, particularly in chaos theory and complexity theory research - and to some researchers, this field is considered well-understood[who?]. The authoritative manner in which NKS presents a vast number of examples and arguments has been criticized as leading the reader to believe that each of these ideas was original to Wolfram, however the notes section at the end of his book acknowledges many of the discoveries made by these other scientists citing their names together with historical facts, although not in the form of a traditional bibliography section. This is generally considered insufficient in scientific literature, however - end notes are normally reserved for only indirectly related material, and lay readers typically ignore end notes, resulting in the impression that the author performed all the work.

In particular, one of the most substantial new technical results presented in the book, that the rule 110 cellular automaton is Turing complete, was not proven by Wolfram, but by his research assistant, Matthew Cook. This is not particularly a surprise since as explained by Wolfram himself, the book was actually a kind of project involving a group of his research assistants led by Wolfram himself, something that means he didn't do every single experiment or contribution directly, as he also acknowledges in the book. The research assistants were however paid for this work as hired by Wolfram's company, not by a university.

Some have argued[who?] that the use of computer simulation is ubiquitous, and instead of starting a paradigm shift NKS just adds justification to a paradigm shift that has been undertaken. Wolfram's NKS might then seem as the book explicitly describing this shift.

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Meeting Room AV

Contents [hide]

1 Computation and its implications

2 Simple programs

3 Mapping and mining the computational universe

4 Systematic abstract science

5 Philosophical underpinnings

5.1 Principle of computational equivalence

6 Applications and results

7 NKS Summer School

8 NKS Conferences

9 Reception

9.1 Scientific philosophy

9.2 Methodology

9.3 Utility

9.4 Principle of computational equivalence

9.5 The fundamental theory (NKS Chapter 9)

9.6 Natural selection

9.7 Originality and self-image

10 See also

11 References

12 External links

[edit]Computation and its implications

The thesis of A New Kind of Science is twofold: that the nature of computation must be explored experimentally, and that the results of these experiments have great relevance to understanding the natural world. Since its crystallization in the 1930s, computation has been primarily approached from two traditions: engineering, which seeks to build practical systems using computations; and mathematics, which seeks to prove theorems about computation (albeit already in the 1970s computing as a discipline was described as being at the intersection of mathematical, engineering, and empirical/scientific traditions[1][2]).

Wolfram describes himself as introducing a third major tradition, which is the systematic, empirical investigation of computational systems for their own sake. This is where the "New" and "Science" parts of the book's title originate. However, in proceeding with a scientific investigation of computational systems, Wolfram eventually came to the conclusion that an entirely new method is needed. Traditional mathematics was failing to describe the complexity seen in these systems meaningfully. Through a combination of experiment and theoretical positioning, the book introduces a new method that Wolfram argues is the most realistic way to make scientific progress with computational systems. This difference in method casts A New Kind of Science as a "kind" of science, and allows its principles to be potentially applicable in a wide range of fields.

[edit]Simple programs

The basic subject of Wolfram's "new kind of science" is the study of simple abstract rules — essentially, elementary computer programs. In almost any class of computational system, one very quickly finds instances of great complexity among its simplest cases. This seems to be true regardless of the components of the system and the details of its setup. Systems explored in the book include cellular automata in one, two, and three dimensions; mobile automata; Turing machines in 1 and 2 dimensions; several varieties of substitution and network systems; primitive recursive functions; nested recursive functions; combinators; tag systems; register machines; reversal-addition; and a number of other systems. For a program to qualify as simple, there are several benchmarks:

Its operation can be completely explained by a simple graphical illustration.

It can be completely explained in a few sentences of human language.

It can be implemented in a computer language using just a few lines of code.

The number of its possible variations is small enough so that all of them can be computed.

Generally, simple programs tend to have a very simple abstract framework. Simple cellular automata, Turing machines, and combinators are examples of such frameworks, while more complex cellular automata do not necessarily qualify as simple programs. It is also possible to invent new frameworks, particularly to capture the operation of natural systems. The remarkable feature of simple programs is that a significant percentage of them are capable of producing great complexity. Simply enumerating all possible variations of almost any class of programs quickly leads one to examples that do unexpected and interesting things. This leads to the question: if the program is so simple, where does the complexity come from? In a sense, there is not enough room in the program's definition to directly encode all the things the program can do. Therefore, simple programs can be seen as a minimal example of emergence. A logical deduction from this phenomenon is that if the details of the program's rules have little direct relationship to its behavior, then it is very difficult to directly engineer a simple program to perform a specific behavior. An alternative approach is to try to engineer a simple overall computational framework, and then do a brute-force search through all of the possible components for the best match.

Simple programs are capable of a remarkable range of behavior. Some have been proven to be universal computers. Others exhibit properties familiar from traditional science, such as thermodynamic behavior, continuum behavior, conserved quantities, percolation, sensitive dependence on initial conditions, and others. They have been used as models of traffic, material fracture, crystal growth, biological growth, and various sociological, geological, and ecological phenomena. Another feature of simple programs is that making them more complicated seems to have little effect on their overall complexity. A New Kind of Science argues that this is evidence that simple programs are enough to capture the essence of almost any complex system.

[edit]Mapping and mining the computational universe

In order to study simple rules and their often complex behaviour, Wolfram believes it is necessary to systematically explore all of these computational systems and document what they do. He believes this study should become a new branch of science, like physics or chemistry. The basic goal of this field is to understand and characterize the computational universe using experimental methods.

The proposed new branch of scientific exploration admits many different forms of scientific production. For instance, qualitative classifications like those found in biology are often the results of initial forays into the computational jungle. On the other hand, explicit proofs that certain systems compute this or that function are also admissible. There are also some forms of production that are in some ways unique to this field of study. For instance, the discovery of computational mechanisms that emerge in different systems but in bizarrely different forms.

Another kind of production involves the creation of programs for the analysis of computational systems — for in the NKS framework, these themselves should be simple programs, and subject to the same goals and methodology. 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. Wolfram believes that programs and their analysis should be visualized as directly as possible, and exhaustively examined by the thousands or more. Since this new field concerns abstract rules, it can in principle address issues relevant to other fields of science. However, in general Wolfram's idea is that novel ideas and mechanisms can be discovered in the computational universe — where they can be witnessed in their clearest forms — and then other fields can pick and choose among these discoveries for those they find relevant.

[edit]Systematic abstract science

While Wolfram promotes simple programs as a scientific discipline, he also insists that its methodology will revolutionize essentially every field of science. The basis for his claim is that the study of simple programs is the most minimal possible form of science, which is equally grounded in both abstraction and empirical experimentation. Every aspect of the methodology advocated in NKS 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 NKS allows computational mechanisms to be studied in their cleanest forms, Wolfram believes the process of doing NKS captures the essence of the process of doing science — and allows that process's strengths and shortcomings to be directly revealed.

Wolfram believes that the computational realities of the universe make science hard for fundamental reasons. But he also argues that by understanding the importance of these realities, we can learn to leverage them in our favor. For instance, instead of reverse engineering our theories from observation, we can simply enumerate systems and then try to match them to the behaviors we observe. A major theme of NKS style research is investigating the structure of the possibility space. Wolfram feels that science is far too ad hoc, in part because the models used are too complicated and/or unnecessarily organized around the limited primitives of traditional mathematics. Wolfram advocates using models whose variations are enumerable and whose consequences are straightforward to compute and analyze.

[edit]Philosophical underpinnings

Wolfram believes that one of his achievements is not just exclaiming, "computation is important!", but in providing a coherent system of ideas that justifies computation as an organizing principle of science. For instance, Wolfram's concept of computational irreducibility — that some complex computations cannot be short-cutted or "reduced" (cf. NP-hard) , is ultimately the reason why computational models of nature must be considered, in addition to traditional mathematical models. Likewise, his idea of intrinsic randomness generation — that natural systems can generate their own randomness, rather than using chaos theory or stochastic perturbations — implies that explicit computational models may in some cases provide more accurate and rich models of random-looking systems.

Based on his experimental results, Wolfram has developed the Principle of Computational Equivalence (see below), which asserts that almost all processes that are not obviously simple are of equivalent sophistication. From this seemingly vague single principle Wolfram draws a broad array of concrete deductions that reinforce many aspects of his theory. Possibly the most important among these is an explanation as to why we experience randomness and complexity: often, the systems we analyze are just as sophisticated as we are. Thus, complexity is not a special quality of systems, like for instance the concept of "heat", but simply a label for all systems whose computations are sophisticated. Understanding this makes the "normal science" of the NKS paradigm possible.

At the deepest level, Wolfram believes that like many of the most important scientific ideas, the Principle allows science to be more general by pointing out new ways in which humans are not special. In recent times, it has been thought that the complexity of human intelligence makes us special — but the Principle asserts otherwise. In a sense, many of Wolfram's ideas are based on understanding the scientific process — including the human mind — as operating within the same universe it studies, rather than somehow being outside it.

[edit]Principle of computational equivalence

The principle states that systems found in the natural world can perform computations up to a maximal ("universal") level of computational power. Most systems can attain this level. Systems, in principle, compute the same things as a computer. Computation is therefore simply a question of translating input and outputs from one system to another. Consequently, most systems are computationally equivalent. Proposed examples of such systems are the workings of the human brain and the evolution of weather systems.

[edit]Applications and results

There are a vast number of specific results and ideas in the NKS book, and they can be organized into several themes. One common theme of examples and applications is demonstrating how little it takes to achieve interesting behavior, and how the proper methodology can discover these cases.

First, there are perhaps several dozen cases where the NKS book introduces the simplest known system in some class that has a particular characteristic. 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 a large number of minimal examples of how simple programs exhibit phenomena like phase transitions, conserved quantities and 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.

Another common theme is taking facts about the computational universe as a whole and using them to reason about fields in a holistic way. For instance, Wolfram discusses how facts about the computational universe inform evolutionary theory, SETI, free will, computational complexity theory, and philosophical fields like ontology, epistemology, and even postmodernism.

Wolfram suggests that the theory of computational irreducibility may provide a resolution to the existence of free will in a nominally deterministic universe. He posits that the computational process in the brain of the being with free will is actually complex enough so that it cannot be captured in a simpler computation, due to the principle of computational irreducibility. Thus while the process is indeed deterministic, there is no better way to determine the being's will than to essentially run the experiment and let the being exercise it.

The book also contains a vast number of individual results — both experimental and analytic — about what a particular automaton computes, or what its characteristics are, using some methods of analysis.

[edit]NKS Summer School

Every year, Wolfram and his group of young top class instructors[3] organizes a summer school.[4] The first four summer schools from 2003 to 2006 were held at Brown university. Later the summer school was hosted by the university of Vermont at Burlington with the exception of the year 2009 that was held at the Istituto di Scienza e Tecnologie dell’Informazione of the CNR in Pisa, Italy. After seven consecutive summer schools more than 200 people have participated, some of which continued developing their 3-week research projects as their Master's or Ph.D thesis.[5] Some of the research done in the summer school has yielded important published results.[6][7]

[edit]NKS Conferences

Wolfram Research has organized three Wolfram Science conferences in Boston, MA, Washington, D.C. and Burlington, VT in the United States in the years 2003, 2006 and 2007 respectively. Two other independent NKS Midwest conferences have been organized at the Indiana University, Bloomington in 2005 and 2008. Other independent workshops related to NKS research have been also organized overseas, such as JOUAL (Just One Universal Algorithm) at the CNR in Pisa, Italy in 2009.

[edit]Reception

A New Kind of Science received extensive media publicity for a scientific book, generating scores of articles in such publications as The New York Times[8], Newsweek[9], Wired[10], and The Economist[11]. It was a best-seller and won numerous awards. NKS was reviewed in a large range of scientific journals, and several themes emerged in these reviews. Many reviewers enjoyed the quality of the book's production and the clear way Wolfram presented many ideas[12]. Even those reviewers who engaged in other criticisms found aspects of the book to be interesting and thought-provoking. On the other hand, many reviewers criticized Wolfram for his lack of modesty, poor editing, lack of mathematical rigor, and the lack of immediate utility of his ideas. Concerning the ultimate importance of the book, a common attitude was that of either skepticism or "wait and see". (Further detail of criticisms provided below.) Many reviewers and the media focused on the use of simple programs (cellular automata in particular) to model nature, rather than the more fundamental idea of systematically exploring the universe of simple programs.

[edit]Scientific philosophy

A key tenet of NKS is that the simpler the system, the more likely a version of it will recur in a wide variety of more complicated contexts. Therefore, NKS argues that systematically exploring the space of simple programs will lead to a base of reusable knowledge. However, many scientists believe that of all possible parameters, only some actually occur in the universe; that, for instance, of all possible variations of an equation, most will be essentially meaningless. NKS has also been criticized for asserting that the behavior of simple systems is somehow representative of all systems.

[edit]Methodology

A common criticism of NKS is that it does not follow established scientific methodology[who?]. NKS does not establish rigorous mathematical definitions[13], nor does it attempt to prove theorems.[14] Along these lines, NKS has also been criticized for being heavily visual, with much information conveyed by pictures that do not have formal meaning. It has also been criticized for not using modern research in the field of complexity, particularly the works that have studied complexity from a rigorous mathematical perspective.

Critics also note that none of the book's contents were published in peer-reviewed journals, the standard method for distributing new results, and complain it insufficiently credited other scientists whose work it is built on. Wolfram relegates all discussion of other people to his lengthy endnotes and thus no one is directly credited in the text. His critics argue that even the endnotes are misleading, glossing over many relevant discoveries and thus making Wolfram's work seem more novel.

[edit]Utility

NKS has been criticized for not providing specific results that would be immediately applicable to ongoing scientific research. There has also been criticism, implicit and explicit, that the study of simple programs has little connection to the physical universe, and hence is of limited value. Steven Weinberg has pointed out that no real world system has been explained using Wolfram's methods in a satisfactory fashion.[15]

[edit]Principle of computational equivalence

The PCE has been criticized for being vague, unmathematical, and for not making directly verifiable predictions; however, Wolfram's group has described the principle as such, not a law, theorem or formula. It has also been criticized for being contrary to the spirit of research in mathematical logic and computational complexity theory, which seek to make fine-grained distinctions between levels of computational sophistication. Others suggest it is little more than a rechristening of the Church-Turing thesis. However, the Church-Turing thesis imposes an upper limit while Wolfram's PCE suggests the nonexistence of intermediate degrees of computation sending a computational system either to the upper level (universal) or to the lowest degree, explained by Klaus Sutner[16] in terms of physics-like computation as a zero-one law claiming that, in practice, constructing actual computers with intermediate degrees is highly artificial and hasn't ever been done, hence endorsing Wolfram's intuition captured in his PCE.

[edit]The fundamental theory (NKS Chapter 9)

Wolfram's speculations of a direction towards a fundamental theory of physics have been criticized as vague and obsolete. Scott Aaronson, Assistant Professor of Electrical Engineering and Computer Science at MIT, also claims that Wolfram's methods cannot be compatible with both special relativity and Bell's theorem violations, which conflicts with the observed results of Bell test experiments.[17]

In a 2002 review of NKS, the Nobel laureate and elementary particle physicist Steven Weinberg wrote, "Wolfram himself is a lapsed elementary particle physicist, and I suppose he can't resist trying to apply his experience with digital computer programs to the laws of nature. This has led him to the view (also considered in a 1981 paper by Richard Feynman) that nature is discrete rather than continuous. 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. Following an idea of Edward Fredkin, he concludes that the universe itself would then be an automaton, like a giant computer. It's possible, but I can't see any motivation for these speculations, except that this is the sort of system that Wolfram and others have become used to in their work on computers. So might a carpenter, looking at the moon, suppose that it is made of wood."[18]

According to NKS Chapter 9, special relativity theory and quantum field theory are merely approximations to a digital network with inaccessible signal propagation below the Planck scale. NKS Chapter 9 and M-theory both attempt to unify general relativity theory and quantum field theory. M-theory postulates that there is a minimum physical wavelength and that vibrating string-like entities can model all of physics. NKS Chapter 9 postulates that there is a finite automaton that builds time, space, and energy from informational substrate below the Planck scale. According to Wolfram, infinities and infinitesimals do not occur in nature, except perhaps for time as a potential infinity. In particular, there is a maximum physical wavelength in addition to the minimum physical wavelength postulated by M-theory. Neither of these scales are accessible to current physics, however, and hence this is untestable speculation until experiments reach those scales.

In the NKS theory, the basic physical realities of time, space, and energy are merely approximations that arise from a few simple rules that operate with hidden determinism below the Planck scale. According to Wolfram, "building on the discovery that even simple programs can yield highly complex behavior, A New Kind of Science shows that with appropriate kinds of rules, simple programs can give rise to behavior that reproduces a remarkable range of known features of our universe — leading to the bold assertion that there could be a simple short program that represents a truly fundamental model of the universe, and which if run for long enough would reproduce the behavior of our world in every detail.”[19]

[edit]Natural selection

Wolfram's claim that natural selection is not the fundamental cause of complexity in biology has led some to state that Wolfram does not understand the theory of evolution[20]. However, some experts have acknowledged that natural selection leaves many unanswered questions, which information theory might be able to explain[21][22]. In this context, Wolfram's work is similar to that of D'Arcy Thompson.

[edit]Originality and self-image

NKS has been heavily criticized as not being original or important enough to justify its title and claims, mostly by people who argue that the book is about simple systems generating complex behavior. However, even though the fact that simple systems are capable of complicated behavior is an important part of the book, the main contribution is the new methodology of mining the computational universe. Edward Fredkin and Konrad Zuse pioneered the idea of a computable universe, the former by writing a line in his book on how the world might be like a cellular automaton, and later further developed by Fredkin using a toy model called Salt[23]. It has been claimed that NKS tries to take these ideas as its own. This has been mainly suggested by people thinking that Wolfram's main thesis is that the universe is a cellular automaton in spite of the fact that Wolfram's proposal as a discrete model of the universe is a trivalent network. Wolfram himself considers that a cellular automaton model is unsuitable to describe quantum and relativistic properties of nature, as explained in his NKS book.

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. However Schmidhuber did his original suggestion in purely algorithmic-theoretic terms, while Wolfram's proposal has been to test every candidate to find the right one in physical terms, in which the known laws of physics actually hold, rather than a theoretical description of what the candidates should algorithmically be.

Additionally, the core idea that very simple rules often generate great complexity is already an established idea in science, particularly in chaos theory and complexity theory research - and to some researchers, this field is considered well-understood[who?]. The authoritative manner in which NKS presents a vast number of examples and arguments has been criticized as leading the reader to believe that each of these ideas was original to Wolfram, however the notes section at the end of his book acknowledges many of the discoveries made by these other scientists citing their names together with historical facts, although not in the form of a traditional bibliography section. This is generally considered insufficient in scientific literature, however - end notes are normally reserved for only indirectly related material, and lay readers typically ignore end notes, resulting in the impression that the author performed all the work.

In particular, one of the most substantial new technical results presented in the book, that the rule 110 cellular automaton is Turing complete, was not proven by Wolfram, but by his research assistant, Matthew Cook. This is not particularly a surprise since as explained by Wolfram himself, the book was actually a kind of project involving a group of his research assistants led by Wolfram himself, something that means he didn't do every single experiment or contribution directly, as he also acknowledges in the book. The research assistants were however paid for this work as hired by Wolfram's company, not by a university.

Some have argued[who?] that the use of computer simulation is ubiquitous, and instead of starting a paradigm shift NKS just adds justification to a paradigm shift that has been undertaken. Wolfram's NKS might then seem as the book explicitly describing this shift.

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