Mathematics

*
A New Kind of Science* by
Stephen Wolfram, hardcover; 1197 pages (Wolfram Media, Inc.) (Publisher
order) (British
order) was born in London and educated at Eton, Oxford, and Caltech. He
received his Ph.D. in theoretical physics in 1979 at the age of 20, having
already made lasting contributions to particle physics and cosmology. In 1981
his work was recognized by a MacArthur award. In the early 1980s he made a
series of classic discoveries about systems known as cellular automata, which
have yielded many new insights in physics, mathematics, computer science,
biology and other fields. In 1986 he founded Wolfram Research, Inc. and began
the creation of *Mathematica*, now the world's leading software system for
technical computing and symbolic programming, and the tool that made *
A New Kind of Science *possible. Over the past decade Wolfram has divided
his time between the leadership of his company and his pursuit of basic science.
Written with exceptional clarity, and illustrated by nearly a thousand original
pictures, this seminal book allows scientists and nonscientists alike to
participate in what promises to be a major intellectual revolution.

Ray Kurzweil challenges the ability of these ideas to fully explain the
complexities of life, intelligence, and physical phenomena.*
Mathematica Book* (4

In *
A New Kind of Science*, Stephen Wolfram presents for the first time a
series of surprising and dramatic discoveries that force a whole new way of
looking at the operation of our universe. Wolfram's discoveries‑which build on
his now‑classic work from the early 1980s‑have been
awaited by the scientific community for more than a decade**. **Wolfram's new science is sure to become an integral part of future
intellectual development in many fields, including physics, biology, computer
science, mathematics, technology, philosophy and the social sciences.

Wolfram is widely regarded as one of the world's most
original scientists, as well as a leading innovator in scientific and technical
computing. Perhaps best known as the inventor and entrepreneur behind *
Mathematica**, *he also helped lay
the groundwork for the field of complexity theory in the early 1980s. Wolfram
has spent the last nearly eleven years developing and refining the radically new
ideas of *
A New Kind of Science*.

In *
A New Kind of Science*, Wolfram shows that by thinking in terms of simple
programs instead of mathematical equations it becomes possible to capture the
essential mechanisms of many systems in nature that have eluded scientific
analysis‑often for centuries. Much as telescopes and microscopes once revealed
great new worlds to science, Wolfram's computer experiments now reveal a strange
new world that overturns some of our most basic intuition about how things ought
to work.

*A
New Kind of Science *provides new insight into a remarkable array of
fundamental questions, including how biology produces complexity, how randomness
arises in physics, what space and time fundamentally are, how there can be both
free will and determinism, how general mathematics really is and what ultimate
limits there are to science. The discoveries Wolfram has made suggest new kinds
of technology, from new strategies for creating computers on atomic scales, to
new forms of cryptography, new concepts for programming, and new methodologies
for creating systems that achieve biological and medical functions.

Wolfram's discoveries have allowed him to address a host of issues that have never before been accessible to science. Through his work, the achievements of earlier initiatives such as cybernetics, chaos theory, fractals, and complexity theorywhose further progress has been stymied by the lack of these key discoveriesare now subsumed as elements in a much larger intellectual structure.

Wolfram's reputation for introducing new concepts in an
accessible and engaging manner is evident throughout *
A New Kind of Science*. The book contains 973 illustrations that both
capture Wolfram's discovery process and form an integral part of the pursuit of
his new science. An extensive notes section illuminates the main text and
provides fascinating historical context.

Science often measures progress in small steps. *
A New Kind of Science *represents a seismic shift‑one that defines a
whole new way of thinking and creates infinite possibilities for scientific and
technological breakthroughs in the future.

For more than three centuries, mathematical equations and
methods such as calculus have been taken as the foundation for the exact
sciences. There have been many profound successes, but a great many important
and obvious phenomena in nature remain unexplained‑especially ones where more
complex forms or behavior are observed. *
A New Kind of Science *builds a framework that shows why equations have
had limitations, and how by going beyond them many new and essential mechanisms
in nature can be captured.

Mathematical equations correspond to particular kinds of
rules. Computer programs can embody far more general rules. *
A New Kind of Science *describes a vast array of remarkable new
discoveries made by thinking in terms of programs‑and how these discoveries
force a rethinking of the foundations of many existing areas of science.

Everyday experience tends to make one think that it is
difficult to get complex behavior, and that to do so requires complicated
underlying rules. A crucial discovery in *
A New Kind of Science *is that among programs this is not true‑and that
even some of the very simplest possible programs can produce behavior that in a
fundamental sense is as complex as anything in our universe. There have been
hints of related phenomena for a very long time, but without the conceptual
framework of *
A New Kind of Science *they have been largely ignored or misunderstood.
The discovery now that simple programs can produce immense complexity forces a
major shift in scientific intuition.

How nature seems so effortlessly to produce forms so much
more complex than in typical human artifacts has long been a fundamental
mystery‑often discussed for example in theological contexts. *
A New Kind of Science *gives extensive evidence that the secret is just
that nature uses the mechanisms of simple programs, which have never been
captured in traditional science.

*
A New Kind of Science *shows that extremely simple programs picked for
example at random can produce behavior that is far more complex than typical
programs intentionally set up by programmers. The fundamental engineering
concept that one must always be able to foresee the outcome of programs one
writes has prevented all but a tiny fraction of all possible programs from being
considered. The idea of allowing more general programs has great potential
significance for technology.

In their times both telescopes and microscopes revealed
vast worlds that had never been seen before. Through the ideas of *
A New Kind of Science**, *
computer experiments now also reveal a vast new world, in many ways more diverse
and surprising even than the world seen in astronomy, or than the flora and
fauna discovered by explorers of the Earth in past centuries. Many of the basic
experiments in *
A New Kind of Science *could in principle have been done by mosaic makers
thousands of years ago. But it took new intuition and new tools to unlock what
was needed to do the right experiments and understand their significance.

Despite attempts from approaches like chaos theory, no
fundamental explanation has ever been found for randomness in physical phenomena
such as fluid turbulence or patterns of fracture. *
A New Kind of Science *presents an explanation based on simple programs
that for example predicts surprising effects such as repeatable randomness.

The Second Law of Thermodynamics (Law of Entropy Increase)
has been a foundational principle in physics for more than a century, but no
satisfactory fundamental explanation for it has ever been given. Using ideas
from studying simple programs, *
A New Kind of Science *gives an explanation, and in doing so shows
limitations of the Second Law.

From traditional intuition one expects that the observed
complexity of biological organisms must have a complex origin‑presumably
associated with a long process of adaptation and natural selection. *
A New Kind of Science *shows how complex features of many biological
organisms can be explained instead through the inevitable behavior of simple
programs associated with their growth and development. This implies that biology
need not just reflect historical accidents, and that a general study of simple
programs can lead to a predictive theory of at least certain aspects of biology.

The underlying mechanism that leads for example to seemingly random fluctuations in prices in markets has never been clear. Discoveries about simple programs‑such as the phenomenon of intrinsic randomness generation‑provide potentially important new insights on such issues.

In its recent history, physics has tried to use
increasingly elaborate mathematical models to reproduce the universe. But
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
single short program that represents a truly fundamental model of the universe,
and which if run for long enough would reproduce the behavior of our universe in
every detail.

Throughout almost the entire history of science, space has
been viewed as something fundamental ‑and typically continuous. *
A New Kind of Science *suggests that space as we perceive it is in fact
not fundamental, but is instead merely the large‑scale limit of an underlying
discrete network of connections. Models constructed on this basis then lead to
new ideas about such issues as the origins of gravity and general relativity,
the true nature of elementary particles and the validity of quantum mechanics.

The standard mathematical formulation of relativity theory
suggests that‑despite our everyday impression‑time should be viewed as a fourth
dimension much like space. *
A New Kind of Science *suggests however that time as we perceive it may
instead emerge from an underlying process that makes it quite different from
space. And through the concept of causal invariance the properties of time seem
to lead almost inexorably to a whole collection of surprising results that agree
with existing observations in physics‑including the special and general theories
of relativity, and perhaps also quantum mechanics.

Seeing the complicated circuitry of existing computers,
one would think that it must take a complicated system to be able to do
arbitrary computation. But *
A New Kind of Science *shows that this is not the case, and that in fact
universal computation can be achieved even in systems with very simple
underlying rules. As a specific example, it gives a proof that the so‑called
rule 110 cellular automaton‑whose rules are almost trivial to describe‑is
universal, so that in principle it can be programmed to perform any computation.
And as a side result, this leads to by far the simplest known universal Turing
machine.

If universal computation required having a system as elaborate as a present‑day computer, it would be inconceivable that typical systems in nature would show it. But the surprising discovery that even systems with very simple rules can exhibit universality implies that it should be common among systems in nature‑leading to many important conclusions about a host of fundamental issues in science, mathematics and technology.

Many of the discoveries in *
A New Kind of Science *can be summarized in the bold new Principle of
Computational Equivalence, which states in essence that processes that do not
look simple almost always correspond to computations of exactly equivalent
sophistication. This runs counter to the implicit assumption that different
systems should do all sorts of different levels and types of computations. But
the Principle of Computational Equivalence has the remarkable implication that
instead they are almost all equivalent‑leading to an almost unprecedentedly
broad unification of statements about different kinds of systems in nature and
elsewhere.

We would normally assume that we as humans are capable of
much more sophisticated computations than systems in nature such as turbulent
fluids or collections of gravitating masses. But the discoveries in *
A New Kind of Science **
*imply that this is not the case, yielding a radically new
perspective on our place in the universe.

Statements like "the weather has a mind of its own" have
usually been considered not scientifically relevant. But the Principle of
Computational Equivalence in *
A New Kind of Science *shows that processes like the flow of air in the
atmosphere are computationally equivalent to minds, providing a major new
scientific perspective, and reopening many debates about views of nature with an
animistic character.

It has usually been assumed that detecting
extraterrestrial signals from a sophisticated mathematical computation would
provide evidence for extraterrestrial intelligence. But the discoveries in *
A New Kind of Science *show that such computation can actually be
produced by very simple underlying rules‑of kinds that can occur in simple
physical systems with nothing like what we normally consider intelligence. The
result is a new view of the character of intelligence, and a collection of ideas
about the nature of purpose, and recognizing it in ultimate extrapolations of
technology.

One might have thought that we would always be able to
recognize signs of the simplicity of an underlying program in any output it
produces. But *
A New Kind of Science *studies all the various common methods of
perception and analysis that we use, and shows that all of them are ultimately
limited to recognizing only specific forms of regularity, which may not be
present in the behavior of even very simple programs‑with implications for
cryptography and for the foundations of fields such as statistics.

We tend to consider behavior complex when we cannot readily reduce it to a simple summary. If all processes are viewed as computations, then doing such reduction in effect requires us as observers to be capable of computations that are more sophisticated than the ones going on in the systems we are observing. But the Principle of Computational Equivalence implies that usually the computations will be of exactly the same sophistication‑providing a fundamental explanation of why the behavior we observe must seem to us complex.

Most of the great successes of traditional exact science
have ultimately come from finding mathematical formulas to describe the outcome
of the evolution of a system. But this requires that the evolution be
computationally reducible, so that the computational work involved can just be
reduced to evaluation of a formula. *
A New Kind of Science *shows however that among most systems
computational reducibility is rare, and computational irreducibility is the
norm. This explains some of the observed limitations of existing science, and
shows that there are cases where theoretical prediction is effectively not
possible, and that observation or experiment must inevitably be used.

For centuries there has been debate about how apparent
human free will can be consistent with deterministic underlying laws in the
universe. The phenomenon of computational irreducibility described in *
A New Kind of Science** *finally
provides a scientifically based resolution of this apparent dichotomy.

The phenomenon of formal undecidability discovered in
mathematics in the 1930s through Godel's Theorem has normally been viewed as
esoteric, and of little relevance to ordinary science. *
A New Kind of Science *shows however that undecidability is not only
possible but actually common in many systems in nature, leading to important
philosophical conclusions about what can and cannot be known in natural science.

Mathematical theorems such as Fermat's Last Theorem that
are easy to state often seem to require immensely long proofs. In *
A New Kind of Science *this fundamental observation about mathematics is
explained on the basis of the phenomenon of computational irreducibility, and is
shown to be a reflection of results like Godel's Theorem being far more
significant and widespread than has been believed before.

Mathematics is often assumed to be very general, in effect
covering any possible abstract system. But the discoveries in *
A New Kind of Science *show that mathematics as it has traditionally been
practiced has actually stayed very close to its historical roots in antiquity,
and has failed to cover a vast range of possible abstract systems‑many of which
are much richer in behavior than the systems actually studied in existing
mathematics. Among new results are unprecedentedly short representations of
existing formal systems such as logic, used to show just how arbitrarily systems
like these have in effect been picked by the history of mathematics. The
framework created in *
A New Kind of Science *provides a major generalization of mathematics,
and shows how fundamentally limited the traditional theorem‑proof approach to
mathematics must ultimately be.

As a vehicle for teaching precise analytical thinking, *
A New Kind of Science *represents a major alternative to existing
mathematics, with such advantages as greater explicitness and visual appeal,
more straightforward applicability to certain issues in natural science, and
side benefits of learning practical computer science and programming.

In existing technology complex tasks tend to be achieved
by systems with elaborately arranged parts. But the discoveries in *
A New Kind of Science *show that complex behavior can be achieved by
systems with an extremely simple underlying structure‑that is for example
potentially easy to implement at an atomic scale. Many specific systems, such as
cellular automata, studied in *
A New Kind of Science *are likely to find their way into a new generation
of technological systems.

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