Mathematical equations do not capture many
of nature’s most essential mechanisms
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.
Thinking in terms of programs rather than
equations opens up a new kind of science
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.
Even extremely simple programs can produce
behavior of immense complexity
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.
Simple programs can yield behavior startlingly
like what we see in nature
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.
Simple programs can do much more than typical
programs written by programmers
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.
Simple computer experiments reveal a vast
world of new phenomena
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.
Randomness in physics can be explained by
mechanisms of simple programs
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.
Thermodynamic behavior can be explained by
mechanisms of simple programs
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.
Complexity in biology can be explained by
mechanisms of simple programs
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.
Simple programs may lay the groundwork for
new insights about financial systems
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.
Our whole universe may be governed by a single
underlying simple program
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.
Underlying space there may be a simple discrete
structure
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 largescale 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.
Time may have a fundamentally different nature
from space
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.
Systems with exceptionally simple rules can
be universal computers
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 socalled 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.
Many systems in nature are capable of universal
computation
If universal computation required having
a system as elaborate as a presentday 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.
The Principle of Computational Equivalence
provides a broad synthesis
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.
Many systems in nature are computationally
equivalent to us as humans
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.
Many systems in nature can show features
like intelligence
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.
Extraterrestrial intelligence is inevitably
difficult to define and recognize
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.
It is easy to make randomness that we cannot
decode
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.
Apparent complexity in nature follows from
computational equivalence
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.
Many important phenomena are computationally
irreducible
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.
Apparent free will can arise from computational
irreducibility
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.
Undecidability occurs in natural science,
not just mathematics
The phenomenon of formal undecidability discovered
in mathematics in the 1930s through Gödel’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.
The difficulty of doing mathematics reflects
computational irreducibility
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 Gödel’s Theorem being far more significant
and widespread than has been believed before.
Existing mathematics covers only a tiny fraction
of all possibilities
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 theoremproof approach to mathematics must
ultimately be.
Studying simple programs can form a basis
for technical education
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.
Mechanisms from simple programs suggest new
kinds of technology
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.
