Tabak J. Mathematics and the Laws of Nature: Developing the Language of Science
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xx MATHEMATICS AND THE LAWS OF NATURE
cause and effect, probabilistic laws apply only to large “ensembles”
or collections of outcomes. By way of example, consider some
hypothetical phenomenon with three possible outcomes, which
we will call A, B, and C. If the phenomenon is repeated many
times, outcome A will occur with a frequency of approximately
f(A); outcome B will have a frequency of approximately f(B); and
outcome C will have a frequency of approximately f(C); and the
more frequently the phenomenon is observed, the closer the
observed frequencies will approach the ideal ones. These kinds
of probabilistic laws have become quite common. The example
examined here is Mendel’s laws of inheritance, an early example of
this type of probabilistic thinking.
In addition to a great deal of new information on modern ideas
about conservation laws, the second edition is further enhanced
by new photographs and an especially interesting interview with
Dr. Renate Hagedorn about her work at the European Centre
for Medium-Range Weather Forecasts (ECMWF), a proving
ground for many of the ideas described in this volume. An updated
chronology, glossary, and suggestions for further reading are also
included.
There are few books written for the general reader that discuss
the subject matter addressed in this volume. Given the importance
of the subject, this is difficult to explain. The ideas described in
this book form the conceptual basis for modern scientific and
engineering research. In many ways, they form the basis for our
modern worldview. They are also intellectually appealing. It is
hoped that in addition to functioning as a valued resource, this
volume will prove to be an enjoyable reading experience.
1
1
the preliminaries
The phrase “law of nature” has come to mean different things
to different people. In this book it is used in a very specific way.
Before this narrative begins to trace the history of natural laws and
their uses, it is worthwhile to provide a definition of the subject
matter of this book. In other words, “What is a law of nature?”
Mathematics versus Science and Engineering
Historically, the development of natural laws and their uses
proceeded in tandem with the development of mathematics.
Mathematics provides a simplified and incomplete model for the
type of reasoning that scientists and engineers must do if they are
to successfully discover and create tomorrow’s world. In contrast
to science and engineering, mathematics depends solely on deduc-
tive reasoning, the type of reasoning that draws specific conclu-
sions from general principles through the use of rules of (logical)
inference. Today, engineers and scientists use many other tools in
addition to deductive reasoning, but deductive reasoning still plays
a central role in most scientific and engineering endeavors.
Formal deductive reasoning is not a “natural” act in the sense
that people are born with the skill. It must be learned, and in fact
there was a time when no one knew how to reason deductively.
The ancient Greeks appear to have been the innovators in this
regard, and the ancient Greek most closely associated with deduc-
tive reasoning today is the mathematician Euclid of Alexandria (fl.
ca. 300 b.c.e.), the author of Elements, one of the most influential
books in history.
2 MATHEMATICS AND THE LAWS OF NATURE
Euclid did not invent deductive reasoning. Elements is a text-
book. Presumably the ideas in it were firmly established before
Euclid began writing his most famous work. Perhaps other books,
just as good as Euclid’s, were available to the mathematicians
of Alexandria even before Euclid’s birth, but for three reasons
Elements has proven to be the most influential of the lot. First, it
is skillfully written; second, it provides a clear example of what
deductive reasoning involves; and, most important, Euclid’s book
survived. Most Greek mathematical texts were lost.
Euclid begins his work by listing five axioms and five postulates.
The axioms and postulates describe the most basic properties of
what we now call Euclidean geometry, although Euclid called
it no such thing. Axioms and postulates were treated differently
by Euclid. Axioms, he believed, were more a matter of common
sense. “The whole is greater than the part” is axiom number five,
for example. Postulates, according to Euclid, were of a more spe-
cialized nature. Postulate number three, for example, states, “A
circle may be described with any point as center and any distance
as radius.” Today, mathematicians make no distinction between
axioms and postulates and usually refer to both as axioms, a prac-
tice that we now adopt.
Euclid’s choice of axioms is crucial because all of the geometrical
discoveries that he describes in his book are logical consequences
of them. Euclid’s axioms are what make Euclidean geometry
Euclidean as opposed to some other type of geometry. Every other
type of geometry depends upon a set of axioms that are logically
different from those chosen by Euclid. Euclid’s axioms are the
Greek solution to a peculiar difficulty associated with deductive
reasoning. To understand the difficulty, imagine logically deduc-
ing one statement, which we will call statement B, from another
statement, which we will call statement A. In other words, we
demonstrate (using logic) that the truth of A implies the truth of
B. Or equivalently, statement B is true whenever statement A is
true. But this type of logical reasoning does not resolve the truth
of statement B. It simply shifts our attention to the truth of state-
ment A because logically all we have established is that if A is true
then B is true. But is A true? In attempting to reason deductively,
The Preliminaries 3
the Greeks had uncovered an endless chain of implications with
the truth of each statement depending on the truth of the preced-
ing one.
Their solution was to list a series of statements, called axioms,
the truth of which cannot be called into question. Once the axi-
oms are chosen, the act of mathematical discovery is reduced to
drawing logical deductions from the axioms and from whatever
statements have been previously shown to be consequences of the
axioms. In short, the axioms are the geometry. They are the jump-
ing-off point, the place at which reasoning begins. Once they have
been specified, all other properties of the geometry are similarly
specified, although most of the other properties of the geometric
system will not be apparent without additional research.
Because the choice of axioms is so important for a deductive
system—geometric or otherwise—a great deal of care goes into
choosing them. Every set of axioms must satisfy certain logical
criteria. Perhaps the most important criterion is that the axioms
should be logically consistent, which means that it should not be
possible to prove a particular statement both true and false using
the same set of axioms. In addition, each axiom should be logically
independent of all the others in the set. Logical independence
means that it should be impossible to prove that one axiom is a
logical consequence of other axioms. Each axiom should, there-
fore, be a stand-alone entity. Finally, the axioms should be com-
plete enough to incorporate those properties of the system that
one wants the system to have at the outset.
Choosing a set of axioms that satisfies these criteria is no easy
task, but once the axioms are determined, all properties of the
resulting system are determined, and all that remains is to discover
what, in a logical sense, is already there for all to see. This for-
mal, logically rigorous method of discovery changed the world. It
enabled those who understood the method to use their minds with
a new precision, to discover new knowledge, and to prove that their
newly discovered conclusions were correct—or at least consistent
with their premises. This is part of the beauty of the mathemati-
cal method. The axioms of each mathematical system are stated
explicitly, the rules of logic are stated explicitly, and so any facts
4 MATHEMATICS AND THE LAWS OF NATURE
that are uncovered by making logical deductions from the axioms
must be mathematically correct, because in mathematics logic is
the sole arbiter of truth. (Today it is recognized that Euclid’s logic
was well developed but not perfect. Contemporary mathematicians
formulate the axioms for what is now called Euclidean geometry in
a different and somewhat more rigorous way, but the differences
between their formulation and Euclid’s are not large. The Greeks
were essentially correct, and we will give no further consideration
to minor flaws in Euclid’s presentation.)
During the 19th century, mathematicians came to see the choice
of axioms as essentially arbitrary—at least from a logical point of
view—and with this change in understanding, Euclidean geometry
ceased to be as central to mathematical and scientific thinking as
it had been for the preceding 2,000 years. More enduring than
Euclid’s geometry was the model for mathematical thought that
Euclidean geometry exemplifies, a system of thought that depends
solely on axioms and rules of logical inference.
In physics and chemistry, the two most successful (and most
mathematical) of all branches of science, and in engineering,
the analogues to mathematical axioms are laws of nature. And as
with mathematical axioms, laws of nature have been used to great
effect. By making logical deductions from one or more laws of
nature, a great deal of new knowledge has been discovered. One
might conclude, therefore, that laws of nature are the axioms of
science and engineering, but there are also important differences
between axioms and laws of nature. Mathematical systems are
highly ordered and logical because they were created to be so.
Mathematical systems are products of the human mind. They con-
tain what their creators put into them and nothing more. By con-
trast, engineers and scientists must work with nature as it exists:
The forces that bind protons and neutrons together within an
atomic nucleus, for example, can be measured with a certain pre-
cision, but alternative forces cannot be chosen to make the work
easier; the locations of mountains and craters on the surface of the
Moon may or may not facilitate the study of lunar geography, but
they are not subject to revision; and the chemical composition of
a given mass of coal is whatever careful measurements reveal it to
The Preliminaries 5
be. Nature is to some degree arbitrary, and every successful model
of nature must accommodate facts that are, on the face of it, both
arbitrary and important.
But in the face of arbitrariness in nature, engineers and scien-
tists also maintain a belief in an underlying order to nature. They
search for, and they sometimes find, statements about nature that
are generally true, and beginning with these statements, they
deduce other statements. They can sometimes accomplish what
mathematicians take for granted, deducing new knowledge from
previously established facts. This is part of what it means to “do”
science and engineering. For scientists and engineers, laws of
nature are the place where logical reasoning begins.
Differences between axioms and laws of nature are, however,
apparent both at the beginning of the reasoning process and at the
end. As previously mentioned, mathematicians are free to choose
whatever axioms they wish provided that the axioms are consistent
and independent. By contrast, engineers and scientists must adopt
laws of nature that reasonably reflect the physical world as it mani-
fests itself through observation and experiment. It is not that they
have no choices with respect to the axioms that they adopt, but in
contrast to mathematicians, their choices are narrower.
Laws of nature are always subject to continual experimental
verification. As experimental results accumulate, confidence in
the validity of a particular law increases—provided no experiment
contradicts the law. But the adoption of any statement as a law of
nature remains provisional. A thousand experiments may support
it, but one reproducible experiment may, if it contradicts the law,
invalidate it or at least restrict its applicability.
The process by which general conclusions are drawn from the
analysis of multiple observations and experiments is called induc-
tive reasoning, and inductive reasoning is also an important part
of all scientific and engineering endeavors. Inductive reasoning
is the bridge between reality and the deductive models of reality
upon which scientists and engineers rely. By contrast, inductive
reasoning plays no part in mathematics except possibly as a way
of identifying problems about which to reason deductively. (For
example, if we made a list of prime numbers, we would find that
6 MATHEMATICS AND THE LAWS OF NATURE
no matter how many prime numbers we listed, we could still find
another prime number that was not on the list. Is the set of prime
numbers infinite? Our inductive experiment suggests the ques-
tion, but cannot be used to answer it. The question of whether or
not the set of prime numbers is infinite can only be answered by
reasoning deductively.)
Specific laws of nature are valid only for specific classes of
phenomena. There is no theory of everything. In other words, a
law of nature can be true in one context and false in another. It is
important, therefore, to specify not just the law of nature but the
context (or system) in which it holds. Logically speaking, laws of
nature must meet the same basic requirements that axioms meet in
mathematics: They should be consistent and independent and rea-
sonably complete. They must also be predictive in the sense that
they are useful for discovering new knowledge. It is, for example, a
true statement that the flag of France is blue, white, and red, but it
is hard to see how this statement can form the basis for a sequence
of logical deductions about anything. By contrast, the statement
that energy can neither be created nor destroyed has proven to be
an extremely useful basis from which to reason deductively about
many phenomena. Although why energy can neither be created
nor destroyed is even less apparent than why the flag of France is
blue, white, and red.
Finally, at the end of the reasoning process, conclusions must
be tested by experiment because scientists and engineers are, in
some ways, held to a higher standard than mathematicians. Reality
matters in the sciences; it has no place in mathematics. When a
mathematician deduces a new statement, that new statement is
true provided that no errors were made in the logic that connected
premise and conclusion. In mathematics, logic alone determines
what is true from what is false. In engineering and science, every
conclusion even if it is logically correct must also coincide with
reality as reality is revealed by experiments. Experimental evi-
dence, not logic, is the arbiter of truth in science. If a logically
deduced conclusion fails to agree with the experimental evidence
then the laws of nature used to deduce the conclusion contain
errors, or they were misapplied.
The Preliminaries 7
Despite these qualifications about the difficulty of using laws of
nature and the limitations on any conclusions derived from them,
history shows that the search for laws of nature and the applica-
tion of laws of nature in the search for scientific truth have been
some of the most important of all human activities. Mathematics
established the model by which natural laws are used to investigate
nature, and natural laws have, in turn, sometimes been adopted
as axioms for mathematical systems. The history of natural laws
as we understand the concept today starts at the beginning of
recorded history.
The Mesopotamians
Five thousand years ago on a hot, flat, largely treeless expanse
of land, a land devoid of stone and other building materials, the
people whom we know as Sumerians began to build a civiliza-
tion. They irrigated land. They built cities. They built schools
and developed one of the first written languages in history, called
cuneiform. Because their land was difficult to defend against
military attack, and because the city-states in the area frequently
attacked one another, the political history of Sumer is complicated
and bloody. Despite the turmoil, however, their culture endured.
Over time Sumerian culture became the foundation for a larger
culture. Slowly the culture of the Sumerians was absorbed and
transformed into the culture of Mesopotamia.
Of special importance to us are the written records of the
Mesopotamians. The system of writing that the Sumerians began,
a system characterized by imprints on clay tablets, was slowly
changed and enriched by those who succeeded them. Long after
the people of Mesopotamia ceased speaking the Sumerian lan-
guage they continued to incorporate elements of the Sumerian
written language into their own written language. The last known
cuneiform texts—which concern astronomy—date from the first
century c.e. That is 3,000 years of cuneiform writing! Over the
next 2,000 years Mesopotamian civilization was largely forgot-
ten. Mesopotamian culture was eventually rediscovered in the
19th century, when archaeological excavations unearthed and
8 MATHEMATICS AND THE LAWS OF NATURE
catalogued hundreds of thou-
sands of clay tablets. In time
scholars translated the tablets
and discovered mathematical
computations, histories of mil-
itary campaigns, letters from
students to parents, invento-
ries of goods, lists of laws,
and records of astronomical
observations. It was an aston-
ishingly complete record of
one of history’s earliest and
longest-lasting civilizations.
The Ancient Sky
The Mesopotamians were
avid astronomers. They seem
to have watched the heavens
almost continuously for thou-
sands of years, carefully doc-
umenting their observations
in the form of astronomical
diaries, forming theories, and
making predictions. Through
Mesopotamian astronomy we can see one of the earliest attempts
to develop a system of “laws” for the purpose of describing natural
phenomena.
Mesopotamian astronomers concentrated on the problem of
predicting future astronomical phenomena. They were less inter-
ested in why things occurred than they were in knowing what
was going to happen next. Making these astronomical predic-
tions successfully required a high level of social organization. For
many years their method of learning about astronomy emphasized
examining the records of past astronomical observations for clues
about future events. To do this they required good educational
institutions, a written language, skilled observers, and careful
Cuneiform tablet. The Mesopotamians
carefully recorded their astronomical
observations on clay tablets that were
then archived and maintained over
many centuries.
The Preliminaries 9
record keepers. They required stable institutions that could col-
lect and preserve records over the course of many generations.
Many cultural and educational barriers had to be overcome before
they could begin a systematic search for the laws of nature. To
understand the kind of astronomical predictions in which the
Mesopotamians were interested (and in which they excelled),
knowing a little about how they understood the sky is helpful.
The modern reader needs patience and imagination to appreci-
ate what it was those astronomers were observing, not just because
we are unfamiliar with Mesopotamian science, but also because we
are unfamiliar with the night sky. In the days before electric lights
the night skies shimmered with the light of a multitude of stars,
both bright and dim. Nebulae were visible to the naked eye. A
brilliant night sky was a familiar sight to people all over the planet.
Today most of us have rarely, if ever, seen a truly dark, clear sky.
We have never seen a nebula except as a picture in a book or an
image on the Web. We cannot. The light of most of these objects
is lost in the glare of streetlights and headlights, stoplights and
illuminated advertisements. We live in a different world. We live
under a different night sky.
But anyone in Earth’s Northern Hemisphere who spends a few
hours out of doors, on a clear night, away from bright lights, care-
fully observing the stars, notices that some stars seem to move
across the sky in great arcs while one star seems to remain in place.
That one star is Polaris, the North Star. The farther in the sky a
star appears to be from Polaris, the larger an arc it makes. Stars
that appear close to Polaris travel in small circles, each centered on
Polaris. The larger arcs dip below the horizon, so over the course
of the evening, stars farther from Polaris may well disappear from
view. If, however, we could trace a large arc over the horizon and
all the way around the sky, we would find that these stars, too,
trace large circles centered on Polaris.
When we see stars move across the night sky in circular arcs, we
know that they only appear to move. We know that their appar-
ent motion is due to the revolution of the Earth about its axis.
Because the stars are many trillions of miles away from us, their
actual motions are too small for us to notice. But whether the