Tabak J. Mathematics and the Laws of Nature: Developing the Language of Science

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70 MATHEMATICS AND THE LAWS OF NATURE

Galileo Galilei

The Italian inventor, mathematician, and physicist Galileo Galilei

(1564–1642) was another central figure in the establishment of the

new sciences. Like Stevin, Galileo was introduced to mathematics

fairly late in life. He received his early education at a monastery

near Florence. Later he enrolled in the University of Pisa to study

medicine. It was while he was a student at the University of Pisa

that he overheard a geometry lesson. Until that time he had no

exposure to mathematics. Though Galileo began his study of

mathematics at the university, he eventually left because he did

not have enough money for the fees. By the time of his departure,

however, he was busy studying mathematics and physics. Soon

he was teaching mathematics and publishing his discoveries in

science. Like Stevin’s, Galileo’s work is characterized by rigor-

ous mathematics and creatively designed and carefully conducted

experiments.

Galileo is best known for the observations of the planets and

stars that he made with the newly invented telescope and for the

resulting persecution that he suffered at the hands of the Catholic

Church. Galileo had appar-

ently been sympathetic to

Copernicus’s ideas about the

geometry of the solar system

from the outset, but with his

telescope he lifted the discus-

sion out of the realm of theory

and provided new and dra-

matic observations to support

these ideas. His observations

are carefully documented and

analyzed in his letters and pub-

lications. Kepler was one of

the people with whom Galileo

corresponded. It is interesting

that despite his contact with

Kepler, Galileo, who in most

respects was a revolutionary

A portrait of Galileo Galilei from

1636: Galileo helped to establish the

discipline of science as we understand

the term today.

New Sciences 71

thinker, never did abandon the idea that the planets move in cir-

cular, rather than elliptical, orbits.

Galileo’s work in science had great immediate impact, not only

because some of his discoveries were dramatic, but also because

he sought the attention. Unlike most of his contemporaries, who

still followed the practice of publishing in Latin, Galileo published

in Italian. He sought to attract the attention of the general pub-

lic and to engage the public in the scientific controversies of the

day. Furthermore, Galileo knew how to write. All of his scientific

works were written with flair. He was not reluctant to respond to

criticism or to criticize those with whom he disagreed. Sometimes

he used reason, but he was not above using humor and sarcasm

as well. His two major works, Dialogues Concerning the Two Chief

World Systems—Ptolemaic and Copernican and Dialogue Concerning

Two New Sciences (also known as Two World Systems and Two New

Sciences, respectively), are important as literature and as science.

They served to attract attention to the new ideas and to popular-

ize them as well. Unlike many scientific treatises, Galileo’s works

generally sold well.

Galileo’s book Two New Sciences is perhaps even more impor-

tant than his writings about astronomy. It was through Two New

Sciences that Galileo helped to invent and then popularize what

has become “modern” science. Galileo wrote Two New Sciences late

in life. He was already under house arrest for the views that he

expressed in Two World Systems. (Galileo spent the last eight years

of his life under house arrest.) As an additional condition of his

punishment, he was ordered to cease writing about science during

his incarceration. He wrote anyway. In addition to his isolation his

writing difficulties were compounded by failing eyesight, which

was perhaps the result of his telescopic observations of the Sun.

When he completed the manuscript, he had it smuggled out and

published in Leiden in 1638. This, too, was a dangerous action to

take, given his relationship with the church. He died four years

after the publication of Two New Sciences.

Two New Sciences is, like Two World Systems, written in the form

of conversations among three characters. The conversations take

place over the course of four days. Each day a different subject is

72 MATHEMATICS AND THE LAWS OF NATURE

discussed. There is also an appendix that contains some additional

theorems on the centers of gravity of solids. Of special interest to us

are the conversations that take place on the third and fourth days.

On the third day the characters discuss “naturally accelerated

motion,” or the motion of bodies undergoing constant accelera-

tion. First, they discuss various experiments meant to illuminate

certain aspects of naturally accelerated motion. The experimental

results disprove various ideas about the motion of falling bodies

that were then popular. They also serve to establish an axiomatic

framework that Galileo then uses to deduce new statements. This

is Galileo at work—his experiments provide a means of demon-

strating the (probable) truth of certain foundational statements

about the nature of motion. These statements are then used as

axioms in the subsequent discussion. The axioms form the basis

for a mathematical model of naturally accelerated motion. With

the help of these axioms Galileo deduces additional statements

about the nature of motion. He accomplishes this via a series of

geometric theorems. Each new result is a logical consequence of

previously established ones, and in this way Galileo forms a long

chain of logical inferences. Galileo’s main character in Two New

Sciences not only creates new knowledge, he demonstrates how

new knowledge can be effectively created.

In the section “Fourth Day” the characters consider projectile

motion. Galileo is especially interested in explaining his ideas

about the motion of a projectile that is thrown or fired on a tra-

jectory that is initially horizontal. It is in this chapter that Galileo

makes some of his most creative observations.

The structure of the chapter is essentially the same as that of the

preceding chapter, except that this chapter begins with some geo-

metric preliminaries. The characters begin by quoting the works

of the great Greek geometers Apollonius and Euclid. Knowledge

of Euclid’s book Elements is clearly a prerequisite for the ensuing

discussion. One of the characters even insists that “all real math-

ematicians assume on the part of the reader perfect familiarity

with at least the Elements of Euclid.”

After the lead character recalls for the reader some basic facts

about Euclidean geometry, the three characters again discuss

New Sciences 73

various experiments. These experiments are intended to elu-

cidate the nature of projectile motion. In this section Galileo

shows how creative a scientist he is by demonstrating how one

can imagine the flight of a projectile as the combination of two

motions. Each motion exists independently of the other. The

first motion is a horizontal motion that continues at constant

velocity. The second motion is a vertical motion that consists of

a “naturally accelerated” motion, the subject of Galileo’s previ-

ous chapter. Galileo’s great insight is that projectile motion can

be mathematically decomposed into a simple, uniform horizontal

motion and the same type of vertical motion experienced by a

body in free fall. This type of analysis, which is original with

Galileo, is important because it is very useful. Galileo follows

this analysis with a series of theorems and proofs about the prop-

erties of projectile motion. In particular Galileo shows that in the

absence of air resistance projectiles follow a parabolic path. (This

is why he began with a discussion of Apollonius, the geometer in

antiquity who made the greatest progress in understanding the

properties of parabolas.)

Another important characteristic of Galileo’s treatment of

motion is that he clearly conceives of a force as something that

changes a motion. This, too, is modern. He is less concerned with

what a force is than with what it does, and over the course of the

chapters “Third Day” and “Fourth Day,” he describes what will

later be known as Newton’s first and second laws of motion.

Even today Galileo’s discoveries are covered in any introductory

physics course. His treatment of the motion of bodies experienc-

ing constant acceleration was remarkably complete and correct.

But as the title indicates, Galileo saw his work in a larger context.

He created new sciences, and he shows the reader the method by

which new sciences can (and perhaps must) be established: He

begins with experimental evidence; he uses the experimental evi-

dence to establish a small number of what we call laws of nature;

he deduces additional statements each of which is a logical conse-

quence of the previously established laws; and he checks to see that

the new statements do not conflict with the available experimental

evidence.

74 MATHEMATICS AND THE LAWS OF NATURE

Perhaps the biggest differ-

ence between Galileo’s work

and modern science is that

modern science depends on

statistics to reveal the rela-

tionship between the scien-

tist’s model of nature and the

experimental evidence. Today,

few researchers would be con-

vinced by an assertion that

the predicted outcome of an

experiment and the measured

outcome “seem to be in good

agreement.” The rigorous use

of statistics makes such state-

ments precise, and modern

science is heavily dependent

on the precision that statisti-

cal analysis makes possible.

In Galileo’s time statistics had

not yet been invented, and

consequently he was unable

to analyze in a rigorous way

the relationship between his

experiments and his math-

ematical model of motion

under constant acceleration.

While the absence of statisti-

cal reasoning is an important difference between contemporary

research and that of Galileo, it is not so important as to invalidate

the oft-repeated claim that the publication of Two New Sciences

marks the beginning of the modern (scientific) era.

Fermat, Descartes, and Wallis

In the years following Galileo’s death, rapid strides were made in

mechanics, the branch of physics concerned with forces and their

Galileo expressed his ideas about the

motion of a projectile with a graph

similar to this one. Suppose a projec-

tile is fired from a cliff at a horizon-

tal trajectory. The equally spaced ver-

tical lines represent the x-coordinate

of the projectile at equally spaced time

intervals. The unequally spaced hori-

zontal lines represent the y-coordinate

of the projectile at equally spaced time

intervals. The horizontal lines are

unequally spaced because the projectile

is accelerating steadily downward as a

result of the force of gravity.

HOM MathLaws-dummy.indd 74 4/1/11 4:49 PM

New Sciences 75

effects on motions. At the same time, important progress was

made in mathematics, and often the same individuals contributed

to both areas of knowledge. There were even cases when two

researchers made the same important discovery independently and

virtually simultaneously.

Two of the most mathematically creative individuals of this

time were the French philosopher, mathematician, and scientist

René Descartes (1596–1650) and the French lawyer, scientist, and

mathematician Pierre de Fermat (1601–65). Fermat, whose pro-

fessional life was devoted to the law, lived an ordered and genteel

life in France. Descartes wandered about Europe for years, finally

settling down in the Netherlands to write about philosophy. They

sometimes wrote to each other about their discoveries, but they

did not collaborate. Each developed and pursued a particular

interest in algebra, and each discovered important connections

between algebra and geometry, perhaps the best known of which

was the discovery of coordinate systems.

Coordinate systems serve as conceptual bridges between algebra

and geometry. Points on a plane are geometric objects. By impos-

ing a coordinate system on a plane, ordered pairs of numbers can

be placed in correspondence with (geometric) points. Once the

correspondence has been established, algebraic equations can be

placed in correspondence with planar curves. In fact, both Fermat

and Descartes discovered that a single equation containing exactly

two unknowns will, under fairly general conditions, describe a

curve. Today, the statement that a single equation relating two

variables can be interpreted as a curve is sometimes called the

fundamental principle of analytic geometry.

To appreciate the importance of the fundamental principle of

analytic geometry, it helps to know that the Greek influence on

mathematics was still very strong during the time of Fermat and

Descartes, and the methods employed by the ancient Greeks for

generating curves were extremely laborious. As a consequence,

the Greeks studied only a dozen or so different types of curves

throughout their long history of mathematical research. In con-

trast to the Greeks, the fundamental principle of analytic geom-

etry enabled anyone with a pencil to write formulas that described

76 MATHEMATICS AND THE LAWS OF NATURE

curves. All that was necessary (almost) was to write a formula in

two variables and set it equal to zero. The formula so obtained

would probably describe a curve. But knowing that a given formu-

la describes a curve is not enough to identify the properties of the

curve. If the curve is graphed on a coordinate system, for example,

where is the curve increasing? Where is it decreasing? Where are

the points at which the curve changes direction?

One method of answering the preceding questions involves the

notion of the tangent, the best straight-line approximation to a

curve. Tangents are useful tools for investigating curves. At each

point where the curve is increasing, the tangent slopes upward.

Where the curve is decreasing, the tangent slopes downward. And

a curve can only change direction in a neighborhood of a point

where the tangent is horizontal. (This situation is now known to

be complicated by the fact that some curves fail to have any tan-

gents, but neither Fermat nor Descartes was aware of the existence

of such curves, and we will give no further consideration to curves

without tangents until chapter 8 of this volume.) Both Descartes

and Fermat discovered how to compute the slope of a tangent,

which is called the derivative, from the formula that described the

curve. Their discovery was later incorporated into the subject now

known as calculus, but they understood the derivative as part of

the subject of infinitesimal analysis, the mathematical precursor

to calculus.

The ability to compute the tangent is extremely important

because the derivative (or slope of the tangent) has a number of

important interpretations. In addition to providing information

about the geometric properties of curves, the derivative can also

be used to deduce important information about the motion of

bodies. Consider, for example, a function that describes the dis-

tance that an object in free fall travels as a function of time. The

derivative of that function can be interpreted as the velocity at

which the object is moving. If, on the other hand, the function

describes the velocity of that same object, then the derivative of

the velocity can be interpreted as the acceleration experienced by

the object. (Derivatives are part of what students learn when they

learn calculus.)

New Sciences 77

There are, however, significant logical hurdles to overcome

in fully understanding derivatives. In fact, a rigorous logical

explanation for derivatives would not be formulated until two

centuries after the deaths of Fermat and Descartes. It was not

that Fermat and Descartes and their successors were unaware of

the difficulties involved in computing derivatives, but they were

unable to resolve them. They were not alone. The great French

mathematician and teacher, Jean le Rond d’Alembert (1717–83),

for example, used to counsel his students to have patience when

learning the techniques of calculus, including derivatives. If they

were patient, he assured them, faith in the techniques would

eventually come. Faith—not understanding. Fortunately, a great

deal of progress could be made in calculus and science without

rigorous mathematical thinking, and both Fermat and Descartes

made many important mathematical and scientific discoveries

The secant line is determined by the points (x

, f(x

)) and (x

+ h, f(x

+ h)).

The tangent line is the limiting position of the secant line as h approaches

zero. As is apparent from the diagram, the tangent line is the best straight-

line approximation to the curve at the point of tangency.

78 MATHEMATICS AND THE LAWS OF NATURE

without ever being clear about why precisely they were obtaining

such good results.

Because derivatives are so important, because they will be used in

this book to represent a larger class of so-called infinite processes,

and because alternatives to the derivative are discussed in chapter

8, we briefly sketch how a derivative is obtained. In what follows,

refer to the accompanying diagram on page 77. Consider the curve

y = f(x) together with the points (x

, f(x

)) and (x

+ h, f(x

+ h)), both

of which are located on the curve. In what follows, we assume that

is some fixed number—in other words, it is not a variable—and

we let h represent a small variable quantity. We will always assume

that h is not zero.

As long as h is small the two points (x

, f(x

)) and (x

+ h,f(x

+ h))

will be located close to one another on the curve. It is well known

that any two points determine a straight line, and the line that is

determined by (x

, f(x

)) and (x

+ h,f(x

+ h)) even has a special

name. It is called the secant line.

The techniques needed to compute the slope of the secant line

were well known to Fermat and Descartes, just as they are well

known to most high school students today. No calculus is neces-

sary. The slope of the secant is a quotient that is formed by divid-

ing the difference in the y-coordinates of the two points by the

difference of their x-coordinates—also known as “the rise over the

run.” In symbols:

(4.1) slope =

f x h f x

x h x

f x h f x

( ) ( )

1 1

+ −

or to write it more simply

slope =

f x h f x

x h x

f x h f x

( ) ( )

1 1

+ −

The fact that h appears alone in the denominator explains why

we must always assume that h is not zero. Division by zero has no

meaning in mathematics. But for any value of h that is not zero,

the quotient makes perfect sense. And as can be inferred from the

New Sciences 79

picture, the smaller that h becomes, the more closely the secant

line approximates the tangent line.

Alternatively, if the function f(x) describes the distance traveled

by a body in free fall as a function of time (in which case the vari-

able x represents time), then the quotient in equation (4.1) above

describes the change in position of the moving body during h units

of time divided by h, the time elapsed. This is just the average

velocity over the interval from x

to x

+ h.

Now here is the deceptively simple part: If we allow the variable

h to approach zero—not equal zero but approach zero—and if as

it approaches zero the quotients in the equation (4.1) approach a

unique number, then that number is defined to be the slope of the

tangent line—also known as the derivative of the curve—at the

point (x

, f(x

)). The derivative can be interpreted as the “rate of

change” of f(x) at the point x = x

. It is important to emphasize

once again that there are logical problems associated with identi-

fying exactly what it means to “approach zero,” and there are addi-

tional logical problems with identifying the unique number that

we have assumed the tangent approaches. But in many important

instances, knowledge about the geometry of the curve could be

obtained without rigorous answers to these questions. Notice, for

example, that the slope of the tangent line in the picture reveals

the direction and magnitude of the rate of change of the curve at

the point of tangency.

The definition of rate of change sketched in the preceding para-

graph is so useful that even today, more than three centuries later,

many scientists and engineers pursue advanced research without

ever using any other definition for rate of change. But the descrip-

tion of derivative given in this section is only one model for describ-

ing the rate of change. And it is important to emphasize that it is

“only” a model. Other concepts of “rate of change” exist. These

were developed in the 20th century when researchers found that

the model just described was inadequate in some situations. But it

took centuries to develop alternatives to the standard derivative, a

testimony to its utility in engineering, science, and mathematics.

The third mathematician who contributed to this especially

fertile time in mathematics history was the British mathematician