Mathematics: How Did It Get to Where It Is Today?
Author(s): Andrew Gleason
Source: Bulletin of the American Academy of Arts and Sciences, Vol. 38, No. 1 (Oct., 1984),
pp. 8-24
Published by: American Academy of Arts & Sciences
Stable URL: http://www.jstor.org/stable/20171741
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http://www.jstor.org
Stated Meeting
How
Mathematics:
Did
Report
It Get
to Where
It
Is Today?
Andrew
we
is perceived
very differently
people, but the at
by different
is hardly un
titude of the pure mathematician
As
all know,
Gleason
mathematics
the profession.
This
evening
the genesis
of the philosophy
that underlies
pure mathematics
today; then
I will try to give you some idea of the abstract
role in modern
that play a major
concepts
derstood
outside
I shall describe
research; and finally, I will talk of the influence
the
of that very practical
modern
machine,
computer.
Even mathematicians
do not agree about a
definition
of the subject,
but here
is a
to at least two mathema
that appeals
definition
verbal
ticians:
Mathematics
is the science
of order.
order in the sense of pattern
and
to
It
is
the
of
mathematics
iden
regularity.
goal
sources of order, kinds of order,
tify and describe
that exist by virtue of logical
and the relations
the various kinds of order that
between
necessity
Here
Imean
occur.
to tradition, mathematics
According
began
who lived roughly from
with Tha?es of Miletus,
640 to 546 BC. It is hard to see how this can be
true since we know
that there was an active
commercial
in the Middle
civilization
East for
at least 2500 years before Tha?es,
and commerce
involves mathematics.
necessarily
Counting
must be nearly as old as language
itself, which
and measuring
in
is very old indeed. Weighing
volve mathematics.
The architects who built the
and temples of ancient Egypt certain
pyramids
must
have
had a working
of geom
knowledge
ly
were
made
in
and
those
old
maps
etry,
days too.
we
in the
On a more
find
level,
sophisticated
Code
reference
of Hammurabi
(about
to commissions,
1700 BC)
interest,
8
and
explicit
propor
tionate
An Egyptian
document
called
dates from about the same
by a scribe named Ahmes
taxation.
the Rhind
papyrus
time. It was written
and contains
in elemen
examples
the
is Ahmes'
tary algebra.
papyrus
Perhaps
? or
or
a
the
textbook
ancient
notes,
equivalent
of Schaum's
Outline
Series.
College
worked
in
mathematics
Thus,
ticed
many
some
sense
was
prac
for thousands
of years before Thaies.
In
can
we say that mathematics
what
sense, then,
was
It may be that Thaies
began with Tha?es?
to
not only
the first
the importance,
recognize
of identifying
mathematical
facts, but of writ
the exact reasons why one believes
ing down
these facts to be true. This
is tremendously
im
to
because
it
one's
criti
opens
portant
reasoning
cism and thus broadens
the scope of the subject.
so we
Tha?es himself
left no identifiable
corpus,
cannot
be
sure of his
about
this
time
that
role, but we know
the concept
of mathematical
to ex
proof arose, and mathematicians
began
amine
the logical
between
interrelationships
are the
various
facts. These
interrelationships
essence of the modern
subject.
Next
I'd like tomention
Pythagoras
(582-507
The first thing to remem
BC, very uncertain).
ber about him is that it is 99 percent certain that
he was not the discoverer
of the famous theorem
on right triangles
that bears his name. There
is
evidence
that this was known at least
persuasive
a thousand
years before his time.
a re
was a mystic who founded
Pythagoras
cen
in
order
the
middle
of
the
sixth
ligious
a
The
for
BC.
order
about
grew
tury
century
and
world.
became
It was
very
influential
in
the Greek
out of ex
eventually
persecuted
istence, but not without
leaving a permanent
on
mark
Greek
intellectual
life. The
order
a mystical
reverence
for mathematics
taught
and
had
a
in certain
interest
geomet
special
and
arrangements
designs
(Figure
1).
most
of this had no mathematical
Although
it encouraged
the study of pure
importance,
ric
mathematics.
Under
the influence
of the
for
the
time
first
in
Pythagoreans,
probably
were
there
substantial
of
groups
peo
history,
for its own
mathematics
sake.
ple studying
9
Pythagoras
may
not
have
discovered
the
but it is very likely that
theorem,
Pythagorean
he or one of his followers
gave the first proof
of the theorem.
Figure 1: The pentagram (left) and the tetractys (right)
had mystical significance for the Pythagoreans.
We know little about the mathematicians
of
names
We
have
the Pythagorean
the
of
age.
a dozen,
but there must
have been
perhaps
no
texts
others.
Almost
remain
many
original
so attributions
are almost en
from that period,
can say for certain,
We
conjectural.
tirely
that
of Plato
the
time
however,
by
(427-347
an
extremely
sophis
BC) there had developed
of mathematical
ticated
notion
and
proof
that may
another
go back to Tha?es:
concept
is not about what's
that mathematics
namely,
in the real world
but is, in fact, about
ideal
When
you
geometry,
you're not
study
things.
on
a
marks
in
the
sand,
chalkboard,
studying
or on paper; you are talking about points
that
are infinitely
lines that are only one
small,
circles
and
that are perfectly
wide,
point
round ?ideal
This
concept had clearly
objects.
mathematics
penetrated
by the time of Plato.
In fact, the abstract
of the geometers
figures
seem to have been the prototype
for Plato's the
The
Book
ory of the ideal world
Republic,
(see
VII,
527).
We
mous
turn next to Euclid.
Euclid wrote his fa
text The Elements
in 13 books about 300
It is easily
the most
textbook
successful
BC.
in history;
it has been
translated
into many
at the
still
different
and
is
available
languages
bookstores
later.
2300
years
nearly
People
as a geometer,
think of Euclid
and it
usually
is his geometry
that I will discuss,
but it is
worth
are
noting
about
Now
years.
teenth
that several books
number
I'm going
In so doing
of The Elements
theory.
to jump
I shall
a century
century,
in
mathematics
portant
10
ahead
two thousand
over the seven
skip
that is extremely
im
I
and all of science.
no disrespect
this with
for seventeenth
but
for lack of time.
century
accomplishments
an
to discuss
I wish
in the
important
change
we
that
underlies
what
in
do
philosophy
It is a change
mathematics.
that grew out of
the concept
of Non-Euclidean
geometry.
In Book
I, Proposition
27, Euclid
proved
that, given a line L and a point P not on it,
do
to L, that is,
there is a line through P parallel
a line which will not meet
L no matter
how
Euclid
far it is produced
proved that
(Figure 2).
such a line exists. He did not prove that there
is only one such line. Maybe
there are more.
If there are at least two, there are infinitely
many
P
-?
L
Figure 2: Euclidean parallel postulate. Through thepoint
P only one line can be drawn parallel to the line L.
Figure 3: Non-Euclidean parallel postulate. Through the
point P at least two lines can be drawn parallel to the line
L (heavy lines). If so, there are many more (light lines).
to Euclid,
This was surely obvious
and I'm
sure he thought, as I'm sure everyone does, that
un
the alternative
picture
(Figure 3) is totally
seem possible
It just doesn't
reasonable.
that
there could be two lines through P which will
never cross L. The intuitive
picture overwhelm
that there is only one. However,
ingly suggests
to have
Euclid may have been the first person
faced up to the fact that this uniqueness
is only
an intuitive
unable
himself
picture.
Finding
to prove that there is only one parallel,
he put
11
a postulate
that says roughly,
"Any
P other
than the one described
in fact cross L if ex
1:27 will
far."
is Euclid's
tended
This
fa
sufficiently
mous
fifth or parallel
Its
postulate.
explicit
is an important
of
formulation
milestone
in his book
line through
in Proposition
mathematics.
the beginning,
From
the fifth postulate
of
Euclid
bothered
Its truth is
mathematicians.
so
clear that people
felt that it was
intuitively
an unnecessary
Another
postulate.
thing that
a
bothered
and
it
bother
people,
legitimate
is the phrase "if extended
far."
was,
sufficiently
to assert
It is awkward
the existence
of some
a
case
this
of
thing
point
(in
intersection)
be
without
any clue as to how itmight
giving
are difficult
found. There
issues in modern
turn on just this kind of
mathematics
which
point.
The
list of distinguished
mathematicians
on the parallel
studied and wrote
postu
late includes Geminus
the as
(first century),
tronomer
Proclus
Ptolemy
(second
century),
the Persian Nasr-ed-din-alTlisi
(fifth century),
who
and Wallace
(thirteenth
century),
(seventeenth
and Legendre
in
Saccheri,
Lambert,
century).
the eighteenth
all tried very hard to
century
was a conse
that the parallel
prove
postulate
rest
of Euclid's
of the
quence
assumptions.
the latter
mathematicians,
Many
particularly
the Non-Euclidean
three, began by assuming
that there are many
lines
parallel
postulate
to L and then pursued
the
through P parallel
as far as they
of this assumption
consequences
could.
curious
Their
intent was
nonintuitive
a strict
eventually
thus making
to find more
theorems,
contradiction
and more
confident
that
would
arise,
a monster
in
the whole
thing
direct proof of Euclid's postulate.
this
However,
ex
to
and doubts
be
did not happen
began
on the matter.
pressed
It was suggested
that the question
should be
to
In
the
Non-Euclidean
subject
experiment.
is al
case, the sum of the angles of a triangle
in the Euclidean
less than 180?, whereas
An extremely
it is always
180?.
exactly
that some tri
show
careful measurement
might
ways
case
12
less than 180? and
totalling
angle had angles
case as the
the Non-Euclidean
thus establish
the other hand,
the inherent
inac
truth. On
curacy of the measurement
process will pre
vent us from ever showing
that any physical
to 180? exactly.
summing
triangle has angles
of being
the first to publish
these
The honor
seems
to
to
who
belong
Kl?gel,
thoughts
of
reviewed
the whole
the
question
parallel
in 1763.
in his thesis
postulate
a very ex
In 1829, Lobatchewski
published
tensive
treatment
Non-Euclidean
of the
of the consequences
in
which
he
parallel postulate,
that no contradiction
would
asserted
firmly
ever be found.
similar
ideas were pub
Quite
in 1838, but his manuscript
lished by Bolyai
seems to have been available
to some as early
as 1821. It is difficult
to assess who was
the
founder
of Non-Euclidean
geometry.
were
Lobatchewski
and Bolyai
Although
convinced
that Non-Euclidean
it was many
years
was
geometry
before Beltrami
legitimate,
a proof that no contradiction
would
published
ever be found
he
speaking,
(1868). Technically
a
that
relative
he
is,
gave
consistency
proof;
in Non-Euclidean
showed how a contradiction
would
lead to a contradiction
in Eu
geometry
as
clidean
hence
if
Non-Eu
well;
geometry
so does Euclid's.
clidean
founders,
geometry
The reception
accorded Non-Euclidean
ge
deserves
careful examination.
ometry
Among
of Europe,
it was
the leading mathematicians
were
soon.
en
In
fact,
many
accepted
fairly
to turn their attention
to geometry,
couraged
and by the end of the century many
signifi
cant papers
had been written.
The
idea of
four-dimensional,
five-dimensional,
studying
even infinite-dimensional
arose. Rie
geometry
mann
a
introduced
generaliza
sophisticated
as Riemannian
tion, now known
geometry,
which
the cornerstone
has become
of modern
theoretical
different
showed how several
physics. Klein
kinds of geometry
could be linked
together.
In the meantime
was
another
revolution
in
mathematics.
the
through
sweeping
Starting
of
under
the
influence
1820s, largely
Cauchy,
13
a new and higher
to
standard
of proof began
text was examined
Euclid's
be required. When
lacu
from this point of view, some important
nae
were
When
discovered.
the idea of Non-Euclidean
geometry
reached
the level
were met with utter
of the schoolmasters,
they
and much
vitriolic
derision,
prose was writ
ten on the subject. Even Lewis Carroll made
some amusing
writers
contributions.
Many
and
the criticism
to the level
Elements
never be
The
of Holy
controversy
Scripture.
came an issue before
the public, however,
per
it was overshadowed
haps because
by Darwin's
seemed
to elevate
of Euclid
Euclid's
Evolution
bombshell.
gave the critics another
at.
to
shoot
target
a
wrote
At the end of the century Hubert
on
in which he made
book
very clear
geometry
for the validity
of
the idea that the criterion
a geometric
is its internal
system
consistency.
no mention
He made
of its truth. It is impor
tant to understand
the distinction
that is be
as to
no
made
here.
There
is
question
ing
a
no
whether
system is true because
geometric
one knows what
to be true.
it means
a system,
Whether
say Non-Euclidean
an
is
of the
geometry,
appropriate
description
real world
is a question
for physicists.
This has
out by many
for
been pointed
people. Gauss,
measure
some
to
tried
the
of
angles
example,
to see whether
perhaps
triangles
they
total less than 180?. Since
the magni
if it exists,
is proportional
tude of the effect,
to the area of the triangle,
it seems unlikely
large
would
in triangles
of less
we
no
and
have
dimensions,
intergalactic
this. Moreover,
when we
direct way of doing
we
in the usual manner,
try to measure
angles
are really
the
behavior
of
investigating
light
that
it could
be measured
than
in nongeometric
issues.
and this brings
waves,
think
such
The more
about
you
questions
it becomes
that the new criterion
the clearer
need not concern
is appropriate.
Mathematics
means
to
if
itself with
truth
truth,
applicable
some Platonic
version
of the real world.
Inter
nal consistency
is a much more
appropriate
criterion.
Another
way
14
of
saying
this
is that
need not concern
mathematics
an ideal version of the universe
and
indeed
that might
philosophy
itself only with
that is, but may
all the universes
consider
should,
is a crucial
be. That
that occurred
during
in
change
the nineteenth
century.
in phys
It is a change of great importance
theoretical
often
ics as well. Modern
physicists
on
make
models
for
the
universe
up
physical
a purely mathematical
basis and then examine
of
the question
whether
they seem to fit the
facts observed.
Einstein
acknowl
explicitly
in conceiving
the the
edged his indebtedness
to the greater
of
ory of relativity
flexibility
new
in
this
inherent
thought
philosophy.
idea that
Another
important mathematical
arose
the
nineteenth
is that of
century
during
an operational
A
is
system.
simple
example
are
a
the
numbers.
Pla
given by
positive
They
of the numbers
tonic idealization
you use to
measure
and
include
very
things. They
weigh
familiar numbers
what
less familiar
like 1, 2, and 3/5 and
ones like the square
some
roots
of 2 and 5, and that famous
number
pi. You
in grade
learned
school
that you can always
add two of these numbers
and get a new one.
or
divide
Or you can multiply
two,
two, or sub
tract the smaller of two unequal
from
numbers
I
the larger, and get a new number.
leave
(Here
out zero and negative
because
numbers,
they
are less familiar
and are Johnny-come-latelies
on the mathematical
scene, dating
only from
the sixteenth
century)
these facts were known
Of course,
from an
But
idea
of
the
the
number
tiquity.
studying
as
an
not
of
arithmetical
system,
assemblage
facts but as a structural whole with an overall
internal organization
?that
is a modern
idea
at roughly
the beginning
of the
century.
are
Inside
the system of positive
numbers
one can carry
smaller
which
systems within
out the arithmetic
If we consider
operations.
originating
nineteenth
numbers
like 2, 3/8,
only rational
(numbers
or 17/59 that can be expressed
as the quotient
we can add, subtract, multi
of two integers),
or
divide
the answers are always again
and
ply,
15
can recall
the
rational.
everyone
(No doubt
to add 5/6 and 4/7.) When
trauma of learning
system has this property,
part of an operational
to the opera
it is said to be closed with respect
or to form a closed
sub
tions of the system,
system.
of ra
that the subsystem
than the
is actually
smaller
of positive
whole
but it is.
numbers,
system
This was discovered
back in the fifth century
it was proved
that the square root
BC, when
of 2 is not a rational number.
One way of put
be
will
which
later, is to say
helpful
ting this,
the integers and perform
that if we start with
to get new numbers,
then
arithmetic
operations
new
these
combine
numbers
arithmetically,
as much
as we please,
and repeat this process
come
root of 2.
we will never
to the square
are
in which
There
other closed subsystems
It is not
tional
obvious
numbers
can do arithmetic
freely, and their analy
of
sis is an active area of research
today. One
the main
lines of attack is to consider how these
are related to one another.
various
subsystems
are consid
Thus whole
systems of arithmetic
ered as single entities.
of the fruits of this new
idea was the
One
we
solution of the three famous construction
prob
to trisect the angle,
to dupli
lems of antiquity:
cate the cube, and to square
the circle. The
to
trisection
means,
given an angle,
problem
a
two
ruler
and
construct,
compass,
using only
into three equal
the angle
lines which
divide
a
the cube means,
angles. Duplicating
given
to
construct
another
cube
cube,
having
exactly
the
of the original.
twice the volume
Squaring
a circle,
a
to
construct
circle means,
given
same
area
as
the
cir
the
square having
exactly
that we are not con
cle. It must be emphasized
solutions.
It is
cerned with good approximate
an
to
into three parts
that
divide
easy
angle
are equal for all practical
the desired
purposes;
must
be exact in the ideal sense.
construction
were
All three of these problems
solved dur
solved
century. They were
ing the nineteenth
are im
that all three constructions
by showing
that is, in each case there is no con
possible;
the desired
result.
struction
that accomplishes
16
re
of mathematics
frequently
Departments
one
of these problems.
ceive "solutions" of
They
are written
that
by people who cannot believe
to prove that a construction
is im
it is possible
as
Their
goes something
reasoning
possible.
follows: "There are clearly an infinite number
a construction
and
of ways to go about making
to have ex
for mathematicians
it is impossible
to deal with
them all." But it is possible
amined
the infinitely many
by the method
possibilities
one could
of treating whole
systems. Certainly
not prove that the square root of 2 is irrational
in turn each of the infinitely
by examining
that its
and observing
numbers
rational
many
not
multi
the
but
is
2;
square
by analyzing
structure of the set of all rational num
plicative
one can easily
that no rational
bers,
prove
number
has
2.
square
The
geo
logic of the proof that the classical
are
metric
construction
problems
impossible
can consider
One
the set
is quite analogous.
a
as
and
in
of all points,
circles
lines,
plane
can
an
two
We
combine
system.
operational
in one way to obtain a line, the
distinct
points
or in another way to
line joining
the points,
the first point as center
obtain
the circle having
and passing
the second. Or we can
through
a
to obtain
two nonparallel
lines
combine
at
in
which
the
lines
the
point
point, namely,
an operation
tersect. We
in the sys
introduce
tem for each kind
in a
of step we make
The
fact that the basic entities
construction.
are of three different
kinds is immaterial.
Hav
we
can
an
system,
inquire
ing
operational
that
whether
it has a smaller closed subsystem,
a
of
the
and
cir
subcollection
lines,
is,
points,
cles with
of these
system.
that combining
the property
any two
the sub
leads always to a result within
it has many,
and we can ex
Indeed
one
in which
each of the
describe
can
be set, but
construction
problems
plicitly
classical
the answers
to the problems
do not fall within
our
the subsystem.
contains
Thus,
subsystem
an angle of 60?, but no angle of 20?. Since no
can get
use of the ruler or compass
legitimate
us out of the subsystem
in which we start, we
can see that there is no construction
that will
17
a 60?
of the
angle. The
description
it
and
the
that
doesn't
proof
subsystem
are more
to the problems
the answers
contain
in
but
principle
they are no
sophisticated,
trisect
closed
the description
of the rational
and the proof
that the rationals
do
root of 2. For us, the
the square
include
different
numbers
not
from
is that, without
the method
fundamental
point
the problems
of systems and closed subsystems,
not have been
would
solved.
kind
of operational
Another
sys
important
tem appears
in connection
with
symmetry.
Look at the Pythagoreans'
pentagram
(Figure
That
symmetry.
10). It has rotational
that if you take two copies of the figure,
one on top of the other, and rotate the top copy,
it all
before you have turned
then somewhere
on the copy be
it fits exactly
the way around,
at one
this happens
low. For the pentagram
fifth of a turn and again at two-fifths,
three
a
turn.
of
full
It
also
has
and
four-fifths
fifths,
means
if
which
that
reflective
you
symmetry,
turn the top copy over (which,
for a plane
amounts
to the same thing as reflect
figure,
on
a
it will again
fit exactly
ing it in mirror)
1, p.
means
the copy
There
below.
that only
is another kind of symmetry
can
some
4
have.
shows
Figure
figures
drawn
fish
M.C.
Escher.
interlocked
flying
by
indefi
the pattern
continued
If we
imagine
it
in
then
has
translational
all
directions,
nitely
two copies of the figure
If you make
symmetry.
and put one on top of the other, you can slide
infinite
it fits the bottom
the top one along until
copy
also
has
rotational
This
sym
design
exactly.
if you rotate the top copy one-third
of
metry;
a turn about a point where
and
three white
the top will once
three black fish-wings
meet,
the bottom.
Lest anyone
think
again match
a design
with
both
that the idea of making
is
and
rotational
translational
symmetry
an
consider
5, painted
modern,
Figure
by
three
thousand
artist more
than
unknown
years
ago.
symmetry
ter
and translational
It has reflective
and rotational
of a quar
symmetry
turn.
18
Figure 4: Flying Fish byM.C. Escher has both transla
tional and rotational symmetry. (Collection of Haags
Gemeentemuseum, The Hague, The Netherlands.)
Figure 5: This wall painting from the Temple of theDead,
Thebes,
has
translational,
rotational,
and
metry. (Permission of Birkh?user Boston,
reflective
sym
Inc.)
How
does
the notion
of symmetry
tie up
with the idea of operational
Two
ways
systems?
a copy of a figure to match
of moving
the origi
a new way. For
to make
nal can be combined
can be slid to
since
the
fish
example,
flying
a match
the right to make
and also up the page
a match,
to make
it follows
that they must
to the right and
match
also if slid diagonally
a
to match
up. In this way the set of all ways
an
and
its
becomes
duplicate
figure
operational
system, and the structure of this system affords
a precise description
of the nature of the sym
of the figure. Operational
metry
systems of this
kind are called groups.
19
been
of symmetry
has certainly
a
the
idea
of
think
known
for
long time, but
in terms of an operational
sys
ing of symmetry
to have
tem seems
in the late
started
only
the theory
of
century.
Strangely,
eighteenth
was
not
in the study of
symmetry
developed
The
idea
with poly
designs but in connection
geometric
That's
what's
nomial
surprising;
equations.
a
about
equation? No
polynomial
symmetrical
one understood
of symmetry
the significance
until the 1770s when
equations
an analysis
in terms of
published
solutions of equations
for the known
up to four.
for polynomial
Lagrange
symmetry
of degrees
It had been known
from ancient
times how
to solve second degree or quadratic
equations,
- 5 =
like x2 + 3x
0. There
is even a formula,
to everyone who has taken high school
familiar
that enables us to write down the solu
algebra,
tions at once. A similar
formula
for solving
or
third
cubic
like
degree
equations,
x3 + 2x2 + 7x
10 = 0, was discovered
in the
century,
early part of the sixteenth
probably
a
named Ferro, but it wasn't
by mathematician
the middle
until
of the century
published
by
The
is complicated,
Cardano.
formula
involv
roots and cube roots. Later
square
ing both
a
in the same
Ferrari
discovered
century
for solving
method
of
the
fourth
equations
his method
could be reduced
degree. While
to a formula,
the formula would
be extremely
never
seen
I
out.
have
it
written
complicated;
Ferrari's method
shows that the
Nevertheless,
can always be found
solutions
ad
by successive
ditions,
subtractions,
divisions,
multiplications,
and taking roots. This
is called a solution
by
radicals. The
search for a solution
by radicals
to fifth
fruitless
and
higher
degree
equations
proved
It was par
for the next two centuries.
to find that some equa
tantalizing
ticularly
five do have
tions of degree
solutions
by
so it was reason
radicals
that are not obvious,
able to hope
that we could always
find such
a solution
if only we were
in
sufficiently
genious.
memoirs
Lagrange's
toward
portant
step
contained
understanding
20
the first
im
higher
but only after sixty more
degree
equations,
of mathemati
of
years
progress
by a number
seen. Every poly
cians was the whole
picture
as its
nomial
has as many
solutions
equation
and Galois
showed that these solutions
degree,
The symmetries
have certain symmetries.
form
as the Galois
a group,
now known
group of
the equation.
Galois
criterion
gave a precise
or not
in terms of this group to decide whether
a given
can
be
solved
radicals.
equation
by
the more
there
speaking,
symmetries
Roughly
it is to solve the equation.
If
are, the harder
is small, then there is a solu
the Galois
group
if it is large, there isn't. It was
tion by radicals;
remarkable
after
this
only
triumph that the im
areas of mathe
in
of
groups
many
portance
was appreciated.
matics
Thus
mathematicians
entered
this century
new
tools
equipped
powerful
conceptual
to apply
to
and philosophical
license
them
no
with
immediate
for
the
problems
regard
with
of whether
in the
they are applicable
question
we
are
real world.
concepts
Today
studying
abstract
than anything
I
that are much more
on
have mentioned,
abstractions
abstrac
piled
tions. Just as in the last century,
there are those
too ab
who
is becoming
cry, "Mathematics
is absolutely
clear: We
stract," but the record
are solving hard down-to-earth
with
problems
the aid of very subtle abstract
ideas. It seems
are essential
to
that highly
abstract
concepts
the subtleties
of nature.
describe
I am often asked what the modern
computer
answer
for mathematics.
The
is doing
is, "A
is not where
great deal," but the contribution
one might
think.
was given
A few years ago much
publicity
to the fact that an old problem,
the four-color
was solved with a
assist from
problem,
major
a computer.
About
two-thousand
hours
of
main-frame
computer
time
was
spent
examin
to the
ing combinatorial
possibilities,
leading
can be colored
result
that any map
in four
colors so that no two states that share a linear
as
to just point contact
boundary
(as opposed
in the case of Colorado
and Arizona)
have the
same color. This had been conjectured
about
21
the at
years ago and had attracted
conse
mathematicians;
many
was
its demonstration
certainly
quently,
we
in
But
retrospect,
although
noteworthy.
was
the
learned a mathematical
fact,
computer
so heavily
involved that we gained
little, if any,
structure
of the
into
the
underlying
insight
structure
is
the
Since
underlying
problem.
essence
mathematical
of
any
prob
really the
is
of the four-color
lem, the solution
problem
as
a
of mathe
not regarded
major milestone
a hundred
of
tention
matics.
has made
the computer
the other hand,
indirect
About
contributions.
important
some
set a
mathematicians
twenty years ago,
On
of
to work calculating
the solutions
computer
a differential
the
called
Korteweg-de
equation
indi
The
Vries
output
computer
equation.
of a completely
cated the existence
unexpected
a challenge
to ex
thus posing
phenomenon,
terms.
in
theoretical
Since
the
results
plain
and a whole
has been met,
then the challenge
new chapter
in the theory of differential
equa
The
final results have
tions has been written.
to do with computers,
the
but without
nothing
no
one
of
the
have
would
computer
thought
question.
As
a final
iteration
descended
in physics.
the
I'd like to mention
example,
Here
is a problem
of functions.
from some very important
problems
in
tell us that the air molecules
Physicists
1029 in number,
the room around us, perhaps
travel
of their time in free flight,
spend most
in
random
lines
in
essentially
straight
ling
in a while,
Once
directions.
say about
every
36
is a relative
10
seconds
term),
(a "while"
the colli
two of these molecules
collide. After
sion they go off again into free flight. The mo
of periods
of
tion of the air, then, consists
by sharply discon
punctuated
orderly motion
events. We
the classical
tinuous
point
adopt
to
of view toward* the mechanics
(as opposed
so
is
whole
motion
the
quantum
mechanics),
to
like
We
would
determinate.
completely
of
behavior
the long-range
about
know more
such systems.
22
a
is no hope
of getting
there
Of
course,
to a problem
of this complex
solution
detailed
can be solved only
in the
At
it
present,
ity.
sense of making
of the
verifiable
predictions
of the air, but one can hope
statistical behavior
some
to achieve
Confronted
mathematician
that.
insight beyond
this kind of situation,
tries to find some
usually
with
the
sim
one essential
that contains
fea
pler problem
ture of the original
In
this
case, let
problem.
us think of bouncing
around
the floor on a
Here we have simple motion
punc
pogo-stick.
tuated by sharp reversals. Our
idealized pogo
stick lands exactly on a single point. Since we
a determinate
are trying to model
process, we
we
land there is a pre
that wherever
imagine
as
to go next. The
to where
cise instruction
we will study is whether
our bounc
question
ing ultimately
process
"goes
in a confined
leads us
to infinity")
far away
very
(the
or forever
keeps us
region.
known
for a long time that the
of
such a process
long-run
depends
in a very subtle way on the parameters
of the
one
the
and
that
boundaries
process
separating
kind of behavior
from another must
be very
that we have
but it is only recently
complicated,
look at the phenomena.
been able to really
With
the aid of color graphics and an unbeliev
we can get very
of computation
able amount
The
have
themselves
pictures
good pictures.
It has been
behavior
the attention
of several mathemati
thus
the
has been at least
cians,
computer
a
of interest
for
resurgence
partly responsible
in the topic of iteration.
attracted
and
Mr.
Gleason
colored
slides
then showed
computer-generated
the iteration of a very
+ c, z and c
simple function
(z-^z2
complex) de
on one parameter, c. Each pixel of the slides
pending
represented one value of c and its color described whether
and how fast the corresponding process goes to infini
illustrating
ty. The boundary between the region where
diverges to infinity and the region where
bounded was seen to consist of remarkably
arabesques. He stated that it has been proved
23
theprocess
it remains
beautiful
that simi
lar arabesques appear at every scale of magnification.
In conclusion, Mr.
Gleason said:
Mathematics
is the science
the full explication
is one of the many
of the patterns
face mathematicians
today.
Gleason
Andrew
ticks
and
fascinating
isHollis
Natural
of patterns
of these
challenges
Professor
Philosophy
University.
24
and
slides
that
ofMathema
at Harvard