270 inflection

to choose, one can always find a larger value of xfor

which is, and remains, less than –Mfor ever

larger values of x.

2. Geometry: In D

ESARGUES

’

S THEOREM

, French

mathematician G

IRARD

D

ESARGUES

(1591–1661) found

it convenient to regard parallel lines as intersecting at

some point of infinity. Thus in the theory of

PROJEC

-

TIVE GEOMETRY

, the notion of points “infinitely far

away” is given meaning and context.

3. Set theory: Italian astronomer and physicist

G

ALILEO

G

ALILEI

(1564–1642) observed that every

counting number can be matched with its square,

showing in some sense that the infinite set of counting

numbers is no more infinite than the subset of square

numbers:

In the 19th century, British algebraist A

UGUSTUS

D

E

M

ORGAN

(1806–71) and German mathematicians

J

ULIUS

W

ILHELM

R

ICHARD

D

EDEKIND

(1831–1916) and

G

EORG

C

ANTOR

(1845–1918) realized that this is a

common property of infinite sets, and that it is appro-

priate to use this property as the definition of what it

means for a set to be infinite:

A set is infinite if its elements can be matched,

without repetition, with the elements of a

proper subset of itself.

This definition has the advantage that it makes clear

what it means for a set to be finite.

A set is finite if it is not infinite.

(A comment should be made on this point. Although we

all have a clear intuitive understanding of what it means

for a set to be

FINITE

, it is not at all easy to provide a

direct mathematical description of this concept. However,

it is possible to show that no set of the form {1,2,3,…,n}

is infinite, that is, there is no means to match the ele-

ments of this set with the elements of a proper subset of

itself without producing repetition. This is done with an

INDUCTION

argument on n. One can use this as a link

between this indirect approach and our intuitive under-

standing.) Cantor went further to develop an astounding

theory of

CARDINALITY

that shows, among other things,

that there are many different types of infinite sets, some

deserving of being called “more infinite” than others.

The notion of the infinite has been studied and used

since antiquity. A

RCHIMEDES OF

S

YRACUSE

(ca.287–212

B

.

C

.

E

.) used the notion of an infinitely small quantity to

develop formulae for the areas and volumes of curved

figures and solids. (In some vague sense, one can view a

circle, for instance, as a regular polygon with infinitely

many sides, all infinitely short in length. It is better,

however, to view the circle as the

LIMIT

figure of a

sequence of regular polygons.) The geometer E

UCLID

(ca.300–260

B

.

C

.

E

.) proved that the set of

PRIME

num-

bers is infinite, and Z

ENO OF

E

LEA

(ca. 490–425

B

.

C

.

E

.)

contemplated the infinite in his studies of time, space,

and motion.

Although the scholars of Greek antiquity utilized

the infinite, they were wary of it. Euclid, for example,

went to great pains to phrase matters in a way that

never made mention of a quantity that was actually infi-

nite. For instance, he proved that from any given finite

collection of primes, one can always construct one more

(rather than state that the set of prime numbers is infi-

nite), and in his famous work T

HE

E

LEMENTS

, he never

made mention of lines that continue indefinitely; he

only spoke of extending line segments further if needed.

(This subtle turn of phrase proved to be important to

G

EORG

F

RIEDRICH

B

ERNHARD

R

IEMANN

with his 19th-

century invention of

SPHERICAL GEOMETRY

.) A

RISTOTLE

(384–322

B

.

C

.

E

.) argued that the “actual infinite” did

not exist, and that one can only argue in terms of

potentiality: given a finite part, one can always provide

more. This point of view held fast for almost two mil-

lennia. Cantor’s work on the actual infinite, inspired by

the beginning ideas of Dedekind and De Morgan, was

deemed revolutionary at its time.

inflection (inflexion) See

CONCAVE UP

/

CONCAVE DOWN

.

inflection point (point of inflection) A point on a

curve at which the tangent line to the curve changes

from rotating in one sense (clockwise or counterclock-

wise) to rotating in the opposite sense. The concavity

of the curve changes at such a point.

See also

CONCAVE UP

/

CONCAVE DOWN

.