200 foundations of mathematics

where rrepresents the radius of the circle. The

QUADRATIC

formula for the roots of a quadratic equa-

tion of the form ax2+ bx + c= 0 is:

foundations of mathematics The branch of mathe-

matics concerned with the justification of mathematical

rules,

AXIOM

s, and modes of inference is called founda-

tions of mathematics. The paradigm for critical mathe-

matical analysis came from the work of the great

geometer E

UCLID

(ca. 300–260

B

.

C

.

E

.) who, in his work

T

HE

E

LEMENTS

, demonstrated that all geometry known

at his time can be logically deduced from a small set of

self-evident truths (axioms). L

EONHARD

E

ULER

(1707–83) produced fundamental results in disparate

branches of mathematics and often saw connections

between those branches. He too searched for small col-

lections of concepts that were fundamental and, hope-

fully, common to all fields. In the late 1800s and at the

turn of the century with the discovery of R

USSELL

’

S

PARADOX

in

SET THEORY

, mathematicians were led to

apparent paradoxes and inconsistencies within the

seemingly very basic notions of “set” and “number.”

This led to the fervent study of the fundamental princi-

ples of elementary mathematics and even to the study of

the process of mathematical thinking itself (

FORMAL

LOGIC

). In the 1930s Austrian mathematician K

URT

G

ÖDEL

(1906–78) stunned the mathematical commu-

nity by proving, essentially, that any formal system of

mathematics that incorporates the principles of arith-

metic will contain statements that can neither be proved

nor disproved, and, in addition, such a system will nec-

essarily be incapable of establishing that it is free from

CONTRADICTION

s. Despite these disturbing conclusions,

the study of the founding principles of mathematics is

still an active area of research today.

See also G

EORG

C

ANTOR

; G

ÖDEL

’

S INCOMPLETE

-

NESS THEOREMS

; B

ERTRAND

A

RTHUR

W

ILLIAM

R

USSELL

;

A

LFRED

N

ORTH

W

HITEHEAD

; E

RNST

F

RIEDRICH

F

ERDI

-

NAND

Z

ERMELO

.

four-color theorem For centuries, cartographers have

known that four colors suffice to color any geographical

map (that is, any division of the plane into regions). It is

required that regions sharing a common length of

boundary be painted different colors (but two regions

meeting at a point, such as the states Arizona and Col-

orado on a map of the United States, may be painted the

same tint). Cartographers had also observed that the

same is true for any map drawn on a

SPHERE

(the globe).

The question of whether this observation could be

proved true mathematically was first posed by English

scholar Francis Guthrie in 1852. Mathematicians

A

UGUSTUS

D

E

M

ORGAN

(1806–71) and A

RTHUR

C

AY

-

LEY

(1821–95) worked to solve the problem and, in

1872, Cayley’s student Alfred Bray Kempe (1849–1922)

produced the first attempt at a proof of the four-color

conjecture. Unfortunately, 11 years later English scholar

Percy Heawood (1861–1955) found that Kempe had

made an error in his work. In 1890 Headwood later

proved that five colors will always suffice to color a pla-

nar map, but the proof that just four will actually suf-

fice eluded him. Heawood also looked beyond just

planar and spherical maps and made a general conjec-

ture that if a surface contains gholes (such as

TORUS

with g= 1 hole, or a sphere with g= 0 holes), then any

map drawn on that surface can be colored with

colors, and that there do exist examples of maps on

these surfaces that do require precisely this many col-

ors. (The brackets indicate to round down to the near-

est integer.)

In 1968 two mathematicians, Gerhard Ringel and

J. W. T. Youngs, proved Heawood to be correct for all

surfaces with two or more holes and for the torus.

Unfortunately, their work did not apply to the case of a

sphere and of a K

LEIN BOTTLE

. It was not until the next

decade when, in 1976, mathematicians Kenneth Appel

and Wolfgang Haken finally established that four col-

ors do indeed suffice to color any map on a sphere (and

hence the plane, since, by placing a small hole in the

center of one of the regions to be painted, a punctured

sphere can be stretched and flattened onto the plane,

and, vice versa, a planar region can be stretched and

molded into a punctured sphere).

Appel and Haken’s proof was deemed controver-

sial at the time, since it used some 1,200 hours of

computer time to check nearly 2,000 complicated spa-