530 well-ordered set

Weierstrass made significant contributions to the

study of

POWER SERIES

and their convergence, and to

F

OURIER SERIES

and their applications to problems in

geometry and mechanics. In providing a precise defini-

tion of a limit, Weierstrass completely revolutionized

mathematical understanding of the founding principles

of

CALCULUS

and modern analysis. In a very real sense,

Weierstrass set the standards of rigor that all mathe-

maticians still follow today.

In the study of

CONVERGENT SERIES

, the compari-

son test shows that the function:

converges for every value of x. (Compare with series

) Weierstrass was able to show, moreover, that

this function is continuous, but that it has no derivative

at any irrational value of x. This was perhaps his most

famous example of a pathological function that is con-

tinuous but possesses no meaningful tangent line at

almost all points. That such functions exist, as Weier-

strass pointed out, indeed demonstrates the need for

uncompromising care in the development of details in

all theoretical work. Jules Henri Poincaré (1854–1912)

admired his “unity of thought” on this matter.

Weierstrass died in Berlin, Germany, on February

19, 1897.

well-ordered set An

ORDERED SET

Ais said to be

well ordered if every nonempty subset of Ahas a small-

est member. For example, the set of natural numbers is

well ordered: for any set of natural numbers one cares

to describe, there will be a smallest member of that set.

On the other hand, the set of all integers is not well

ordered. For example, the subset of negative integers

has no smallest member.

In 1904 German mathematician E

RNST

F

RIEDRICH

F

ERDINAND

Z

ERMELO

(1871–1953) proved that any

ordered set can be made well ordered if one is willing

to change the ordering on that set. For example, the set

of all integers can be made well ordered if one arranges

them as follows:

0, 1, –1, 2, –2, 3, –3, 4,…

(The “smallest” element in the set of all negative inte-

gers, for instance, is now –1.) Zermelo needed to invoke

a controversial

AXIOM OF CHOICE

to prove this claim.

Whitehead, Alfred North (1861–1947) British Logic,

Theoretical physics, Philosophy Born on February 15,

1861, in Kent, England, Alfred Whitehead is best remem-

bered in mathematics for his three-volume work Prin-

cipia Mathematica, written in collaboration with English

logician B

ERTRAND

A

RTHUR

W

ILLIAM

R

USSELL

(1872–1970). Whitehead and Russell were hoping to

derive the whole of mathematics from purely logical prin-

ciples.

Whitehead entered Trinity College, Cambridge, in

1880 to study applied mathematics. He worked and

taught in this field for 12 years before his interests even-

tually turned toward pure mathematics and the founda-

tional principles of the subject. In 1891 he commenced

work on Treatise on Universal Algebra, a project that

took him seven years to complete. At the same time,

Bertrand Russell entered the college as an undergradu-

ate, and Whitehead immediately recognized him as a

brilliant philosopher and capable mathematical scholar.

Their collaboration on Principia Mathematica began in

1900 and was initially intended to be a one-volume

work. Russell’s 1901 discovery of his famous set-theory

paradoxes, however, forced the two men to reevaluate

the project and extend the work to a three-volume trea-

tise. These volumes were published from 1910 to 1913.

During this time Whitehead also published other

material, including texts on projective and descriptive

geometry and a general popular overview of mathemat-

ics, An Introduction to Mathematics. Whitehead also

wrote pieces on the philosophy of science, and offered

an alternative viewpoint to A

LBERT

E

INSTEIN

’s theory

of relativity in his 1922 book The Principle of Relativ-

ity. The work was not well understood and has been

largely ignored.

Whitehead worked as a lecturer at Cambridge for

over 30 years before accepting a position as chair of phi-

losophy at Harvard University in 1924. He remained

there until his retirement in 1937. He died in Cambridge,

Massachusetts, 10 years later on December 30, 1947.

whole number A

NATURAL NUMBER

is sometimes

referred to as a whole number. Matters are confusing