this in his treatise, An Investigation of the Laws of Thought, on Which Are Founded
the Mathematical Theories of Logic and Probabilities(1854). Today, this is called
Boolean algebra. (For more information about Boole, see “Algebra”; for more informa-
tion about symbolic logic, see “Foundations of Mathematics.”)
MODERN MATHEMATICS
Who first developed set theory?
German mathematician George (Georg) Ferdinand Ludwig Philipp Cantor
(1845–1918) was not only known for his work on transfinite numbers, but also for his
development of set theory, which is the basis of modern mathematical analysis (for
more information on set theory, see “Foundations of Mathematics”). His Mathematis-
che Annalenwas a basic introduction to set theory. Unlike most long evolutionary his-
tories of mathematical subjects, Cantor’s set theory was his creation alone. In the late
19th century, Cantor also developed the Continuum Hypothesis. He realized that
there were many different sized infinities, further conjecturing that two particular
infinities constructed by different processes were the same size.
What was the Principia Mathematica?
In 1910 the first volume of the Principia Mathematicawas published by Welsh mathe-
matician and logician Bertrand Arthur William Russell (1872–1970) and English
mathematician and philosopher Alfred North Whitehead (1861–1947). This book was
an attempt to put mathematics on a logical foundation, developing logic theory as a
basis for mathematics. It gave detailed derivations of many major theorems in set the-
ory, examined finite and transfinite arithmetic, and presented elementary measure
theory. The two mathematicians published three volumes, but the fourth, on geome-
try, was never completed.
On their own, both men did a great deal to advance mathematics, too. Russell dis-
covered the Russell paradox (see below), introduced the theory of types, and popular-
ized first-order predicate calculus. Russell’s logic consisted of two main ideas: that all 31
HISTORY OF MATHEMATICS
Why was non-Euclidean geometry important to Albert Einstein?
N
on-Euclidian geometry, especially the form suggested by Bernhard Rie-
mann, enabled Albert Einstein (1879–1955) to work on his general relativity
theory (1916), showing that the true geometry of space may be non-Euclidean.
(For more information about mathematics and Einstein, see “Math in the Physi-
cal Sciences.”)