a philosopher, he is also remembered as the first to cast
doubts on the existence of
TRANSCENDENTAL NUMBER
s
and on hierarchies of infinite sets, notions first put for-
ward by K
ARL
T
HEODOR
W
ILHELM
W
EIERSTRASS
(1815–97) and G
EORG
C
ANTOR
(1845–1918). At one
point, Kronecker even went as far as to deny the exis-
tence of
IRRATIONAL NUMBER
s.
Kronecker entered Berlin University in 1841 and
received a doctoral degree in mathematics after complet-
ing a thesis on the topic of algebraic number theory 4
years later. Rather than pursue an academic career, Kro-
necker decided to return to Liegitz to help with the fam-
ily banking business. Over the 10 years that followed, he
became quite wealthy and then had the luxury to pursue
mathematics on his own without the need to ever accept
a university position. In 1855 he returned to Berlin and
published a series of influential papers in quick succes-
sion, which garnered him national attention. In 1861
Kronecker was elected as a member of the prestigious
Berlin Academy, which earned him the right to teach at
Berlin University, even though he was not a faculty
member. (In 1883, some 20 years later, Kronecker was
awarded an official position with the university.) Kro-
necker was also elected to the Paris Academy in 1868,
and to the R
OYAL
S
OCIETY
of London in 1884.
During the 1870s Kronecker took a keen interest in
the new, indirect approaches mathematicians were
using to prove the existence of certain types of num-
bers. But he soon came to feel that such nonconstruc-
tive practices were philosophically flawed. For
instance, Cantor had demonstrated that the set of real
numbers is “more infinite” than the set of
ALGEBRAIC
NUMBER
s, and hence, numbers that are not algebraic
(so-called transcendental numbers) must exist.
In Kronecker’s view, mathematics could only ever
be based on finite quantities with a finite number of
operations applied to them. An argument such as
Cantor’s, therefore, in Kronecker’s thinking, had no
meaning. Even when C
ARL
L
OUIS
F
ERDINAND VON
L
INDEMANN
(1852–1939) proved in 1882 that the
specific number πis transcendental, Kronecker com-
plimented Lindemann on his beautiful proof but
added that, in fact, he had accomplished nothing,
since transcendental numbers do not exist.
Kronecker was adamant in his views and, in his
1887 piece Über den Zahlbergriff (On number theory),
he attempted to persuade the mathematical community
of the absolute necessity of only using direct and finite
techniques. He is famous for once having remarked that
“God created the integers. All else is the work of man” as
an attempt to bring mathematical thinking back to con-
crete principles. Both Weierstrass and Cantor felt under
personal attack by Knonecker and thought that he was
deliberately undermining their own research programs.
Although many mathematicians today would not
agree with Kronecker’s views, they were taken seri-
ously at the time, and the ideas that Kronecker put
forward were later expanded upon in the century that
followed. The practice of using only direct and finite
approaches to prove results in mathematics is today
called constructivism.
Kronecker died on December 29, 1891, in Berlin,
Germany.
Kruskal’s count In the early 1980s, Princeton physi-
cist Martin Kruskal discovered a remarkable mathe-
matical property that all passages of written text seem
to possess. This phenomenon is now referred to as
Kruskal’s count.
To illustrate the principle, review, for example, the
familiar nursery rhyme:
Twinkle twinkle little star,
How I wonder what you are,
Up above the world so high,
Like a diamond in the sky.
Twinkle twinkle little star,
How I wonder what you are.
Perform the following steps:
1. Select any word from the first or second line and
count the number of letters it contains.
2. Count that many words forward through the pas-
sage to land on a new word. (For example, choosing
the word star, with four letters, will transport you to
the word what.)
3. Count the number of letters in the new word, and
move forward again that many places.
4. Repeat this procedure until you can go no further
(that is, counting forward will take you off the
nursery rhyme).
5. Observe the final word on which you have landed.
Surprisingly, no matter on which word you start this
counting task, the procedure always takes you to the
same word in the final line, namely, the word you.
Kruskal’s count 297