Today, we wonder about counting to infinity. The
University of Houston's College of Engineering
presents this series about the machines that make
our civilization run, and the people whose
ingenuity created them.
When I was in grade school,
Time magazine ran an article that snatched
my imagination. Someone proposed a new number
called the googol -- a one followed by a
hundred zeros. Later I learned that it wouldn't
help very much in counting real objects, because
there aren't that many real objects in the whole
universe -- even atoms.
But still, we've all wondered where counting ends
and infinity begins. And we have good reason for
asking about infinity. Every engineering student
soon learns that infinity is not just the end of
numbers. If we ask how real systems behave when
velocity, or time, or force becomes infinite -- if
we ask about the character of infinity -- we get
unexpected and profoundly useful answers.
Georg Cantor also wondered about infinity. He was
born in Russia in 1845. He wanted to become a
violinist like his mother. But his father, a
worldly merchant, wouldn't have it. When he was
seventeen, his father died, and Cantor went on to
finish a doctorate in mathematics, in Berlin. He
was only twenty-two when he did.
His career didn't last long -- he burned out before
he was forty and spent the rest of his life in and
out of mental illness. But what he did was
spectacularly important, and it arose out of an
innocent counting question. He began with an idea
we find even in nursery rhymes. Do you remember
1-potato, 2-potato, 3-potato, 4
5-potato, 6-potato, 7-potato, more?
Counting is really matching one set of things with
another -- in this case, numbers with potatoes.
Cantor asked if counting all the infinite number of
points on a line would be like counting all the
points in a surface. He identified orders of
infinity. Count all the whole numbers and you get
the first order of infinity. Raise two to that
power and you get the next. And that happens to be
the number of sets you can form from the first
infinity.
By the time he was done, Cantor had invented what
we call the transfinite numbers -- numbers that go
beyond infinity. And to do that he'd had to invent
set theory. And set theory has become a building
block of modern mathematics. Cantor had fallen into
an odyssey of the mind -- a journey through a
strange land.
Along the
way, he had to overcome the doubts of his father,
objections of the great mathematicians, and his own
doubts as well. When he was 33 he wrote:
"The essence of mathematics is freedom." Cantor had to
value freedom very highly -- freedom coupled with
the iron discipline of mathematics, freedom
expressed as the driving curiosity of a bright
child, freedom to pursue innocent fascination until
it finally touched the world we all live in.
Cantor lived his troubled life until 1918. But that
was long enough. He finally saw set theory
accepted. He finally saw himself vindicated for
that soul-scarring voyage of the mind.
I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.
(Theme music)
Meschkowski, H., Cantor, Georg. Dictionary of
Scientific Biography, Vol. III, (C.C. Gillespie,
ed.). New York: Charles Scribner's Sons, 1971, pp
52-58.
I am grateful to Professor James Casey, formerly at
UH and presently at the University of California at
Berkeley, for suggesting the topic.
This is a reworked version of Episode 90.
For more on Cantor, see the website (in German)
http://www.mathematik.uni-halle.de/history/cantor/
The Engines of Our Ingenuity is
Copyright © 1988-1999 by John H.
Lienhard.
