Today, we meet the man who showed us how to count
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
we'd be hard pressed to find 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
knows that infinity isn't just the end of numbers.
If we ask how real systems behave when velocities,
or time, or force become infinite -- if we ask
about the character of infinity, we get some very
unexpected answers.
The mathematician Georg Cantor also wondered about
infinity. He was born in Russia in 1845 and was
taught by a father who wouldn't let him become a
violinist and who didn't want him studying
mathematics, either. But when he was 17, his father
died. Cantor went on to finish a doctorate in
mathematics in Berlin while he was still only 22.
His career wasn't long -- he burned out before he
was 40 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 mother goose. Do
you remember
1-potato, 2-potato, 3-potato, 4
5-potato, 6-potato, 7-potato, more?
Counting is like 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 was like counting all the points
in a surface. To answer the question, he had to
invent something called transfinite numbers --
numbers that go beyond infinity. And to do that he
had to invent set theory. And set theory has become
a basic building block of modern mathematics.
Cantor fell into an odyssey of the mind -- a
journey through a strange land. He had to overcome
the resistance of his father, of the great
mathematicians of his day -- even of his own
doubts.
When he was 33, he wrote: "The essence of
mathematics is freedom." To do what he did, he had
to value freedom very highly -- freedom coupled
with iron discipline -- freedom expressed through
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, and that
was long enough for him to finally see set theory
accepted and himself vindicated for his
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)
