Engines of Our Ingenuity

No. 90:

by John H. Lienhard

Click here for audio of Episode 90.

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.


(Theme music)

Dictionary of Scientific Biography (C.C. Gillespie, ed.). New York: Charles Scribner's Sons, 1974.

This episode has been revised as Episode 1484.


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