He had learned in school that Cantor was the sole founder of set theory — and that it all started with a proof he published in 1874. In that proof, Cantor showed that there are different sizes of infinity, putting to bed the notion that infinity was merely a piece of mathematical trickery.
Goos began research for a podcast about Cantor’s discovery. But he soon found that the true story was more complicated than he’d been told.
“My approach originally was to tell the story everybody tells. It’s a beautiful story,” he said. “But it’s a wrong story. It’s not really what happened.”
The Trojan Horse
The true story was that Cantor wasn’t a lone genius. He had a partner — at least for a time.
Whenever Cantor met like-minded mathematicians, he was known to court them eagerly. He would show up at a collaborator’s residence at daybreak, excited to discuss some new idea he’d had, sometimes waiting for hours until they woke up. So it was with Dedekind. After their 1872 encounter in Gersau, Cantor took every opportunity to ask the older mathematician for advice.
In November 1873, Cantor began an exchange that would forever alter the course of human knowledge. “Allow me to put a question to you,” he wrote to Dedekind in a hastily penned letter. “It has a certain theoretical interest for me, but I cannot answer it myself; perhaps you can.”
Cantor had found an outlet for the zealous drive his father had instilled: the infinite nature of the number line. “He had a very strong sense of mission,” said José Ferreirós, a historian and philosopher of mathematics at the University of Seville in Spain. “He was convinced that the introduction of actual infinity was going to change not only mathematics, but science in general.” To Cantor, this kind of infinity didn’t contradict God’s supremacy. It just meant that rather than being remote and unknowable, God was everywhere, residing between all things.
DEDEKIND TO CANTOR 14 July 1873 “I have often and with great pleasure recalled the beautiful day in Gersau and Beckenried when I had the pleasure of making your personal acquaintance.”
He began studying the real numbers as a single, infinite package, asking questions no one had thought to ask before. Was there a difference between the infinity signaled by the three dots in 1, 2, 3, … , and the one built into the mysterious continuum of the number line? In other words, were there more real numbers than whole numbers?
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