Over the last month, we spent a lot of time talking about infinity. Pretty much everything we discussed was the work of one man, Georg Cantor.
You might think that such a gifted mathematician, who laid the basis for all our modern understanding of infinity would have been universally respected by his contemporaries.
Not so!
Cantor’s ideas of infinity, as well as his methods of proof, infuriated many prominent mathematicians.
Cantor was born in 1845 in St. Petersburg, though his family soon moved to Germany, seeking milder winters. His exceptional mathematical skills were noted early, and he studied with some of the famous mathematicians of the day, such as Kronecker and Weierstrass. (If you’re not a mathematician, you’ve probably never heard those names before, but, I promise, they’re well-known mathematicians!)
After graduation, he took up a professorship at the University of Halle. This was a good position, and he advanced to full professor at a young age.
Cantor’s early work was on number theory, but his famous work was on the nature of infinity, and the sizes of sets. In an early paper he proved that the set of real numbers is larger than the set of counting numbers (see this post). This was the first instance of showing that infinite sets could have different sizes.
Before Cantor, infinity was a vague thing, and often rejected as a proper thing. For instance, one of the most famous mathematicians of all time, Gauss, had the view that, “Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn’t belong in mathematics.”^{1}
Indeed, many of Cantor’s contemporaries hated Cantor’s ideas. One of the most prominent opponents to his ideas was Leopold Kronecker, head of the math department at the University of Berlin. Berlin was more prestigious than Halle, and Cantor wanted to get a position there, but Kronecker’s opposition nixed any opportunity for that.
Kronecker’s position is often put under the label “finitism.” He once said, “God created the natural [counting] numbers; all else is the work of man.” The viewpoint is that the counting numbers clearly exist, but any other mathematical object should be able to be derived by a finite number of steps from this basic foundation. Kronecker rejected Cantor’s original proof that the set of real numbers is bigger than the set of counting numbers on this ground. Cantor developed the proof we discussed in response to his criticisms.
Kronecker at one point said, “I don’t know what predominates in Cantor’s theory – philosophy or theology, but I am sure that there is no mathematics there.” Among other things, he called Cantor a “scientific charlatan,” a “renegade” and a “corrupter of youth.”
Kronecker was not so fond of Cantor.
These criticisms, among others^{2}, stung Cantor deeply, leading to several episodes of depression. It sucked the joy of math out of him for several years. For a while, he turned to, of all things, the Shakespeare authorship question, on which he wrote a couple of pamphlets.
He did eventually return to math, though his later work was not as transformative as his earlier work. Cantor eventually retired in 1913, and lived in poverty due to World War I. He died in 1918 at the age of 72.
Many mathematicians realized the importance of Cantor’s work. In 1904, Cantor was awarded the Sylvester medal, the highest honor of the Royal Society. Cantor’s research into infinity have influenced mathematicians for generations. The influential and important mathematician Hilbert said of his work, “No one will drive us from the paradise which Cantor created for us.”^{3}
- Dunham, William. Journey through Genius: The Great Theorems of Mathematics. Penguin. p. 254. Gauss lived 1777 to 1855, dying when Cantor was 10 years old. ↩
- At one point, Cantor tried to publish a paper in Acta Mathematica, which was run by an acquaintance, Mittag-Leffler. Mittag-Leffler asked him to withdraw the paper, as the paper was “one hundred years too soon.” That… hurt their relationship. ↩
- Though, apparently, another mathematician replied, “If one person can see it as a paradise of mathematicians, why should not another see it as a joke?” ↩
Great piece!
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Great article. A detailed account of Cantor and Infinity is also covered in John D Barrow’s “The Infinite Book”.
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