Do you know how many natural numbers there are? If we start counting: 1, 2, 3, 4… will we finish one day? No, because they are infinite. Now, if we take the infinite even numbers, is there more, less or the same number as natural? We can believe that there are more naturals than pairs because the latter are a subset of the former. It is not like that: there are as many natural numbers as even numbers . The clearest way to see it is by lining up the pairs (2, 4, 6, 8, 10…) and below the natural ones (1, 2, 3, 4, 5…) Can we join each even number with every natural number? Yes, and there are none left. Then there are as many even numbers as there are natural numbers. That a part of a set has as many elements as the whole set is so rare that, realizing it, Galileo said that the infinite was “intrinsically incomprehensible”.
The concept of infinite numbers was somewhat puzzling until 1874, when the German Georg Cantor showed that infinity could be treated mathematically . He defined an infinite number in this way: one that can be paired with a certain part of itself, as we have seen before. He found a series of surprising results: there are as many natural numbers as there are fractions (or rational numbers in mathematical language) and as many points on a straight line – obviously infinite – as there are on a plane or in space. “I see it but I don’t believe it,” he wrote in 1877.
But he also discovered that there were more points on a line than natural numbers, which means that the infinity of the line is greater than that of the natural numbers . I call the smallest infinity aleph-0, where the letter alef is the first of the Hebrew, Arabic, and Persian alphabets. This infinity is that of the natural numbers. The next one, aleph-1, is the number of points on a line, and from there follows an endless series that Cantor named transfinite numbers.
His ideas did not find immediate acceptance among his colleagues, for they were more willing to dispense with infinity altogether than to speak of an infinite number of infinities, which was what Cantor did. One of his former teachers, Leopold Kronecker, was a harsh critic. He called him mathematically insane and made every effort to prevent Cantor from getting a teaching position at the University of Berlin. Another even more famous mathematician, the Frenchman Henri Poincaré, said that Cantor’s mathematical theory of infinity was something that later generations would regard as “an illness from which one has recovered.”
Such attacks by the first spades of European mathematics had a tremendous emotional effect on Cantor, a man already somewhat paranoid. He began to see conspiracies everywhere to such an extent that he stopped collaborating with the only mathematical journal that published his work because he was convinced that its director was part of a conspiracy plotted against him. In the spring of 1884 all this tension broke loose and Cantor suffered a nervous breakdown . Once recovered, he abandoned mathematics and dedicated himself to writing philosophical texts. He died in 1918 in a mental hospital.
Mathematical logicians have little logic
Of course, it is strange to discover that the most outstanding logicians of the 20th century, those who have laid the foundations of methodical and rational thought, have passed through the asylum at some point in their lives. One of the most sane was Alonzo Church although, at times, his behavior could not be described as normal. Mathematician Gian-Carlo Rota wrote of him: “Physically, Alonzo looked like a cross between a panda and an owl . He spoke slowly, building his speech in long paragraphs that seemed to be taken from a book. He never talked just to talk. For example, he would never say, “It’s raining.” Instead he would have said: I must postpone my walk to Nassau Street because it is raining, a fact that I can verify by looking out the window.
Church spoke slowly and evenly, without emotion. His classes were quite a spectacle. For the first 10 minutes I meticulously erased the blackboard until there was not the slightest trace of chalk left on it. Sometimes he used soap and water, and although his students tried to save him the trouble, it didn’t matter: he continued with his ritual. His classes were the transcription into spoken language of his logic texts, point by point. And on those rare occasions when what he said in class differed in some way from the written text, Church gave advance notice. He could always be found in the corridors of his workplace at any time of the day or night. Even one Christmas day his students found him sitting on the stairs.
And there we have a great among greats, the logician Kurt Gödel . A great friend of Einstein, the German physicist once commented that he was going to work at the Institute for Advanced Study in Princeton instead of working at home just to walk home with Gödel. Shy, withdrawn and eccentric, he always wore warm clothes, even in the height of summer. In winter he left all the windows in his house open, believing that his enemies were trying to assassinate him using poison gas. He was obsessed with the disease but paid no attention to the prescriptions and recommendations of his doctors. Towards the end of his life he was convinced that they wanted to eliminate him by poisoning his food, so he only ate what his wife cooked; he didn’t even trust the one he could prepare himself. And this was the reason for his death. At the end of 1977 his wife fell seriously ill and stopped cooking. Gödel refused to eat and died of starvation on January 14, 1978. He then weighed 30 kilos.
Referencias:
Bruno, Leonard C (1999) Math and mathematicians, UXL
Dauben, Joseph W. (1979). Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press
Rota, G. (2009) Indiscrete thoughts, Birkhäuser Boston
