I do wish that the authors of the aptly named ‘science fiction’ blog io9 would do some basic checking of facts if they are going to write about the history of science and mathematics. Today an Esther Inglis-Arkell has posted an article about Georg Cantor, which in two very essential points is, to put it mildly, a total disaster. I have copied part of Ms Inglis-Arkell’s ignorance below.
Imagine a thin line, almost a thread, stretching to infinity in both directions. It runs to the end of the universe. It is, in essence, infinite. Now look at the space all around it. That also runs to the end of the universe. It’s also infinite. Both are infinite, yes, but are they the same? Isn’t one infinity bigger than the other?
That’s the question that Georg Cantor, a German mathematician who died shortly before World War I ended, grappled with throughout his life. Infinity was supposed to be an absolute number, especially in mathematics, where dividing infinity by a billion or multiplying it by a billion results, invariably and always, in infinity. Cantor thought about it and came up with aleph-nought, a ‘number’ that counts all the integers — whole numbers without fractions — that there are in existence. Aleph-nought has to be infinity, since there are an infinite quantity of whole numbers. But what about all the ‘real numbers’? Rational numbers include whole numbers, but they also include fractions of whole numbers. And since a whole number can be divided into an infinite number of fractions in and of itself, the set of rational numbers had to be larger than aleph-nought. But then what about real numbers? Real numbers include rational numbers, and irrational numbers (like the square root of five), and integers. This has to be a greater infinite number than all the other infinite numbers.
This was when mathematicians started, metaphorically, booing and hissing when Cantor went by. The idea simply didn’t make sense to them. Infinity was infinity. That was the end of it. Cantor called his various sets of different quantities of infinity ‘transfinite’ numbers — they are also known as cardinal numbers — and designated aleph-nought as the smallest transfinite number in existence. The controversy his views stirred up cost him an appointment at the University of Berlin. It also cost him his sanity on many different occasions. Throughout his later life he stayed in mental hospitals regularly.
Ms Inglis-Arkell’s first major error is in the first emphasised section above. The set of rational numbers is not larger than aleph-nought! One of the counter intuitive results that Cantor demonstrated in his investigations into the infinite is that although the set of the natural numbers is a subset of the set of rational numbers, the two set actually have the same cardinality i.e. they are the same size. This is not a trivial mistake on the part of Ms Inglis-Arkell but display a fundamental ignorance of Cantorian theory.
Her second error is on a human level historically possibly even worse and one of the most persistent and ignorant myths in the history of maths. Georg Cantor suffered from bi-polar disorder and whilst the stress caused by the serious objections to his work by a number of his colleagues probably aggravated his illness it was almost certainly not its cause.
In the second paragraph above Ms Inglis-Arkell propagates an unfortunate deception that has crept into the history of maths. It is implied that those such as Kronecker, Poincaré or Brouwer who raised objections to Cantors work did so out of ignorance or spite, whereas the objections raised by the constructivists and intuitionalists are based on solid mathematical and philosophical grounds. Objections, I would point out that have not been adequately dealt with till this day. It should be pointed out that the opening paragraph posted above supposedly outlining Cantor’s motivation is historically false but the true grounds for his work are much to complex to deal with here.