# Infinity did not drive Cantor crazy.

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.

Filed under History of Mathematics, Myths of Science

### 28 responses to “Infinity did not drive Cantor crazy.”

1. Vít Tuček

Would you be so kind and provide a reference for your claims? I am especially intersted in the objections to set theory.

• Vit, the Stanford Encyclopedia of Philosophy and Wikipedia are good places to start. Look up Jan Egbertius Brouwer, intuitionism, constructivism, measure theory, NBG, ZFC, or naive set theory. Present-day intellectuals are still thinking about this, eg http://www.paultaylor.eu/ASD. Although the question is about the size of a pinhead among human concerns in general, it’s kind of an enormous topic.

2. The idea that Cantor went insane from thinking too much about infinity reminds me of the equally common notion that Nietzsche’s philosophy drove him mad. I won’t speak to Cantor problems—I understand he suffered from depressive episodes at intervals, but I’ve never made a study of his life. Nietzsche’s case, on the other hand, seems pretty clear to me and I have studied that. While he may not have been the mental health poster child during his productive years, his terminal breakdown was almost certainly the result of something organic such as paresis (end-stage syphilis) since the psychiatric manifestations were accompanied by physical paralysis and other somatic problems that don’t go along with functional psychosis. Of course fictional forms of insanity can take any form you like, so cooking up the myth of the mad philosopher doesn’t require you to consult the DSM.

3. Does anyone else suspect that the popularity of the canard of Cantor’s infinity-induced madness is a form of anti-intellectualism in disguise?

• I think it’s is more probably an example of the mad genius cliché.

• I think the mad genius cliché is part and parcel of anti-intellectualism, as it suggests that intellectual brilliance is tied to some kind of mental disorder and hence not a goal worth pursuing. The way scientists are traditionally portrayed in children’s books and cartoons generally teaches the same lesson.

• Interesting thought. I think you are probably right. Scientists are very clever but definitely suspect.

• +1

4. Leibniz steered clear of the unresolvable “labyrinth” of the nature of continuum. Cantor did not.

5. As I pointed out in a comment on io9 — and quoted from David Foster Wallace’s excellent Everything and More — “Saying that ∞ drove Cantor mad is sort of like mourning St. George’s loss to the dragon: it’s not only wrong, but insulting.” For a larger chunk: http://drmathochist.wordpress.com/2008/10/11/dfw-on-cantor-and-mathematical-insanity/

• Michael Weiss

Nice quote, especially the longer excerpt. But I do wonder about one remark in the longer quote:

The real irony is that the view of ∞ as some forbidden zone or road to insanity—which view was very old and powerful and haunted math for 2000+ years—is precisely what Cantor’s own work overturned.

Now, Gauss, famously, objected to the “completed infinity” in his correspondence, and I’m sure this uneasiness could be replicated many times over throughout mathematical history. It lies at the root of intuitionism, a serious philosophy of mathematics. So the “forbidden zone” part is fine. But when did any one actually intone against it as a “road to insanity”? Does DFW provide examples in his book? And insanity is the issue with this Cantor legend.

A side-light on intuitionism. On the one hand, it lead to significant work in mathematical logic and theoretical computer science, work generally accepted by experts in those fields of all philosophical stripes. (Gödel, a hard-core Platonist, did work on intuitionistic logic. Paul Cohen’s method of forcing has connections with this too.) On the other hand, I doubt there are many adherents to the philosophy of intuitionism these days, at least among professional mathematicians.

• I got around to reading DFW’s Everything and More a year or so ago. “Excellent” it is not. I endorse the judgement expressed in Rudy Rucker’s review:

David Foster Wallace is a great writer…[his] work is revelatory, funny, and post ironic. I fully expected to enjoy Everything and More. But it’s a train wreck of a book, a disaster. Nonmathematicians will find Everything and More unreadable, and mathematicians will view it with at best, sardonic amusement. Crippling errors abound.

The book does have all of DFW’s signature wit and flair. But it’s obvious that his grasp of the material is quite shaky. It’s like having as a tour guide in a strange city, someone who just arrived there a week before you did.

DFW hired Ph.D. student Erica Neely (now a professor of philosophy) as a technical adviser, and she gave an interview in 2015 about the experience. From the preface to this interview (not written by Neely):

Despite undergoing extensive editing (including a manuscript review by a Norton-commissioned mathematician who had ‘serious reservations and pointed to errors, some small, some larger in it’, Everything and More was published in October 2003 to hostile reviews from the scientific community… [this prompted] the staff at Norton to take note of suggested revisions, especially from Wallace’s mathematically-trained contributors Jordan Ellenberg and Prabhakar Ragde, to the point of requesting that Ragde subject the text to yet another proof-reading prior to a paperback edition. However, Wallace opposed many of Ragde’s suggestions…</blockquote.

6. Being a mathematician, not a typographer, I tend to think of “infinity” as a number. It is sometimes appended to the real numbers to form what is known as the extended real numbers. Furthermore, Cantor’s transfinite numbers and J.H. Conway’s surreal numbers include infinite quantities, though they do not use the “lazy eight” to represent them.

7. It would be self-contradictory for infinity to be exceeded, true? (True dat) So on what grounds is there anything larger than infinity? Just because one set contains another does not mean that it is numerically larger in number of elements. And just because a convincing argument is presented showing something larger than infinity (that is only convincing if one overlooks a flawed premise in the argument), does not mean that there can be something larger than infinity: there can’t be!

There are a number of errors IN MATHEMATICAL FOUNDATIONS that all populate the arena around whether or not there is something larger than infinity (a non-starter and a self-contradiction) vs. whether or not infinite numbers actually can or must exist. For some reason, orthodoxy in mathematics has not yet corrected the errors here ( http://limitstomaths.com ) which would result in showing that there is nothing larger than infinity (which is manifest merely by definition) and that infinite numbers have to exist (which is errantly disclaimed due to self-contradictory attempted definition — errant definition that is not really definition at all).

• mathfan

You’re a moron. Take an upper level math class before you bash things you don’t understand, thank you.

• Your first suggestion is false. There are an ∞ number of points between my thumb and my nose, but also ∞ points between my thumb and Bogotá. Yet the latter distance is greater, so ∞ > ∞. No contradiction.

To give a purely mathematical example the infinite limit of 3x / x = 3∞/∞ = 3 whereas the infinite limit of x / xx = ∞ / ∞ = 0. Clearly infinities can be different sizes, that’s why ∞/∞ is indeterminate.

• Eric

I agree with you – infinity is a concept, not a number.

In essence, it simply means that a pattern will continue to repeat without any apparent reason to end.

The best anyone can do is create relative relationships between patterns that seemingly go on forever.

8. Perhaps Cantor was somewhat ahead of his time? Perhaps he came up with a hypothesis which hypothetically wasn’t meant to be tackled yet? There have been philosophical, mathematical and scientific quandaries which remain unsolved until well after the original inquirer’s death, due to the invention/discovery of new scientific tools and resources.

Perhaps there is an accepted (though incorrect) assumption that Cantor’s study of Infinity drove him insane, because fate sought to deter him from making too much progress on a concept which may have not been appropriate for the time period?

9. Are you really calling Brouwer’s objections to Cantor Mathematically and philosophically solid? If anyone in this story is insane, it’s Brouwer, not Cantor. It’s a tragedy – the man revolutionized topology for long enough to get tenure and then switched to a subject he had no skill in…

• The real tragedy is that you understand absolutely nothing about the philosophy of mathematics.

• It’s no traagedy

10. Great article.

11. slam

As somebody, who has struggled with mental illness for much of my life, I don’t tolerate people who spout crap about mental illness in my comments column, so comment removed