Someone on Twitter drew my attention to a BBC4 television documentary by David Malone from 2007 about mathematics. Interested I thought I would give it a whirl, I wish I hadn’t. It’s a sort of biography of Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing so what could go wrong? It’s called Dangerous Knowledge but Dangerous Twaddle would have been more appropriate.

In my opinion it starts off with a real humdinger: atmospheric images with the following dramatic voice over:

*Beneath the surface of the world are the rules of science but beneath them there is a far deeper set of rules. A matrix of pure mathematics, which explains the nature of the rules of science and how it is we can understand them in the first place.*

Ignoring the fact that I don’t actually agree with this piece of trite metaphysics, the author completely blows it in my opinion because another even more dramatic voice over follows this with the following quote:

*To see a World in a Grain of Sand*

*And a Heaven in a Wild Flower*

*Hold Infinity in the palm of your hand*

*And Eternity in an hour*

This is of course one of the most well known quotes by William Blake taken from his *Auguries of Innocence*. Malone is obviously ignorant of Blake’s opinion of the mathematical description of the world.

*God forbid that Truth should be confined to Mathematical Demonstrations*! (Written as a marginal note to Reynolds’ *Discourses*.)

Before we get down to the real reason that I’m writing this a couple of things that annoyed me whilst watching this documentary. The author-narrator mispronounces the names of both Leibniz and Dedekind. One would think that if somebody is making a documentary about mathematics and mathematicians they would at least take the trouble to get the names of famous mathematicians right. At the end of the section about Cantor he describes him as the greatest mathematician of his century! Regular readers of this blog will know that I intensely dislike such superlatives in the history of science. Even if I didn’t, is Cantor really the greatest mathematician of the nineteenth century? There’s an awful lot of competition. In the section on Gödel, we get told about his friendship with his fellow Austrian mathematician, Albert Einstein.

Now, I know that trying to keep track of Einstein’s nationality is rather difficult for the non-historian; he had a total of eight different ones including being stateless for five years. However, he was only a citizen of the Austrian Empire from April 1911 to July 1912, as professor at the University of Prague. Eleven months out of a life of 76 years hardly justifies calling him an Austrian.

My real beef with the documentary is contained in a further piece of voice over from the introduction:

*… pursued the questions to the brink of insanity and over it*.

Basically Malone spends eighty minutes telling the world that if brilliant mathematicians think outside the box it can and will drive them insane! This is quite simply bullshit!

He devotes the largest part of the documentary to Georg Cantor and the invention of set theory. I found his explanations of what Cantor achieved and why he did it totally opaque and I spent quite a lot of time at university studying and understanding it. Malone gives a totally bogus explanation of the continuum hypothesis, which suggests very strongly that he simply doesn’t understand it, and then goes on to explain that it was Cantor’s inability to prove the continuum hypothesis drove him insane. I will return to this.

We then move on to Ludwig Boltzmann and his championing of a probablistic atomic theory when the majority of physicists and philosophers opposed the real existence of atoms. Once again Malone tells us that it was Boltzmann’s science that drove him mad and led him to commit suicide.

Although it is always dubious to make historical diagnoses of illnesses, in particular mental ones, Both Cantor and Boltzmann displayed all of the symptoms of a severe bipolar disorder. Bipolar disorder is not caused by mathematical research or any other work for that matter. A stressful working situation might well aggravate an existing bipolar disorder it won’t cause it. This is as I said, dangerous twaddle.

Malone now accelerates his gallop into the realm of total crap with his segment on Kurt Gödel. Following the usually incorrect statement of Gödel’s incompleteness theorem. People almost invariably leave off the very important final “within the system” in their accounts. What Gödel showed in that we cannot produce a formal logical system within which all true mathematical statements are provable. However this does not mean that the statements that are unprovable within the given system are fundamentally unprovable, as Malone claims in his statement of Gödel. However this is a minor quibble compared to Malone’s central claim. He states that Gödel took up the continuum hypothesis and because he like Cantor was unable to prove it, he too went insane. Now, it is well known that Gödel displayed serious symptoms of mental illness that got increasingly worse as he got older, until he quite literally starved himself to death due to his paranoid belief that somebody was trying to poison him. I’m not a clinical psychiatrist, but I’m more that willing to state that Gödel’s ability or lack of it to solve the continuum hypothesis did not cause his mental illness. Malone, however, seem to be totally unaware that Gödel in fact showed the continuum hypothesis was consistent, i.e. cannot be proved false, with the axioms of Zermelo-Fraenkel set theory. This is one of the major breakthroughs in the history of set theory; far from being frustrated by the continuum hypothesis Gödel produced one of his most important results with it.

Malone closes out his trip through the insane and suicidal mathematical geniuses with none other than Alan Turing. Following up on the usually false claim that Turing invented the computer and a very confusing explanation of Turing’s achievements in meta-mathematics, Malone takes us forward to Turing’s death. He has the British secret service responsible for Turing’s chemical castration following his conviction for indecency, which is just simply crap. Turing was offered a choice between a prison sentence or probation with the hormonal treatment as a condition by the court. He freely chose the later. I’m not even going to enter the discussion of whether he committed suicide or not and if he did why. There has already been enough ink spilt on that particular topic.

Malone made a documentary about four major figures in the history of mathematics, logic and mathematical physics and presents the quite honestly laughable thesis that it was their intellectual audacity and the opposition that they experienced to their theories that drove them insane. This is quite simple put, bullshit. As someone who has experienced, at time quite serious, mental illness I find it quite frightening that an organisation such as the BBC is not only prepared to air such crap but to finance it with obviously comparatively large sums of money. We live in a society where it is extremely difficult to explain to people what mental illness is and such pseudo-psychological bullshit as Malone’s documentary does nothing to help with this problem.

Oy veh

The theme of the genius driven mad by too much thought is a folklore motif that goes back a long way. In Acts 26:24, Festus tells Paul “Your great learning is driving you insane,” an accusation/diagnosis that Christian versions of the most excellent Festus have more recently launched against the memory of Nietzsche. Hard, sustained thinking really is a strain, though, so the mathematicians, physicists, and philosophers who have actually experienced it don’t all dismiss the possibility that you can drive yourself over the edge. It’s a running joke among students of thermodynamics. My copy of David Goodstein’s States of Matter is one of those Dover reprints of a classic textbook that pros keep around to consult. It begins with a couple of sentences that various scientists and engineers have quoted to me or alluded to over the years: “Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics.”

Thing is, the notion that the Continuum hypothesis (or whatever) will drive you insane is mostly nonsense, but the idea is pretty much ineradicable so I figure it helps to keep a sense of humor about it . “I laugh for fear of being obliged to weep.”

Thanks for posting.

I think I watched part of that documentary a couple of years ago. And I was underwhelmed. I agree that it is twaddle.

“However, he was only a citizen of the Austrian Empire from April 1911 to July 1912, as professor at the University of Prague. Eleven months out of a life of 76 years hardly justifies calling him an Austrian.”If I recall correctly, this happened automatically and perhaps without Einstein’s knowledge when he became a professor in Prague.

Later, there was some debate about who should congratulate him on winning the Nobel Prize, which depended on his nationality. Via a similar argument, since he was employed at the Kaiser-Wilhelm-Institut, someone exclaimed “Einstein ist Reichsdeutscher”, thus settling the matter.

Einstein once said that if his theory was true, then Germany would say he is German and France would say he is a citizen of the world. I false, France would say he is German and Germany would say he is a Jew.

If I understand it correctly becoming a citizen of the Austrian Empire was a requirement for the Prague professorship.

Nope, he knew. As Isaacson’s bio puts it:

As it turned out, Einstein’s desire for the job was greater than his ornery impracticality. He agreed to write “Mosaic” as his faith, and he also accepted Austro-Hungarian citizenship, with the proviso that he was allowed to remain a Swiss citizen as well. Along with the German citizenship that he had forsaken but that would soon be foisted back on him, that meant he had held, off and on, three citizenships by the age of 32.The faculty had chosen Einstein as their first choice for the post, but they were overruled by the ministry, who preferred Jaumann, the second place choice, because he was Austrian and not Jewish. Jaumann rejected the offer in a fit of pique at not being first choice, and so Einstein got the job. It would have been pushing his luck for him to refuse Austrian citizenship.

He was also voluntarily stateless for five years between his initial birth citizenship Württemberg and his Swiss citizenship.

While I agree that the mental illnesses in these cases had nothing to do with maths, while Einstein was in Prague, his office looked onto an insane asylum. He once remarked “There are the crazy people who don’t do quantum mechanics”.

“At the end of the section about Cantor he describes him as the greatest mathematician of his century!”

The exclamation mark is well-deserved. I would like to have seen his argument that Cantor was a greater mathematician than both Gauss (1777-1855) and Poincaré (1854-1912). I certainly will not be watching this programme on iPlayer.

Arthur Cayley, J.J. Sylvester, George Boole, Charles Hermite, Henri Lebesgue, David Hilbert, Felix Klein, Karl Weierstrass, Guiseppe Peano, Niels Abel, Sophus Lie, Sofia Kovalevskaya… just to name some of the more obvious candidates

I know that the lives and work of both Cantor and Hilbert spanned the turn of the century, but it seems odd to put them in the same bucket, century-wise. Anyway, HIlbert is someone who might have agreed with the ranking of Cantor. Famously he said, “No one will drive us out of the paradise that Cantor has created for us!” He also wrote about Cantor’s set theory: “It is, I think, the finest product of mathematical genius, and one of the supreme achievements of purely intellectual human activity.”

I don’t find all of your obvious candidates obvious at all. Kovalevskaya and Sylvester, really? And you left out Dirichlet. And Dedekind! And Riemann!!!

Kovalevskaya qualifies due to the obstacles that she had to overcome in her career. You obviously know far too little about Sylvester. As for Dedekind, Dirichlet and Riemann I did say some…

OK, I’ll bite. What do you imagine I don’t know about Sylvester? His most important work was on matrices, but he had a lot of company there—Cayley, Hamilton, Grassman, Cauchy, Jacobi, Vandermonde,… Because of his significance to the history of math in the U.S., I’ve also read several articles about him over the years.

What Kovalevskaya accomplished in the face of obstacles was impressive, yes, but are we grading on a curve? Judging the work

on its own merits, can you really say with a straight face that it comes close to Cantor’s achievemnts?Set theory had an

enormousimpact on math in the 20th century, and Cantor’s work was a huge driving force, though of course he hardly stands alone.I agree completely with your objection to calling Cantor “the greatest mathematician of his century”. (Using 1900 as a dividing line is also problematical.) But we can still sort figures into tiers, with fuzzy boundaries. Abel, Poincaré, Cantor, yes. Sylvester, Kovalevskaya in that tier—no.

I’m still amazed that when you think of pivotal figures in the history of math, Sylvester pops into your head before Riemann. Should I jump to the conclusion (as you apparently did with me) that you don’t know that much about Riemann’s work?

As for forgetting/overseeing Riemann, I threw my list together in ten seconds between cups of tea and it was not intended to be either definitive or complete. Sylvester together with Cayley laid the foundations of matrix algebra and also possibly more important the two laid the foundations of invariant theory. Whilst I am willing to concede that maybe Sylvester is not a totally serious candidate for greatest mathematician, I will probably surprise or even shock you when I say that I think that both matrix algebra and invariant theory are more important to the development of mathematics than Cantorial set theory.

I think the impact of Cantor’s set theory is vastly overrated and I say that as somebody who studied the history of metamathematics and the so-called foundations crisis in mathematics at university under one of the world’s leading experts on the subject. Yes, anybody who has enough maths to understand it gets blown away by Cantor’s proof that the rational numbers are countable and the real numbers aren’t and if they go further that there is an infinite hierarchy of infinite sets but in the end so what. These things play in everyday maths and physics almost no role whatsoever. Yes, maths students have to prove that between any two rational numbers there is a real number and similar things but does one ever use these things later?

Maybe the most important thing about Cantor’s work is the use of the diagonal proof argument, which was then used by Gödel and Turing amongst others. However, Cantor was not the first to use it, Paul du Bois-Reymond was.

As far as the so-called foundations crisis goes, in the end it was/is for 99.99999% of mathematicians a lot of hot air; they just don’t care. The Gödel/Cohen proofs that the continuum hypothesis is independent of the axioms of ZF are maybe fascinating for set theorists and metamathematicians but for normal mathematicians and others who use mathematics they use ZFC and don’t care one way or the other.

The set theorists who actually work in Cantor’s transfinite arithmetic are a very small group of mostly warped loonies who publish their obscure discoveries in journals with a readership of seventeen and meet up a couple of times in the year at conferences where they spout the gobble-de-gook at each other for a few days. Their impact on the everyday world of mathematics tends towards zero.

He now sits back and waits for the outraged set theorist and metamathematicians to launch their virulent protest against this infamous characterisation of their discipline.

I think the impact of Cantor’s set theory is vastly overrated and I say that as somebody who studied the history of metamathematics and the so-called foundations crisis in mathematics at university under one of the world’s leading experts on the subject.This sentence could have been written only by someone who has never encountered a significant chunk of modern math at the graduate level. I am not talking only, or even primarily, about the so-called foundational crisis. The whole language of math changed. Just to take invariant theory, here is the opening sentence from the preface of Fogarty’s

Invariant Theory(not even that recent a book, 1969):The present notes are devoted to a reasonably exhaustive discussion of the basic qualitative problem of algebraic invariant theory, viz., given a ring R and a group G of automorphisms of R, to describe the ring of G-invariant elements of R.No 19th C. mathematician would have written anything like that. A 20th C. mathematician almost automatically frames his/her subject area in terms of sets with structures on them and maps between them. (Lots of 21st c. mathematicians go directly to categories.) While Sylvester and his contemporaries thought in terms of substituting a rational expression into an algebraic form, nowadays we talk about the action of a linear group on a polynomial ring (or even more abstract formulations). And as Hilbert and others made clear, Cantor’s work was very largely responsible for this shift in viewpoint. That’s what Hilbert meant by “Cantor’s Paradise”.

Sylvester together with Cayley laid the foundations of matrix algebra and also possibly more important the two laid the foundations of invariant theory.Also a revealing comment. I didn’t mention invariant theory separately, because their work in this area was intertwined with matrix theory. After all, the invariants are invariant under linear transformations!

These things [countability arguments] play in everyday maths and physics almost no role whatsoever.Again, a statement that could be written only by someone who has never had a graduate (or even advanced undergraduate) course in modern analysis or algebra or topology. (As for physics, that’s a different story.)

The argument you

shouldhave made: this transformation in the framework of math wasn’t single-handedly engineered by Cantor. True enough, and the proper rejoinder to what I said about Sylvester. On this basis, one could have a reasonable debate on their relative merits.You wrote:

You obviously know far too little about Sylvester.Yeah, right.

Almost all of your textbook quote on invariant theory has nothing what so ever to do with Cantor or his work.

As I said, it’s a question of how things are couched—sets with structures and maps between them. That’s the standard framework, well established by the mid-20th C, across nearly the whole landscape of mathematics. The only serious question is how much we should attribute this to Cantor (and Dedekind). That’s open to reasonable differences of opinion.

Invariant theory (and more generally the theory of transformation groups) is of course one of the major themes of modern math (and physics). Again, the question is how much of this is due to Sylvester (and Cayley)? I feel that the decisive push came from Klein and Lie. Based on Hawkins’ book

Emergence of the Theory of Lie Groupsand various things I’ve read over the years, I would say that their influence, while not insignificant, was not so central.Nothing I’ve written should be construed as a defense of the Malone documentary, which I haven’t seen and won’t waste my time on. It does indeed sound like dangerous twaddle.

By “their influence”, I meant Sylvester’s and Cayley’s influence, not Klein’s and Lie’s, of course.

Well, some of the religious fundamentalists in the USA are taking no chances. A major “Christian” publishing house in the USA states:

But then most mathematicians are platonists (with a small p) and believe in the existence of numbers independently of human creation. As a physicist, I only care that when I use a mathematical tool like calculus it gives me the right result. It is of some, but not overriding, interest to me that both Newton’s and Liebniz’s formulations of calculus were insufficiently rigorous and it took the genius of Cauchy to reformulate calculus in a rigorous form (at least as far as the 19th Century mathematicians were concerned).

For anyone interested in the philosophy and history of mathematics, Rich Cochrane and Andrew McGettigan teach this subject at City Lit in London (next term is about Gödel) and their slides and readings are on their website: http://artgeometry.com/resources.html

And even the formalists don’t believe that “anything goes” in math. You can’t just decide that statement X will be a theorem of formal theory T; you have to provide a proof from the axioms of T. If someone says, “OK, add X to T as a new axiom”, you run the risk of an inconsistent theory.

Actually the rigorous form of calculus owes more to the efforts of Weierstraß than Cauchy. Interestingly the convergence of infinite series the central problem worked on by Cauchy and Weierstraß was exactly what Cantor was working on that led him to develop his set theory and transfinite arithmetic.

@MW you are of course quite right but on the question of inconsistency and Gödel there are two points that I find of interest.

Firstly Wittgenstein, and I’m not especially a Wittgenstein fan, said, in my opinion quite correctly (paraphrase), “and if you do find an inconsistency some where in a distant corner of arithmetic, you are not going to simple abandon all of the useful stuff you’ve used up till now”.

Secondly when talking about Gödel people forget the condition, using the methods of the system. Gerhard Genzen proved the consistency of Peano axiom arithmetic using other methods. A proof that is even accepted by most constructivists.

Actually the rigorous form of calculus owes more to the efforts of Weierstraß than Cauchy.Not disagreeing. I will pull out a Hermann Weyl quote, “Sufficient unto the day is the rigor thereof.”

As several studies have shown, the “rigorization” of calculus was a long drawn-out process. It’s not like all mathematicians before Cauchy just ignored the issues, nor did Weierstraß write the last word on the topic. But if one has to name a single name, Weierstraß pops into my head before Cauchy.

No doubt this is partly influenced by the fact that Cauchy “proved” that the limit of a pointwise convergence sequence of continuous functions is continuous (which is false). But also because Cauchy still refers to infinitesimals, which Weierstraß eschews.

Regarding the Wittgenstein quote, nice.

As a practical or social matter, of course he’s right. But it’s not like an inconsistency would be ignored, either, since under the rules of logic that the majority of all mathematicians use, such an inconsistency (presumably a theorem of ZFC) would make

everystatement in ZFC provable. That’s nearly all statements the mathematicians care about. “Here’s a proof of the Riemann hypothesis. Here’s a disproof. Ditto for the ABC conjecture, Poincaré’s conjecture, the hailstone conjecture, the Feit-Thompson theorem…”We have a perfect model of what would almost certainly happen in the response to the paradoxes in the early 20th C. Namely, Zermelo and others found a way to wall it off. (I suppose I should mention Russell and Whitehead, but they didn’t find the “right”, i.e., useful solution.)

By the way, I am aware of the work on paraconsistent logic. That’s just another approach to quarantining contradictions. Instead of changing axioms, we change the rules of logic. If that was the “best” way out of the quandary, future mathematicians would get used to it.

But then most mathematicians are platonists (with a small p) and believe in the existence of numbers independently of human creation.I’ve certainly heard this claim often enough. Who knows, it may be true. I’m not aware of any surveys or polls supporting or opposing this.

Speaking from personal experience, most mathematicians just don’t worry about these issues. Why should they? If your theorems can be proved in ZFC, you don’t care if there is a unique “real universe of sets”. But if your theorem uses, say the continuum hypothesis, you just do what Erdős did in his beautiful solution to Wetzel’s problem: you state it as “CH implies that pointwise countable families of analytic functions need not be countable.” (Actually, Erdős proved an if-and-only-if, not a one-way implication.)

As for those who

docare about these philosophical questions, most would regard platonist vs. non-platonist as an oversimplified framing.But then most mathematicians are platonists (with a small p) and believe in the existence of numbers independently of human creation.On second reading, I noticed the choice of “numbers” rather than “sets”. The number of mathematicians skeptical about the existence of “all” the natural numbers is probably very small. Not that I’ve done a survey or anything…

A proof that is even accepted by most constructivists.Interesting. How do you “construct” ε_0? But the sequent calculi and cut-elimination are eminently constructivist. Are you referring to the work of Troelstra and his collaborators?

I am quoting Christian Thiel, my professor, who is both a constructivist, Erlanger Schule, and a Genzen expert. I never went that far down that road.

Why would Gödel go mad for failing to prove CH? He was a Platonist and believed that CH is false.

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