The story of a paradox

[This is roughly the text of my story of Bertrand Russell, given at The Laborastory, a monthly science storytelling event in Melbourne. In the interests of brevity and entertainment, I took a little licence; but a little stretch can sometimes yield a greater truth.]


I would like to thank the organisers of the Laborastory, for their love of science, and bringing it to the people; the Spotted Mallard, for their love in hosting it; and I acknowledge the traditional owners of this land, the Wurundjeri people of the Kulin nation, and pay respect to their elders past and present.

Well, I’m here to talk about Bertrand Russell. It’s an impossible task. Because this is a man who really lived half a dozen lives. He was a mathematician, a philosopher, an educationalist, a social critic, a best-selling author – he was a towering figure of humanity in the 20th century. Since we’re at the laborastory, not the philoso-story or many other o-stories, I’m going to focus on Russell the mathematician. Hopefully my omissions aren’t completely unforgiveable.

* * *

Bertrand Russell was born in 1872 into the upper reaches of the British aristocracy; his grandfather had twice been Prime Minister of Britain. But his parents died before his fourth birthday and he was brought up by his puritanical grandmother. She would not entrust her grandson’s education to a school, and so young Bertrand was educated by governesses and tutors. He learned fast. And he fucking loved mathematics.

But the most crucial component of grandmother’s curriculum was religion, and she refused to entrust the teaching of that subject to anyone but herself. Her theology was old-school fire and brimstone.

Her indoctrination however did not quite have the desired effect. Instead, it pushed an inquiring young mind into a lifelong attitude of scepticism. Teenage Bertrand Russell kept a secret diary – not about girls, because he didn’t know any – but about the nature of the soul, human mortality, and similarly uplifting matters. It was complete heresy to grandma, so he wrote it in Greek.

Doubting the certainties of religious faith, Russell developed an overwhelming desire to know what can be known with certainty. And this desire drove him to mathematics.

In mathematics, there is a sort of certainty. One plus one really is two. Mathematical statements can be proved. And mathematical proofs are true by sheer force of logic.

These days, the only contact most people have with the notion of mathematical proof is in year 8 or 9 geometry. You may remember being made to explain such things as why two triangles were similar; this was called a “proof”. The intention may have been for you to appreciate the certainty of mathematical proof, but the effect in practice is much more heartwarming – it brings students together with a topic they can all hate in unison.

Luckily, Russell did not learn his geometry from the year 9 curriculum. He convinced his brother Frank to explain it to him. And in those days they learnt geometry old-school, from the ancient Greek text, Euclid’s Elements.

It’s full of amazing theorems and ingenious deductions, but Euclid has to begin somewhere. He begins from axioms – basic starting assumptions, which are supposed to be completely obvious, things that no person would question. Or, no sane person would question.

Of course Russell questioned them.

Exasperated, Frank declared that if Bertrand did not accept the axioms then he could not go on.

Bertrand was not happy. He would come back to that later, with a vengeance.

But for now he relented. Because geometry made his miserable youth bearable. At the depths of his despair, he contemplated suicide. He wrote, “I did not, however, commit suicide, because I wished to know more mathematics.”

That is possibly the only time in the history of the world anybody has thought that thought.

* * *

Eventually Russell left for Cambridge and a stellar academic career. But he had forged a habit of solitary, deep thought, developing strong opinions and ideas – always logically watertight, usually brilliant, sometimes eccentric, occasionally insane. As he wrote,

[t]hought is subversive and revolutionary … merciless to privilege, established institutions, and comfortable habits… anarchic and lawless, indifferent to authority, careless of the well-tried wisdom of the ages.

Well, Russell certainly was. And as a naturally gifted writer, and general troublemaker, he eventually published his ideas on pretty much everything he thought about.

Even when his views were controversial or outrageous. Especially when his views were controversial or outrageous. Because those questions are often the most important, and he was absolutely fearless.

For instance, he was to become known as a notorious atheist for his incendiary essay “Why I am not a Christian”, and an unorthodox socialist for his book “Roads to Freedom”, examining the best social system for a good society.

Perhaps surprisingly, the vast majority of his writing still stands up pretty well today. Not all, to be sure. But his provocations have often become our common sense – and often, in part at least, because he argued so effectively.

Take, for instance, the First World War. When the war came Russell campaigned against it tirelessly, giving speeches and writing pamphlets – eventually losing his job and going to jail for it. It was not to be the only time.

But his arguments today seem positively tepid. Britain’s alliances were unwise, and should stay out of it, he said. Today it’s common sense, even inadequate. But back then, it was enough to see him suffer criminal prosecution.

Except, “suffer” is not quite the right word.

Russell was elated to face prosecution. Finally he’d discharged his moral responsibilities and could get some maths done.

But he was disappointed. The magistrate deciding his case was far too reasonable. He was sentenced to only 6 months jail.

Imprisonment is, of course, not very pleasant. But Russell had reading and writing privileges, provided he didn’t mention the war. This suited him perfectly, as he had been neglecting other topics like mathematics. And it gave him an opportunity to mix with his fellow prisoners, who he found were no worse than the rest of the population, although, he wrote,

they were on the whole slightly below the usual level of intelligence, as was shown by their having been caught.

In six months jail he read two hundred books and wrote two.

* * *

Now for mathematics, the turn of the 20th century was a period of unbridled optimism. To many mathematicians like the German David Hilbert, it seemed that soon it would be possible to apply mathematical logic, mechanically, to answer any mathematical question. Mathematics could become an infinitely powerful machine.

Others, like the French mathematician Henri Poincare, thought human understanding and intuition played the central role in mathematics. Poincare hated the thought of his beautiful French mathematical culture reduced to Hilbert’s German sausage machine.

But history appeared to be on Hilbert’s side. Logicians like George Boole – he of the Boolean search – Gottlob Frege, and Georg Cantor, had shown that much of mathematics could be mechanised, reduced to pure logic, and sets.

Today, mathematicians still love the joy of sets.

Russell did too. He dug into the foundations of mathematics – and what he found broke the foundations and destroyed it all.

What did he do? He discovered a paradox, now known as Russell’s paradox.

Let me try to explain it.

I invited some friends here tonight. I told them about this great event and said they should come along. Some time later, I wasn’t sure if they’d booked themselves a table.

So, I told them, if you haven’t booked a table, I’ll book one for you.

I said, I will book a table for everyone who doesn’t book a table themselves.

And then I felt very pleased with myself, as I usually forget about all this kind of practical stuff and then panic at the last minute.

But then I thought – hang on a minute. Should I book myself a table?

Well, I was to book a table for everyone who doesn’t book a table themselves.

So if I don’t book myself a table myself, then I should book a table for myself.

And if I do book a table for myself, then I shouldn’t have.

I was stuck in a terminal loop. I do if I don’t and I don’t if I do.

At this point, thankfully the Laborastory organisers emailed me and resolved my ineptitude by telling me that actually there is a separate speakers’ table.

But in mathematics there are no organisers to resolve your ineptitude.

Russell’s paradox is, in essence, the Laborastory table-booking paradox. Russell just wrote it in the language of sets. A set in mathematics is just a collection of objects, which could be anything – numbers, letters, your missing socks. A set can also contain other sets. You could even have the set of all sets. A set can even contain itself. Russell said to consider a particular set – the set of all sets which do not contain themselves.

Russell asked: Does this set contain itself?

I leave that question for you to discuss over your next beer.  You will probably get a headache.

Even if your head doesn’t explode, well, set theory does in fact explode with this paradox and, sets being a foundational idea in mathematics, the whole of mathematics falls apart.

Mathematicians were devastated by this discovery. Russell’s colleague Frege had just finished his book claiming to reduce mathematics to logic. Upon hearing the news, he was forced to add one of the most abjectly sad appendices in scientific history, admitting that his magnum opus was actually completely flawed and could not work.

* * *

Speaking of things which are completely flawed and cannot work, Russell gained greatest notoriety not for his work on mathematics, or philosophy, but… marriage.

Russell wrote a book, Marriage and Morals, in which he argues for birth control, liberalised divorce laws, and gender equality. By the standards of contemporary feminist theory, it’s pretty tame. But that is only because it’s now common sense.

All respectable opinion was outraged.

At the time, he was about to teach a class in formal symbolic logic at the City University of New York. A mother of a student, fearing her daughter’s indoctrination into – perhaps enjoying sex? – by taking this class from a, quote, “lecherous erotomaniac”,  sued the university. He was promptly dismissed.

If you’ve ever doubted the allure of formal symbolic logic, bear this in mind.

* * *

But, back to mathematics. Having ruined it for everyone, Russell, together with his colleague Alfred North Whitehead, tried to put it back together. Their project was to start over from the very beginning, and build up, step by step, without paradox, the mathematics that we all know and love. Well, that some of us know and love.

The result was the 2000-page 3-volume work, Principia Mathematica. It took them 10 years and, being written mostly in formal logic symbols, it looks like alien hieroglyphics.

The scale of the work is awe-inspiring. It might inspire other thoughts too, like, yawning, “WTF is this Klingon poetry?”, or admiration at the sheer bloody-minded persistence.

The high point comes after 360 pages, when they prove a stunning result: 1 + 1 = 2. That’s right, it takes them 360 pages to prove 1 + 1 = 2. You can now set your mind at rest.

Every academic library in the world has a copy of Principia Mathematica. I don’t think anyone has ever read it all the way through.

* * *

Russell was exhausted after writing the Principia. He’d had enough of mathematics. He wrote that

In universities, mathematics is taught mainly to men who are going to teach mathematics to men who are going to teach mathematics to… Sometimes, it is true, there is an escape from this treadmill. Archimedes used mathematics to kill Romans, Galileo to improve the [Tuscan] artillery, modern physicists to exterminate the human race. It is usually on this account that… mathematics is commended… as worthy of State support.

Accordingly, much of his subsequent work was devoted to promoting peace and nuclear disarmament. The Bertrand Russell Peace Foundation still exists today. Its reports are still worth reading and still ignored by the mainstream.

Russell had one joint publication with Albert Einstein. It was a manifesto on the abolition of nuclear weapons. But it wasn’t in an A* journal, so it would count for nothing today.

He lived so long – 98 years – that he saw his most of his opinions on sex, marriage and war become mainstream. He won a Nobel Prize – but not for anything I’ve been talking about tonight – in literature. Did I mention he also wrote a monumental History of Western Philosophy? Or arguably literally saved the world during the Cuban Missile Crisis? Such were his accomplishments that, in a talk of this length, saving the world must be a mere footnote. As I said, there’s a lot more which, alas, I don’t have time to share.

In the end he became respectable. He was never happy about this.

* * *

Let me finish by saying something about the legacy of Russell and Whitehead’s Principia Mathematica.

There is actually at least one person who read it cover to cover: Kurt Godel, upstart logician.

Godel noticed that Russell and Whitehead had missed a crucial point. They had assumed that what is true, and what is provable, are the same.

But they are not. Godel showed there are mathematical statements which are true, but which cannot be formally proved. Basically, he showed that mathematics can never be a sausage machine.

It is still, however, too much of a sausage fest, unfortunately.

Others wondered what could actually be done with the formal procedures and logic developed by Russell and others. A young man named Alan Turing made machines to do them, now known as computers. They were used to assist war, then to assist business, and finally, today they are used to watch cat videos. I think Russell would probably have approved of this progression.


Written by dan

September 10th, 2016 at 11:55 am

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