“From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2.” —Principia Mathematica Volume I, 1st edition, page 379
Bertrand Russell and Alfred North Whitehead spent the best part of a decade writing Principia Mathematicia, an attempt to describe a set of axioms from which all mathematical truths could be proven. This three volume work was so abstract that the proof that 1+1=2 spans two volumes: it starts on page 379 of Volume I, but the proof is only completed on page 86 of Volume 2.
I am interested in the sociology of mathematics. Mathematicians don’t tend to resolve their differences about proof by burning each other at the stake. I have never read a newspaper report of a Middle Eastern sect of group theorists throwing logicians off the roof of a building.
You would think then, that mathematicians could resolve their differences about everything with logic. But it isn’t so.
Russell and Whitehead fell out, because of their different views on the First World War. Whitehead a supporter of the war, was happy to see his son, North*, enlist and fight in France. Russell famously went to Brixton Prison, because of his opposition to the war. Whitehead’s war time correspondence to Russell shows that both men thought the gulf between them was unbridgeable. Whitehead wrote to Russell in 1916 that “on the whole, men who refuse military service are avoiding a plain, though painful moral duty – often no doubt with the excuse of a mistaken conscience.” Meanwhile Russell’s protege Wittgenstein was fighting on the opposite side.
What is happening here? I think probably that Whitehead was using a heuristic, such as “go with the crowd”, “don’t break ranks” or perhaps just “but THEY started it!!!” Russell on the other hand, probably thought about the situation more deeply, in part because he was married to a non-conformist Quaker Alys Pearsall Smith, and in part because he had spent a year in Berlin, in 1895 studying economics and writing about German social democracy. Perhaps too, the fact that Russell was an orphan encouraged him to think for himself, rather than to trust authority.
In any case, the example shows that heuristics can be dangerous, if wrongly applied. Not every button should be pushed. Most of the time a button on a wall is likely to be a light switch, but occasionally pushing a button might set off an alarm. Quiz: can you think of the sort of room where a button is likely to be an alarm, not a light switch?
I think not even Gerd Gigerenzer would say that one simple rule of thumb “always” does better. The trick is to choose the appropriate rule of thumb from the tool box: a heuristic is that is suited to the environment. For instance, if you go into a restaurant, it is fine to display ignorance and ask the waiter for his recommendations. On the other hand, if you go to a used car salesman, or a bank and let them know that you are ignorant of the basics, the situation is completely different. Then if you ask for advice expecting it to be anything other than sales, then that is a recipe for being sold a “lemon”.
Whitehead was using a heuristic to form his opinions, and then using his considerable intellect to reasonably justify his beliefs. So we have to recognise that heuristics are how the mind works, and to help the mind work in the way it has evolved, rather than use logic to justify irrational beliefs.
I also can’t help thinking that again and again Russell did avoid using the wrong heuristic. And that is why investors like Buffett, Soros, Keynes and Taleb are all such fans. To be a great investor, you need to use heuristics, but also understand where they break down. That is why Buffett likes to tell the story about paying 3x book value for See’s Candy, rather than buying distressed businesses at below their value. That is why Soros likes to say that “stock market bubbles don’t grow out of thin air. They have a solid basis in reality, but reality as distorted by a misconception.”
The worst thing you can do is form a prejudice and then fool yourself by building a sophisticated model to justify your misconception. Remember that the heliocentric model of the universe, with the planets revolving around the sun, was much simpler than the Ptolemic model. The model was more mathematically elegant, but the Church didn’t like this way of thinking about the universe, because it questioned their authority. Russell once wrote:
“We all have a tendency to think that the world must conform to our prejudices. The opposite view involves some effort of thought, and most people would die sooner than think – in fact they do so.”
He wrote this, not in an anti war essay, but in The ABC of Relativity. At least, that is according to wiki quote, I couldn’t actually find it in my version of the book.
* Did that mean his full name was North North Whitehead? Sadly not, the father’s middle name was North, and he is usually referred to as AN Whitehead to distinguish him from his father, who was also called Alfred Whitehead. The son, who was called “North” by the family, was actually christened Thomas North Whitehead.