There is now a whole genre of books demonstrating that logical reasoning seldom comes easily to human beings.
A classic example is Nobel prizewinner Daniel Kahneman’s Thinking, Fast and Slow (2011), which explores in vivid detail the “systematic errors in the thinking of normal people”. Our memories are distorted, we see significant links between chance events, we grossly overestimate the risks of certain things happening – and, according to Kahneman, even “experts show many of the same biases as the rest of us in attenuated form”.
Yet he seems accepting rather than downhearted by what he has discovered: “Considering how little we know, the confidence we have in our beliefs is preposterous – and it is also essential.” Some of the “bugs” in our thinking are probably useful, in that they allow us to take quick decisions in situations of uncertainty. But in so far as they lead us astray, they may be almost impossible to eradicate.
Rather more gung-ho are the books that attempt to cure us of our innumeracy. Mathematics may not come naturally to most of us, but, they suggest, we can all learn to be a bit more sceptical about the statistics thrown at us every day, immunising ourselves against the constant false claims of politicians and advertisers. That is very much the argument of Brian W. Kernighan, professor of computer science at Princeton University, in his new book Millions, Billions, Zillions: Defending Yourself in a World of Too Many Numbers (Princeton University Press).
Whenever we have to “estimate something from incomplete information”, he argues, we should avoid instant recourse to a search engine and, instead, “make [our] own estimate”. This “won’t take long and you’ll quickly get good at it. Practice will arm you for a lifetime of being wary about what other people are telling you.”
Some very simple arithmetic (and a little basic information such as the rough population of the US at the time) would have enabled readers of a 1996 “Dear Abby” advice column to realise that it couldn’t possibly be true that “Americans receive almost two million tons of junk mail daily” (and perhaps to speculate that “daily” was a slip for “per year”).
Kahneman makes clear that our intuitive understanding of very large and very small numbers is highly erratic. Yet it is still striking and depressing how many examples Kernighan can provide of leading newspapers issuing corrections stating that a reference to “millions” in an article should really have been “billions” (or vice versa).
Another obvious source of error is getting systems of measurement wrong. Millions, Billions, Zillions cites the extraordinary case of the Mars Climate Orbiter space probe, which in 1999 disintegrated in Mars’ atmosphere because “one part of the software used data in conventional English units, while a different part used standard metric units”. Similarly, an Air Canada flight ran out of fuel in 1983 because the amount loaded “had been measured in pounds when it should have been in kilograms”. When it comes to basic maths, it seems, pretty much all of us “could do better”.