peter_norvig's picture
Director of Research, Google Inc.; Fellow of the AAAI and the ACM; co-author, Artificial Intelligence: A Modern Approach

John McCarthy, the late co-founder of the field of artificial intelligence, wrote, "He who refuses to do arithmetic is doomed to talk nonsense." It seemed incongruous that a professor who worked with esoteric high level math would be touting simple arithmetic, but he was right; in fact in many cases all we need to avoid nonsense is the simplest form of arithmetic: counting.  

In 2008, the US government approved a $700 billion bank bailout package. A search for the phrase "$700 million bailout" reveals hundreds of writers who were eager to debate whether this was a prudent or rash, but who couldn't distinguish the difference between $2 per citizen and $2,000 per citizen. Knowing the difference is crucial to understanding the efficacy of the deal, and is just a matter of counting.

Consider the case of a patient who undergoes a routine medical screening and tests positive for a disease that affects 1% of the population. The screening test is known to be 90% accurate. What is the chance that the patient actually has the disease? When a group of trained physicians were asked, their average answer was 75%. They reasoned that it should be somewhat less than the 90% accuracy of the test, because the disease is rare. But if they had bothered to count, they would have reasoned like this: on average, out of every 100 people, 1 will have the disease, and 99 won't. The 1 will have a positive test result, and so will about 10 of the 99 (because the test is 10% inaccurate). So we have a pool of 11 people who test positive, of which only 1 actually has the disease, so the chance is about 1/11 or 9%. That means that highly trained physicians are reasoning very poorly, scaring patients with an estimate that is much too high, all because they didn't bother to count. The physicians might say they are trained to do medicine, not probability theory, but as Pierre Laplace said in 1812, "Probability theory is nothing but common sense reduced to calculation," and the basis of probability theory is just counting: "The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible."

In their report on climate change, the IPCC stated the consensus view that "most of the observed warming over the last 50 years is likely to have been due to the increase in greenhouse gas concentrations." Yet some criticized this consensus, saying scientists are still uncertain. Who's right? Naomi Oreskes took it upon herself to resolve the question, by counting. She searched a scientific database using the keywords "climate change" and scrutinized the 928 abstracts that matched. She found that 25% of the articles did not address the consensus (because, for example, they were about regional climate rather than global), but that none of the 928 rejected the consensus. This is a powerful form of counting. What's more, I don't need to take Prof. Oreskes's word for it: I did my own experiment, sampling 25 abstracts (I didn't have the patience to do 928) and I too found that none of them rejected the consensus.

This is a powerful tool. When faced with a complex issue, you have the capability to resolve the question, not by examining your political predispositions and arguing for whatever agrees with them, but by examining the evidence and counting the number of cases that are favorable, and comparing to those that aren't. You don't need to be a mathematical wizard; just apply what you learned as a toddler: count!