NYT published a paper Basics: Gut Instinct’s Surprising Role in Math on relations between symbolic/approxiamte arithmetics in human mind. A quote caught my eye:
“When mathematicians and physicists are left alone in a room, one of the games they’ll play is called a Fermi problem, in which they try to figure out the approximate answer to an arbitrary problem,” said Rebecca Saxe, a cognitive neuroscientist at the Massachusetts Institute of Technology who is married to a physicist. “They’ll ask, how many piano tuners are there in Chicago, or what contribution to the ocean’s temperature do fish make, and they’ll try to come up with a plausible answer.”
“What this suggests to me,” she added, “is that the people whom we think of as being the most involved in the symbolic part of math intuitively know that they have to practice those other, nonsymbolic, approximating skills.”
This is wrong. “Fermi problems” are indeed a well known class of brain teasers; but Rebecca Saxe is wrong, they are not answered by “non-symbolic skills” or ”gut instinct”. Some explanations follow.
“Fermi problems” became part of maths/physics folklore after Enrico Fermi was using them to test recruits to his lab (without revelaing to them what his lab was doing: A-bomb). His co-workers were expected to do something absolutely new; prior to setting up an experiment for measuring something which was not measured before they had to try to estimate theoretically the range of possible values and then choose appropropriate equipment and set-up. Therefore, they are very practical stuff, Fermi problems, and solutions are found not by gut feeling, but by a rather sophisticated analysis.
Tradition claims that “what is the number of piano tuners in your city” was one of the original Enrico Fermi’s questions. I frequently used it in selection interviews to Summer School, an intermediate step to selection to the FMSh, Preparatory Boarding School of Novosibirsk University l in Siberia in 1970s. RecentlyI I used the same question again in mock “Cambridge” interviews I run for A level students from my local school in Manchester. The way how a child thinks is much more interesting than an answer — and I can confidently claim that a successful solution requires a highly structured “symbolic” approximating skills.
When asked about the number of piano tuners in the city of Novosibirsk, almost all people who solved the problem were in a surprising agreement that the city perhaps had two professional (that is, earning their living by tuning pianos) piano tuners . The answer was hard to check until I gave the problem to a colleague of mine who instantly answered: “two”.
“Why?”- asked I.
“Because I know both of them”. And my friend added: “well, I mean professional piano tuners — there are many more amateurs. I am myself an amateur tuner: I have special keys for tuning and tune my own piano”.
I had a chance to work with the famous mathematician Israel Gelfand who, among many things he did in his long life, in 1950-s was a mathematical advisor to Andrei Sakharov (which meant that Gelfand was solving for Sakharov differential equation that Sakharov could not solve himself — admittedely, this was setting the plank quite high). I once asked Gelfand about Sakharov. Gelfand thought in silence for a while and then said: ”Sakharov was a great physicist and was always thinking about some physical problem. Once he told me: “you know, on my way to our meeting I made some calculations in my head — it appears that our Sun produces, per unit of mass, less energy than a heap of rotting manure”".
I passed this story to astrophysicist colleagues from my Department. They were extremely surprised and did some calculation on napkins in our coffee lounge — indeed, the power output per unit of mass of rotting manure happened (if I remember correctly–I am too lazy to repeat calculations) to be two order of magnitudes higher that that of the Sun. “This is why the Sun was lasting for so long” — commented one of my colleagues philosophically.
(With thanks to muriel for a link to NYT.)