Everybody knows the story of the turkey appearing in the title of this blog: on his first day at the farm, a turkey observed he was fed at 9:00 a.m., and the same happened the following day, and the following, and the following again, and so on. Day by day he collected a large number of observations, under the most different circumstances. Every day, were it cold or warm, rainy or sunny, winter or spring, he was fed at 9:00 a.m. He was a good logician, thus he concluded: “I am always fed at 9:00 a.m.”. Well, the day after it was Thanksgiving and he had his throat cut.
The story was originally invented by Bertrand Russell in his The Problems of Philosophy (in his version the protagonist was a chicken, not a turkey) to show that induction can offer no evidence in prediction of future events. From then, the inductivist turkey story has been variously quoted and referred by many philosophers.
Karl Raimund Popper mentioned our turkey to argue against the neopositivistic image of science. Beside our turkey, he presented another similar case of failing induction: the likewise famous black swan. Before the discovery of Australia, everybody was absolutely certain that all swans are white. They were so sure that they usually quoted Juvenal’s verse “rara avis in terris nigroque simillima cygno” to refer to something impossible or at least very improbable. But when the first black swan was observed in Australia, their absolute certainty felt down as a house of cards.
More recently, Nassim Nicholas Taleb, in his book The Black Swan, adopted this metaphor to talk about the impact of highly unpredictable events in our life. He defines a Black Swan event as an event that
- lies outside the realm of regular expectations;
- carries an extreme impact on our lives;
- can be explained only after it has occurred.
His conclusion is that we should always keep our mind open to novelties and never trust in those who pretends to rule the unpredictable. Anyway, it must be noted that the discovery of black swans in Australia was not a “Black Swan event”! For it does not satisfy the second condition. Taleb had better call this category of events the “Inductivist Turkey event”! It’s sure that the failure of turkey’s prediction had an extreme impact at least on his life!
All this has to do with induction in science, or in everyday life, but what if we come to induction in mathematics? It seems that our inductivist turkey has nothing to worry here: induction in mathematics is completely different from induction in science, both if we interpret it à la Russell or à la Poincaré. However, there is an interesting episode about history of mathematics that should be always kept in mind not to forget Taleb’s lesson. It is mentioned by Freeman J. Dyson in his article “Mathematics in the Physical Sciences” (Scientific American, 211:3, 1984):
In the 1910 the mathematician Oswald Veblen and the physicist James Jeans were discussing the reform of the mathematical curriculum at Princeton University. “We may as well cut out group theory,” said Jeans. “That is a subject which will never be of any use in physics”. (p. 128)
Well, today group theory is one of the most important contributions of mathematics to science, being widely applied to particle physics! Fortunately, nobody gave heed to Jeans, and group theory continued being taught in Princeton.