On induction
24 Feb 2024
In his essay by the same name1, Russell uses what I believe is, probably, the best example of the use of induction (at least at first glance):
We are all convinced that the sun will rise tomorrow.
It is a great example because it easily shows what induction is about, to both literate and illiterate people.
Under closer inspection, however, the proposition “the sun will rise tomorrow” appears to be a tautology.
What does “tomorrow” mean? It simply means “the next day”. Traditionally, by definition “day” spanned the period from sunrise to sunset. It is not until more recently that the more technical definition of “a 24 hour period starting at midnight” or “a full rotation of the Earth around its axis” was introduced. Even nowadays, we may still say “see you tomorrow” even though it is 1am, and we are already in the new day.
As such, “the sun will rise tomorrow” is a tautology, as there is no morrow without sunrise.
In the same essay, on the principle of induction Russell states that induction shouldn’t be seen as “proving” that the next instance will occur, but merely rendering it more probable:
[T]he fact that two things have been found often together and never apart does not, by itself, suffice to prove demonstratively that they will be found together in the next case we examine. The most we can hope is that the oftener things are found together, the more probable it becomes that they will be found together another time.
However, conceding that having many instances of a rule does not prove the next instance will follow the rule, but does show it is more probable, runs into the same problem of induction: this perceived rule of “increase in probability” is itself an inductive observation of our past experiences.
In fact, in some cases the probability of the next instance following the rule actually decreases with further positive observations. Consider a bag of red and blue balls. If I remove a number of balls, one by one, and they’re all red, Russell’s interpretation of the Principle of Induction would suggest that although these observations do not prove that the next ball that is removed will be red again, it does show that the probability that the next ball is red has increased.
The reality, however, is that not only is this conclusion wrong, but the probability of the next ball being red has actually decreased, as the number of red balls relative to blue balls has decreased.
This is related to Russell’s later change of view about induction2. He points out that “induction used without common sense leads more often to false conclusions than true ones”. This is of importance nowadays in Machine Learning for example, where blind induction leads to useless overfitting.
Induction without common sense leads to ridiculous spurious correlations such as the correlation of the name “Johnnie” as a baby name and the burglary rate in New Hampshire, milk consumption and divorce rate in Colorado or the distance between Neptune and the Sun, and air pollution in Washington, DC.
Given enough data, it is not difficult to find such coincidences one might apply induction to, unless common sense is used.
Notes
1 In B. Russell, The Problems of Philosophy.
2 B. Russell, Note on non-demonstrative inference and induction.