Hume's problem of induction

The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge understood in the classic philosophical sense, since it focuses on the alleged lack of justification for either:

The problem calls into question all empirical claims made in everyday life or through the scientific method, and, for that reason, the philosopher C. D. Broad said that "induction is the glory of science and the scandal of philosophy." Although the problem arguably dates back to the Pyrrhonism of ancient philosophy, as well as the Carvaka school of Indian philosophy, David Hume introduced it in the mid-18th century, with the most notable response provided by Karl Popper two centuries later.

In inductive reasoning, one makes a series of observations and infers a new claim based on them. For instance, from a series of observations that a woman walks her dog by the market at 8am on Monday, it seems valid to infer that next Monday she will do the same, or that, in general, the woman walks her dog by the market every Monday. That next Monday the woman walks by the market merely adds to the series of observations, it does not prove she will walk by the market every Monday. First of all, it is not certain, regardless of the number of observations, that the woman always walks by the market at 8am on Monday. In fact, Hume would even argue that we cannot claim it is "more probable", since this still requires the assumption that the past predicts the future.

Second, the observations themselves do not establish the validity of inductive reasoning, except inductively. Bertrand Russell illustrated this point in The Problems of Philosophy:

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