George Pólya on Inductive reasoning

“The examples of plausible reasoning collected in this book... may throw light upon a much agitated philosophical problem: the problem of induction. The crucial question is: Are there rules for induction? ...the question should be... treated... in closer touch with the practice of scientists. ...older writers, such as Euler and Laplace, clearly perceived... that the role of inductive evidence in mathematical investigation is similar to its role in physical research. ...so the door opens to investigating induction inductively.”

“Induction is the process of discovering general laws by the observation and combination of particular instances. It is used in all sciences, even in mathematics. ...Induction tries to find regularity and coherence behind the observations. Its most conspicuous instruments are generalization, specialization, analogy. Tentative generalization starts from an effort to understand the observed facts; it is based on analogy, and tested by further facts. ...many mathematical results are found by induction first and proved later. Mathematics presented with rigor is a systematic deductive science but mathematics in the making is an experimental inductive science. ...In the physical sciences, there is no higher authority than observation and induction but in mathematics there is such an authority: rigorous proof.”