My intention is not to replace one set of general rules by another suc — Inductive reasoning

"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."

"Induction... proposes to have to do with things; and, as a mode or principle of argumentation, it may perhaps be correctly defined as a process of reasoning from particulars to a general: a method which requires a scrupulous, accurate, and comprehensive examination of all the cases which come within the range of the subject of inquiry, and from these instances infers the great axiomatic truth, or the universal and invariable law, in which they are found to meet, and which they will be always found to obey. ...This is unquestionably the nature of the principle of induction as proposed by Lord Bacon. Its useful and successful application, however, to the various departments of knowledge,—and there is scarcely any department to which, under suitable modifications, it may not be advantageously applied,—requires much care, attention, and assiduous patience."

"There is no inductive method which could lead to the fundamental concepts of physics. Failure to understand this fact constituted the basic philosophical error of so many investigators of the nineteenth century. It was probably the reason why the molecular theory, and Maxwells theory were able to establish themselves only at a relatively late date. Logical thinking is necessarily deductive; it is based upon hypothetical concepts and axioms. How can we hope to choose the latter in such a manner as to justify us in expecting success as a consequence?"

"Induction, I maintain, may or may not employ hypothesis, but what is essential to it is the inference from the particular to the general, from the known to the unknown, and the nature of this inference it is impossible to represent adequately by reference to the forms of deduction."

"There is... no special uncertainty attaching to the truths arrived at by induction. They are, indeed, like all other truths, relative to the present constitution of nature and the present constitution of the human mind, but this is a limitation to which all our knowledge alike is subject, and which it is vain for us to attempt to transcend."

"It is the authority of the less general case, which most commonly prevails, inasmuch as it generally precedes the interpretation of the more general case in the order of investigation, and is more immediately and more essentially connected with the first principles of the science. Assuming therefore the correctness of the interpretation of the less general case, it is by an inductive process of reasoning only, that we pass from it to the interpretation of the more general case, and the existence of one does not determine, in the mathematical sense of the term, the existence of the other: but it is the necessary connection which exists between the interpretation of the more general result and those which are subordinate to it, which makes it so important to examine and ascertain the latter..."