On Induction (1875-1899)

"Whatever lies beyond the limits of experience, and claims another origin than that of induction and deduction from established data, is illegitimate." (George H Lewes, "The Foundations of a Creed", 1875)

"In the Theory of Numbers it happens rather frequently that, by some unexpected luck, the most elegant new truths spring up by induction." (Carl Friedrich Gauss, Werke, 1876)

"The general mental qualification necessary for scientific advancement is that which is usually denominated 'common sense', though added to this, imagination, induction, and trained logic, either of common language or of mathematics, are important adjuncts." (Joseph Henry, [Address] 1877)

"Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics." (Benjamin Peirce, "Linear Associative Algebra", American Journal of Mathematics Vol. 4, 1881)

"I am convinced that it is impossible to expound the methods of induction in a sound manner, without resting them on the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge, - knowledge mingled with ignorance, producing doubt." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"There is as great a distinction between mathematics and the mathematical sciences as there is between induction and the inductive sciences. Practically, few cases of induction do not involve, to a greater or less extent, deductions; so few mathematical processes do not involve some strictly logical procedure." (Charles C Everett, "The Science of Thought", 1890)

"In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction." (Giuseppe Peano, "Notations de Logique Mathématique", 1894)

"Observe, finally, that this induction is possible only if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science: the different moves of the game do not resemble one another." (Henri Poincaré, "On the Nature of Mathematical Reasoning", 1894)

"If men of science owe anything to us, we may learn much from them that is essential. For they can show how to test proof, how to secure fulness and soundness in induction, how to restrain and to employ with safety hypothesis and analogy." (Lord John Acton, [Lecture] "The Study of History", 1895)

"As for me (and probably I am not alone in this opinion), I believe that a single universally valid principle summarizing an abundance of established experimental facts according to the rules of induction, is more reliable than a theory which by its nature can never be directly verified; so I prefer to give up the theory rather than the principle, if the two are incompatible." (Ernst Zermelo, "Über mechanische Erklärungen irreversibler Vorgänge. Eine Antwort auf Hrn. Boltzmann’s ‘Entgegnung’" Annalen der Physik und Chemie 59, 1896)

"The ordinary logic has a great deal to say about genera and species, or in our nineteenth century dialect, about classes. Now a class is a set of objects compromising all that stand to one another in a special relation of similarity. But where ordinary logic talks of classes the logic of relatives talks of systems. A system is a set of objects compromising all that stands to one another in a group of connected relations. Induction according to ordinary logic rises from the contemplation of a sample of a class to that of a whole class; but according to the logic of relatives it rises from the contemplation of a fragment of a system to the envisagement of the complete system." (Charles S Peirce, "Cambridge Lectures on Reasoning and the Logic of Things: Detached Ideas on Vitally Important Topics", 1898)