"A system is not so important as a method. A system is of significance because it brings order and clearness into our knowledge, but he who hopes by its help to reach something more, he who thinks to extend his knowledge by means of a system is self-deceived." (Harald Høffding, "A history of modern philosophy", 1900)
"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments." (David Hilbert, "Mathematical Problems", Bulletin American Mathematical Society Vol. 8, 1901-1902)
"Man's determination not to be deceived is precisely the origin of the problem of knowledge. The question is always and only this: to learn to know and to grasp reality in the midst of a thousand causes of error which tend to vitiate our observation." (Federigo Enriques, "Problems of Science", 1906)
"If the fresh facts which come to our knowledge all fit themselves into the scheme, then our hypothesis may gradually become a solution." (Arthur C Doyle, "The Adventure of Wisteria Lodge", 1908)
"Knowledge is the distilled essence of our intuitions, corroborated by experience." (Elbert Hubbard, "A Thousand & One Epigrams, 1911)
"It is experience which has given us our first real knowledge of Nature and her laws. It is experience, in the shape of observation and experiment, which has given us the raw material out of which hypothesis and inference have slowly elaborated that richer conception of the material world which constitutes perhaps the chief, and certainly the most characteristic, glory of the modern mind." (Arthur J Balfour, "The Foundations of Belief", 1912)
"The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. They reveal the kinship between other facts, long known, but wrongly believed to be strangers to one another." (Henri Poincaré, 1913)
"By intuition is frequently understood perception, or the knowledge of actual reality, the apprehension of something as real. […] Intuition is the undifferentiated unity of the perception of the real and of the simple image of the possible. " (Benedetto Croce, "The Essence of Æsthetic", 1921)
"Observed facts must be built up, woven together, ordered, arranged, systematized into conclusions and theories by reflection and reason, if they are to have full bearing on life and the universe. Knowledge is the accumulation of facts. Wisdom is the establishment of relations. And just because the latter process is delicate and perilous, it is all the more delightful." (Gamaliel Bradford, "Darwin", 1926)
"Hypothesis, however, is an inference based on knowledge which is insufficient to prove its high probability." (Frederick L Barry, "The Scientific Habit of Thought", 1927)
"In mathematics as in the other sciences, the same processes can be observed again and again. First, new questions arise, for internal or external reasons, and draw researchers away from the old questions. And the old questions, just because they have been worked on so much, need ever more comprehensive study for their mastery. This is unpleasant, and so one is glad to turn to problems that have been less developed and therefore require less foreknowledge - even if it is only a matter of axiomatics, or set theory, or some such thing." (Felix Klein,"Development of Mathematics in the 19th Century", 1928)
"With fuller knowledge we should sweep away the references to probability and substitute the exact facts." (Sir Arthur S Eddington, "The Nature of the Physical World", 1928)
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