"The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself." (Bertrand Russell, "Principles of Mathematics", 1903)
"A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as exactly as possible a set of experimental laws. […] Thus a true theory is not a theory which gives an explanation of physical appearances in conformity with reality; it is a theory which represents in a satisfactory manner a group of experimental laws. A false theory is not an attempt at an explanation based on assumptions contrary to reality; it is a ·group of propositions which do not agree with the experimental laws. Agreement with experiment is the sole criterion of truth for a physical theory." (Pierre-Maurice-Marie Duhem, "La théorie physique. Son objet, sa structure", 1906)
"Today it is no longer questioned that the principles of the analysts are the more far-reaching. Indeed, the synthesists lack two things in order to engage in a general theory of algebraic configurations: these are on the one hand a definition of imaginary elements, on the other an interpretation of general algebraic concepts. Both of these have subsequently been developed in synthetic form, but to do this the essential principle of synthetic geometry had to be set aside. This principle which manifests itself so brilliantly in the theory of linear forms and the forms of the second degree, is the possibility of immediate proof by means of visualized constructions." (Felix Klein, "Riemannsche Flächen", 1906)
"Banishing fundamental facts or problems from science merely because they cannot be dealt with by means of certain prescribed principles would be like forbidding the further extension of the theory of parallels in geometry because the axiom upon which this theory rests has been shown to be unprovable. Actually, principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all." (Ernst Zermelo, "Neuer Beweis für die Möglichkeit einer Wohlordnung", Mathematische Annalen 65, 1908)
"Now even in mathematics unprovability, as is well known, is in no way equivalent to nonvalidity, since, after all, not everything can be proved, but every proof in turn presupposes unproved principles. Thus, in order to reject such a fundamental principle, one would have to ascertain that in some particular case it did not hold or to derive contradictory consequences from it; but none of my opponents has made any attempt to do this." (Ernst Zermelo, "Neuer Beweis für die Möglichkeit einer Wohlordnung", Mathematische Annalen 65, 1908)
"Every one knows there are mathematical axioms. Mathematicians have, from the days of Euclid, very wisely laid down the axioms or first principles on which they reason. And the effect which this appears to have had upon the stability and happy progress of this science, gives no small encouragement to attempt to lay the foundation of other sciences in a similar manner, as far as we are able." (William K Clifford et al, "Scottish Philosophy of Common Sense", 1915)
"A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details." (Hermann Weyl, "Unterrichtsblätter für Mathematik und Naturwissenschaften", 1932)
"It goes without saying that the laws of nature are in themselves independent of the properties of the instruments with which they are measured. Therefore in every observation of natural phenomena we must remember the principle that the reliability of the measuring apparatus must always play an important role." (Max Planck,"Where is Science Going?", 1932)
"I think that we shall have to get accustomed to the idea that we must not look upon science as a 'body of knowledge,' but rather as a system of hypotheses; that is to say, as a system of guesses or anticipations which in principle cannot be justified, but with which we work as long as they stand up to tests, and of which we are never justified in saying that we know they are 'true' or 'more or less certain' or even 'probable’." (Karl R Popper, "The Logic of Scientific Discovery", 1934)
"The fundamental gospel of statistics is to push back the domain of ignorance, prejudice, rule-of-thumb, arbitrary or premature decisions, tradition, and dogmatism and to increase the domain in which decisions are made and principles are formulated on the basis of analyzed quantitative facts." (Robert W Burgess, "The Whole Duty of the Statistical Forecaster", Journal of the American Statistical Association , Vol. 32, No. 200, 1937)
"When an active individual of sound common sense perceives the sordid state of the world, desire to change it becomes the guiding principle by which he organizes given facts and shapes them into a theory. The methods and categories as well as the transformation of the theory can be understood only in connection with his taking of sides. This, in turn, discloses both his sound common sense and the character of the world. Right thinking depends as much on right willing as right willing on right thinking." (Max Horkheimer, "The Latest Attack on Metaphysics", 1937)
"The question of the origin of the hypothesis belongs to a domain in which no very general rules can be given; experiment, analogy and constructive intuition play their part here. But once the correct hypothesis is formulated, the principle of mathematical induction is often sufficient to provide the proof." (Richard Courant & Herbert Robbins, "What Is Mathematics?: An Elementary Approach to Ideas and Methods" , 1941)
"It is to be hoped that in the future more and more theoretical physicists will command a deep knowledge of mathematical principles; and also that mathematicians will no longer limit themselves so exclusively to the aesthetic development of mathematical abstractions." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)
"Of course we have still to face the question why these analogies between different mechanisms - these similarities of relation-structure - should exist. To see common principles and simple rules running through such complexity is at first perplexing though intriguing. When, however, we find that the apparently complex objects around us are combinations of a few almost indestructible units, such as electrons, it becomes less perplexing." (Kenneth Craik, "The Nature of Explanation", 1943)
"We can put it down as one of the principles learned from the history of science that a theory is only overthrown by a better theory, never merely by contradictory facts." (James B Conant, "On Understanding Science", 1947)
"It is always more easy to discover and proclaim general principles than it is to apply them." (Winston Churchill, "The Second World War: The gathering storm", 1948)
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