On Principles (1975-1999)

"No theory ever agrees with all the facts in its domain, yet it is not always the theory that is to blame. Facts are constituted by older ideologies, and a clash between facts and theories may be proof of progress. It is also a first step in our attempt to find the principles implicit in familiar observational notions." (Paul K Feyerabend, "Against Method: Outline of an Anarchistic Theory of Knowledge", 1975)

"The conception of the mental construction which is the fully analysed proof as being an infinite structure must, of course, be interpreted in the light of the intuitionist view that all infinity is potential infinity: the mental construction consists of a grasp of general principles according to which any finite segment of the proof could be explicitly constructed." (Michael Dummett, "The philosophical basis of intuitionistic logic", 1975)

"A good theorem will almost always have a wide-ranging influence on later mathematics, simply by virtue of the fact that it is true. Since it is true, it must be true for some reason; and if that reason lies deep, then the uncovering of it will usually require a deeper understanding of neighboring facts and principles." (Ian Richards, "Number theory", 1978)

"Real progress in understanding nature is rarely incremental. All important advances are sudden intuitions, new principles, new ways of seeing." (Marilyn Ferguson, "The Aquarian Conspiracy: Personal and Social Transformation in the 1980s", 1980)

"For the great majority of mathematicians, mathematics is […] a whole world of invention and discovery - an art. The construction of a new theorem, the intuition of some new principle, or the creation of a new branch of mathematics is the triumph of the creative imagination of the mathematician, which can be compared to that of a poet, the painter and the sculptor." (George F J Temple, "100 Years of Mathematics: a Personal Viewpoint", 1981)

"Both religion and science must preserve their autonomy and their distinctiveness. Religion is not founded on science nor is science an extension of religion. Each should possess its own principles, its pattern of procedures, its diversities of interpretation and its own conclusions." (Pope John Paul II, [Letter to Father George V Coyne], 1988)

"[…] mathematics does not come to us written indelibly on Nature’s Tablets, but rather is the product of a controlled search governed by metaphorical considerations, the premier instance being the heuristics of the conservation principles." (Philip Mirowski, "More Heat than Light: Economics as Social Physics: Physics as Nature’s Economics", 1989)

"Principles are the territory. Values are maps. When we value correct principles, we have truth - a knowledge of things as they are." (Stephen R Covey, "The 7 Habits of Highly Effective People", 1989)

"The ‘objective reality’, or the territory itself, is composed of ‘lighthouse’ principles that govern human growth and happiness - natural laws that are woven into the fabric of every civilized society throughout history and comprise the roots of every family and institution that has endured and prospered. The degree to which our mental maps accurately describe the territory does not alter its existence." (Stephen Covey, "The 7 Habits of Highly Effective People", 1989)

"A distinctive feature of mathematics, that feature in virtue of which it stands as a paradigmatically rational discipline, is that assertions are not accepted without proof. […] By proof is meant a deductively valid, rationally compelling argument which shows why this must be so, given what it is to be a triangle. But arguments always have premises so that if there are to be any proofs there must also be starting points, premises which are agreed to be necessarily true, self-evident, neither capable of, nor standing in need of, further justification. The conception of mathematics as a discipline in which proofs are required must therefore also be a conception of a discipline in which a systematic and hierarchical order is imposed on its various branches. Some propositions appear as first principles, accepted without proof, and others are ordered on the basis of how directly they can be proved from these first principle. Basic theorems, once proved, are then used to prove further results, and so on. Thus there is a sense in which, so long as mathematicians demand and provide proofs, they must necessarily organize their discipline along lines approximating to the pattern to be found in Euclid's Elements." (Mary Tiles,"Mathematics and the Image of Reason" , 1991)

"The word theory, as used in the natural sciences, doesn’t mean an idea tentatively held for purposes of argument - that we call a hypothesis. Rather, a theory is a set of logically consistent abstract principles that explain a body of concrete facts. It is the logical connections among the principles and the facts that characterize a theory as truth. No one element of a theory [...] can be changed without creating a logical contradiction that invalidates the entire system. Thus, although it may not be possible to substantiate directly a particular principle in the theory, the principle is validated by the consistency of the entire logical structure." (Alan Cromer, "Uncommon Sense: The Heretical Nature of Science", 1993)

"Within image theory, it is suggested that important components of decision-making processes are the different 'images' that a person may use to evaluate choice options. Images may represent a person's principles, goals, or plans. Decision options may then match or not match these images and be adopted, rejected, considered further, depending on circumstances." (Deborah J Terry & Michael A Hogg, "Attitudes, Behavior, and Social Context: The Role of Norms and Group Membership", 1999)

On Principles (1950-1974)

 "It is more important to have a clear understanding of general principles, without, however, thinking of them as fixed laws, than to load the mind with a mass of detailed technical information which can readily be found in reference books or card indexes." (William I B Beveridge, "The Art of Scientific Investigation", 1950)

"[…] the chief reason in favor of any theory on the principles of mathematics must always be inductive, i.e., it must lie in the fact that the theory in question enables us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence the early deductions, until they reach this point, give reasons rather from them, than for believing the premises because true consequences follow from them, than for believing the consequences because they follow from the premises." (Alfred N Whitehead, "Principia Mathematica", 1950)

"The principle of complementarity states that no single model is possible which could provide a precise and rational analysis of the connections between these phenomena [before and after measurement]. In such a case, we are not supposed, for example, to attempt to describe in detail how future phenomena arise out of past phenomena. Instead, we should simply accept without further analysis the fact that future phenomena do in fact somehow manage to be produced, in a way that is, however, necessarily beyond the possibility of a detailed description. The only aim of a mathematical theory is then to predict the statistical relations, if any, connecting the phenomena." (David Bohm, "A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables", 1952)

"We have thus assigned to pure reason and experience their places in a theoretical system of physics. The structure of the system is the work of reason: the empirical contents and their mutual relations must find their representation in the conclusions of the theory. In the possibility of such a representation lie the sole value and justification of the whole system, and especially of the concepts and fundamental principles which underlie it. Apart from that, these latter are free inventions of human intellect, which cannot be justified either by the nature of that intellect or in any other fashion a priori." (Albert Einstein, "Ideas and Opinions", 1954)

"For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created." (Freeman Dyson, "Mathematics in the Physical Sciences", Scientific American, 1964)

"The method of guessing the equation seems to be a pretty effective way of guessing new laws. This shows again that mathematics is a deep way of expressing nature, and any attempt to express nature in philosophical principles, or in seat-of-the-pants mechanical feelings, is not an efficient way." (Richard Feynman, "The Character of Physical Law", 1965)

"Traditional organizational theories have tended to view the human organization as a closed system. This tendency has led to a disregard of differing organizational environments and the nature of organizational dependency on environment. It has led also to an over-concentration on principles of internal organizational functioning, with consequent failure to develop and understand the processes of feedback which are essential to survival." (Daniel Katz, "The Social Psychology of Organizations", 1966)

"The parallelism of general conceptions or even special laws in different fields therefore is a consequence of the fact that these are concerned with 'systems' and that certain general principles apply to systems irrespective of their nature. Hence principles such as those of wholeness and sum, mechanization, hierarchic order, approached to steady states, equifinality, etc., may appear in quite different disciplines. The isomorphism found in different realms is based of the existence of general system principles, of a more or less well-developed ‘general system theory’." (Ludwig von Bertalanffy, "General System Theory", 1968)

"Thus, there exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular kind, the nature of their component elements, and the relations or 'forces' between them. It seems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principles applying to systems in general. In this way we postulate a new discipline called General System Theory. Its subject matter is the formulation and derivation of those principles which are valid for ‘systems’ in general." (Ludwig von Bertalanffy, „General System Theory: Foundations, Development, Applications", 1968)

"Modern scientific principle has been drawn from the investigation of natural laws, technology has developed from the experience of doing, and the two have been combined by means of mathematical system to form what we call engineering." (George S Emmerson, "Engineering Education: A Social History", 1973)

On Principles (1900-1949)

 "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)

On Principles (1850-1899)

 "In the original discovery of a proposition of practical utility, by deduction from general principles and from experimental data, a complex algebraical investigation is often not merely useful, but indispensable; but in expounding such a proposition as a part of practical science, and applying it to practical purposes, simplicity is of the importance: - and […] the more thoroughly a scientific man has studied higher mathematics, the more fully does he become aware of this truth – and […] the better qualified does he become to free the exposition and application of principles from mathematical intricacy." (William J M Rankine, "On the Harmony of Theory and Practice in Mechanics", 1856)

"It is easily seen from a consideration of the nature of demonstration and analysis that there can and must be truths which cannot be reduced by any analysis to identities or to the principle of contradiction but which involve an infinite series of reasons which only God can see through." (Gottfried W Leibniz, "Nouvelles lettres et opuscules inédits", 1857)

"It is, after all, a principle of logic not to multiply entities unnecessarily." (Antoine-Laurent Lavoisier, "Réflexions sur le phlogistique", 1862)

"The more man inquires into the laws which regulate the material universe, the more he is convinced that all its varied forms arise from the action of a few simple principles. These principles themselves converge, with accelerating force, towards some still more comprehensive law to which all matter seems to be submitted. Simple as that law may possibly be, it must be remembered that it is only one amongst an infinite number of simple laws: that each of these laws has consequences at least as extensive as the existing one, and therefore that the Creator who selected the present law must have foreseen the consequences of all other laws." (Charles Babbage, "Passages From the Life of a Philosopher", 1864)

"As in the experimental sciences, truth cannot be distinguished from error as long as firm principles have not been established through the rigorous observation of facts." (Louis Pasteur, "Étude sur la maladie des vers à soie", 1870)

"Some definite interpretation of a linear algebra would, at first sight, appear indispensable to its successful application. But on the contrary, it is a singular fact, and one quite consonant with the principles of sound logic, that its first and general use is mostly to be expected from its want of significance. The interpretation is a trammel to the use. Symbols are essential to comprehensive argument." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)

"I say that a manifold (a collection, a set) of elements that belong to any conceptual sphere is well-defined, when on the basis of its definition and as a consequence of the logical principle of excluded middle it must be regarded as internally determined, both whether an object pertaining to the same conceptual sphere belongs or not as an element to the manifold, and whether two objects belonging to the set are equal to each other or not, despite formal differences in the ways of determination." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"It is of the nature of true science to take nothing on trust or on authority. Every fact must be established by accurate observation, experiment, or calculation. Every law and principle must rest on inductive argument. The apostolic motto, ‘Prove all things, hold fast that which is good’, is thoroughly scientific. It is true that the mere reader of popular science must often be content to take that on testimony which he cannot personally verify; but it is desirable that even the most cursory reader should fully comprehend the modes in which facts are ascertained and the reasons on which the conclusions are based." (Sir John W Dawson, "The Chain of Life in Geological Time", 1880)

"[…] not only a knowledge of the ideas that have been accepted and cultivated by subsequent teachers is necessary for the historical understanding of a science, but also that the rejected and transient thoughts of the inquirers, nay even apparently erroneous notions, may be very important and very instructive. The historical investigation of the development of a science is most needful, lest the principles treasured up in it become a system of half-understood prescripts, or worse, a system of prejudices." (Ernst Mach, "The Science of Mechanics", 1883)

"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)

On Principles (1800-1849)

"Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. It's chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them." (J B Joseph Fourier, "The Analytical Theory of Heat", 1822)

"The aim of every science is foresight. For the laws of established observation of phenomena are generally employed to foresee their succession. All men, however little advanced make true predictions, which are always based on the same principle, the knowledge of the future from the past." (Auguste Compte, "Plan des travaux scientifiques nécessaires pour réorganiser la société", 1822)

"It is true that of far the greater part of things, we must content ourselves with such knowledge as description may exhibit, or analogy supply; but it is true likewise, that these ideas are always incomplete, and that at least, till we have compared them with realities, we do not know them to be just. As we see more, we become possessed of more certainties, and consequently gain more principles of reasoning, and found a wider base of analogy." (Samuel Johnson, 1825)

"To invent without scruple a new principle to every new phenomenon, instead of adapting it to the old; to overload our hypothesis with a variety of this kind, are certain proofs that none of these principles is the just one, and that we only desire, by a number of falsehoods, to cover our ignorance of the truth." (David Hume, "Of the passions", 1826)

"In Pure Mathematics, where all the various truths are necessarily connected with each other, (being all necessarily connected with those hypotheses which are the principles of the science), an arrangement is beautiful in proportion as the principles are few; and what we admire perhaps chiefly in the science, is the astonishing variety of consequences which may be demonstrably deduced from so small a number of premises." (Dugald Stewart, "Elements of the Philosophy of the Human Mind" Vol. 3, 1827)

"For one person who is blessed with the power of invention, many will always be found who have the capacity of applying principles." (Charles Babbage, "Reflections on the Decline of Science in England, and on Some of Its Causes", 1830)

"A maxim is a conclusion upon observation of matters of fact, and is merely speculative; a ‘principle’ carries knowledge within itself, and is prospective." (Samuel T Coleridge, "The Table Talk and Omniana of Samuel Taylor Coleridge", 1831)

"The function of theory is to put all this in systematic order, clearly and comprehensively, and to trace each action to an adequate, compelling cause. […] Theory should cast a steady light on all phenomena so that we can more easily recognize and eliminate the weeds that always spring from ignorance; it should show how one thing is related to another, and keep the important and the unimportant separate. If concepts combine of their own accord to form that nucleus of truth we call a principle, if they spontaneously compose a pattern that becomes a rule, it is the task of the theorist to make this clear." (Carl von Clausewitz, "On War", 1832)

"The insights gained and garnered by the mind in its wanderings among basic concepts are benefits that theory can provide. Theory cannot equip the mind with formulas for solving problems, nor can it mark the narrow path on which the sole solution is supposed to lie by planting a hedge of principles on either side. But it can give the mind insight into the great mass of phenomena and of their relationships, then leave it free to rise into the higher realms of action." (Carl von Clausewitz, "On War", 1832)

"Algebra, as an art, can be of no use to any one in the business of life; certainly not as taught in the schools. I appeal to every man who has been through the school routine whether this be not the case. Taught as an art it is of little use in the higher mathematics, as those are made to feel who attempt to study the differential calculus without knowing more of the principles than is contained in books of rules." (Augustus de Morgan, "Elements of Algebra", 1837)

"These sciences, Geometry, Theoretical Arithmetic and Algebra, have no principles besides definitions and axioms, and no process of proof but deduction; this process, however, assuming a most remarkable character; and exhibiting a combination of simplicity and complexity, of rigour and generality, quite unparalleled in other subjects." (William Whewell, "The Philosophy of the Inductive Sciences", 1840)

"[…] in order to observe, our mind has need of some theory or other. If in contemplating phenomena we did not immediately connect them with principles, not only would it be impossible for us to combine these isolated observations, and therefore to derive profit from them, but we should even be entirely incapable of remembering facts, which would for the most remain unnoted by us." (Auguste Comte, "Cours de Philosophie Positive", 1830-1842)

"There are, undoubtedly, the most ample reasons for stating both the principles and theorems [of geometry] in their general form […] But, that an unpractised learner, even in making use of one theorem to demonstrate another, reasons rather from particular to particular than from the general proposition, is manifest from the difficulty he finds in applying a theorem to a case in which the configuration of the diagram is extremely unlike that of the diagram by which the original theorem was demonstrated. A difficulty which, except in cases of unusual mental powers, long practice can alone remove, and removes chiefly by rendering us familiar with all the configurations consistent with the general conditions of the theorem." (John S Mill, "A System of Logic", 1843)

"In truth, ideas and principles are independent of men; the application of them and their illustration is man's duty and merit." (Edward Forbes, 1847)

On Principles (1799-1799)

"[…] for the saving the long progression of the thoughts to remote and first principles in every case, the mind should provide itself several stages; that is to say, intermediate principles, which it might have recourse to in the examining those positions that come in its way. These, though they are not self-evident principles, yet, if they have been made out from them by a wary and unquestionable deduction, may be depended on as certain and infallible truths, and serve as unquestionable truths to prove other points depending upon them, by a nearer and shorter view than remote and general maxims. […] And thus mathematicians do, who do not in every new problem run it back to the first axioms through all the whole train of intermediate propositions. Certain theorems that they have settled to themselves upon sure demonstration, serve to resolve to them multitudes of propositions which depend on them, and are as firmly made out from thence as if the mind went afresh over every link of the whole chain that tie them to first self-evident principles." (John Locke, "The Conduct of the Understanding", 1706)

"[Mathematics] guides our minds in an orderly way, and furnishes us simple and rational principles by means of which ambiguities are clarified, disorder is converted into order, and complexities are analyzed into their component parts." (Johann B Mencken, "The Charlatanry of the Learned", 1715)

"Principles taken upon trust, consequences lamely deduced from them, want of coherence in the parts, and of evidence in the whole, these are every where to be met with in the systems of the most eminent philosophers, and seem to have drawn disgrace upon philosophy itself." (David Hume, "A Treatise of Human Nature", 1739-40)

"In mathematics it [sophistry] had no place from the beginning: Mathematicians having had the wisdom to define accurately the terms they use, and to lay down, as axioms, the first principles on which their reasoning is grounded. Accordingly we find no parties among mathematicians, and hardly any disputes." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"Men are often led into errors by the love of simplicity, which disposes us to reduce things to few principles, and to conceive a greater simplicity in nature than there really is." (Thomas Reid, "Essays on the Intellectual Powers of Man", 1785)

"Metaphysical truths can only be established by producing effects from corresponding causes; and though we may confront such demonstrative evidence with the immutable laws of mathematical decision, we must be sensible that there will still remain some pretense for doubt; thus the basis of that knowledge, which on these principles we have been long labouring to accomplish, will become an endless toil, an endless force for controversy: and having the passions and the prejudices of mankind to combat, which mathematical certainty can alone effectually suppress, we must content ourselves only with making converts of those who have minds sufficiently expansive without the shackles of Euclid, and the vanity of displaying their own learning and pedantry." (James Douglas, "A Dissertation on the Antiquity of the Earth", 1785)

"Every science has for its basis a system of principles as fixed and unalterable as those by which the universe is regulated and governed. Man cannot make principles; he can only discover them." (Thomas Paine, "The Age of Reason", 1794)

On Principles (-1699)

"In all disciplines in which there is systematic knowledge of things with principles, causes, or elements, it arises from a grasp of those: we think we have knowledge of a thing when we have found its primary causes and principles, and followed it back to its elements." (Aristotle, "Physics", cca. 350 BC)

"The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. And since these (e.g. order and definiteness) are obviously causes of many things, evidently these sciences must treat this sort of causative principle also (i.e. the beautiful) as in some sense a cause." (Aristotle, "Metaphysica", cca. 350 BC)

"Reflexion is careful and laborious thought, and watchful attention directed to the agreeable effect of one's plan. Invention, on the other hand, is the solving of intricate problems and the discovery of new principles by means of brilliancy and versatility." (Marcus Vitruvius Pollio, "De architectura" ["On Architecture], cca. 15BC)

"[…] the least initial deviation from the truth is multiplied later a thousand-fold. Admit, for instance, the existence of a minimum magnitude, and you will find that the minimum which you have introduced, small as it is, causes the greatest truths of mathematics to totter. The reason is that a principle is great rather in power than in extent; hence that which was small at the start turns out a giant at the end." (St. Thomas Aquinas, "De Ente et Essentia", cca. 1252)

"All that is required between cognizer and cognized is a likeness in terms of representation, not a likeness in terms of an agreement in nature. For it's plain that the form of a stone in the soul is of a far higher nature than the form of a stone in matter. But that form, insofar as it represents the stone, is to that extent the principle leading to its cognition." (Thomas Aquinas, "Quaestiones disputatae de veritate", cca. 1256-1259)

"It is superfluous to suppose that what can be accounted for by a few principles has been produced by many." (Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"Reason may be employed in two ways to establish a point: first for the purpose of furnishing sufficient proof of some principle, as in natural science, where sufficient proof can be brought to show that the movement of the heavens is always of uniform velocity. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results […]" (Saint Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried." (Francis Bacon, "Novum Organum", 1620)

"Reality cannot be found except in One single source, because of the interconnection of all things with one another. […] It is a good thing to proceed in order and to establish propositions (principles). This is the way to gain ground and to progress with certainty." (Gottfried Leibniz, 1670)

On Principles (Unsourced)

 "A small error in the beginning (or in principles) leads to a big error in the end (or in conclusions)." (ancient axiom)

"A theory of physics is not an explanation; it is a system of mathematical oppositions deduced from a small number of principles the aim of which is to represent as simply, as completely, and as exactly as possible, a group of experimental laws." (Pierre-Maurice-Marie Duhem)

"It [science] has as its highest principle and most coveted aim the solution of the problem to condense all natural phenomena which have been observed and are still to be observed into one simple principle, that allows the computation of past and more especially of future processes from present ones. [...] Amid the more or less general laws which mark the achievements of physical science during the course of the last centuries, the principle of least action is perhaps that which, as regards form and content, may claim to come nearest to that ideal final aim of theoretical research." (Max Planck)

"No mathematical exactness without explicit proof from assumed principles – such is the motto of the modern geometer." (George Bruce Halsted)

"The aim of every science is foresight (prevoyance). For the laws of established observation of phenomena are generally employed to foresee their succession. All men, however little advanced make true predictions, which are always based on the same principle, the knowledge of the future from the past." (Auguste Compte)

"The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God." (John W N Sullivan)

"The most general law in nature is equity - the principle of balance and symmetry which guides the growth of forms along the lines of the greatest structural efficiency." (Herbert Read)

"[…] the mathematician learns early to accept no fact, to believe no statement, however apparently reasonable or obvious or trivial, until it has been proved, rigorously and totally by a series of steps proceeding from universally accepted first principles." (Alfred Adler)

"There is a great difference between the spirit of Mathematics and the spirit of Observation. In the former, the principles are palpable, but remote from common use; so that from want of custom it is not easy to turn our head in that direction; but if it be thus turned ever so little, the principles are seen fully confessed, and it would argue a mind incorrigibly false to reason inconsequentially on principles so obtrusive that it is hardly possible to overlook them." (Blaise Pascal)

"There is no law of physics that does not lend itself to most economical derivation from a symmetry principle. However, a symmetry principle hides from view any sight of the deeper structure that underpins that law and therefore also prevents any immediate sight of how in each case that mutability comes about." (John A Wheeler)

"We consider it a good principle to explain the phenomena by the simplest hypothesis possible." (Ptolemy)

"What makes a great mathematician? A feel for form, a strong sense of what is important. Möbius had both in abundance. He knew that topology was important. He knew that symmetry is a fundamental and powerful mathematical principle. The judgment of posterity is clear: Möbius was right." (Ian Stewart)

Henri-Frédéric Amiel - Collected Quotes

"An error is the more dangerous in proportion to the degree of truth which it contains." (Henri-Frédéric Amiel, [journal entry] 1852)

"Common sense is the measure of the possible; it is composed of experience and prevision; it is calculation applied to life." (Henri-Frédéric Amiel, [journal entry] 1852)

"True poetry is truer than science, because it is synthetic, and seizes at once what the combination of all the sciences is able, at most, to attain as a final result. The soul of nature is divined by the poet; the man of science, only serves to accumulate materials for its demonstration." (Henri-Frédéric Amiel, [journal entry] 1852)

"The man who insists upon seeing with perfect clearness before he decides, never decides." (Henri-Frédéric Amiel, [journal entry] 1856)

"To know how to suggest is the great art of teaching. To attain it we must be able to guess what will interest." (Henri-Frédéric Amiel, 1864)

"Nature does at least what she can to translate into visible form the wealth of the creative formula. By the vastness of the abysses into which she penetrates, in the effort - the unsuccessful effort - to house and contain the eternal thought, we may measure the greatness of the divine mind." (Henri-Frédéric Amiel, [journal entry] 1866)

"What we call little things are merely the causes of great things; they are the beginning, the embryo, and it is the point of departure which, generally speaking, decides the whole future of an existence. One single black speck may be the beginning of gangrene, of a storm, of a revolution." (Henri-Frédéric Amiel, [journal entry] 1868)

"Pure truth cannot be assimilated by the crowd; it must be communicated by contagion." (Henri-Frédéric Amiel, [journal entry] 1875)

"Everything which is, is thought, but not conscious and individual thought. The human intelligence is but the consciousness of being. It is what I have formulated before: Everything is a symbol of a symbol, and a symbol of what? of mind." (Henri-Frédéric Amiel, 1882)

"Time is but the measure of the difficulty of a conception. Pure thought has scarcely any need of time, since it perceives the two ends of an idea almost at the same moment." (Henri-Frédéric Amiel, 1883)

"Time is but the space between our memories; as soon as we cease to perceive this space, time has disappeared." (Henri-Frédéric Amiel, [journal entry]  1884)

"Every situation is an equilibrium of forces; every life is a struggle between opposing forces working within the limits of a certain equilibrium." (Henri-Frédéric Amiel, [journal entry] 1885)

"Understanding [...] must begin by saturating itself with facts and realities. [...] Besides, we only understand that which is already within us. To understand is to possess the thing understood, first by sympathy and then by intelligence. Instead of first dismembering and dissecting the object to be conceived, we should begin by laying hold of it in its ensemble. The procedure is the same, whether we study a watch or a plant, a work of art or a character." (Henri-Frédéric Amiel, [journal entry] 1886)

"The art which is grand and yet simple is that which presupposes the greatest elevation both in artist and in public." (Henri-Frédéric Amiel, [journal entry] 1893)

"The test of every religious, political, or educational system is the man that it forms." (Henri-Frédéric Amiel, [journal entry] 1893)

"Time is the supreme illusion. It is but the inner prism by which we decompose being and life, the mode under which we perceive successively what is simultaneous in idea." (Henri-Frédéric Amiel, [journal entry] 1893)

"Our systems, perhaps, are nothing more than an unconscious apology for our faults, a gigantic scaffolding whose object is to hide from us our favorite sin." (Henri-Frédéric Amiel, [journal entry] 1896)

"Analysis kills spontaneity. The grain once ground into flour springs and germinates no more." (Henri-Frédéric Amiel)

"Minds accustomed to analysis never allow objections more than half-value, because they appreciate the variable and relative elements which enter in." (Henri-Frédéric Amiel)

"No forms of error are so erroneous as those that have the appearance without the reality of mathematical precision." (Henri-Frédéric Amiel)

"Time and space are fragments of the infinite for the use of finite creatures." (Henri-Frédéric Amiel)

"To do easily what is difficult for others is the mark of talent. To do what is impossible for talent is the mark of genius." (Henri-Frédéric Amiel)

"Wisdom consists in rising superior both to madness and to common sense, and in lending one's self to the universal delusion without becoming its dupe." (Henri-Frédéric Amiel)

Misquoted: Gauss on the Queen of Mathematics

The following two quotes from Carl Friedrich Gauss are frequently met in books on Mathematics and Science (sometimes with slight differences in formulation as they were translated from German):

"Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics." 
"Mathematics is the Queen of the Sciences, and Number Theory the Queen of Mathematics."

The earliest occurrence of the quotes I met first in Eric T Bell's "Mathematics: Queen & Servant of Science" (1951), which is probably one of the most quoted sources for the quotes:

"Mathematics is Queen of the Sciences and Arithmetic the Queen of Mathematics. She often condescends to render service to astronomy and other natural sciences, but under all  circumstances the first place is her due."

However, doing a little research I found an earlier reference in John T Merz' "European Thought in the Nineteenth Century" Vol II (1903), who cites Sartorius von Waltershausen's "Gauss zum Gedächtniss" (1856) as source:

"Mathematics is the Queen of the Sciences, and arithmetic the Queen of Mathematics. She frequently condescends to do service for astronomy and other natural sciences, but to her belongs, under all circumstances, the foremost place."

Arithmetic, at least the way introduced in schools, is the study of numbers and their properties in report with the basic operations (addition, subtraction, multiplication, division, exponentiation and extraction of roots). In exchange, Number Theory appeared while studying the intrinsic properties of integers and integer-valued functions, what is known also as Higher Arithmetic. Even if the earliest discoveries date from early antiquity, the basis were put by Fermat, Euler, Gauss and others mathematicians, incorporating over time domains like Complex Analysis, Group theory or Galois theory.

According to Merriam-Webster, "Number Theory" was used for the first time around 1864 and it took some time until it was incorporated in mathematical texts. Arithmetic and Number Theory become synonymous in the 20th century (which might be strange for some). Thus, in some occurrence of the quote "Arithmetic" was replaced by "Number Theory" to reflect nowadays’ interpretation of Gauss' words. The two quotes seem to mean the same thing, though they probably need to be interpreted upon case through the historical context.  

'Queen' is used here metaphorically as an analogy between the meanings associated with a queen (ruler, fertility, authority), respectively the roles taken by Mathematics and Arithmetic in science. Therefore, a science considered as a queen could be seen as having a high fecundity of ideas and authority in another field. As "Queen of Science" were considered also Metaphysics (Immanuel Kant), Philosophy and even Theology. Also Physics and Chemistry have a statute of royalty.

The importance of Higher Arithmetic in respect to Mathematics, is stressed by Gauss in a letter to Dirichlet from 1838: 

"I place this part of Mathematics above all others (and have always done so)."

The idea is reiterated by David Hilbert half of century later:

"We thus see how arithmetic, the queen of mathematical science, has conquered large domains and has assumed the leadership. That this was not done earlier and more completely, seems to me to depend on the fact that the theory of numbers has only in quite recent times arrived at maturity." (David Hilbert, "Theorie der Algebraischen Zahlkörper", Bericht der Mathematiker-Vereinigung,' vol. IV, 1897)

Paraphrasing Godfrey H Hardy, even if Number Theory began as an experimental science, its role in the further development of Mathematics is important, worthily of the status of a 'queen'.

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On Science: The Queen(s) of Science

"Time was, when she [metaphysics] was the queen of all the sciences; and, if we take the will for the deed, she certainly deserves, so far as regards the high importance of her object-matter, this title of honor. Now, it is the fashion of the time to heap contempt and scorn upon her." (Immanuel Kant, "Critique of Pure Reason", 1781)

"We thus see how arithmetic, the queen of mathematical science, has conquered large domains and has assumed the leadership. That this was not done earlier and more completely, seems to me to depend on the fact that the theory of numbers has only in quite recent times arrived at maturity." (David Hilbert, "Theorie der Algebraischen Zahlkörper", Bericht der Mathematiker-Vereinigung,' vol. IV, 1897)

"It seems to me that no one science can so well serve to co-ordinate and, as it were, bind together all of the sciences as the queen of them all, mathematics." (Edward W Davis, "Publications of the Nebraska State Historical Society" Vol. 7, 1898)

"The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results." (Carl G Hempel, "Geometry and Empirical Science", 1945)

"Needless to say that once the universal approach exists we can go from one field to another and use the results of one field to promote another field. However, we should never forget limitations of 'universal approaches'. It is hiqhly dangerous to apply such an approach, if it has worked in a certain domain, to other domains as a dogma. Using any universal approach you must again and again check whether the prepositions made are fulfilled by the objects to which these approaches are applied. Going to more and more abstractions where we must heavily rely on mathematics which, after all, is the Queen of science." (Hermann Haken, 1979)

"Thus, astronomy was probably the first exact science, practiced long before the concept of science as such had been formulated. (Mathematics may have been earlier, but I do not consider it a natural science: the mother of many kings is not necessarily a queen.)" (Erwin Chargaff, "Serious Questions", Nature, 1986)

"Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems - general and specific statements - can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences." (Sir Erik C Zeeman, "Private Games", 1988)

"Intellectual inquiry begins with myth, religion and philosophy. Originally, philosophy (or perhaps theology or metaphysics) is the queen of the sciences, other intellectual disciplines having only a highly subservient, specialized role to play within philosophy. [...] Instead of being the queen of the sciences, overarching all other sciences, philosophy has been transformed into a highly specialized, technical, somewhat meagre enterprise, concerned not with improving our knowledge and understanding of the world - for that is the business of the empirical sciences - but rather with clarifying concepts and solving conceptual problems." (Nicholas Maxwell, "Karl Popper, Science and Enlightenment", 2017)

"Mathematics is Queen of the Sciences and Arithmetic the Queen of Mathematics. She often condescends to render service to astronomy and other natural sciences, but under all  circumstances the first place is her due." (Carl Friedrich Gauss)

"The strongest arguments prove nothing so long as the conclusions are not verified by experience. Experimental science is the queen of sciences and the goal of all speculation." (Roger Bacon)