On Hypotheses: The Riemann Hypothesis

"Whoever proves or disproves [the Riemann Hypothesis] will cover himself in glory..." (Eric T Bell, 1937)

"[...] the Riemann hypothesis remains one of the outstanding challenges of mathematics, a prize which has tantalized and eluded some of the most brilliant mathematicians of this century...Hilbert is reputed to have said that the first comment he would make after waking at the end of a thousand year sleep would be, 'Is the Riemann hypothesis established yet?'" (Richard E  Bellman, A Brief Introduction of Theta Functions, 1961)

"At this point, it is not possible to remain silent on what is probably the most intriguing unsolved problem in the theory of the zeta function and actually in all of number theory - and most likely even one of the most important unsolved problems in contemporary mathematics, namely the famous Riemann hypothesis. [...] Still, the problem is open and fascinates and teases the best contemporary minds." (Emil Grosswald, "Topics in the Theory of Numbers", 1966)

"The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers. This fact alone singles out the Riemann hypothesis as the main open question of prime number theory." (Enrico Bombieri,  "Prime Territory", The Sciences,  1992)

"The Riemann hypothesis [...] is still widely considered to be one of the greatest unsolved problems in mathematics, sure to wreath its conqueror with glory." (Bruce Schechter, "143-year-old problem still has mathematicians guessing", 2002)

"The dependence of so many results on Riemann's challenge is why mathematicians refer to it as a hypothesis rather than a conjecture. The word 'hypothesis' has the much stronger connotation of a necessary assumption that a mathematician makes in order to build a theory. 'Conjecture', in contrast, represents simply a prediction of how mathematicians believe their world behaves. Many have had to accept their inability to solve Riemann's riddle and have simply adopted his prediction as a working hypothesis. If someone can turn the hypothesis into a theorem, all those unproven results would be validated." (Marcus du Sautoy, "The Music of the Primes", 2003)

"The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes." (David M Burton, "Elementary Number Theory", 2006)

"Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician. I read and reread proofs in much the same way as I listen to a piece of music: understanding how themes are established, mutated, interwoven and transformed. What people don't realise about mathematics is that it involves a lot of choice: not about what is true or false (I can't make the Riemann hypothesis false if it's true), but from deciding what piece of mathematics is worth ‘listening to’." (Marcus du Sautoy, "Listen by numbers: music and maths", 2011)

"If [the Riemann Hypothesis is] not true, then the world is a very different place. The whole structure of integers and prime numbers would be very different to what we could imagine. In a way, it would be more interesting if it were false, but it would be a disaster because we've built so much round assuming its truth." (P  Sarnak)

"The Riemann Hypothesis is a precise statement, and in one sense what it means is clear, but what it's connected with, what it implies, where it comes from, can be very unobvious." (M Huxley)

"[...] the Riemann Hypothesis will be settled without any fundamental changes in our mathematical thoughts, namely, all tools are ready to attack it but just a penetrating idea is missing." (Y Motohashi)

"The consequences [of the Riemann Hypothesis] are fantastic: the distribution of primes, these elementary objects of arithmetic. And to have tools to study the distribution of these of objects." (H Iwaniec)

On Arithmetic (Unsourced)

"Arithmetic, then, means dealing logically with certain facts that we know, about numbers, with a view to arriving at knowledge which as yet we do not possess." (Philosophy & Fun of Algebra)

"As arithmetic and algebra are sciences of great clearness, certainty, and extent, which are immediately conversant about signs, upon the skillful use whereof they entirely depend, so a little attention to them may possibly help us to judge of the progress of the mind in other sciences, which, though differing in nature, design, and object, may yet agree in the general methods of proof and inquiry." (George Berkeley)

"I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths." (F L Gottlob Frege)

"Music is the arithmetic of sounds as optics is the geometry of light." (Claude Debussy)

"Music is the hidden arithmetical exercise of a soul unconscious that it is calculating." (Gottfried W Leibniz)

"The human mind has never invented a labor-saving machine equal to algebra." (J Willard Gibbs)

"The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods; the gods are there, behind the wall, at play with numbers." (Le Corbusier)

"The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic." (Gottfried W Leibniz)

"You cannot ask us to take sides against arithmetic." (Winston S Churchill)

On Arithmetic (1976 - 1999)

"Mathematics then becomes the ladder by which we all may climb into the heaven of perfect insight and eternal satisfaction, and the solution of arithmetic and algebraic problems is connected with the salvation of our souls." (Scott Buchanan, "Poetry and Mathematics", 1975)

"Dividing by zero is the closest thing there is to arithmetic blasphemy." (William Dunham, "The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities", 1978)

"[...] an intellectual is a highly educated man who can’t do arithmetic with his shoes on, and is proud of his lack." (Robert A Heinlein, "The Cat Who Walks Through Walls: A comedy of manners", 1985)

"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)

"The letter ‘i’ originally was meant to suggest the imaginary nature of this number, but with the greater abstraction of mathematics, it came to be realized that it was no more imaginary than many other mathematical constructs. True, it is not suitable for measuring quantities, but it obeys the same laws of arithmetic as do the real numbers, and, surprisingly enough, it makes the statement of various physical laws very natural." (John A Paulos, "Beyond Numeracy", 1991)

"He who refuses to do arithmetic is doomed to talk nonsense." (John McCarthy, "Progress and its Substanability", 1995)

On Arithmetic (1950 - 1074)

"Arithmetic is where numbers fly like pigeons in and out of your head." (Carl Sandburg, "Arithmetic" [in "Complete Poems"], 1950)

"Arithmetic is numbers you squeeze from your head to your hand to your pencil to your paper till you get the answer." (Carl Sandburg, "Arithmetic" [in "Complete Poems"], 1950)

"Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number - there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method." (John von Neumann, "Various techniques used in connection with random digits", 1951)

"[…] arithmetic is a calculus which starts only from certain conventions but floats as freely as the solar system and rests on nothing." (Friedrich Waismann, "Introduction to Mathematical Thinking", 1951)

"Historically, Statistics is no more than State Arithmetic, a system of computation by which differences between individuals are eliminated by the taking of an average. It has been used - indeed, still is used - to enable rulers to know just how far they may safely go in picking the pockets of their subjects." (Michael J Moroney, "Facts from Figures", 1951)

"The first acquaintance which most people have with mathematics is through arithmetic. [...] Arithmetic, therefore, will be a good subject to consider in order to discover, if possible, the most obvious characteristic of the science. Now, the first noticeable fact about arithmetic is that it applies to everything, to tastes and to sounds, to apples and to angels, to the ideas of the mind and to the bones of the body." (Alfred N Whitehead, "An Introduction to Mathematics", 1958)

"The territory of arithmetic ends where the two ideas of 'variables' and of 'algebraic form' commence their sway." (Alfred N Whitehead,  "An Introduction to Mathematics", 1958)

"The statistician has no use for information that cannot be expressed numerically, nor generally speaking, is he interested in isolated events or examples. The term ' data ' is itself plural and the statistician is concerned with the analysis of aggregates." (Alfred R Ilersic, "Statistics", 1959)

"The statistics themselves prove nothing; nor are they at any time a substitute for logical thinking. There are […] many simple but not always obvious snags in the data to contend with. Variations in even the simplest of figures may conceal a compound of influences which have to be taken into account before any conclusions are drawn from the data." (Alfred R Ilersic, "Statistics", 1959)

"There are good statistics and bad statistics; it may be doubted if there are many perfect data which are of any practical value. It is the statistician's function to discriminate between good and bad data; to decide when an informed estimate is justified and when it is not; to extract the maximum reliable information from limited and possibly biased data." (Alfred R Ilersic, "Statistics", 1959)

"While it is true to assert that much statistical work involves arithmetic and mathematics, it would be quite untrue to suggest that the main source of errors in statistics and their use is due to inaccurate calculations." (Alfred R Ilersic, "Statistics", 1959)

"Science will never be able to reduce the value of a sunset to arithmetic. Nor can it reduce friendship or statesmanship to a formula. Laughter and love, pain and loneliness, the challenge of accomplishment in living, and the depth of insight into beauty and truth; these will always surpass the scientific mastery of nature." (Louis Orr, [Commencement Address] 1960)

"Poor arithmetic will make the bridge fall down just as surely as poor physics, poor metallurgy, or poor logic will." ( Thomas T Woodson, "Introduction to Engineering Design", 1966)

"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

"[...] there is an essential element in science that is cold, objective, and nonhuman…the laws of nature are as impersonal and free of human values as the rules of arithmetic.[...] Nowhere do we see human value or human meaning. (Steven Weinberg, "Reflections of a Working Scientist", Daedalus Vol. 103, 1974)

On Arithmetic (1925 - 1949)

"[…] extensions beyond the complex number domain are possible only at the expense of the principle of permanence. The complex number domain is the last frontier of this principle. Beyond this either the commutativity of the operations or the rôle which zero plays in arithmetic must be sacrificed." (Tobias Dantzig, "Number: The Language of Science", 1930)

"For it is true, generally speaking, that mathematics is not a popular subject, even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers." (David Hilbert, "Anschauliche Geometrie", 1932)

"Will anyone seriously assert that the existence of negative numbers is guaranteed by the fact that there exist in the world hot assets and cold, and debts? Shall we refer to these things in the structure of arithmetic? Who does not see that thereby an entirely foreign element enters into arithmetic, which endangers the pureness and clarity of its concepts?" (Friedrich Waismann, "Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics", 1936)

"If you find my arithmetic correct, then no amount of vapouring about my psychological condition can be anything but a waste of time. If you find my arithmetic wrong, then it may be relevant to explain psychologically how I came to be so bad at my arithmetic, and the doctrine of the concealed wish will become relevant - but only after you have yourself done the sum and discovered me to be wrong on purely arithmetical grounds. It is the same with all thinking and all systems of thought. If you try to find out which are tainted by speculating about the wishes of the thinkers, you are merely making a fool of yourself. You must first find out on purely logical grounds which of them do, in fact, break down as arguments. Afterwards, if you like, go on and discover the psychological causes of the error." (Clive S Lewis, "Bulverism", 1941)

"In other words, without a theory, a plan, the mere mechanical manipulation of the numbers in a problem does not necessarily make sense just because you are using Arithmetic!" (Lillian R Lieber, "The Education of T.C. MITS", 1944)

"If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability." (Vannevar Bush, "As We May Think", Atlantic Monthly, 1945)

"With a literature much vaster than those of algebra and arithmetic combined, and as least as extensive as that of analysis, geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics." (Eric T Bell, "The Development of Mathematics", 1945)

On Arithmetic (1875 - 1899)

 "I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetical act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding […]" (Richard Dedekind, "On Continuity and Irrational Numbers", 1872)

"The most distinct and beautiful statement of any truth [in science] must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both." (Henry Thoreau, "A Week on the Concord and Merrimack Rivers", 1873)

"Thought is symbolical of Sensation as Algebra is of Arithmetic, and because it is symbolical, is very unlike what it symbolises. For one thing, sensations are always positive; in this resembling arithmetical quantities. A negative sensation is no more possible than a negative number. But ideas, like algebraic quantities, may be either positive or negative. However paradoxical the square of a negative quantity, the square root of an unknown quantity, nay, even in imaginary quantity, the student of Algebra finds these paradoxes to be valid operations. And the student of Philosophy finds analogous paradoxes in operations impossible in the sphere of Sense. Thus although it is impossible to feel non-existence, it is possible to think it; although it is impossible to frame an image of Infinity, we can, and do, form the idea, and reason on it with precision." (George H Lewes "Problems of Life and Mind", 1873)

"Thus numbers may be said to rule the whole world of quantity, and the four rules of arithmetic may be regarded as the complete equipment of the mathematician." James C Maxwell, "Electricity and Magnetism", 1873)

"The rules of Arithmetic operate in Algebra; the logical operations supposed to be peculiar to Ideation operate in Sensation, There is but one Calculus, but one Logic; though for convenience we divide the one into Arithmetic the calculus of values, and Algebra the calculus of relations; the other into the Logic of Feeling and the Logic of Signs." (George H Lewes "Problems of Life and Mind", 1873)

"Algebra is but written geometry and geometry is but figured algebra." (Sophie Germain, "Mémoire sur la surfaces élastiques", 1880)

"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction." (Gottlob Frege, "The Foundations of Arithmetic", 1884)

"In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in […] that part of logic which deals with the theory of numbers. In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number concept entirely independent of the notions of intuition of space and time, that I consider it an immediate result from the laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

"Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories." (George B Mathews, "Theory of Numbers", 1892)

"I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths." (Gottlob Frege, "Grundgesetze der Arithmetik", 1893) 

"Considering the remarkable elegance, generality, and simplicity of the method [Homer’s Method of finding the numerical values of the roots of an equation], it is not a little surprising that it has not taken a more prominent place in current mathematical textbooks. [...] As a matter of fact, its spirit is purely arithmetical; and its beauty, which can only be appreciated after one has used it in particular cases, is of that indescribably simple kind, which distinguishes the use of position in the decimal notation and the arrangement of the simple rules of arithmetic. It is, in short, one of those things whose invention was the creation of a commonplace." (George Chrystal, "Algebra", 1893)

"The object of all arithmetical operations is to save direct enumeration, by utilizing the results of our old operations of counting. Our endeavor is, having done a sum once, to preserve the answer for future use [...]. Such, too, is the purpose of algebra, which, substituting relations for values, symbolizes and definitely fixes all numerical operations which follow the same rule." (Ernst Mach, "The Science of Mechanics", 1893)

"The science of arithmetic may be called the science of exact limitation of matter and things in space, force, and time." (Francis W Parker, "Talks on Pedagogics", 1894),

"The best review of arithmetic consists in the study of algebra." (Florian Cajori, "Teaching and History of Mathematics in U. S.", 1896)

"Anyone who understands algebraic notation, reads at a glance in an equation results reached arithmetically only with great labour and pains." (A Augustin Cournot, "Researches Into the Mathematical Principles of the Theory of Wealth", 1897)

"In order to comprehend and fully control arithmetical concepts and methods of proof, a high degree of abstraction is necessary, and this condition has at times been charged against arithmetic as a fault. I am of the opinion that all other fields of knowledge require at least an equally high degree of abstraction as mathematics, - provided, that in these fields the foundations are also everywhere examined with the rigour and completeness which is actually necessary." (David Hilbert, "Die Theorie der algebraischen Zahlkorper", 1897)

"Mathematics in its pure form, as arithmetic, algebra, geometry, and the applications of the analytic method, as well as mathematics applied to matter and force, or statics and dynamics, furnishes the peculiar study that gives to us, whether as children or as men, the command of nature in this its quantitative aspect; mathematics furnishes the instrument, the tool of thought, which we wield in this realm." (William T  Harris, "Psychologic Foundations of Education", 1898)

"The laws of algebra, though suggested by arithmetic, do not depend on it. They depend entirely on the conventions by which it is stated that certain modes of grouping the symbols are to be considered as identical. This assigns certain properties to the marks which form the symbols of algebra. The laws regulating the manipulation of algebraic symbols are identical with those of arithmetic. It follows that no algebraic theorem can ever contradict any result which could be arrived at by arithmetic; for the reasoning in both cases merely applies the same general laws to different classes of things. If an algebraic theorem can be interpreted in arithmetic, the corresponding arithmetical theorem is therefore true." (Alfred N Whitehead, "Universal Algebra", 1898)

"The method of arithmetical teaching is perhaps the best understood of any of the methods concerned with elementary studies." (Alexander Bain, "Education as a Science",  1898)

On Arithmetic (1800 - 1849)

 "Number theory is revealed in its entire simplicity and natural beauty when the field of arithmetic is extended to the imaginary numbers" (Carl F Gauss, "Disquisitiones arithmeticae" ["Arithmetical Researches"], 1801)

"The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. […] The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." (Carl F Gauss, "Disquisitiones Arithmeticae" ["Arithmetical Researches"], 1801)

"I am convinced more and more that the necessary truth of our geometry cannot be demonstrated, at least not by the human intellect to the human understanding. Perhaps in another world we may gain other insights into the nature of space which at present are unattainable to us. Until then we must consider geometry as of equal rank not with arithmetic, which is purely a priori, but with mechanics." (Carl F Gauss, [Letter to Olbers] 1817)

"It is characteristic of higher arithmetic that many of its most beautiful theorems can be discovered by induction with the greatest of ease but have proofs that lie anywhere but near at hand and are often found only after many fruitless investigations with the aid of deep analysis and lucky combinations." (Carl F Gauss, 1817)

"Our general arithmetic, so far surpassing in extent the geometry of the ancients, is entirely the creation of modern times. Starting originally from the notion of absolute integers, it has gradually enlarged its domain. To integers have been added fractions, to rational quantities the irrational, to positive the negative .and to the real the imaginary. This advance, however, has always been made at first with timorous and hesitating step. The early algebraists called the negative roots of equations false roots, and these are indeed so when the problem to which they relate has been stated in such a form that the character of the quantity sought allows of no opposite. But just as in general arithmetic no one would hesitate to admit fractions, although there are so many countable things where a fraction has no meaning, so we ought not to deny to, negative numbers the rights accorded to positive simply because innumerable things allow no opposite. The reality of negative numbers is sufficiently justified since in innumerable other cases they find an adequate substratum. This has long been admitted, but the imaginary quantities - formerly and occasionally now, though improperly, called impossible-as opposed to real quantities are still rather tolerated than fully naturalized, and appear more like an empty play upon symbols to which a thinkable substratum is denied unhesitatingly by those who would not depreciate the rich contribution which this play upon symbols has made to the treasure of the relations of real quantities." (Carl F Gauss, "Theoria residuorum biquadraticorum, Commentatio secunda", Göttingische gelehrte Anzeigen, 1831)

"What would life be like without arithmetic, but a scene of horrors?" (Sydney Smith, "Letter 692]  [in "The Letters of Sydney Smith" Vol.2] 1835)

"The science of algebra, independently of any of its uses, has all the advantages which belong to mathematics in general as an object of study, and which it is not necessary to enumerate. Viewed either as a science of quantity, or as a language of symbols, it may be made of the greatest service to those who are sufficiently acquainted with arithmetic, and who have sufficient power of comprehension to enter fairly upon its difficulties." (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)

"Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space." (James J Sylvester, "A Probationary Lecture on Geometry", 1844)

"The Higher Arithmetic presents us with an inexhaustible storehouse of interesting truths - of truths, too, which are not isolated but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions which have the stamp of simplicity upon them but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process while the simpler methods of proof long remain hidden from us." (Carl F Gauss, [introduction to Gotthold Eisenstein’s "Mathematische Abhandlungen"] 1847)

"Geometrical reasoning, and arithmetical process, have each its own office: to mix the two in elementary instruction, is injurious to the proper acquisition of both." (Augustus De Morgan, "Trigonometry and Double Algebra", 1849)

On Arithmetic (1850 - 1874)

"[Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions. […] We will briefly say that Algebra is the Calculus of functions, and Arithmetic is the Calculus of Values." (Auguste Comte, "Philosophy of Mathematics", 1851)

"For what is the theory of determinants? It is an algebra upon algebra; a calculus which enables us to combine and foretell the results of algebraical operations, in the same way as algebra itself enable us to dispense with the performance of the special operations of arithmetic. All analysis must ultimately clothe itself under this form." (James J Sylvester, "On the Relation Between the Minor Determinants of Linearly Equivalent", [in "The Collected Mathematical Papers of James Joseph Sylvester" Vol. I)  1851)

"That arithmetic is the basest of all mental activities is proved by the fact that it is the only one that can be accomplished by means of a machine." (Arthur Schopenhauer, "Psychological Observations" [in  "Parerga and Paralipomena"] 1851)

"The ideas which these sciences, Geometry, Theoretical Arithmetic and Algebra involve extend to all objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics." (William Whewell, "The Philosophy of the Inductive Sciences" , 1858)

"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", 1858)

"It is better to teach the child arithmetic and Latin grammar than rhetoric and moral philosophy, because they require exactitude of performance it is made certain that the lesson is mastered, and that power of performance is worth more than knowledge." (Ralph W Emerson, "Lecture on Education", 1853)

"Let him [the author] be permitted also in all humility to add ... that in consequence of the large arrears of algebraical and arithmetical speculations waiting in his mind their turn to be called into outward existence, he is driven to the alternative of leaving the fruits of his meditations to perish (as has been the fate of too many foregone theories, the still-born progeny of his brain, now forever resolved back again into the primordial matter of thought), or venturing to produce from time to time such imperfect sketches as the present, calculated to evoke the mental co-operation of his readers, in whom the algebraical instinct has been to some extent developed, rather than to satisfy the strict demands of rigorously systematic exposition." (James J Sylvester, Philosophic Magazine, 1863) 

"Arithmetic, like the sea, is an undulation without any possible end." (Victor Hugo, "The Toilers of the Sea", 1866)

"[Arithmetic] is another of the great master-keys of life. With it the astronomer opens the depths of the heavens; the engineer, the gates of the mountains; the navigator, the pathways of the deep. The skillful arrangement, the rapid handling of figures, is a perfect magician’s wand." (Edward Everett, "Orations and Speeches", 1868)

"We do not listen with the best regard to the verses of a man who is only a poet, nor to his problems if he is only an algebraist; but if a man is at once acquainted with the geometric foundation of things and with their festal splendor, his poetry is exact and his arithmetic music." (Ralph W Emerson, "Society and Solitude", 1870)

On Arithmetic (1300 - 1799)

"Music submits itself to principles which it derives from arithmetic." (St. Thomas d'Aquin," Summa theologica", 1485)

"[...] if the worth of the arts were measured by the matter with which they deal, this art - which some call astronomy, others astrology, and many of the ancients the consummation of mathematics - would be by far the most outstanding. This art which is as it were the head of all the liberal arts and the one most worthy of a free man leans upon nearly all the other branches of mathematics. Arithmetic, geometry, optics, geodesy, mechanics, and whatever others, all offer themselves in its service." (Nicolaus Copernicus, "On the Revolutions of the Heavenly Spheres", 1543)

"Mathematic is either Pure or Mixed: To Pure Mathematic belong those sciences which handle Quantity entirely severed from matter and from axioms of natural philosophy. These are two, Geometry and Arithmetic; the one handling quantity continued, the other dissevered. [...] Mixed Mathematic has for its subject some axioms and parts of natural philosophy, and considers quantity in so far as it assists to explain, demonstrate and actuate these." (Francis Bacon, "De Augmentis", 1623)

"Music is a hidden arithmetic exercise of the soul, which does not know that it is counting."
["Musica est exercitium arithmeticae occultum nescientis se numerare animi."] (Gottfried Leibniz, [Letter to Christian Goldbach], 1712)

"As arithmetic and algebra are sciences of great clearness, certainty, and extent, which are immediately conversant about signs, upon the skillful use whereof they entirely depend, so a little attention to them may possibly help us to judge of the progress of the mind in other sciences, which, though differing in nature, design, and object, may yet agree in the general methods of proof and inquiry." (George Berkeley, "Alciphorn: or, the Minute Philosopher", 1732)

"Now as to what pertains to these Surd numbers (which, as it were by way of reproach and calumny, having no merit of their own are also styled Irrational, Irregular, and Inexplicable) they are by many denied to be numbers properly speaking, and are wont to be banished from arithmetic to another Science, (which yet is no science) viz. algebra." (Isaac Barrow, "Mathematical Lectures", 1734)

"Arithmetic and geometry, those wings on which the astronomer soars as high as heaven." (Robert Boyle, "Usefulness of Mathematics to Natural Philosophy", 1744)

"[...] the ideas which these sciences [Geometry, Theoretical Arithmetic and Algebra] involve extend to all objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics." (William Whewell, "The Philosophy of the Inductive Sciences Founded Upon Their History" Vol. 1, 1747)

"Algebra is a general Method of Computation by certain Signs and Symbols which have been contrived for this Purpose, and found convenient. It is called an Universal Arithmetic, and proceeds by Operations and Rules similar to those in Common Arithmetic, founded upon the same Principles." (Colin Maclaurin, "A Treatise on Algebra", 1748)

"The operations of symbolic arithmetick seem to me to afford men one of the clearest exercises of reason that I ever yet met with, nothing being there to be performed without strict and watchful ratiocination, and the whole method and progress of that appearing at once upon the paper, when the operation is finished, and affording the analyst a lasting and, as it were, visible ratiocination." (Robert Boyle, "The Works of the Honourable Robert Boyle" Vol. III, 1772)

"The scientific part of Arithmetic and Geometry would be of more use for regulating the thoughts and opinions of men than all the great advantage which Society receives from the general application of them: and this use cannot be spread through the Society by the practice; for the Practitioners, however dextrous, have no more knowledge of the Science than the very instruments with which they work. They have taken up the Rules as they found them delivered down to them by scientific men, without the least inquiry after the Principles from which they are derived: and the more accurate the Rules, the less occasion there is for inquiring after the Principles, and consequently, the more difficult it is to make them turn their attention to the First Principles; and, therefore, a Nation ought to have both Scientific and Practical Mathematicians." (James Williamson, "Elements of Euclid with Dissertations", 1781)

"[…] direction is not a subject for algebra except in so far as it can be changed by algebraic operations. But since these cannot change direction (at least, as commonly explained) except to its opposite, that is, from positive to negative, or vice versa, these are the only directions it should be possible to designate. […] It is not an unreasonable demand that operations used in geometry be taken in a wider meaning than that given to them in arithmetic. " (Casper Wessel, „On the Analytical Representation of Direction", 1787)

"An ancient writer said that arithmetic and geometry are the wings of mathematics; I believe one can say without speaking metaphorically that these two sciences are the foundation and essence of all the sciences which deal with quantity. Not only are they the foundation, they are also, as it were, the capstones; for, whenever a result has been arrived at, in order to use that result, it is necessary to translate it into numbers or into lines; to translate it into numbers requires the aid of arithmetic, to translate it into lines necessitates the use of geometry." (Joseph-Louis de Lagrange, "Leçons élémentaires sur les mathématiques", 1795)

"So long as algebra and geometry proceeded separately their progress was slow and their application limited, but when these two sciences were united, they mutually strengthened each other, and marched together at a rapid pace toward perfection." (Joseph-Louis de Lagrange, "Leçons élémentaires sur les mathématiques", 1795)

On Arithmetic (-1299)

 "[Arithmetic] has a very great and elevating effect, compelling the soul to reason about abstract numbers, and rebelling against the introduction of visible or tangible objects into the argument." (Plato, "The Republic", cca. 375 BC)

"[...] arithmetic is a kind of knowledge in which the best natures should be trained, and which must not be given up." (Plato, "The Republic", cca. 375 BC)

"[...] the art of calculation (logistika) and arithmetic are both concerned with number; those who have a natural gift for calculating have, generally speaking, a talent for learning of all kinds, and even those who are slow are, by practice in it, made smarter. But the art of calculation is only preparatory to the true science; those who are to govern the city are to get a grasp of logistilca, not in the popular sense with a view to use in trade, but only for the purpose of knowledge, until they are able to con- template the nature of number in itself by thought alone." (Plato, "The Republic", cca. 375 BC)

"[...] those who have a natural talent for calculation are generally quick at every other kind of knowledge; and even the dull, if they have had an arithmetical training, although they may derive no other advantage from it, always become much quicker than they would otherwise have been [...]" (Plato, "The Republic", cca. 375 BC)

"Can we deny that a warrior should have a knowledge of arithmetic?" (Plato, "The Republic", cca. 375 BC)

"No single instrument of youthful education has such mighty power, both as regards domestic economy and politics, and in the arts, as the study of arithmetic. Above all, arithmetic stirs up him who is by nature sleepy and dull, and makes him quick to learn, retentive, shrewd, and aided by art divine he makes progress quite beyond his natural powers." (Plato, "Laws", cca. 360 BC)

"[...] if arithmetic, mensuration, and weighing be taken away from any art, that which remains will not be much." (Plato, "Philebus", cca. 360 - 347 BC)

"The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantity as such, music the relations between quantities, geometry magnitude at rest, spherics magnitude inherently moving." (Diadochus Proclus)

"Mathematical science […] has these divisions: arithmetic, music, geometry, astronomy. Arithmetic is the discipline of absolute numerable quantity. Music is the discipline which treats of numbers in their relation to those things which are found in sound." (Cassiodorus, cca. 6th century)

On Arithmetic (1900 - 1924)

"All knowledge must be recognition, on pain of being mere delusion; Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians…. Whatever can be thought of has being and its [arithmetic] being is a precondition, not a result, of its being thought of." (Bertrand A W Russell, "Is Position in Space and Time Absolute or Relative", Mind Vol. X, 1901)

"Great numbers are not counted correctly to a unit, they are estimated; and we might perhaps point to this as a division between arithmetic and statistics, that whereas arithmetic attains exactness, statistics deals with estimates, sometimes very accurate, and very often sufficiently so for their purpose, but never mathematically exact." (Arthur L Bowley, "Elements of Statistics", 1901)

"I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetical act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding [...] (Richard Dedekind, "Essays on the Theory of Numbers", 1901)

"Symbolism is useful because it makes things difficult. Now in the beginning everything is self-evident, and it is hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we must invent a new and difficult symbolism in which nothing is obvious. [...] Thus the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions." Bertrand Russell, International Monthly, 1901)

"Arithmetical symbols are written diagrams and geometrical figures are graphic formulas." (David Hilbert, Bulletin of the American Mathematical Society Mathematical Problems Vol. 8, 1902)

"Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians." (Bertrand Russell, "The Principles of Mathematics", 1903)

"For we do not listen with the best regard to the verses of a man who is only a poet, nor to his problems if he is only an algebraist; but if a man is at once acquainted with the geometric foundations of things and with their festal splendor, his poetry is exact and his arithmetic musical." (Ralph Waldo Emerson, "Works and Days" [in "Society and solitude: Twelve chapters", 1903])

"We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new; not from it alone can any science issue. In one sense these philosophers are right; to make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition. But how many different ideas are hidden under thi4s same word?" (Henri Poincaré , "Intuition and Logic in Mathematics", 1905)

"Arithmetic does not present to us that feeling of continuity which is such a precious guide; each whole number is separate from the next of its kind and has in a sense individuality; each in a manner is an exception and that is why general theorems are rare in the theory of numbers; and that is why those theorems which may exist are more hidden and longer escape those who are searching for them." (Henri Poincaré, "Annual Report of the Board of Regents of the Smithsonian Institution", 1909)

"The science of arithmetic may be called the science of exact limitation of matter and things in space, force, and time." (Francis W Parker, "Talks on Pedagogics: An outline of the theory of concentration", 1909)

"The student of arithmetic who has mastered the first four rules of his art, and successfully striven with money sums and fractions, finds himself confronted by an unbroken expanse of questions known as problems." (Stephen Leacock, "Literary Lapses", 1911)

"Geometry formerly was the chief borrower from arithmetic and algebra, but it has since repaid its obligation with abundant usury; and if I were asked to name, in one word, the pole-star round which the mathematical firmament revolves, the central idea which pervades as a hidden spirit the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this!" (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul." (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions." (George H Howison, Journal of Speculative Philosophy Vol. 5, 1914)

"The way to enable a student to apprehend the instrumental value of arithmetic is not to lecture him on the benefit it will be to him in some remote and uncertain future, but to let him discover that success in something he is interested in doing depends on ability to use numbers." (John Dewey, "Democracy and Education: An Introduction to the Philosophy of Education", 1916)