On Approximation (1925-1949)

"Science does not aim at establishing immutable truths and eternal dogmas; its aim is to approach the truth by successive approximations, without claiming that at any stage final and complete accuracy has been achieved." (Bertrand Russell, "The ABC of Relativity", 1925)

"The scientist is a practical man and his are practical aims. He does not seek the ultimate but the proximate. He does not speak of the last analysis but rather of the next approximation. […] On the whole, he is satisfied with his work, for while science may never be wholly right it certainly is never wholly wrong; and it seems to be improving from decade to decade." (Gilbert N Lewis, "The Anatomy of Science", 1926)

"The different premises will possibly allow equally good explanations with respect to the imprecise nature of our sensory perception, because the sensory perception is, in fact, not concerned with issues of precision mathematics but of approximation mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The mistake from which todays’ science suffers is that the theoreticians are concerned too unilaterally with precision mathematics, while the practitioners use a sort of approximate mathematics, without being in touch with precision mathematics through which they could reach a real approximation mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"The weak point in all such reflections is that they depend on an arbitrary preference of certain ideas and concepts of precision mathematics, while observations in nature always have only limited precision and can be related in very different manners to topics of precision mathematics. It is more generally questionable whether we should be looking for the essence of a correct explanation of nature on the basis of precision mathematics, and whether we could ever go beyond a skillful use of approximation mathematics." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"Although this may seem a paradox, all exact science is dominated by the idea of approximation. When a man tells you that he knows the exact truth about anything, you are safe in inferring that he is an inexact man." (Bertrand Russell, "The Scientific Outlook", 1931)

"Is the density anywhere near that corresponding to the static universe, or is it so small that we can consider the empty universe as a good approximation?" (Willem de Sitter, "Kosmos", 1932)

"The unsolved problems of Nature have a distinctive fascination, though they still far outnumber those which have even approximately been resolved."(Henry N Russell, "The Solar System and Its Origin", 1935)

"The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration of semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form." (Leonid V Kantorovich, "On one class of functional equations", 1936)

"[…] reality is a system, completely ordered and fully intelligible, with which thought in its advance is more and more identifying itself. We may look at the growth of knowledge […] as an attempt by our mind to return to union with things as they are in their ordered wholeness. […] and if we take this view, our notion of truth is marked out for us. Truth is the approximation of thought to reality […] Its measure is the distance thought has travelled […] toward that intelligible system […] The degree of truth of a particular proposition is to be judged in the first instance by its coherence with experience as a whole, ultimately by its coherence with that further whole, all comprehensive and fully articulated, in which thought can come to rest." (Brand Blanshard, "The Nature of Thought" Vol. II, 1939)

"Nature does not consist entirely, or even largely, of problems designed by a Grand Examiner to come out neatly in finite terms, and whatever subject we tackle the first need is to overcome timidity about approximating." (Sir Harold Jeffreys & Bertha S Jeffreys, "Methods of Mathematical Physics", 1946)

"I think that it is a relatively good approximation to truth - which is much too complicated to allow anything but approximations - that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is […] governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much ‘abstract’ inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas." (John von Neumann,  "The Mathematician", The Works of the Mind Vol. I (1), 1947)

"The principle of bounded rationality [is] the capacity of the human mind for formulating and solving complex problems is very small compared with the size of the problems whose solution is required for objectively rational behavior in the real world - or even for a reasonable approximation to such objective rationality." (Herbert A Simon, "Administrative Behavior", 1947)

Previous Post <<||>> Next Post

On Approximation (1900-1924)

"We see that experience plays an indispensable role in the genesis of geometry; but it would be an error thence to conclude that geometry is, even in part, an experimental science. If it were experimental it would be only approximative and provisional. And what rough approximation!" (Henri Poincaré, "Science and Hypothesis", 1901)

"The laws of thermodynamics, as empirically determined, express the approximate and probable behavior of systems of a great number of particles, or, more precisely, they express the laws of mechanics for such systems as they appear to beings who have not the fineness of perception to enable them to appreciate quantities of the order of magnitude of those which relate to single particles, and who cannot repeat their experiments often enough to obtain any but the most probable results." (Josiah W Gibbs, "Elementary Principles in Statistical Mechanics", 1902)

"So completely is nature mathematical that some of the more exact natural sciences, in particular astronomy and physics, are in their theoretic phases largely mathematical in character, while other sciences which have hitherto been compelled by the complexity of their phenomena and the inexactitude of their data to remain descriptive and empirical, are developing towards the mathematical ideal, proceeding upon the fundamental assumption that mathematical relations exist between the forces and the phenomena, and that nothing short, of the discovery and formulations of these relations would constitute definitive knowledge of the subject. Progress is measured by the closeness of the approximation to this ideal formulation." (Jacob W A Young, "The Teaching of Mathematics", 1907)

"An exceedingly small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation 'approximately'. If that enabled us to predict the succeeding situation with 'the same approximation', that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon. (Jules H Poincaré, "Science and Method", 1908)

"The ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experience, and deduces from these axioms the rigid mathematical conclusions. In applying this method it must not be forgotten that the mathematical developments transcending the limits of exactness of the science are of no practical value. It follows that a large portion of abstract mathematics remains without finding any practical application, the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in the science. Thus, while the astronomer can put to use a wide range of mathematical theory, the chemist is only just beginning to apply the first derivative, i.e. the rate of change at which certain processes are going on; for second derivatives he does not seem to have found any use as yet." (Felix Klein, "Lectures on Mathematics", 1911)

"Much of the skill of the true mathematical physicist and of the mathematical astronomer consists in the power of adapting methods and results carried out on an exact mathematical basis to obtain approximations sufficient for the purposes of physical measurements. It might perhaps be thought that a scheme of Mathematics on a frankly approximative basis would be sufficient for all the practical purposes of application in Physics, Engineering Science, and Astronomy, and no doubt it would be possible to develop, to some extent at least, a species of Mathematics on these lines. Such a system would, however, involve an intolerable awkwardness and prolixity in the statements of results, especially in view of the fact that the degree of approximation necessary for various purposes is very different, and thus that unassigned grades of approximation would have to be provided for. Moreover, the mathematician working on these lines would be cut off from the chief sources of inspiration, the ideals of exactitude and logical rigour, as well as from one of his most indispensable guides to discovery, symmetry, and permanence of mathematical form. The history of the actual movements of mathematical thought through the centuries shows that these ideals are the very life-blood of the science, and warrants the conclusion that a constant striving toward their attainment is an absolutely essential condition of vigorous growth. These ideals have their roots in irresistible impulses and deep-seated needs of the human mind, manifested in its efforts to introduce intelligibility in certain great domains of the world of thought." (Ernest W Hobson, [address] 1910)

"The ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experience, and deduces from these axioms the rigid mathematical conclusions. In applying this method it must not be forgotten that the mathematical developments transcending the limits of exactness of the science are of no practical value. It follows that a large portion of abstract mathematics remains without finding any practical application, the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in the science. Thus, while the astronomer can put to use a wide range of mathematical theory, the chemist is only just beginning to apply the first derivative, i. e. the rate of change at which certain processes are going on; for second derivatives he does not seem to have found any use as yet." (Felix Klein, "Lectures on Mathematics", 1911)

"The objects of abstract Geometry possess in absolute precision properties which are only approximately realized in the corresponding objects of physical Geometry." (Ernest W Hobson, "Squaring the Circle", 1913)

"It is well to notice in this connection [the mutual relations between the results of counting and measuring] that a natural law, in the statement of which measurable magnitudes occur, can only be understood to hold in nature with a certain degree of approximation; indeed natural laws as a rule are not proof against sufficient refinement of the measuring tools." (Luitzen E J Brouwer, "Intuitionism and Formalism", Bulletin of the American Mathematical Society, Vol. 20, 1913)

"[…] as the sciences have developed further, the notion has gained ground that most, perhaps all, of our laws are only approximations." (William James, "Pragmatism: A New Name for Some Old Ways of Thinking", 1914)

"Human knowledge is not (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed (transformed one-sidedly) into an independent, complete, straight line [...]" (Vladimir I Lenin, "On the Question of Dialectics", 1915)

"It would be a mistake to suppose that a science consists entirely of strictly proved theses, and it would be unjust to require this. […] Science has only a few apodeictic propositions in its catechism: the rest are assertions promoted by it to some particular degree of probability. It is actually a sign of a scientific mode of thought to find satisfaction in these approximations to certainty and to be able to pursue constructive work further in spite of the absence of final confirmation." (Sigmund Freud, "Introductory Lectures on Psycho-Analysis", 1916)

"It is the nature of a real thing to be inexhaustible in content; we can get an ever deeper insight into this content by the continual addition of new experiences, partly in apparent contradiction, by bringing them into harmony with one another. In this interpretation, things of the real world are approximate ideas. From this arises the empirical character of all our knowledge of reality." (Hermann Weyl, "Space-Time-Matter", 1918)

"It can, you see, be said, with the same approximation to truth, that the whole of science, including mathematics, consists in the study of transformations or in the study of relations." (Cassius J Keyser. "Mathematical Philosophy: A Study of Fate and Freedom", 1922)

Previous Post <<||>> Next Post

On Approximation (-1899)

"Those who devised the eccentrics seen thereby in large measure to have solved the problem of apparent motions with approximate calculations. But meanwhile they introduced a good many ideas which apparently contradict the first principles of uniform motion. Nor could they elicit or deduce from the eccentrics the principal consideration, that is, the structure of the universe and the true symmetry of its parts."  (Nicolaus Copernicus, "De revolutionibus orbium coelestium", 1543)

"Systems in physical science […] are no more than appropriate instruments to aid the weakness of our organs: they are, properly speaking, approximate methods which put us on the path to the solution of the problem; these are the hypotheses which, successively modified, corrected, and changed in proportion as they are found false, should lead us infallibly one day, by a process of exclusion, to the knowledge of the true laws of nature." (Antoine L Lavoisier, "Mémoires de l’Académie Royale des Sciences", 1777)

"[It] may be laid down as a general rule that, if the result of a long series of precise observations approximates a simple relation so closely that the remaining difference is undetectable by observation and may be attributed to the errors to which they are liable, then this relation is probably that of nature." (Pierre-Simon Laplace, "Mémoire sur les Inégalites Séculaires des Planètes et des Satellites", 1787)

"Force is not a fact at all, but an idea embodying what is approximately the fact." (William K Clifford, "The Common Sense of the Exact Sciences", 1823)

"The first steps in the path of discovery, and the first approximate measures, are those which add most to the existing knowledge of mankind." (Charles Babbage, "Reflections on the Decline of Science in England, And on Some of Its Causes", 1830)

"Experimental science hardly ever affords us more than approximations to truth; and whenever many agents are concerned we are in great danger of being mistaken." (Sir Humphry Davy, cca. 1836)

"With certain limited exceptions, the laws of physical science are positive and absolute, both in their aggregate, and in their elements, - in their sum, and in their details; but the ascertainable laws of the science of life are approximative only, and not absolute." (Elisha Bartlett, "An Essay on the Philosophy of Medical Science", 1844)

"Science gains from it [the pendulum] more than one can expect. With its huge dimensions, the apparatus presents qualities that one would try in vain to communicate by constructing it on a small [scale], no matter how carefully. Already the regularity of its motion promises the most conclusive results. One collects numbers that, compared with the predictions of theory, permit one to appreciate how far the true pendulum approximates or differs from the abstract system called 'the simple pendulum'." (Jean-Bernard-Léon Foucault, "Demonstration Experimentale du Movement de Rotation de la Terre", 1851)

"But in practical science, the question is - What are we to do? - a question which involves the necessity for the immediate adoption of some rule of working. In doubtful cases, we cannot allow our machines and our works of improvement to wait for the advancement of science; and if existing data are insufficient to give an exact solution of the question, that approximate solution must be acted upon which the best data attainable show to be the most probable. A prompt and sound judgment in cases of this kind is one of the characteristics of a Practical Man in the right sense of that term." (W J Macquorn Rankine, "On the Harmony of Theory and Practice in Mechanics", 1856)

"We live in a system of approximations. Every end is prospective of some other end, which is also temporary; a round and final success nowhere. We are encamped in nature, not domesticated." (Ralph W Emerson, "Essays", 1865)

"Man’s mind cannot grasp the causes of events in their completeness, but the desire to find those causes is implanted in man’s soul. And without considering the multiplicity and complexity of the conditions any one of which taken separately may seem to be the cause, he snatches at the first approximation to a cause that seems to him intelligible and says: ‘This is the cause!’" (Leo Tolstoy, "War and Peace", 1867)

"[...] very often the laws derived by physicists from a large number of observations are not rigorous, but approximate." (Augustin-Louis Cauchy, "Sept leçons de physique" ["Seven lessons of Physics"], Bureau du Journal Les Mondes, 1868)

"Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming empirical evidence that this hypothesis is very approximately correct, even if not rigidly true." (Bertrand Russell, "Foundations of Geometry", 1897)

"Science is about finding ever better approximations rather than pretending you have already found ultimate truth." (Friedrich Nietzsche)

Previous Post <<||>> Next Post

Music and Mathematics III

“Music is an order of mystic, sensuous mathematics. A sounding mirror, an aural mode of motion, it addresses itself on the formal side to the intellect, in its content of expression it appeals to the emotions.” (James Huneker, “Chopin: The Man and His Music”, 1900)

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” (Bertrand Russell, 'The Study of Mathematics”, 1902)

“It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true.” (Bertrand Russell, “Autobiography”, 1967)

“The syntax and the grammar of the language of music are not capricious; they are dictated by the texture and organization of the deep levels of the mind, so with mathematics.”( H E Huntley, “The Divine Proportion”, 1970)

“Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience.” (Keith Devlin,”Mathematics: The Science of Patterns”, 1994)

“Music and math together satisfied a sort of abstract 'appetite', a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical.” (Edward Rothstein, “Emblems of Mind: The Inner Life of Music and Mathematics”, 1995)

“Like musicians who can read and write complicated scores in a world without sounds, for us mathematics is a source of delight, excitement, and even controversy which are hard to share with non-mathematicians. In our small micro-cosmos we should ever seek the right balance between competition and solidarity, criticism and empathy, exclusion and inclusion.” (Gil Kalai, "Combinatorics with a Geometric Flavor: Some Examples", 2000)

“Mathematics can be as effortless as humming a tune, if you know the tune. But our culture does not prepare us for appreciation of mathematics as it does for appreciation of music. Though we start hearing music very early in life, the same cannot be said of mathematics, even though the two subjects are twins. This is a shame; to know music without knowing its mathematics is like hearing a melody without its accompaniment.” (Gareth Loy, “Musimathics: The Mathematical Foundations of Music” Vol. 1, 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)

See also:
Music and Mathematics
Music and Mathematics II
The Music of Numbers

Music and Mathematics II

“And so they have handed down to us clear knowledge of the speed of the heavenly bodies and their risings and settings, of geometry, numbers and, not least, of the science of music. For these sciences seem to be related.” (Archytas of Tarentym, 4th c. BC)

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

“Musicke I here call that Science, which of the Greeks is called Harmonie. Musicke is a Mathematical Science, which teacheth, by sense and reason, perfectly to judge, and order the diversities of soundes hye and low.” (John Dee, 1570)

“For many parts of Nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated to use with sufficient dexterity, without the aid and intervention of Mathematic: of which sort are Perspective, Music, Astronomy, cosmography, Architecture, Machinery, and some others.” (Sir Francis Bacon, De Augmentis, Bk. 3 [The Advancement of Learning], 1605)

“Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures.” (John Arbuthnot, “An Essay on the Usefulness of Mathematical Learning”, 1701)

“Music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics. Notwithstanding all the experience I may have acquired in music from being associated with it for so long, I must confess that only with the aid of mathematics did my ideas become clear and did light replace a certain obscurity of which I was unaware before.” (Jean-Philippe Rameau, “Treatise on Harmony reduced to its natural principles”, 1722)

“Mathematics and music, the most sharply contrasted fields of scientific activity which can be found, and yet related, supporting each other, as if to show forth the secret connection which ties together all the activities of our mind, and which leads us to surmise that the manifestations of the artist's genius are but the unconscious expressions of a mysteriously acting rationality.” (Hermann von Helmholtz, “Vorträge und Reden”, Bd. 1, 1884)

“I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra) music.” (James J Sylvester, [Presidential Address to the British Association] 1869)

“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)

“The mind of man may be compared to a musical instrument with a certain range of notes, beyond which in both directions we have an infinitude of silence. The phenomena of matter and force lie within our intellectual range, and as far as they reach we will at all hazards push our inquiries. But behind, and above, and around all, the real mystery of this universe [Who made it all?] lies unsolved, and, as far as we are concerned, is incapable of solution.” (John Tyndall, “Fragments of Science for Unscientific People”, 1871)

See also:
Music and Mathematics
Music and Mathematics III
The Music of Numbers 

The Music of Numbers

“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)

“Music is fashioned wholly in the likeness of numbers. […] Whatever is delightful in song is brought about by number. Sounds pass quickly away, but numbers, which are obscured by the corporeal element in sounds and movements, remain.“ (Anon, "Scholia Enchiriadis", cca. 900)

“Sound is generated by motion, since it belongs to the class of successive things. For this reason, while it exists when it is made, it no longer exists once it has been made. […] All music, especially mensurable music, is founded in perfection, combining in itself number and sound." (Jean de Muris, “Ars novae musicae”, 1319)

“The length of strings is not the direct and immediate reason behind the forms [ratios] of musical intervals, nor is their tension, nor their thickness, but rather, the ratios of the numbers of vibrations and impacts of air waves that go to strike our eardrum.” (Galileo Galilei, "Two New Sciences", 1638)

“We must distinguish carefully the ratios that our ears really perceive from those that the sounds expressed as numbers include.“ (Leonhard Euler, "Conjecture into the reasons for some dissonances generally heard in music", 1760)

“Music is like geometric figures and numbers, which are the universal forms of all possible objects of experience.” (Friedrich Nietzsche, “Birth of Tragedy”, 1872)

“In addition to this it [mathematics] provides its disciples with pleasures similar to painting and music. They admire the delicate harmony of the numbers and the forms; they marvel when a new discovery opens up to them an unexpected vista; and does the joy that they feel not have an aesthetic character even if the senses are not involved at all? […] For this reason I do not hesitate to say that mathematics deserves to be cultivated for its own sake, and I mean the theories which cannot be applied to physics just as much as the others.” (Henri Poincaré, 1897)

“Architecture is geometry made visible in the same sense that music is number made audible.” (Claude F Bragdon, “The Beautiful Necessity: Seven Essays on Theosophy and Architecture”, 1910)

“Through and through the world is infected with quantity: To talk sense is to talk quantities. It is not use saying the nation is large - How large? It is no use saying the radium is scarce - How scarce? You cannot evade quantity. You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves.” (Alfred N Whitehead, “The Aims of Education and Other Essays”, 1917)

“It is not surprising that the greatest mathematicians have again and again appealed to the arts in order to find some analogy to their own work. They have indeed found it in the most varied arts, in poetry, in painting, and in sculpture, although it would certainly seem that it is in music, the most abstract of all the arts, the art of number and of time, that we find the closest analogy.” (Havelock Ellis, “The Dance of Life”, 1923)

See also:
Music and Mathematics
Music and Mathematics II
Music and Mathematics III

On Numbers: Zero

"When sunya [zero] is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by sunya becomes sunya." (Brahmagupta, 628)

"Every number arises from One, and this in turn from the Zero. In this lies a great and sacred mystery - in hoc magnum latet sacramentum: HE is symbolized by that which has neither beginning nor end; and just as the zero neither increases nor diminishes another number to which it is added or from which it is subtracted so does HE neither wax nor wane. And as the zero multiplies by ten the number behind which it is placed so does HE increase not tenfold, but a thousand fold - nay, to speak more correctly, HE creates all out of nothing, preserves and rules it  - omnia ex nichillo creat, conservat atque gubernat." ("Salem Codex", 12th century)

"The whole science of mathematics depends upon zero. Zero alone determines the value in mathematics. Zero is in itself nothing. Mathematics is based upon nothing, and, consequently, arises out of nothing." (Lorenz Oken, "Elements of Physiophilosophy", 1847)

"A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined." (George B Mathews, "Theory of Numbers", 1892)

"The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought." (Alfred N Whitehead, "An Introduction to Mathematics", 1911)

"In the history of culture the discovery of zero will always stand out as one of the greatest achievements of the human race." (Tobias Danzig, "Number: The Language of Science", 1930)

"The zero is the most important digit. It is a stroke of genius, to make something out of noting by giving it a name and inventing a symbol for it." (B L van der Waerden, "Science Awakening", 1962)

"[…] it took men about five thousand years, counting from the beginning of number symbols, to think of a symbol for nothing." (Isaac Asimov, "Of Time and Space and Other Things", 1965)

"[zero is] A mysterious number, which started life as a space on a counting board, turned into a written notice that a space was present, that is to say that something was absent, then confused medieval mathematicians who could not decide whether it was really a number or not, and achieved its highest status in modern abstract mathematics in which numbers are defined anyway only by their properties, and the properties of zero are at least as clear, and rather more substantial, than those of many other numbers." (David Wells, "The Penguin Dictionary of Curious and Interesting Numbers", 1986)

"Clearly, however, a zero probability is not the same thing as an impossibility; […] In systems that are now called chaotic, most initial states are followed by nonperiodic behavior, and only a special few lead to periodicity. […] In limited chaos, encountering nonperiodic behavior is analogous to striking a point on the diagonal of the square; although it is possible, its probability is zero. In full chaos, the probability of encountering periodic behavior is zero." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Zero is behind all of the big puzzles in physics. The infinite density of the black hole is a division by zero. The big bang creation from the void is a division by zero. The infinite energy of the vacuum is a division by zero. Yet dividing by zero destroys the fabric of mathematics and the framework of logic - and threatens to undermine the very basis of science. […] The universe begins and ends with zero." (Charles Seife ."Zero, the Biography of a Dangerous Idea", 2000)

"Mathematics is an activity about activity. It hasn't ended - has hardly in fact begun, although the polish of its works might give them the look of monuments, and a history of zero mark it as complete. But zero stands not for the closing of a ring: it is rather a gateway." (Robert Kaplan, "The Nothing that Is: A Natural History of Zero", 2000)

"Zero was at the heart of the battle between East and West. Zero was at the center of the struggle between religion and science. Zero became the language of nature and the most important tool in mathematics. And the most profound problems in physics - the dark core of a black hole and the brilliant flash of the big bang - are struggles to defeat zero." (Charles Seife ."Zero, the Biography of a Dangerous Idea", 2000)

"The concept of zero is so familiar that it takes a great deal of effort to recapture how mysterious, subtle, and contradictory the idea really is." (William Byers, "How Mathematicians Think", 2007)

"Zero is the mathematically defined numerical function of nothingness. It is used not for an evasion but for an apprehension of reality. Zero is by far the most interesting number among all the others: It is a symbol for what is not there. It is an emptiness that increases any number it's added to. Zero is an inexhaustible and indispensable paradox. Zero is the only number which can be divided by every other number. Zero is also only number which can divide no other number. It seems zero is also the most debated number in mathematics. We know that mathematicians are involved in heated philosophical and logical discussions around the issues of zero: Can we divide a number by zero? Is the result of this division infinity or not? Is zero a positive or a negative number? Is it even or is it odd?" (Fahri Karakas, "Reflections on zero and zero-centered spirituality in organizations", 2008)

"However, in contrast to one, which is singularly straightforward, zero is secretly peculiar. If you pierce the obscuring haze of familiarity around it, you’ll see that it is a quantitative entity that, curiously, is really the absence of quantity. It took people a long time to get their minds around that." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Zero is not a point of non-existence. Zero is always a balance point of existents. The human understanding of 'zero' must undergo the most radical of all transformations. Most people, especially scientists, associate it with absolute nothingness, with non-existence. This is absolutely untrue. Or, to put it another way, we can define it in two ways: 1) nothing as non-existence, in which case it has absolutely no consequences but leads to all manner of abstract paradoxes and contradictions, or 2) nothing as existence, in which case it is always a mathematical balance point for somethings. It is purely mathematical, not scientific, or religious, or spiritual, or emotional, or sensory, or mystical. It is analytic nothing and whenever you encounter it you have to establish the exact means by which it is maintaining its balance of zero." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

Numbers and Inquiry

“[…] we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.” (Leonhard Euler)

 "There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations." (Auguste Comte, “The Positive Philosophy”, 1830)

"When you can measure what you are talking about and express it in numbers, you know something about it." (Lord Kelvin)

“It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.” (Sir Arthur C Doyle, “The Adventures of Sherlock Holmes”, 1892)

“[…] numerous samples collected without a clear idea of what is to be done with the data are commonly less useful than a moderate number of samples collected in accordance with a specific design.” (William C Krumbein)

“The purpose of computing is insight, not numbers […] sometimes […] the purpose of computing numbers is not yet in sight.” (Richard Hamming, [Motto for the book] “Numerical Methods for Scientists and Engineers”, 1962)

“Data in isolation are meaningless, a collection of numbers. Only in context of a theory do they assume significance […]” (George Greenstein, “Frozen Star”, 1983)

“The value of having numbers - data - is that they aren't subject to someone else's interpretation. They are just the numbers. You can decide what they mean for you.” (Emily Oster, “Expecting Better”, 2013)

“Torture numbers and they’ll confess to anything.” (Gregg Easterbrook)

“If the statistics are boring, you've got the wrong numbers.” (Edward Tufte)

On Numbers: From Zero to Infinity

“A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.” (Bhaskara II, “Bijaganita”, 12th century)

 “The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits.” (George H. Lewes, “Problems of Life and Mind”, 1873)

“One microscopic glittering point; then another; and another, and still another; they are scarcely perceptible, yet they are enormous. This light is a focus; this focus, a star; this star, a sun; this sun, a universe; this universe, nothing. Every number is zero in the presence of the infinite.” (Victor Hugo, “The Toilers of the Sea”, 1874)

"A great deal of misunderstanding is avoided if it be remembered that the terms infinity, infinite, zero, infinitesimal must be interpreted in connexion with their context, and admit a variety of meanings according to the way in which they are defined." (George B Mathews, "Theory of Numbers", 1892)

“Infinity is the land of mathematical hocus pocus. There Zero the magician is king. When Zero divides any number he changes it without regard to its magnitude into the infinitely small [great?], and inversely, when divided by any number he begets the infinitely great [small?]. In this domain the circumference of the circle becomes a straight line, and then the circle can be squared. Here all ranks are abolished, for Zero reduces everything to the same level one way or another. Happy is the kingdom where Zero rules!” (Paul Carus, “The Nature of Logical and Mathematical Thought”; Monist Vol 20, 1910)

“I do not say that the notion of infinity should be banished; I only call attention to its exceptional nature, and this so far as I can see, is due to the part which zero plays in it, and we must never forget that like the irrational it represents a function which possesses a definite character but can never be executed to the finish If we bear in mind the imaginary nature of these functions, their oddities will not disturb us, but if we misunderstand their origin and significance we are confronted by impossibilities.” (Paul Carus, “The Nature of Logical and Mathematical Thought”; Monist Vol 20, 1910)

"Each act of creation could be symbolized as a particular product of infinity and zero. From each such product could emerge a particular entity of which the appropriate symbol was a particular number." (Srinivasa Ramanujan)

“If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things.” (Robert Kaplan, “The Nothing that Is: A Natural History of Zero”, 2000)

“Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity.” (Charles Seife, “Zero: The Biography of a Dangerous Idea”, 2000)

“Zero seems as diaphanous as a fairy’s wing, yet it is as powerful as a black hole. The obverse of infinity, it’s enthroned at the center of the number line - at least as the line is usually drawn - making it a natural center of attention. It has no effect when added to other numbers, as if it were no more substantial than a fleeting thought. But when multiplied times other numbers it seems to exert uncanny power, inexorably sucking them in and making them vanish into itself at the center of things. If you’re into stark simplicity, you can express any number (that is, any number that’s capable of being written out) with the use of zero and just one other number, one.” (David Stipp, “A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics”, 2017)

On Graphics I: Go, Figures!

"If one, with the scientist, studies the works of nature, which are made up of elements or matter and form, his reasoning is dependent on the data provided by sense-experience. And if one, with the mathematician, abstracts figures or calculates numerically, he must, in order to gain assent, accurately adduce many examples of both differentiated plurality and quantitative extension. The like holds true of the philosopher, whose domain is [abstract] reasoning, and who is the client of both the scientist and the mathematician. For the philosopher, too, begins with those things which are based on the evidence of the senses and contribute to the knowledge of immaterial intelligibles." (John of Salisbury, "Metalogicon", 1159)

"A mathematician, as good as he may be, without the support of a good drawing, is nothing but a half-mathematician, but also a man without eyes." (Lodovico Cardi, [letter to Galileo Galilei] 1611)

"Algebra is but written geometry and geometry is but figured algebra." (Sophie Germain, "Mémoire sur les Surfaces Élastiques", 1880)

"The visible figures by which principles are illustrated should, so far as possible, have no accessories. They should be magnitudes pure and simple, so that the thought of the pupil may not be distracted, and that he may know what features of the thing represented he is to pay attention to." (National Education Association, 1894)

"[…] geometry is the art of reasoning well from badly drawn figures; [...]" (Henri Poincaré, 1895)

"Nothing is so illuminating as a set of properly proportioned diagrams." (Allan C Haskell, "How to Make and Use Graphic Charts", 1919)

"Mathematics is not a compendium or memorizable formula and magically manipulated figures." (Scott Buchanan, "Poetry and Mathematics", 1929)

"The essential fact is simply that all the pictures which science now draws of nature, and which alone seem capable of according with observational facts, are mathematical pictures." (Sir James Jeans, "The Mysterious Universe", 1930)

"The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings." (Alan M Turing, "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society Vol 45 (2), 1939)

"Figures alone prove or disprove nothing. Only the conclusions drawn from the collected material can do this. And these are theoretical." (Ludwig von Mises) 

"Figures and symbols are closely connected with mathematical thinking, their use assists the mind. […] At any rate, the use of mathematical symbols is similar to the use of words. Mathematical notation appears as a sort of language, une langue bien faite, a language well adapted to its purpose, concise and precise, with rules which, unlike the rules of ordinary grammar, suffer no exception." (George Pólya, "How to solve it", 1945)

"Geometry is the science of correct reasoning on incorrect figures." (George Pólya, "How to Solve It", 1945)

"Many people think of mathematics itself as a static art - a body of eternal truth that was discovered by a few ancient, shadowy figures, and upon which engineers and scientists can draw as needed." (Paul R Halmos, "Innovation in Mathematics", Scientific American Vol. 199 (3) , 1958)

"The art of using the language of figures correctly is not to be over-impressed by the apparent air of accuracy, and yet to be able to take account of error and inaccuracy in such a way as to know when, and when not, to use the figures. This is a matter of skill, judgment, and experience, and there are no rules and short cuts in acquiring this expertness." (Ely Devons, "Essays in Economics", 1961)

"If it can’t be expressed in figures, it is not science; it is opinion." (Robert Heinlein, "Time Enough For Love: the Lives of Lazarus Long", 1973)

"To remember simplified pictures is better than to forget accurate figures." (Otto Neurath, "Empiricism and Sociology", 1973)

"The greatest value of a picture is when it forces us to notice what we never expected to see." (John W Tukey, "Exploratory Data Analysis", 1977) 

"In information graphics, what you show can be as important as what you hide." (Alberto Cairo, "The Functional Art", 2011)

"[geometry is] the art of reasoning well from ill-drawn figures"  (Henri Poincaré)

"It is very helpful to represent these things in this fashion since nothing enters the mind more readily than geometric figures." (René Descartes)

More on Prime Numbers

“Since primes are the basic building blocks of the number universe from which all the other natural numbers are composed, each in its own unique combination, the perceived lack of order among them looked like a perplexing discrepancy in the otherwise so rigorously organized structure of the mathematical world.” (H Peter Aleff, “Prime Passages to Paradise”)

"The seeming absence of any ascertained organizing principle in the distribution or the succession of the primes had bedeviled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case?" (Apostolos Doxiadis, “Uncle Petros and Goldbach's Conjecture”, 2000)

“As archetypes of our representation of the world, numbers form, in the strongest sense, part of ourselves, to such an extent that it can legitimately be asked whether the subject of study of arithmetic is not the human mind itself. From this a strange fascination arises: how can it be that these numbers, which lie so deeply within ourselves, also give rise to such formidable enigmas? Among all these mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant." (Gerald Tenenbaum & Michael M France, “The Prime Numbers and Their Distribution”, 2000)

“Prime numbers belong to an exclusive world of intellectual conceptions. We speak of those marvelous notions that enjoy simple, elegant description, yet lead to extreme - one might say unthinkable - complexity in the details. The basic notion of primality can be accessible to a child, yet no human mind harbors anything like a complete picture. In modern times, while theoreticians continue to grapple with the profundity of the prime numbers, vast toil and resources have been directed toward the computational aspect, the task of finding, characterizing, and applying the primes in other domains." (Richard Crandall & Carl Pomerance, “Prime Numbers: A Computational Perspective”, 2001)

"[Primes] are full of surprises and very mysterious […]. They are like things you can touch […] In mathematics most things are abstract, but I have some feeling that I can touch the primes, as if they are made of a really physical material. To me, the integers as a whole are like physical particles." (Yoichi Motohashi, “The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics”, 2002)

“The primes have tantalized mathematicians since the Greeks, because they appear to be somewhat randomly distributed but not completely so. […] Although the prime numbers are rigidly determined, they somehow feel like experimental data." (Timothy Gowers, “Mathematics: A Very Short Introduction”, 2002)

“Our world resonates with patterns. The waxing and waning of the moon. The changing of the seasons. The microscopic cell structure of all living things have patterns. Perhaps that explains our fascination with prime numbers which are uniquely without pattern. Prime numbers are among the most mysterious phenomena in mathematics.” (Manindra Agrawal, 2003)

“The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes.” (David Ruelle, “The Mathematician's Brain”, 2007)

 “Mathematicians call them twin primes: pairs of prime numbers that are close to each other, almost neighbors, but between them there is always an even number that prevents them from truly touching. […] If you go on counting, you discover that these pairs gradually become rarer, lost in that silent, measured space made only of ciphers. You develop a distressing presentiment that the pairs encountered up until that point were accidental, that solitude is the true destiny. Then, just when you’re about to surrender, you come across another pair of twins, clutching each other tightly.” (Paolo Giordano, “The Solitude of prime numbers”, 2008)

“[…] if all sentient beings in the universe disappeared, there would remain a sense in which mathematical objects and theorems would continue to exist even though there would be no one around to write or talk about them. Huge prime numbers would continue to be prime, even if no one had proved them prime.” (Martin Gardner, “When You Were a Tadpole and I Was a Fish”, 2009)

On Prime Numbers

"The prime numbers are useful in analyzing problems concerning divisibility, and also are interesting in themselves because of some of the special properties which they possess as a class. These properties have fascinated mathematicians and others since ancient times, and the richness and beauty of the results of research in this field have been astonishing." (Carl H Denbow & Victor Goedicke, “Foundations of Mathematics”, 1959)

 “No branch of number theory is more saturated with mystery than the study of prime numbers: those exasperating, unruly integers that refuse to be divided evenly by any integers except themselves and 1. Some problems concerning primes are so simple that a child can understand them and yet so deep and far from solved that many mathematicians now suspect they have no solution. Perhaps they are 'undecideable'. Perhaps number theory, like quantum mechanics, has its own uncertainty principle that makes it necessary, in certain areas, to abandon exactness for probabilistic formulations." (Martin Gardner, "The remarkable lore of the prime numbers", Scientific American, 1964)

“There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers belong to the most arbitrary and ornery objects studied by mathematicians: they grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision.” (Don Zagier, “The First 50 Million Prime Numbers”, The Mathematical Intelligencer Vol. 0, 1977)

"Prime numbers have always fascinated mathematicians, professional and amateur alike. They appear among the integers, seemingly at random, and yet not quite: there seems to be some order or pattern, just a little below the surface, just a little out of reach." (Underwood Dudley, “Elementary Number Theory”, 1978)

“Prime numbers. It was all so neat and elegant. Numbers that refuse to cooperate, that don’t change or divide, numbers that remain themselves for all eternity.” (Paul Auster, “The Music of Chance”, 1990)

“If we imagine mathematics as a grand orchestra, the system of whole numbers could be likened to a bass drum: simple, direct, repetitive, providing the underlying rhythm for all the other instruments. There surely are more sophisticated concepts - the oboes and French horns and cellos of mathematics - and we examine some of these in later chapters. But whole numbers are always at the foundation.” (William Dunham, “The Mathematical Universe”, 1994)

"Prime numbers are the most basic objects in mathematics. They also are among the most mysterious, for after centuries of study, the structure of the set of prime numbers is still not well understood […]" (Andrew Granville, 1997)

"To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people." (Andreas Knauf, "Number Theory, Dynamical Systems and Statistical Mechanics", 1998)

“Rather mathematicians like to look for patterns, and the primes probably offer the ultimate challenge. When you look at a list of them stretching off to infinity, they look chaotic, like weeds growing through an expanse of grass representing all numbers. For centuries mathematicians have striven to find rhyme and reason amongst this jumble. Is there any music that we can hear in this random noise? Is there a fast way to spot that a particular number is prime? Once you have one prime, how much further must you count before you find the next one on the list? These are the sort of questions that have tantalized generations.” (Marcus du Sautoy, “The Music of the Primes”, 1998)