On Semiotics: On Signs (trivia)

"Science will persevere just as long as we retain a faculty we show no signs of losing: the ability to conceive — in no matter how imperfect or rudimentary a form - what the truth might be and retain also the inclination to ascertain whether our imaginings correspond to real life or not." (Peter B Medawar, "The Limits of Science". 1984)

"Nature is a vast tablet, inscribed with signs, each of which has its own signifi cancy, and becomes poetry in the mind when read; and geology is simply the key by which myriads of these signs, hitherto indecipherable, can be unlocked and perused, and thus a new province added to the poetical domain." (Hugh Miller, "Sketch-Book of Popular Geology", 1880)

"In a universe whose size is beyond human imagining, where our world floats like a dust mote in the void of night, men have grown inconceivably lonely. We scan the time scale and the mechanisms of life itself for portents and signs of the invisible. As the only thinking mammals on the planet - perhaps the only thinking animals in the entire sidereal universe - the burden of consciousness has grown heavy upon us. We watch the stars, but the signs are uncertain. We uncover the bones of the past and seek for our origins. There is a path there, but it appears to wander. The vagaries of the road may have a meaning, however; it is thus we torture ourselves." (Loren C Eiseley, "The Immense Journey", 1957)

"We seem to be entering a new world of technology, but the vehicle which is carrying us - science - shows dangerous signs of inadequacy for the voyage ahead. (Barry Commoner, "Science and Survival", 1966)

"[...] the picture of the universe presented by astronomy is one of dismal stretches of time and space and unparalled desolation. In the eternal abyss of space - bleak, cold, and dark - there are no signs of a Cosmic Consciousness." (Woolsey Teller, "The Atheism of Astronomy", 1972)

"Nature talks in signs and, to understand its language, one has to pay attention to similarities in form." (Jeremy Narby, "The Cosmic Serpent: DNA and the Origins of Knowledg", 1998)

"This notion of 'being due' - what is sometimes called the gambler’s fallacy - is a mistake we make because we cannot help it. The problem with life is that we have to live it from the beginning, but it makes sense only when seen from the end. As a result, our whole experience is one of coming to provisional conclusions based on insufficient evidence: read ing the signs, gauging the odds." (John Haigh," Taking Chances: Winning With Probability", 1999)

On Learning: On Recreation in Math

"Mathematics should be learned through recreational games, the way the Egyptians do, through amusement and pleasure." (Plato, "Laws" VII, cca 360 BC) [attributed to John A Comenius]

"Science is the labor and handicraft of the mind; poetry can only be considered its recreation." (Francis Bacon, "The Proficience and Advancement of Learning, Divine and Human", 1605)

"But leaving those of the Body, shall proceed to such Recreations as adorn the Mind; of which those of the Mathematicks are inferior to none." (William Leybourn," Pleasure with Profit", 1694) 

"And as the ideal in the whole of Nature moves in an infinite process toward an Absolute Perfection, we may say that art is in strict truth the apotheosis of Nature. Art is thus at once the exaltation of the natural toward its destined supernatural perfection, and the investiture of the Absolute Beauty with the reality of natural existence. Its work is consequently not a means to some higher end, but is itself a final aim; or, as we may otherwise say, art is its own end. It is not a mere recreation for man, a piece of by-play in human life, but is an essential mode of spiritual activity, the lack of which would be a falling short of the destination of man. It is itself part and parcel of man's eternal vocation." (George H Howison, "The Limits of Evolution, and Other Essays, Illustrating the Metaphysical Theory of Personal Idealism", 1901)

"Chess combines the beauty of mathematical structure with the recreational delights of a competitive game." (Martin Gardner, "Mathematics, Magic, and Mystery", 1956)

"Recreational mathematics is a splendid hobby which young and old can equally enjoy. The popularity of Sudoku shows that an aptitude for recreational mathematics is widespread in the population. From Sudoku it is easy to ascend to mathematical pursuits that offer more scope for imagination and originality." (Freeman Dyson, 2011)

"Discrete Mathematics is a branch of mathematics dealing with finite or countable processes and elements. Graph Theory is an area in Discrete Mathematics which studies configurations involving a set of vertices interconnected by edges (called graphs). From humble beginnings and almost recreational type problems, Graph Theory has found its calling in the modern world of complex systems and especially of the computer. Graph Theory and its applications can be found not only in other branches of mathematics, but also in scientific disciplines such as engineering, computer science, operational research, management sciences and the life sciences." (Khee Meng Koh et al, " Graph theory: Undergraduate mathematics", 2015)

On Principles (1925-1949)

"A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details." (Hermann Weyl, "Unterrichtsblätter für Mathematik und Naturwissenschaften", 1932)

"It goes without saying that the laws of nature are in themselves independent of the properties of the instruments with which they are measured. Therefore in every observation of natural phenomena we must remember the principle that the reliability of the measuring apparatus must always play an important role." (Max Planck,"Where is Science Going?", 1932)

"I think that we shall have to get accustomed to the idea that we must not look upon science as a 'body of knowledge,' but rather as a system of hypotheses; that is to say, as a system of guesses or anticipations which in principle cannot be justified, but with which we work as long as they stand up to tests, and of which we are never justified in saying that we know they are 'true' or 'more or less certain' or even 'probable’." (Karl R Popper, "The Logic of Scientific Discovery", 1934)

"The fundamental gospel of statistics is to push back the domain of ignorance, prejudice, rule-of-thumb, arbitrary or premature decisions, tradition, and dogmatism and to increase the domain in which decisions are made and principles are formulated on the basis of analyzed quantitative facts." (Robert W Burgess, "The Whole Duty of the Statistical Forecaster", Journal of the American Statistical Association , Vol. 32, No. 200, 1937)  

"When an active individual of sound common sense perceives the sordid state of the world, desire to change it becomes the guiding principle by which he organizes given facts and shapes them into a theory. The methods and categories as well as the transformation of the theory can be understood only in connection with his taking of sides. This, in turn, discloses both his sound common sense and the character of the world. Right thinking depends as much on right willing as right willing on right thinking." (Max Horkheimer, "The Latest Attack on Metaphysics", 1937)

"The question of the origin of the hypothesis belongs to a domain in which no very general rules can be given; experiment, analogy and constructive intuition play their part here. But once the correct hypothesis is formulated, the principle of mathematical induction is often sufficient to provide the proof." (Richard Courant & Herbert Robbins, "What Is Mathematics?: An Elementary Approach to Ideas and Methods", 1941)

"A formula is simply a mathematical statement of a principle or a rule describing the relation between two or more quantities. This mathematical statement shows that there is an equality between certain quantities; in other words, the formula translates a verbal rule into algebraic symbols. Thus a formula is very similar to an equation." (William L Schaaf, "Mathematics for Mechanics", 1942)

"It is to be hoped that in the future more and more theoretical physicists will command a deep knowledge of mathematical principles; and also that mathematicians will no longer limit themselves so exclusively to the aesthetic development of mathematical abstractions." (George D Birkhoff, "Mathematical Nature of Physical Theories" American Scientific Vol. 31 (4), 1943)

"Of course we have still to face the question why these analogies between different mechanisms - these similarities of relation-structure - should exist. To see common principles and simple rules running through such complexity is at first perplexing though intriguing. When, however, we find that the apparently complex objects around us are combinations of a few almost indestructible units, such as electrons, it becomes less perplexing." (Kenneth Craik, "The Nature of Explanation", 1943)

"We can put it down as one of the principles learned from the history of science that a theory is only overthrown by a better theory, never merely by contradictory facts." (James B Conant, "On Understanding Science", 1947)

"It is always more easy to discover and proclaim general principles than it is to apply them." (Winston Churchill, "The Second World War: The gathering storm", 1948)

William L Schaaf - Collected Quotes

"A formula is simply a mathematical statement of a principle or a rule describing the relation between two or more quantities. This mathematical statement shows that there is an equality between certain quantities; in other words, the formula translates a verbal rule into algebraic symbols. Thus a formula is very similar to an equation." (William L Schaaf, "Mathematics for Mechanics", 1942)

"A graph, as the name itself suggests, can go a step further than the formula - it can make visible what the formula represents - it can give an actual picture of the mathematical relationship. The relationship literally becomes more graphic; the relative magnitudes of the variables become apparent to the eye, as do extreme maximum and minimum values, if any; so do the rates at which they change; trends become clear; extrapolation and interpolation become more meaningful; any special features of the relationship are emphasized; general types of relationships are recognizable; two or more relationships can frequently be directly compared with one another." (William L Schaaf, "Mathematics for Mechanics", 1942)

"A type of picture-graph less commonly used than formerly is the pictorial representation of an object which has been arbitrarily subdivided to show certain numerical relationships; as, for example, the pictorial representation of the food values of beefsteak. This is a very poor type of graphic representation, and should definitely be avoided. The irregular outline of the picture as a whole, and of each of the shaded areas, makes a comparison of the areas difficult, if not altogether impossible; the shading only to the confusion." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"Geometric lines are fictions in the sense that, while we draw them, we think of them as having no width, simply length. Lines may be curved or straight." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"Graphs showing time changes, or the increases and decreases in the amount of something over a period of time, are generally of two kinds: (1) vertical bar graphs, and (2) broken- or smooth-line graphs. Both kinds differ from the categorical charts [...] in that they have two scales instead of only one; that is why it is preferable to call them graphs rather than charts, although these terms are used rather freely and interchangeably, and there is no standard convention." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"If a number or a quantity is thought of as being broken up, or subdivided into any number of equal parts, and then a certain number of those parts is considered separately in relation to the total number of such parts, we arrive at the idea of a fraction." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"In [...] horizontal bar-charts, showing comparisons between different kinds of things, or between different places, only one numerical scale is required, viz., the scale representing the amounts involved. No other numerical scale is needed, since we are dealing with various categories. While not always the most effective device for exhibiting such comparisons, horizontal bar-charts are simple and convenient." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"In mathematics, a definite quantitative relation between two or more variables, whether expressed verbally, by a formula, or by a graph, is called a functional relationship, or simply a mathematical function. Each variable is said to be a function of the other. The word function, as used here, has nothing to do with use or purpose; it simply calls attention to the fact that the quantities in question are quantitatively related to each other in a definite manner." (William L Schaaf, "Mathematics for Mechanics", 1942)

"It is clear that for any given point on a graph, its horizontal distance from the vertical scale (abscissa) represents the magnitude of the independent variable, while the vertical distance above or below the horizontal scale (ordinate) represents the corresponding magnitude of the dependent variable. Thus the position of the curve with respect to the axes depicts the actual magnitudes of the variables. But in studying changing variables and functional relationships, it is frequently desirable to inquire as to the rate at which a quantity is changing, i.e., how fast it is increasing or decreasing, rather than how large or how small it is. Rate implies a ratio; a rate of change means the amount of change in the function (or dependent variable) per unit change in the independent variable." (William L Schaaf, "Mathematics for Mechanics", 1942)

"Many forms in Nature exhibit the geometric property of symmetry. Anything symmetrical, whether natural or manmade, is usually pleasing in appearance, since it is 'balanced', and appeals to the eye. Symmetry is one of the most important principles of ornament, design and architecture. It is not, however, the only one; others are repetition and rhythm." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"The mathematical concept of chance, or probability, must not be confused with the psychological notion of likelihood. If a coin upon being tossed six times in succession has come up 'heads' each of the six times, we may be impelled to feel that upon the seventh toss it is 'more likely' to turn 'tails' in view of the six previous heads; but this is only an emotional reaction and not a mathematical probability. Mathematically [...] on any single throw, the chances are even for heads or tails, irrespective of what the previous trials may have been. In the long run, the greater the number of trials, the more nearly equal will become the number of heads and tails." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"The methods of algebra are essentially an extension of arithmetic. In other words, the numbers, symbols and operations used in algebra are the same as those used in arithmetic, only they are more general in character. This means (1) that letters as well as numbers are are “used te to represent quantities, and (2) that ‘numbers are re‘garded as having quality as well as quantity." (William L Schaaf, "Mathematics for Mechanics", 1942)

"There is no such thing as an absolute or perfect measurement. An object can be thought of as having an actual, real, or 'true' length; but that length can never be found completely, it can only be found approximately. How 'exact' any particular measurement happens to be depends upon the nature of the instruments used, the skill of the operator, and the conditions under which it is made. The difference between the true length and the measured length is technically known as the error. An error is not a mistake. The careless use, or the misuse, of a measuring instrument leads to mistakes. The proper use of a measuring instrument always involves errors. The errors may be large or small; they can never be completely eliminated. The extent of the approximation is known as the degree of accuracy of the measurement; a numerical measure of the extent of the error is known as the precision of the measurement. What particu. lar degree of accuracy is sought depends chiefly upon the purpose for which the measurement is made, or the "use to which the object is to be put." (William L Schaaf, "Mathematics for Mechanics", 1942)

"Practical geometry deals with the nature and properties of various geometric forms, such as rectangles, triangles, circles, etc., and emphasizes especially the measurement of such figures. Hence the chief value of practical geometry is in problems of design and construction." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"The operation of adding two numbers is essentially a process of grouping, or, more accurately, regrouping. When we add two numbers we do not increase anything; we regroup the numbers in accordance  with the standard pattern or number system based on groups of ten." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"When statistical data are of such a nature that it is permissible to assume that 'in-between values' vary continuously and uniformly (or very nearly so) from one observed or measured value to the next, a modification of the broken-line graph may be used. Instead of connecting the plotted points with straightline segments, a 'smooth' curved line is drawn between the points [...]. Such curvedline graphs may be drawn either 'free hand' or with the aid of drafting instruments known as French curves." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"You may expect to find graphs anywhere: in books, in periodicals, in newspapers, in pamphlets, on show cards in advertisements, in business reports, and so on. Their use, however, is sometimes limited. For one thing, they are of necessity less accurate than the figures on which they are based, which, of course, doesn’t matter too much in many cases. In the second place, they are sometimes misleading, which may or may not be intentional. It is also possible that the reader of a chart or graph may misinterpret it." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"Mathematics is on the artistic side a creation of new rhythms, orders, designs, harmonies, and on the knowledge side, is a systematic study of various rhythms, orders." (William L Schaaf, "Mathematics: Our Great Heritage: Essays on the Nature and Cultural Significance of Mathematics", 1948)

"A maximum value of a function is one that is greater than any values immediately preceding or following; a minimum value of a function is one that is less than any values immediately preceding or following."  (William L Schaaf, "The Calculus, a college course guide", 1963)

"Emerson once said that it didn’t matter much where a man was, so long as you knew the direction in which he was moving. In somewhat the same way, the significance of a graph or curve often lies not so much in what height a point on the curve has reached, as it does in how fast its height is changing, and whether it is increasing or decreasing." (William L Schaaf, "The Calculus, a college course guide", 1963)

"If a function increases as the independent variable increases, or decreases as the independent variable decreases, the function is said to be increasing; if the function decreases as the independent variable increases, or increases as the independent variable decreases, the function is decreasing." (William L Schaaf, "The Calculus, a college course guide", 1963)

"In other words, we may think of the instantaneous speed at a certain instant as the limiting value which the average speed would approach tf the interval were indefinitely shortened, while always including the instant in question." (William L Schaaf, "The Calculus, a college course guide", 1963)

"It hardly need be pointed out that to use the calculus skillfully requires considerable practice with standard formulas for differentiating various functions. Among the most commonly used formulas are those for the power function, for a product, and for a quotient. The exercise below affords further practice in the use of these formulas" (William L Schaaf, "The Calculus, a college course guide", 1963)

"It is apparent, therefore, that for a function to have a maximum or a minimum value, it is necessary for the value of f’(x) to be zero or infinite, but that this alone is not a sufficient condition. In addition to f’(x) having a zero or infinite value, f’ (x) must change in sign as it passes through zero (or infinity)." (William L Schaaf, "The Calculus, a college course guide", 1963)

"Once more we must remind the reader not to confuse the amount of change with the rate of change. A function may increase or decrease by a very small amount in a short interval, and yet be changing very rapidly - just as a bullet may travel only a small distance in one thousandth of a second and yet be moving at a very high speed." (William L Schaaf, "The Calculus, a college course guide", 1963)

"The creation of Analytic Geometry by Descartes in the early part of the seventeenth century was a milestone of tremendous significance. Indeed, it was the beginning of modern mathematics in the broad perspective of history. ('Modern' mathematics in the sense of contemporary mathematics did not commence until about 1900, with the advent of functional analysis, abstract spaces, set theory, and symbolic logic.) This great step of recognizing the relation between the numbers of algebra and the entities of geometry once having been taken, it is perhaps not surprising that further advances were soon to follow, culminating in the invention of the Calculus by Isaac Newton and by Gottfried Leibniz. This was a classic illustration of nearly simultaneous, but presumably independent creation." (William L Schaaf, "The Calculus, a college course guide", 1963)

"Thus far we have regarded integration as the inverse of the operation of differentiation, and the integral was thought of as an anti-derivative. It is possible, however, to consider integration from another point of view, namely, as a process of summation, or as the addition of many similar elements. Indeed, it is largely from this point of view that the Integral Calculus developed historically, growing out of early attempts to determine the area bounded by various curves. A given area was subdivided into many small parts, and these 'infinitesimal parts' were then added." (William L Schaaf, "The Calculus, a college course guide", 1963)

"When comparing infinitesimals, we refer to their order. This is a relative term, suggesting comparative degree of smallness. If the limit of the quotient of two infinitesimals is a constant, not zero, they are said to be of the same order; if this limit is zero, the first differential (the numerator) is said to be of higher order than the second, and the second of lower order than the first. If the limit is infinite, the first differential (the numerator) is said to be of lower order than the second, and the second of higher order than the first." (William L Schaaf, "The Calculus, a college course guide", 1963)

"Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number π." (William L Schaaf, "Nature and History of π", 1967)

On Semiotics: On Signs (-1799)

"It was ordained at the beginning of the world that certain signs should prefigure certain events." (Marcus Tullius Cicero, "De Divinatione" ["Concerning Divination"], 44 BC) 

"It would be a sign of great simplicity to think that the world was created in six days, or indeed at all in time; [...] Time is a thing posterior to the world. Therefore it would be correctly said that the world was not created in time, but that time had its existence in consequence of the world. For it is the motion of the heaven that has displayed the nature of time." (Philo Judaeus, "Allegories of the Sacred Laws" ["Legum allegoriae"], 1st century AD) 

"All things are filled full of signs, and it is a wise man who can learn about one thing from another." (Plotinus, "Enneads", cca. 270 AD)

"We are meant to take them [the words ‘increase and multiply’] in a figurative sense. […] It is only in the case of signs outwardly given that we find increase and multiplication in the sense that a single truth can be expressed by several different means […] that a single expression can be interpreted in several different ways." (St. Augustine,"Confessions", 397- 400)

"Letters are signs of things, symbols of words, whose power is so great that without a voice they speak to us the words of the absent; for they introduce words by the eye, not by the ear." (Isidore of Seville, "Etymologiae", cca. 800-625)

"The Imagination that is raised in man (or any other creature imbued with the faculty of imagining) by words, or other voluntary signs, is that we generally call Understanding; and is common to Man and Beasts." (Thomas Hobbes, "Leviathan: Or, The Matter, Forme, & Power of a Common-wealth Ecclesiasticall and Civill", 1651)

"The signs of science are, some certain and infallible; some, uncertain. Certain, when he that pretendeth the science of anything, can teach the same; that is to say, demonstrate the truth thereof perspicuously to another; uncertain, when only some particular events answer to his pretence, and upon many occasions prove so as he says they must. " (Thomas Hobbes, "Leviathan: Or, The Matter, Forme, & Power of a Common-wealth Ecclesiasticall and Civill", 1651)

"It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters, insofar as they are subject to reasoning, all that we can do in arithmetic and geometry." (Gottfried W Leibniz, 1677) 

"In signs, one sees an advantage for discovery that is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed, the labor of thought is wonderfully diminished” (Gottfried W Leibniz, [letter to Tschirnhaus] cca. 1686)

"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, "Alciphron: or the Minute Philosopher", 1732)

"[…] the sciences that are expressed by numbers or by other small signs, are easily learned; and without doubt this facility rather than its demonstrability is what has made the fortune of algebra." (Julien Offray de La Mettrie,"Man a Machine", 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)

"All abstract sciences are nothing but the study of relations between signs." (Denis Diderot, "Lettre sur les sourds et muets" ["Letter on the Deaf and Dumb"], 1751) 

On Semiotics: On Signs (1950-)

"By a symbol I do not mean an allegory or a sign, but an image that describes in the best possible way the dimly discerned nature of the spirit. A symbol does not define or explain; it points beyond itself to a meaning that is darkly divined yet still beyond our grasp, and cannot be adequately expressed in the familiar words of our language." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960)

"Language, in its origin and essence, is simply a system of signs or symbols that denote real occurrences or their echo in the human soul." (Carl G Jung, "The Structure And Dynamics Of The Psyche", 1960))

"The symbol is the tool which gives man his power, and it is the same tool whether the symbols are images or words, mathematical signs or mesons." (Jacob Bronowski, "The Reach of Imagination", 1967) 

"If 'model' is taken to mean visual representation or analogy with familiar experience, then clearly not every theory involves a model. Thus field theories, whether classical or quantal, are hardly visualisable. And if 'model' is taken to mean mechanism - either in a narrow mechanical sense or in a wide sense including nonmechanical mechanisms such as the meson field mechanism of nuclear forces - then some theories do contain models of this kind while others do not. [...] On the other hand in a third sense every physical theory is a model, namely of the underlying mathematical formalism. Moreover a physical theory is twice a model in the model-theoretic sense: once because every one of its basic signs has a particular interpretation within mathematics, another time because the same sign may have a physical interpretation as well - as is the case with all the referential primitives." (Mario Bunge, "Philosophy of Physics", 1973)

"Information is carried by physical entities, such as books or sound waves or brains, but it is not itself material. Information in a living system is a feature of the order and arrangement of its parts, which arrangement provides the signs that constitute a ‘code’ or ‘language’." (John Z Young, "Programs of the Brain", 1978)

"In a modern professional vocabulary a hypothesis is an imaginative preconception of what might be true in the form of a declaration with verifiable deductive consequences. It no longer tows ‘gratuitous’, ‘mere’, or ‘wild’ behind it, and the pejorative usage" (‘Evolution is a mere hypothesis’, ‘It is only a hypothesis that smoking causes lung cancer’) is one of the outward signs of little learning." (Sir Peter B Medawar, "Pluto’s Republic: Incorporating the Art of the Soluble and Induction Intuition in Scientific Thought", 1982)

"Signs are not empirical objects. Empirical objects become signs (or they are looked at as signs) only from the point of view of a philosophical decision." (Umberto Eco, "Semiotics and the Philosophy of Language", 1984)

"When semiotics posits such concepts as 'sign', it does not act like a science; it acts like philosophy when it posits such abstractions as subject, good and evil, truth or revolution."  (Umberto Eco, "Semiotics and the Philosophy of Language", 1984)

"Modeling underlies our ability to think and imagine, to use signs and language, to communicate, to generalize from experience, to deal with the unexpected, and to make sense out of the raw bombardment of our sensations. It allows us to see patterns, to appreciate, predict, and manipulate processes and things, and to express meaning and purpose. In short, it is one of the most essential activities of the human mind. It is the foundation of what we call intelligent behavior and is a large part of what makes us human. We are, in a word, modelers: creatures that build and use models routinely, habitually – sometimes even compulsively – to face, understand, and interact with reality. " (Jeff Rothenberg, "The Nature of Modeling. In: Artificial Intelligence, Simulation, and Modeling", 1989)

"Maps, due to their melding of scientific and artistic approaches, always involve complex interaction between the denotative and the connotative meanings of signs they contain." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"When visualization tools act as a catalyst to early visual thinking about a relatively unexplored problem, neither the semantics nor the pragmatics of map signs is a dominant factor. On the other hand, syntactics (or how the sign-vehicles, through variation in the visual variables used to construct them, relate logically to one another) are of critical importance." (Alan M MacEachren, "How Maps Work: Representation, Visualization, and Design", 1995)

"In both quantum theory and general relativity, we encounter predictions of physically sensible quantities becoming infinite. This is likely the way that nature punishes impudent theorists who dare to break her unity. […] If infinities are signs of missing unification, a unified theory will have none. It will be what we call a finite theory." (Lee Smolin, "The Trouble with Physics: The Rise of String Theory, The Fall of a Science and What Comes Next", 2006)

"[…] in all things that live there are certain irregularities and deficiencies which are not only signs of life, but sources of beauty. No human face is exactly the same in its lines on each side, no leaf perfect in its lobes, no branch in its symmetry. All admit irregularity as they imply change; […]" (John Ruskin,"The Stones of Venice: The Sea Stories", 2013)

"Using maps as communication tools masks their complexity as a mode of thinking. Maps act like language: we attribute the signs or marks in the map to a natural extension of thought. But post-structuralism exposed maps (like language) as artificial signs whose meaning is tethered to time, place, culture, gesture, smell - in short, a plethora of cognitive and phenomenal attributes of our communication ecology." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"When dealing with meaningful visual representation, aspects of a representation's meaning can be altered by modifying its visual characteristics; these characteristics are extensively explored in semiotics, the study of signs and symbols and their use or interpretation." (Vidya Setlur & Bridget Cogley, "Functional Aesthetics for data visualization", 2022)

 

On Semiotics: On Signs (1850-1899)

"Observe this: the abstraction of the philosopher is meant to keep the object itself, with its perturbing suggestions, out of sight, allowing only one quality to fill the field of vision; whereas the abstraction of the poet is meant to bring the object itself into more vivid relief, to make it visible by means of the selected qualities. In other words, the one aims at abstract symbols, the other at picturesque effects. The one can carry on his deductions by the aid of colourless signs, X or Y. The other appeals to the emotions through the symbols which will most vividly express the real objects in their relations to our sensibilities." (George H Lewes, "The Principles of Success in Literature", 1865)

"The degree in which each mind habitually substitutes signs for images will be, CETERIS PARIBUS [with other conditions remaining the same], the degree in which it is liable to error. This is not contradicted by the fact that mathematical, astronomical, and physical reasonings may, when complex, be carried on more successfully by the employment of signs; because in these cases the signs themselves accurately represent the abstractness of the relations. Such sciences deal only with relations, and not with objects; hence greater simplification ensures greater accuracy. But no sooner do we quit this sphere of abstractions to enter that of concrete things, than the use of symbols becomes a source of weakness. Vigorous and effective minds habitually deal with concrete images." (George H Lewes, "The Principles of Success in Literature", 1865)

"I believe, therefore, that there can be no possible sense at all in speaking of any other truth for our representations except a practical [truth]. Our representations of things can be nothing else at all except symbols, naturally given signs for things, that we learn to use for the regulation of our motions and actions. When we have correctly learned to read such a symbol, we are then capable of so adjusting our actions with its help that they have the desired result, that is, the expected new sensations occur. Another comparison between representations and things not only fails to exist in actuality – here all schools agree – but any other kind of comparison is in no way thinkable and has no sense at all." (Hermann von Helmholtz, "Handbuch der Physologieschen Optik", 1867)

"The being of a sign is merely being represented. Now really being and being represented are very different. Giving to the word sign the full scope that reasonably belongs to it for logical purposes, a whole book is a sign; and a translation of it is a replica of the same sign. A whole literature is a sign." (Charles S Peirce," What Is a Sign?", 1894) 

"In algebra we perform, as far as possible, all numerical operations which are identical in form once for all, so that only a remnant of work is left for the individual case. The use of the signs of algebra and analysis, which are merely symbols of operations to be performed, is due to the observation that we can materially disburden the mind in this way and spare its powers for more important and more difficult duties, by imposing all mechanical operations upon the hand." (Ernst Mach, "The Economical Nature of Physical Enquiry", Popular Scientific Lectures, 1895)

"Strange as it may sound, the power of mathematics rests upon its evasion of all unnecessary thought and on its wonderful saving of mental operation. Even those arrangement-signs which we call numbers are a system of marvelous simplicity and economy. When we employ the multiplication-table in multiplying numbers of several places, and so use the results of old operations of counting instead of performing the whole of each operation anew; when we consult our table of logarithms, replacing and saving thus new calculations by old ones already performed; when we employ determinants instead of always beginning afresh the solution of a system of equations; when we resolve new integral expressions into familiar old integrals; we see in this simply a feeble reflexion of the intellectual activity of a Lagrange or a Cauchy, who, with the keen discernment of a great military commander, substituted new operations for whole hosts of old ones. No one will dispute me when I say that the most elementary as well as the highest mathematics are economically-ordered experiences of counting, put in forms ready for use." (Ernst Mach, "Popular Scientific Lectures", 1895)

"Mathematics gives the young man a clear idea of demonstration and habituates him to form long trains of thought and reasoning methodically connected and sustained by the final certainty of the result; and it has the further advantage, from a purely moral point of view, of inspiring an absolute and fanatical respect for truth. In addition to all this, mathematics, and chiefly algebra and infinitesimal calculus, excite to a high degree the conception of the signs and symbols - necessary instruments to extend the power and reach of the human mind by summarizing an aggregate of relations in a condensed form and in a kind of mechanical way. These auxiliaries are of special value in mathematics because they are there adequate to their definitions, a characteristic which they do not possess to the same degree in the physical and mathematical [natural?] sciences. There are, in fact, a mass of mental and moral faculties that can be put in full play only by instruction in mathematics; and they would be made still more available if the teaching was directed so as to leave free play to the personal work of the student." (M P Berthelot, "Science as an Instrument of Education", Popular Science Monthly, 1897)

"All our ideas and concepts are only internal pictures, or if spoken, combinations of sounds. The task of our thinking is so to use and combine them that by their means we always most readily hit upon the correct actions and guide others likewise. In this, metaphysics follows the most down-to-earth and practical point of view, so that extremes meet. The conceptual signs that we form thus exist only within us, we cannot measure external phenomena by the standard of our ideas. We can therefore pose such formal questions as whether only matter exists and force is a property of it, or whether force exists independently of matter or conversely whether matter is a product of force but none of these questions are significant since all these concepts are only mental pictures whose purpose is to represent phenomena correctly." (Ludwig Boltzmann, 1899)

"[…] no theory can be objective, actually coinciding with nature, but rather that each theory is only a mental picture of phenomena, related to them as sign is to designatum. From this it follows that it cannot be our task to find an absolutely correct theory but rather a picture that is, as simple as possible and that represents phenomena as accurately as possible. One might even conceive of two quite different theories both equally simple and equally congruent with phenomena, which therefore in spite of their difference are equally correct. (Ludwig Boltzmann, "On the development of the methods of theoretical physics", 1899)

On Semiotics: On Signs (1800-1849)

"It has been already observed, that demonstration ultimately depends on observations made on individual objects, and that a conclusion expressed by certain characters and signs, if general, must be true 'in each particular case that presents itself, on assigning specific values to the signs." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"Algebra is a species of short-hand writing; a language, or system of characters or signs, invented for the purpose of facilitating the comparison and combination of ideas." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"It has been already observed, that demonstration ultimately depends on observations made on individual objects, and that a conclusion expressed by certain characters and signs, if general, must be true 'in each particular case that presents itself, on assigning specific values to the signs." (Robert Woodhouse," On the necessary Truth of certain Conclusions obtained by Means of imaginary Quantities", 1801)

"The theory of which we have just given an overview may be considered from a point of view apt to set aside the obscure in what it presents, and which seems to be the primary aim, namely: to establish new notions on imaginary quantities. Indeed, putting to one side the question of whether these notions are true or false, we may restrict ourselves to viewing this theory as a means of research, to adopt the lines in direction only as signs of the real or imaginary quantities, and to see, in the usage to which we have put them, only the simple employment of a particular notation. For that, it suffices to start by demonstrating, through the first theorems of trigonometry, the rules of multiplication and addition given above; the applications will follow, and all that will remain is to examine the question of didactics. And if the employment of this notation were to be advantageous? And if it were to open up shorter and easier paths to demonstrate certain truths? That is what fact alone can decide." (Jean-Robert Argand, "Essai sur une manière de représenter les quantités imaginaires, dans les constructions géométriques", Annales Tome IV, 1813)

"The diversity of languages is not a diversity of signs and sounds but a diversity of views of the world." (Wilhelm von Humboldt, 1820)

"Science sees signs: poetry, the thing signified." (Julius C Hare, "Guesses at Truth", 1827)

"The mutual interdependence of thought and word illuminates clearly the truth that languages are not really means for representing already known truths, but are rather instruments for discovering previously unrecognised ones. The differences between languages are not those of sounds and signs but those of differing  worldviews […] objective truth always rises from the entire energy of subjective individuality." (Wilhelm von Humboldt, "Über die Verschiedenheit des menschlichen Sprachbaues und ihren Einfluss auf die geistige Entwickelung des Menschengeschlechts" ["On the Diversity of Human Language Structure and Its Influence on the Mental Development of the Human Race"], 1836) 

On Semiotics: On Signs (1900-1949)

"All our thinking is performed upon signs of some kind or other, either imagined or actually perceived. The best thinking, especially on mathematical subjects, is done by experimenting in the imagination upon a diagram or other scheme, and it facilitates the thought to have it before one’s eyes." (Charles S Peirce, “The Principles of Mathematics”, cca. 1902) 

"A sign is a thing which is the representative, or deputy, of another thing for the purpose of affecting a mind. […] The utility of icons is evidenced by the diagrams of the mathematician, whether they involve continuity, like geometrical figures, or are arrays of discrete objects like a body of algebraical formulae, all of which are icons. Icons have to be used in all thinking." (Charles S Peirce, [manuscript] 1903)

"Many diagrams resemble their objects not at all in looks; it is only in respect to the relations of their parts that their likeliness consists. […] When, in algebra, we write equations under one another in a regular array, especially when we put resembling letters for corresponding coefficients, the array is an icon. […] In fact, every algebraic equation is an icon, in so far as it exhibits, by means of the algebraic signs (which are not themselves icons), the relations of the quantities concerned." (Charles S Peirce, "New Elements", 1904)

"The edifice of science is not raised like a dwelling, in which the foundations are first firmly laid and only then one proceeds to construct and to enlarge the rooms. Science prefers to secure as soon as possible comfortable spaces to wander around and only subsequently, when signs appear here and there that the loose foundations are not able to sustain the expansion of the rooms, it sets about supporting and fortifying them. This is not a weakness, but rather the right and healthy path of development." (David Hilbert, "The Logical Principles of Mathematical Thinking", 1905)

"It is important to understand what I mean by semiosis. All dynamic action, or action of brute force, physical or psychical, either takes place between two subjects, - whether they react equally upon each other, or one is agent and the other patient, entirely or partially, — or at any rate is a resultant of such actions between pairs. But by 'semiosis' I mean, on the contrary, an action, or influence, which is, or involves, a cooperation of three subjects, such as a sign, its object, and its interpretant, this tri-relative influence not being in any way resolvable into actions between pairs." (Charles S Peirce,"Pragmatism", 1907) 

"Language, in its origin and essence, is simply a system of signs or symbols that denote real occurrences or their echo in the human soul." (Carl G Jung, "Symbols for the transformation", 1912)

"I call the combination of a concept and a sound-image a sign, but in current usage the term generally designates only a sound-image, a word, for example" (arbor, etc.). One tends to forget that arbor is called a sign only because it carries the concept ‘tree’, with the result that the idea of the sensory part implies the idea of the whole." (Ferdinand de Saussure, "Course in General Linguistics", 1915

"The logic of things, i.e., of the material concepts and relations on which the structure of a science rests, cannot be separated by the logic of signs. For the sign is no mere accidental cloak of the idea, but its necessary and essential organ. It serves not merely to communicate a complete and given thought content, but is an instrument, by means of which this content develops and fully defines itself. […] Consequently, all truly strict and exact thought is sustained by the symbolic and semiotics on which it is based." (Ernst Cassirer, "The Philosophy of Symbolic Forms", 1923)

"The words in a poem, (or more exactly, syllables) are vocal signs that convey an intangible essence (the pattern of feeling) that vanishes the moment we approach it with an analytical intelligence." (Herbert Read, "What is a Poem", 1926)

"The words of the language, as they are written or spoken, do not seem to play any role in any mechanism of thought. The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be 'voluntarily' reproduced or combined. […] But taken from a psychological viewpoint, this combinatory play seems to be the essential feature in productive thought - before there is any connection with logical construction in words or other kinds of signs which can be communicated to others. The above-mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will." (Albert Einstein, [letter to Hadamard, in" (Jacques Hadamard, "The Psychology of Invention in the Mathematical Field,1945)])

🎲On Probability Theory (2010-2019)

"At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate." (Terence Tao, "Topics in Random Matrix Theory", 2012)

"Descriptive statistics are built on the assumption that we can use a single value to characterize a single property for a single universe. […] Probability theory is focused on what happens to samples drawn from a known universe. If the data happen to come from different sources, then there are multiple universes with different probability models. If you cannot answer the homogeneity question, then you will not know if you have one probability model or many. [...] Statistical inference assumes that you have a sample that is known to have come from one universe." (Donald J Wheeler, "Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"The four questions of data analysis are the questions of description, probability, inference, and homogeneity. [...] Descriptive statistics are built on the assumption that we can use a single value to characterize a single property for a single universe. […] Probability theory is focused on what happens to samples drawn from a known universe. If the data happen to come from different sources, then there are multiple universes with different probability models.  [...] Statistical inference assumes that you have a sample that is known to have come from one universe." (Donald J Wheeler," Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"Probability theory provides the best answer only when the rules of the game are certain, when all alternatives, consequences, and probabilities are known or can be calculated. [...] In the real game, probability theory is not enough. Good intuitions are needed, which can be more challenging than calculations. One way to reduce uncertainty is to rely on rules of thumb." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"When statisticians, trained in math and probability theory, try to assess likely outcomes, they demand a plethora of data points. Even then, they recognize that unless it’s a very simple and controlled action such as flipping a coin, unforeseen variables can exert significant influence." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

"Logic provides a set of formal rules for determining what propositions are implied to be true or false given the assumption that some other set of propositions is true or false. Probability theory provides a set of formal rules for determining the likelihood of a proposition being true given the likelihood of other propositions." (Ian Goodfellow et al, "Deep Learning", 2015)

"Probability theory is not the only tool for rationality. In situations of uncertainty, as opposed to risk, simple heuristics can lead to more accurate judgments, in addition to being faster and more frugal. Under uncertainty, optimal solutions do not exist (except in hindsight) and, by definition, cannot be calculated. Thus, it is illusory to model the mind as a general optimizer, Bayesian or otherwise. Rather, the goal is to achieve satisficing solutions, such as meeting an aspiration level or coming out ahead of a competitor."  (Gerd Gigerenzer et al, "Simply Rational: Decision Making in the Real World", 2015)

"New information is constantly flowing in, and your brain is constantly integrating it into this statistical distribution that creates your next perception (so in this sense 'reality' is just the product of your brain’s ever-evolving database of consequence). As such, your perception is subject to a statistical phenomenon known in probability theory as kurtosis. Kurtosis in essence means that things tend to become increasingly steep in their distribution [...] that is, skewed in one direction. This applies to ways of seeing everything from current events to ourselves as we lean 'skewedly' toward one interpretation, positive or negative. Things that are highly kurtotic, or skewed, are hard to shift away from. This is another way of saying that seeing differently isn’t just conceptually difficult - it’s statistically difficult." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017)

"Bootstrapping provides an intuitive, computer-intensive way of assessing the uncertainty in our estimates, without making strong assumptions and without using probability theory. But the technique is not feasible when it comes to, say, working out the margins of error on unemployment surveys of 100,000 people. Although bootstrapping is a simple, brilliant and extraordinarily effective idea, it is just too clumsy to bootstrap such large quantities of data, especially when a convenient theory exists that can generate formulae for the width of uncertainty intervals." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

"But [bootstrap-based] simulations are clumsy and time-consuming, especially with large data sets, and in more complex circumstances it is not straightforward to work out what should be simulated. In contrast, formulae derived from probability theory provide both insight and convenience, and always lead to the same answer since they don’t depend on a particular simulation. But the flip side is that this theory relies on assumptions, and we should be careful not to be deluded by the impressive algebra into accepting unjustified conclusions." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)