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 Education: 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)

"A mathematical point is an idea - an abstraction - something we can think about. It cannot be defined logically. Fortunately, our common experience with physical objects makes it relatively certain that each of us has the same idea in mind when he uses the word point, and so the lack of a logical definition presents no particular difficulty." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"Having brought our story down to the present, we may well ask: What is geometry? Is it a system of logic, of pure mathematics, or is it a practical art to serve as the handmaiden to science, engineering, astronomy, and kindred technical fields? The answer is simple: it is both a logical science and a practical art." (William L Schaaf, "Plane and solid geometry for home study", [forward] 1944)

"No step should ever be taken, and no statement should ever be made, without giving a legitimate reason therefor. A 'legitimate' reason means a reference to a definition, to an assumption, or to a proposition previously proved; it may also include reference to a statement in the hypothesis, or to a line arbitrarily drawn at the outset of the demonstration." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"To define a term adequately means to describe it in such a way that no doubt or ambiguity arises as to its meaning when that term is used. A good definition should indicate to what larger group of objects the object defined belongs, as well as how that object differs from other objects of that group. The larger group is sometimes called the genus, and the smaller, special group is called the species. Geometric objects, i.e., figures, are for the most part easy to define with precision." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"Whenever we discuss a geometric statement in order to prove it, we refer to previously proved statements for 'evidence'. In this process of referring back to earlier statements that have already been established by proof, it is clear that some few of the earliest statements must of necessity remain unproved - that is, they have to be taken for granted, since we must begin our chain of reasoning somewhere. It is not possible to prove every statement. Those statements which are thus taken for granted without proof are called assumptions. Hence in all logical reasoning, two considerations are of prime importance: (1) the meanings of the terms used, and (2) the basic assumptions made." (William L Schaaf, "Plane and solid geometry for home study", 1944)

"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 cylinder is a quadric surface if and only if its right section is a conic. As will be seen shortly, a quadric surface is one whose equation is of the second degree. A quadric cylinder is called parabolic, elliptic, or hyperbolic according as its right section is a parabola, an ellipse, or a hyperbola. The cylinder of revolution, or circular cylinder, is a special case of the elliptic cylinder." (William L Schaaf, "Analytic Geometry; a college course guide", 1962) 

"A cylindrical surface is any surface generated by a moving straight line which constantly intersects a given fixed curve and which remains parallel to a fixed straight line. The fixed curve is called the directrix, and the generating line is known as the generatrix. Thus a cylinder is a ruled surface, that is, the surface is comprised of straight lines (the generators), all of which are parallel." (William L Schaaf, "Analytic Geometry; a college course guide", 1962) 

"A transformation by translation has the effect of moving the origin to a new position without changing the directions of the axes. A transformation by rotation has the effect of turning the axes through an angle about the origin as a pivot, but without changing the position of the origin. Either transformation may be effected by itself, or both may be effected jointly; although, in such case, the rotation is normally performed first and the translation second." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"If each segment of a broken line in space be given the direction determined in passing continuously from one terminal to the other, then the algebraic sum of the projections of the segments upon any directed line equals the projection of the closing line. If the broken line in question should be a closed polygon, the sum of the projections of the sides upon any directed line is zero." (William L Schaaf, "Analytic Geometry; a college course guide", 1962) 

"In algebra, when we plot the graph of an equation, the curve so obtained represents the locus of the equation. In analytic geometry, the word 'curve' refers to any locus; it may be a straight line, a curved line, or a group of lines. We therefore see that: (1) The locus of an equation is a curve containing all those points, and only those points, whose coordinates satisfy the given equation. (2) The equation of a locus is an equation such that the coordinates of every point on the locus satisfy the equation, and every ordered pair of numbers which satisfy the equation are the coordinates of a point on the locus." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"It is possible, however, to determine the general shape and salient features of a curve by a more searching method. This is accomplished by examining more closely the equation of the curve, rather than by computing a more extensive table of values. Certain characteristic features of curves can be determined by inspection of the form of the equation, and by computing only certain special values of the variables. These characteristic properties include the intercepts, extent, symmetry, and asymptotes of the curve." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"It is possible, however, to express geometric facts by using algebraic methods. In other words, the properties of figures and the relationships between them can be described by means of numbers and equations. Conversely, algebraic expressions and equations can be represented or interpreted geometrically. When we translate from the language of geometry to the language of algebra, or vice versa, we are using analytical methods. Geometry when studied in this way is called analytic geometry." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"It is to be understood that by an algebraic equation is meant an equation which involves only integral and fractional powers of the variables x and y. If the equation is of degree higher than two, the equation represents an algebraic higher plane curve. Any equation which is not algebraic is known as a transcendental equation; functions defined by such equations are said to be transcendental functions. For example, y = cos x, y = a log x, and y = k"" are transcendental functions."  (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"The empirical data are usually available in the form of a table exhibiting pairs of corresponding values of the variables. Since the given values do not lie exactly along a straight hne, a parabola, an exponential curve or a power curve, it is necessar}' to find a curve that fits the given data approximately. This process is called curve fitting, and the equation of the curve which fits the data approximately is called an empirical equation. When fitting a curve to a set of given points, we must assume a general form for the equation which we are going to use to represent the curve." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"Very little can be learned about the nature of a surface merely by locating points on the surface. Instead, we study (1) its intercepts and traces in the coordinate planes; (2) its extent; (3) its symmetry, and (4) plane sections." (William L Schaaf, "Analytic Geometry; a college course guide", 1962) 

"When either the translation or the rotation transformations are applied to an equation, they change the coordinates of all points in the plane (except the origin in the case of a rotation). These transformations actually move the axes with relation to the curve, but they do not alter the shape of the curve represented by the original equation and by the transformed equation. Beside this constant or unvarying feature, i.e., the preservation of the geometric form of the curve, there are also certain algebraic expressions whose values remain unchanged in both equations. Mathematical forms which are preserved or do not vary when other changes take place are called invariants." (William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"When plotting or drawing the locus of a given equation, the position selected for the coordinate axes is, of course, immaterial. Hence, after the locus has been drawn, we may change the position of the axes arbitrarily, as we wish. Naturally, if the locus remains where it was drawn, but the position of the axes is changed, then the equation will have to be changed to correspond to the new position of the axes if the equation is to describe the same locus. In other words, a given locus may be described by more than one equation; each of the equations describes the locus in question with reference to a different set of axes."(William L Schaaf, "Analytic Geometry; a college course guide", 1962)

"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)

"When a curve approaches an axis or any other straight line in this fashion, the line is said to be an asymptote of the curve. As a working definition, we may say: An asymptote of a curve is any straight line which a curve approaches continuously as the curve moves on to infinity." (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)

🕸️On Graph Theory: On Complexity

"As with any graphic, networks are used in order to discover pertinent troups of to inform others of the groups and structures discovered. It is a good means of displaying structures, However, it ceases to be a means of discovery when the elements are numerous. The figure rapidly becomes complex, illegible and untransformable." (Jacques Bertin, "Graphics and graphic information processing", 1977)

"The more complex the network is, the more complex its pattern of interconnections, the more resilient it will be." (Fritjof Capra, "The Web of Life: A New Scientific Understanding of Living Systems", 1996)

"[…] networks are the prerequisite for describing any complex system, indicating that complexity theory must inevitably stand on the shoulders of network theory. It is tempting to step in the footsteps of some of my predecessors and predict whether and when we will tame complexity. If nothing else, such a prediction could serve as a benchmark to be disproven. Looking back at the speed with which we disentangled the networks around us after the discovery of scale-free networks, one thing is sure: Once we stumble across the right vision of complexity, it will take little to bring it to fruition. When that will happen is one of the mysteries that keeps many of us going." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"At an anatomical level - the level of pure, abstract connectivity - we seem to have stumbled upon a universal pattern of complexity. Disparate networks show the same three tendencies: short chains, high clustering, and scale-free link distributions. The coincidences are eerie, and baffling to interpret." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"A first step in understanding complex systems is trying to understand patterns and regularities of interactions in a way which might make it possible to break the systems down into possible subcomponents. To do so, it is necessary to find a way of representing complex systems. […] A convenient way to represent complex systems is through graphs or networks." (Jörg Reichardt, "Structure in Complex Networks", 2009)

"For the study of the topology of the interactions of a complex system it is of central importance to have proper random null models of networks, i.e., models of how a graph arises from a random process. Such models are needed for comparison with real world data. When analyzing the structure of real world networks, the null hypothesis shall always be that the link structure is due to chance alone. This null hypothesis may only be rejected if the link structure found differs significantly from an expectation value obtained from a random model. Any deviation from the random null model must be explained by non-random processes." (Jörg Reichardt, "Structure in Complex Networks", 2009)

"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)

"The exploding interest in network science during the first decade of the 21st century is rooted in the discovery that despite the obvious diversity of complex systems, the structure and the evolution of the networks behind each system is driven by a common set of fundamental laws and principles. Therefore, notwithstanding the amazing differences in form, size, nature, age, and scope of real networks, most networks are driven by common organizing principles. Once we disregard the nature of the components and the precise nature of the interactions between them, the obtained networks are more similar than different from each other." (Albert-László Barabási, "Network Science", 2016)

"[...] the Game of Life, in which a few simple rules executed repeatedly can generate a surprising degree of complexity. Recall that the game treats squares, or pixels, as simply on or off (filled or blank) and the update rules are given in terms of the state of the nearest neighbours. The theory of networks is closely analogous. An electrical network, for example, consists of a collection of switches with wires connecting them. Switches can be on or off, and simple rules determine whether a given switch is flipped, according to the signals coming down the wires from the neighbouring switches. The whole network, which is easy to model on a computer, can be put in a specific starting state and then updated step by step, just like a cellular automaton. The ensuing patterns of activity depend both on the wiring diagram (the topology of the network) and the starting state. The theory of networks can be developed quite generally as a mathematical exercise: the switches are called ‘nodes’ and the wires are called ‘edges’. From very simple network rules, rich and complex activity can follow." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

"Most of the complexity indeed arises from the presence of entities that appear only once or very few times, but still generate cliques within the graph. Such entities are not very informative to capture patterns and provide insights. Besides, they are possibly strongly affected by statistical variability. On the other hand, we should focus on strong correlations that are supported by larger occurrences and provide more reliable statistical results." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

🕸️On Graph Theory: Connectedness

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

"Networks are not en route from a random to an ordered state. Neither are they at the edge of randomness and chaos. Rather, the scale-free topology is evidence of organizing principles acting at each stage of the network formation process." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"In a random network the loss of a small number of nodes can cause the overall network to become incoherent - that is, to break into disconnected subnetworks. In a scale-free network, such an event usually won’t disrupt the overall network because most nodes don’t have many links. But there’s a big caveat to this general principle: if a scale-free network loses a hub, it can be disastrous, because many other nodes depend on that hub." (Thomas Homer-Dixon, "The Upside of Down: Catastrophe, Creativity, and the Renewal of Civilization", 2006)

"If a network is solely composed of neighborhood connections, information must traverse a large number of connections to get from place to place. In a small-world network, however, information can be transmitted between any two nodes using, typically, only a small number of connections. In fact, just a small percentage of random, long-distance connections is required to induce such connectivity. This type of network behavior allows the generation of 'six degrees of separation' type results, whereby any agent can connect to any other agent in the system via a path consisting of only a few intermediate nodes." (John H Miller & Scott E Page, "Complex Adaptive Systems", 2007)

"A graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)

"Some connected graphs are 'more connected' than others. That is, a connected graph’s vulnerability to disconnection by edge- or vertex-deletion varies. Two numerical parameters, vertex-connectivity and edge-connectivity, are useful in measuring a graph’s connectedness. Intuitively, a network’s vulnerability should be closely related also to the number of  alternative paths between each pair of nodes. There is a rich body of mathematical results concerning this relationship, many of which are variations of a classical result of Menger, and some of these extend well beyond graph theory." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)

"Spanning trees capture the connectedness of a graph in the most efficient way, and they provide a foundation for a systematic analysis of the cycle structure of a graph. Mathematicians regard the algebraic structures underlying the collection of cycles and edge-cuts of agraph as beautiful in their own right. Establishing connections between linear algebra and graph theory provides some powerful analytical tools for understanding a graph’s structure." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)

🕸️On Graph Theory: On Randomness

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

"Networks are not en route from a random to an ordered state. Neither are they at the edge of randomness and chaos. Rather, the scale-free topology is evidence of organizing principles acting at each stage of the network formation process." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"Like regular networks, random ones are seductive idealizations. Theorists find them beguiling, not because of their verisimilitude, but because they're the easiest ones to analyze. [...] Random networks are small and poorly clustered; regular ones are big and highly clustered." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"In a random network the loss of a small number of nodes can cause the overall network to become incoherent - that is, to break into disconnected subnetworks. In a scale-free network, such an event usually won’t disrupt the overall network because most nodes don’t have many links. But there’s a big caveat to this general principle: if a scale-free network loses a hub, it can be disastrous, because many other nodes depend on that hub." (Thomas Homer-Dixon, "The Upside of Down: Catastrophe, Creativity, and the Renewal of Civilization", 2006)

"If a network is solely composed of neighborhood connections, information must traverse a large number of connections to get from place to place. In a small-world network, however, information can be transmitted between any two nodes using, typically, only a small number of connections. In fact, just a small percentage of random, long-distance connections is required to induce such connectivity. This type of network behavior allows the generation of 'six degrees of separation' type results, whereby any agent can connect to any other agent in the system via a path consisting of only a few intermediate nodes." (John H Miller & Scott E Page, "Complex Adaptive Systems", 2007)

"Thus, nonlinearity can be understood as the effect of a causal loop, where effects or outputs are fed back into the causes or inputs of the process. Complex systems are characterized by networks of such causal loops. In a complex, the interdependencies are such that a component A will affect a component B, but B will in general also affect A, directly or indirectly. A single feedback loop can be positive or negative. A positive feedback will amplify any variation in A, making it grow exponentially. The result is that the tiniest, microscopic difference between initial states can grow into macroscopically observable distinctions." (Carlos Gershenson, "Design and Control of Self-organizing Systems", 2007)

"For the study of the topology of the interactions of a complex system it is of central importance to have proper random null models of networks, i.e., models of how a graph arises from a random process. Such models are needed for comparison with real world data. When analyzing the structure of real world networks, the null hypothesis shall always be that the link structure is due to chance alone. This null hypothesis may only be rejected if the link structure found differs significantly from an expectation value obtained from a random model. Any deviation from the random null model must be explained by non-random processes." (Jörg Reichardt, "Structure in Complex Networks", 2009)

"Self-organizing networks suffer various types of random damage. Therefore, if the network remained static, it would soon become dysfunctional. Some networks have developed highly specific screening systems which recognize and repair random damage. On the one hand, this process requires energy, which arrives in the form of perturbations or noise. On the other hand, noise-triggered network restructuring will repeat a few steps of the original self-organization and therefore constitutes a much cheaper way of providing a continuous repair function, with the additional advantage that it is always adaptive with respect to the actual environment of the network." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

"In the telephone system a century ago, messages dispersed across the network in a pattern that mathematicians associate with randomness. But in the last decade, the flow of bits has become statistically more similar to the patterns found in self-organized systems. For one thing, the global network exhibits self-similarity, also known as a fractal pattern. We see this kind of fractal pattern in the way the jagged outline of tree branches look similar no matter whether we look at them up close or far away. Today messages disperse through the global telecommunications system in the fractal pattern of self-organization." (Kevin Kelly, "What Technology Wants", 2010)

"Random search can be more efficient than nonrandom search - something that Good and Turing had discovered at Bletchley Park. A random network, whether of neurons, computers, words, or ideas, contains solutions, waiting to be discovered, to problems that need not be explicitly defined." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Although cascading failures may appear random and unpredictable, they follow reproducible laws that can be quantified and even predicted using the tools of network science. First, to avoid damaging cascades, we must understand the structure of the network on which the cascade propagates. Second, we must be able to model the dynamical processes taking place on these networks, like the flow of electricity. Finally, we need to uncover how the interplay between the network structure and dynamics affects the robustness of the whole system." (Albert-László Barabási, "Network Science", 2016)

"It would be wrong of me to give the impression that information flow in biology is restricted to gene regulatory networks. Unfortunately, the additional complexity of some other networks makes them even harder to model computationally, especially as the simple version of 0s and 1s (off and on) mostly won’t do. On top of that, the number of components skyrockets when it comes to more finely tuned functions like metabolism. The general point remains: biology will ‘stand out’ from random complexity in the manner of its information patterning and processing, and though complex, the software account of life will still be vastly simpler than the underlying molecular systems that support it, as it is for electronic circuits." (Paul Davies, "The Demon in the Machine: How Hidden Webs of Information Are Solving the Mystery of Life", 2019)

🕸️On Graph Theory: On Structure (-1999)

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

"[…] semantic nets [are defined] as graphical analogues of data structures representing 'facts' in a computer system for understanding natural language." (Lenhart K Schubert," "Extending the Expressive Power of Semantic Networks", Artificial Intelligence 7, 1976)

"The concepts a person uses are represented as points, and the causal links between these concepts are represented as arrows between these points. This gives a pictorial representation of the causal assertions of a person as a graph of points and arrows. This kind of representation of assertions as a graph will be called a cognitive map. The policy alternatives, all of the various causes and effects, the goals, and the ultimate utility of the decision maker can all be thought of as concept variables, and represented as points in the cognitive map. The real power of this approach ap pears when a cognitive map is pictured in graph form; it is then relatively easy to see how each of the concepts and causal relation ships relate to each other, and to see the overall structure of the whole set of portrayed assertions." (Robert Axelrod, "The Cognitive Mapping Approach to Decision Making" [in "Structure of Decision: The Cognitive Maps of Political Elites"], 1976)

"A semantic network or net represents knowledge as a net-like graph. An idea, event, situation or object almost always has a composite structure; this is represented in a semantic network by a corresponding structure of nodes (drawn as circles or boxes) representing conceptual units, and directed links (drawn as arrows between the nodes) representing the relations between the units. […] An abstract (graph-theoretic) network can be diagrammed, defined mathematically, programmed in a computer, or hard-wired electronically. It becomes semantic when you assign a meaning to each node and link. Unlike specialized networks and diagrams, semantic networks aim to represent any kind of knowledge which can be described in natural language. A semantic network system includes not only the explicitly stored net structure but also methods for automatically deriving from that a much larger structure or body of implied knowledge." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"The essential idea of semantic networks is that the graph-theoretic structure of relations and. abstractions can be used for inference as well as understanding. […] A semantic network is a discrete structure as is any linguistic description. Representation of the continuous 'outside world' with such a structure is necessarily incomplete, and requires decisions as to which information is kept and which is lost." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"There is a multilayering of global networks in the key strategic activities that structure and destructure the planet. When these multilayered networks overlap in some node, when there is a node that belongs to different networks, two major consequences follow. First, economies of synergy between these different networks take place in that node: between financial markets and media businesses; or between academic research and technology development and innovation; between politics and media." (Manuel Castells, "The Rise of the Network Society", 1996)

"Graphs are one of the unifying themes of computer science - an abstract representation that describes the organization of transportation systems, human interactions, and telecommunication networks. That so many different structures can be modeled using a single formalism is a source of great power to the educated programmer." (Steven S Skiena, "The Algorithm Design Manual", 1997)

"Neural networks conserve the complexity of the systems they model because they have complex structures themselves. Neural networks encode information about their environment in a distributed form. […] Neural networks have the capacity to self-organise their internal structure." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"The internal structure of a connectionist network develops through a process of self-organisation, whereas rule-based systems have to search through pre-programmed options that define the structure largely in an a priori fashion. In this sense, learning is an implicit characteristic of neural networks. In rule-based systems, learning can only take place through explicitly formulated procedures." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998) 

🕸️On Graph Theory: On Structure (2000-2009)

"Average path length reflects the global structure; it depends on the way the entire network is connected, and cannot be inferred from any local measurement. Clustering reflects the local structure; it depends only on the interconnectedness of a typical neighborhood, the inbreeding among nodes tied to a common center. Roughly speaking, path length measures how big the network is. Clustering measures how incestuous it is." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Nodes and connectors comprise the structure of a network. In contrast, an ecology is a living organism. It influences the formation of the network itself." (George Siemens, "Knowing Knowledge", 2006)

"For the study of the topology of the interactions of a complex system it is of central importance to have proper random null models of networks, i.e., models of how a graph arises from a random process. Such models are needed for comparison with real world data. When analyzing the structure of real world networks, the null hypothesis shall always be that the link structure is due to chance alone. This null hypothesis may only be rejected if the link structure found differs significantly from an expectation value obtained from a random model. Any deviation from the random null model must be explained by non-random processes." (Jörg Reichardt, "Structure in Complex Networks", 2009)

"Self-organizing networks suffer various types of random damage. Therefore, if the network remained static, it would soon become dysfunctional. Some networks have developed highly specific screening systems which recognize and repair random damage. On the one hand, this process requires energy, which arrives in the form of perturbations or noise. On the other hand, noise-triggered network restructuring will repeat a few steps of the original self-organization and therefore constitutes a much cheaper way of providing a continuous repair function, with the additional advantage that it is always adaptive with respect to the actual environment of the network." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009) 

"To understand, how noise is related to scale-freeness, we have to do some mathematics again. Noise is usually characterized by a mathematical trick. The seemingly random fluctuation of the signal is regarded as a sum of sinusoidal waves. The components of the million waves giving the final noise structure are characterized by their frequency. To describe noise, we plot the contribution (called spectral density) of the various waves we use to model the noise as a function of their frequency. This transformation is called a Fourier transformation [...]" (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)

🕸️On Graph Theory: On Structure (2010-2019)

"As it turns out, complex networks are everywhere. Or, to be more precise, it turns out that if we model real-world situations in terms of networks, we often discover new things. What is striking, is that many real-world net- works look alike: the structure of the Internet resembles the organization of our brain, but also the organization of online social communities. Where these similarities come from is still a mystery, just as it is often very difficult to understand how certain networks were actually structured." (Maarten van Steen, "Graph Theory and Complex Networks: An Introduction", 2010)

"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)

"[...] the contexts in which a social network is embedded will generally have significant effects on its structure. Each individual in a social network has a distinctive set of personal characteristics, and similarities and compatibilities between two people’s characteristics can strongly influence whether a link forms between them. Each individual also engages in a set of behaviors and activities that can shape the formation of links within the network. These considerations suggest what we mean by a network’s surrounding contexts: factors that exist outside the nodes and edges of a networks, but which nonetheless affect how the network’s structure evolves." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"When people talk about the 'connectedness' of a complex system, in general they are really talking about two related issues. One is connectedness at the level of structure – who is linked to whom – and the other is connectedness at the level of behavior – the fact that each individual’s actions have implicit consequences for the outcomes of everyone in the system."(David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)

"Cybernetics is the study of systems which can be mapped using loops (or more complicated looping structures) in the network defining the flow of information. Systems of automatic control will of necessity use at least one loop of information flow providing feedback." (Alan Scrivener, "A Curriculum for Cybernetics and Systems Theory", 2012)

"All living systems are networks of smaller components, and the web of life as a whole is a multilayered structure of living systems nesting within other living systems - networks within networks." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"The great strength of node–link layouts is that for sufficiently small networks they are extremely intuitive for supporting many ofthe abstract tasks that pertain to network data. They particularly shine for tasks that rely on understanding the topological structure of the network, such as path tracing and searching local topological neighborhoods a small number of hops from a target node, and can also be very effective for tasks such as general overview or finding similar substructures. The effectiveness of the general idiom varies considerably depending on the specific visual encoding idiom used [...]" (Tamara Munzner, "Visualization Analysis and Design", 2014)

"Although cascading failures may appear random and unpredictable, they follow reproducible laws that can be quantified and even predicted using the tools of network science. First, to avoid damaging cascades, we must understand the structure of the network on which the cascade propagates. Second, we must be able to model the dynamical processes taking place on these networks, like the flow of electricity. Finally, we need to uncover how the interplay between the network structure and dynamics affects the robustness of the whole system." (Albert-László Barabási, "Network Science", 2016)

🕸️On Graph Theory: On Structure (2020-2029)

"Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlinear systems is mathematically unclear. […] The emergence of exponential growth from a multivariable nonlinear network is not mathematically intuitive. This indicates that the network structure and the flux functions of the modeled system must be subjected to constraints to result in long-term exponential dynamics." (Wei-Hsiang Lin et al, "Origin of exponential growth in nonlinear reaction networks", PNAS 117 (45), 2020

"A graph is a simple and quite old mathematical concept: a data structure consisting of a set of vertices (or nodes/points) and edges (or relationships/lines) that can be used to model relationships among a collection of objects." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)

"Graphs are useful for representing how things are either physically or logically linked in simple or complex structures. A graph in which we assign names and meanings to the edges and vertices becomes what is known as a network. In these cases, a graph is the mathematical model for describing a network, whereas a network is a set of relations between objects, which could include people, organizations, nations, items found in a Google search, brain cells, or electrical transformers." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)

"In many areas of machine learning, graphs are used to model local relationships between data elements and to build global structures from local information. Building graphs is sometimes necessary for dealing with problems arising from applications in machine learning or data mining, and at other times, it's helpful for managing data. It's important to note that the transformation from the original data to a graph data representation can always be performed in a lossless manner. The opposite is not always true." (Alessandro Negro, "Graph-Powered Machine Learning", 2021)

"Spanning trees capture the connectedness of a graph in the most efficient way, and they provide a foundation for a systematic analysis of the cycle structure of a graph. Mathematicians regard the algebraic structures underlying the collection of cycles and edge-cuts of agraph as beautiful in their own right. Establishing connections between linear algebra and graph theory provides some powerful analytical tools for understanding a graph’s structure." (Jonathan L Gross et al, "Topics in Graph Theory", 2023)

"A small-world network is characterized by a high clustering coefficient and a short average path length, meaning that most nodes can be reached from any other node through a small number of intermediate connections. This structure often mirrors real-world social networks, where individuals are typically connected through a few mutual acquaintances, allowing for rapid information dissemination." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

"Assortativity is used to quantify the tendency of nodes being connected to similar nodes, which can impact the network’s ability to withstand failures or 'attacks'. High assortativity indicates that nodes of similar degrees are more likely to be connected, leading to a resilient structure where the failure of some nodes does not significantly disrupt overall connectivity. Conversely, networks with low assortativity tend to have nodes connecting with dissimilar degrees, making them more vulnerable to targeted attacks on high-degree nodes [...]" (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

"The concept of temporal graphs is useful in all the real-world problems that can be represented as a graph, where the nodes and edges of the graph may change over time. For example, temporal graphs are extensively applied in modeling social networks. By capturing the evolving relationships between individuals, temporal graphs enable a more accurate representation of social dynamics. This is particularly useful for predicting changes in friendships, community structures, and the information diffusion over time." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)