Michael Barnsley - Collected Quotes

"Elementary functions, such as trigonometric functions and rational functions, have their roots in Euclidean geometry. They share the feature that when their graphs are 'magnified' sufficiently, locally they 'look like' straight lines. That is, the tangent line approximation can be used effectively in the vicinity of most points. Moreover, the fractal dimension of the graphs of these functions is always one. These elementary 'Euclidean' functions are useful not only because of their geometrical content, but because they can be expressed by simple formulas. We can use them to pass information easily from one person to another. They provide a common language for our scientific work. Moreover, elementary functions are used extensively in scientific computation, computer-aided design, and data analysis because they can be stored in small files and computed by fast algorithms." (Michael Barnsley, "Fractals Everwhere", 1988)

"For a map to be useful it must have information marked on it, such as heights above sea-level, population densities, roads, vegetation, rainfall, types of underlying rock, ownership, names, incidence of volcanoes, malarial infesta- and so on. A good way of providing such information is with colors. For example, if we use blue for water and green for land then we can 'see' the land on the map and we can understand some geometrical relationships. We can estimate overland distances between points, land areas of islands, the shortest sea passage from Llanellian Bay to Amylwch Harbour, the length of the coastline, etc. All this is achieved through the device of marking some colors on a blank map!" (Michael Barnsley, "Fractals Everwhere", 1988)

"Fractal geometry is concerned with the description, classification, analysis, and observation of subsets of metric spaces (X, d). The metric spaces are usually, but not always, of an inherently 'simple' geometrical character; the subsets are typically geometrically 'complicated'. There are a number of general properties of subsets of metric spaces, which occur over and over again, which are very basic, and which form part of the vocabulary for describing fractal sets and other subsets of metric spaces. Some of these properties, such as openness and closedness, which we are going to introduce, are of a topological character. That is to say, they are invariant under homeomorphism." (Michael Barnsley, "Fractals Everwhere", 1988)

"How big is a fractal? When are two fractals similar to one another in some sense? What experimental measurements might we make to tell if two different fractals may be metrically equivalent? [...] There are various numbers associated with fractals which can be used to compare them. They are generally referred to as fractal dimensions. They are attempts to quantify a subjective feeling which we have about how densely the fractal occupies the metric space in which it lies. Fractal dimensions provide an objective means for comparing fractals." (Michael Barnsley, "Fractals Everwhere", 1988)

"In deterministic fractal geometry the focus is on those subsets of a space which are generated by, or possess invariance properties under, simple geometrical transformations of the space into itself. A simple geometrical transformation is one which is easily conveyed or explained to someone else. Usually they can be completely specified by a small set of parameters." (Michael Barnsley, "Fractals Everwhere", 1988)

"In deterministic geometry, structures are defined, communicated, and analysed, with the aid of elementary transformations such as affine transfor- transformations, scalings, rotations, and congruences. A fractal set generally contains infinitely many points whose organization is so complicated that it is not possible to describe the set by specifying directly where each point in it lies. Instead, the set may be defined by "the relations between the pieces." It is rather like describing the solar system by quoting the law of gravitation and stating the initial conditions. Everything follows from that. It appears always to be better to describe in terms of relationships." (Michael Barnsley, "Fractals Everwhere", 1988)

"In fractal geometry we are especially interested in the geometry of sets, and in the way they look when they are represented by pictures. Thus we use the restrictive condition of metric equivalence to start to define mathematically what we mean when we say that two sets are alike. However, in dynamical systems theory we are interested in motion itself, in the dynamics, in the way points move, in the existence of periodic orbits, in the asymptotic behavior of orbits, and so on. These structures are not damaged by homeomorphisms, as we will see, and hence we say that two dynamical systems are alike if they are related via a homeomorphism." (Michael Barnsley, "Fractals Everwhere", 1988)

"The observation by Mandelbrot of the existence of a "Geometry of Nature" has led us to think in a new scientific way about the edges of clouds, the profiles of the tops of forests on the horizon, and the intricate moving arrangement of the feathers on the wings of a bird as it flies. Geometry is concerned with making our spatial intuitions objective. Classical geometry provides a first approximation to the structure of physical objects; it is the language which we use to communicate the designs of technological*products, and, very approximately, the forms of natural creations. Fractal geometry is an extension of classical geometry. It can be used to make precise models of physical structures from ferns to galaxies. Fractal geometry is a new language. Once you can speak it, you can describe the shape of a cloud as precisely as an architect can describe a house." (Michael Barnsley, "Fractals Everwhere", 1988)

"We must be careful how we interpret a map. Geographical maps are complicated by the real number system and the unphysical notion of infinite divisibility. Mathematically, the map is an abstract place. A point on the map cannot represent a certain physical atom in the real world, not just because of inaccuracies in the map, but because of the dual nature of matter: according to current theories one cannot know the exact location of an atom, at a given instant." (Michael Barnsley, "Fractals Everwhere", 1988)

Daniel Fleisch - Collected Quotes

"Among the differences that will always be with you are the small overshoots and oscillations just before and after the vertical jumps in the square waves. This is called 'Gibbs ripple' and it will cause an overshoot of about 9% at the discontinuities of the square wave no matter how many terms of the series you add. But [...] adding more terms increases the frequency of the Gibbs ripple and reduces its horizontal extent in the vicinity of the jumps." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"An understanding of complex numbers can make the study of waves consid erably less mysterious, and you probably already have an idea that complex numbers have real and imaginary parts. Unfortunately, the term 'imaginary' often leads to confusion about the nature and usefulness of complex numbers." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"As the mechanical wave source moves through the medium, it pushes on a nearby segment of the material, and that segment moves away from the source and is compressed (that is, the same amount of mass is squeezed into a smaller volume, so the density of the segment increases). That segment of increased density exerts pressure on adjacent segments, and in this way a pulse (if the source gives a single push) or a harmonic wave (if the source oscillates back and forth) is generated by the source and propagates through the material." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"Before considering the wave equation for mechanical waves, you should understand the difference between the motion of individual particles and the motion of the wave itself. Although the medium is disturbed as a wave goes by, which means that the particles of the medium are displaced from their equilibrium positions, those particles don’t travel very far from their undisturbed positions. The particles oscillate about their equilibrium positions, but the wave does not carry the particles along – a wave is not like a steady breeze or an ocean current which transports material in bulk from one location to another. For mechanical waves, the net displacement of material produced by the wave over one cycle, or over one million cycles, is zero. So, if the particles aren’t being carried along with the wave, what actually moves at the speed of the wave? […] the answer is energy." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"But the presence of √−1 (the rotation operator between the two perpendicular numbe rlines in the complex plane) in the exponent causes the expression e^ix to move from the real to the imaginary number line. As it does so, its real and imaginary parts oscillate in a sinusoidal fashion […] So the real and imaginary parts of the expression e^ix oscillate in exactly the same way as the real and imaginary components of the rotating phasor […]" (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"So a very useful way to think about i (√−1) is as an operator that produces a 90◦ rotation of any vector to which it is applied. Thus the two perpendicular number lines form the basis of what we know today as the complex plane. Unfortunately, since multiplication by √−1 is needed to get from the horizontal to the vertical number line, the numbers along the vertical number line are called 'imaginary'. We say 'unfortunately' because these numbers are every bit as real as the numbers along the horizontal number line. But the terminology is pervasive, so when you first learned about complex numbers, you probably learned that they consist of a “real” and an 'imaginary' part." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"That’s where boundary conditions come in. A boundary condition 'ties down' a function or its derivative to a specified value at a specified location in space or time. By constraining the solution of a differential equation top satisfy the boundary condition(s), you may be able to determine the value of the function or its derivatives at other locations. We say “may” because boundary conditions that are not well-posed may provide insufficient or contradictory information." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"The 'disturbance' of such waves involves three things: the longitudinal displacement of material, changes in the density of the material, and variation of the pressure within the material. So pressure waves could also be called 'density waves' or even 'longitudinal displacement waves', and when you see graphs of the wave disturbance in physics and engineering textbooks, you should make sure you understand which of these quantities is being plotted as the 'displacement' of the wave." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"The easiest way to think about the shape of a wave is to imagine taking a snapshot of the wave at some instant of time. To keep the notation simple, you can call the time at which the snapshot is taken t = 0; snapshots taken later will be timed relative to this first one. At the time of that first snapshot […] can be written as y = f(x, 0) […] Many waves maintain the same shape over time – the wave moves in the direction of propagation, but all peaks and troughs move in unison, so the shape does not change as the wave moves." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"This equation is considered by some mathematicians and physicists to be the most important equation ever devised. In Euler’s relation, both sides of the equation are expressions for a complex number on the unit circle. The left side emphasizes the magnitude (the 1 multiplying e^iθ ) and direction in the complex plane (θ), while the right side emphasizes the real (cos θ) and imaginary (sin θ) components. Another approach to demonstrating the equivalence of the two sides of Euler’s relation is to write out the power-series representation of each side; [...]" (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"When you encounter the classical wave equation, it’s likely to be accompanied by some or all of the words 'linear, homogeneous, second-order partial differential equation'. You may also see the word 'hyperbolic' included in the list of adjectives. Each of these terms has a very specific mathematical meaning that’s an important property of the classical wave equation. But there are versions of the wave equation to which some of these words don’t apply, so it’s useful to spend some time understanding them." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

"Why are boundary conditions important in wave theory? One reason is this: Differential equations, by their very nature, tell you about the change in a function (or, if the equation involves second derivatives, about the change in the change of the function). Knowing how a function changes is very useful, and may be all you need in certain problems. But in many problems you wish to know not only how the function changes, but also what value the function takes on at certain locations or times." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)

Emile Cheysson - Collected Quotes

"If statistical graphics, although born just yesterday, extends its reach every day, it is because it replaces long tables of numbers and it allows one not only to embrace at glance the series of phenomena, but also to signal the correspondence or anomalies, to find the causes, to identify the laws." (Émile Cheysson, circa 1877)

"It is this combination of observation at the foundation and geometry at the summit that I wished to express by naming this method Geometric Statistics. It cannot be subject to the usual criticisms directed at the use of pure mathematics in economic matters, which are said to be too complex to be confined within a formula." (Emile Cheysson, "La Statistique géométrique", 1888)

"It then becomes a method of graphical interpolation or extrapolation, which involves hypothetically extending a curve within or beyond the range of known data points, assuming the continuity of its pattern. In this way, one can fill in gaps in past observations and even probe the depths of the future." (Emile Cheysson, "La Statistique géométrique", 1888)

"This method is what I call Geometric Statistics. But despite its somewhat forbidding name-which I’ll explain in a moment - it is not a mathematical abstraction or a mere intellectual curiosity accessible only to a select few. It is intended, if not for all merchants and industrialists, then at least for that elite who lead the masses behind them. Practice is both its starting point and its destination. It was inspired in me more than fifteen years ago by the demands of the profession, and if I’ve decided to present it today, it’s because I’ve since verified its advantages through various applications, both in private industry and in public service." (Emile Cheysson, "La Statistique géométrique", 1888)

"Graphical statistics thus possess a variety of resources that it deploys depending on the case, in order to find the most expressive and visually appealing way to depict the phenomenon. One must especially avoid trying to convey too much at once and becoming obscure by striving for completeness. Its main virtue - or one might say, its true reason for being - is clarity. If a diagram becomes so cluttered that it loses its clarity, then it is better to use the numerical table it was meant to translate." (Emile Cheysson, "Albume de statistique graphique", 1889)

"This method not only has the advantage of appealing to the senses as well as to the intellect, and of illustrating facts and laws to the eye that would be difficult to uncover in long numerical tables. It also has the privilege of escaping the obstacles that hinder the easy dissemination of scientific work - obstacles arising from the diversity of languages and systems of weights and measures among different nations. These obstacles are unknown to drawing. A diagram is not German, English, or Italian; everyone immediately grasps its relationships of scale, area, or color. Graphical statistics are thus a kind of universal language, allowing scholars from all countries to freely exchange their ideas and research, to the great benefit of science itself." (Emile Cheysson, "Albume de statistique graphique", 1889)

"Today, there is hardly any field of human activity that does not make use of graphical statistics. Indeed, it perfectly meets a dual need of our time: the demand for information that is both rapid and precise. Graphical methods fulfill these two conditions wonderfully. They allow us not only to grasp an entire series of phenomena at a glance, but also to highlight relationships or anomalies, identify causes, and extract underlying laws. They advantageously replace long tables of numbers, so that - without compromising the precision of statistics - they broaden and popularize its benefits." (Emile Cheysson, "Albume de statistique graphique", 1889)

"When a law is contained in figures, it is buried like metal in an ore; it is necessary to extract it. This is the work of graphical representation. It points out the coincidences, the relationships between phenomena, their anomalies, and we have seen what a powerful means of control it puts in the hands of the statistician to verify new data, discover and correct errors with which they have been stained." (Emile Cheysson, "Les methods de la statistique", 1890)

Sources: Bibliothéque Nationale de la France [>>] 

Siméon-Denis Poisson - Collected Quotes

"For each of the elements into which we have divided the amount of fluid matter, its shape will be altered during the time dt, and also its volume will change if the fluid is compressible; but, since its mass must remain unaltered, it follows that, if we seek to determine its volume and its density at the end of time t + dt, their product will necessarily be the same as after time t. (Siméon-Denis Poisson, "Traité de Méecanique" vol. II, 1811)

"In many different fields, empirical phenomena appear to obey a certain general law, which can be called the Law of Large Numbers. This law states that the ratios of numbers derived from the observation of a very large number of similar events remain practically constant, provided that these events are governed partly by constant factors and partly by variable factors whose variations are irregular and do not cause a systematic change in a definite direction." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"In ordinary life, the words chance and probability are almost synonymous and most often used indifferently. However, if necessary to distinguish their meaning, we attach here the word chance to events taken independently from our knowledge, and retain its previous definition [!] for the word probability. Thus, by its nature an event has a greater or lesser chance, known or unknown, whereas its probability is relative to our knowledge about it." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"[...] in the game of heads or tails, the arrival of heads results from the constitution of the tossed coin. It can be regarded as physically impossible that the chances of both outcomes are the same; however, if that constitution is unknown to us, and we did not yet try out the coin, the probability of heads is for us absolutely the same as that of tails. Actually, we have no reason to believe in one of these events rather than in the other one. This will not be the same after many tosses of the coin: the chance of each side does not change during the trials, but for someone who knows their results, the probability of the future occurrence of heads and tails varies in accord with the number of times they happened." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The measure of the probability of an event is the ratio of the number of cases favourable for it to the total number of favourable andcontrary cases, all of them equally possible or having the same chance. That proposition signifies that when this ratio is the same for two events, we have the same reason to believe in the occurrence of either of them. Otherwise, we have more reason to believe in the arrival of that event for which it is larger." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The probability of an event is our reason to believe that it will occur or occurred. [...] Probability depends on our knowledge about an event; for the same event it can differ for different persons. Thus, if a person only knows that an urn contains white and black balls, whereas another person alsoknows that there are more white balls than black ones, the latter has more grounds to believe in the extraction of a white ball. In other words, for him, that event has a higher probability than for the former." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"[...] the rules for establishing the probability of an observed event given the probability of its cause, which are the basis of the theory under consideration, require taking into account all the presumptions prior to the observation, if only they are thought to exist, or if proven that they are not absent." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The calculus of probability is equally applicable to things of all kinds, moral and physical and, if only in each case observations provide the necessary numerical data, it does not at all depend on their nature." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The constancy of ratios between the number of times that an event had occurred and the very large number of trials, which establishes itself and is manifested in spite of the variations of the chance of that event during these trials, tempts us to attribute this remarkable regularity to some ceaselessly acting occult cause. However, the theory of probability determines that the constancy of those ratios is a natural state of things belonging to physical and moral categories and maintains all by itself without any aid by some alien cause. On the contrary, it can only be hindered or disturbed by an intervention of a similar [alien] cause." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"[...] the law of large numbers governs phenomena produced by known forces acting together with accidental causes whose effect lacks any regularity." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The law of large numbers is noted in events which are attributed to pure chance since we do not know their causes or because they are too complicated. Thus, games, in which the circumstances determining the occurrence of a certain card or certain number of points on a die infinitely vary, can not be subjected to any calculus. If the series of trials is continued for a long time, the different outcomes nevertheless appear in constant ratios. Then, if calculations according to the rules of a game are possible, the respective probabilities of eventual outcomes conform to the known Jakob Bernoulli theorem. However, in most problems of contingency a prior determination of chances of the various events is impossible and, on the contrary, they are calculated from the observed result." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen)." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The phenomena of any kind are subject to a general law, which one can call the Law of Large Numbers. It consists in the fact, that, if one observes very large numbers of phenomena of the same kind depending on constant or irregularly changeable causes, however not progressively changeable, but one moment in the one sense, the other moment in the other sense; one finds ratios of these numbers which are almost constant." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"The probability of an event is the reason we have to believe that it has taken place, or that it will take place." (Siméon-Denis Poisson, "Règles générales des probabilités", 1837)

"Things of all kinds are subject to a universal law which may be called the law of large numbers. It consists in the fact that, if one observes very considerable numbers of events of the same nature, dependent on constant causes and causes which vary irregularly, sometimes in one direction, sometimes in the other, it is to say without their variation being progressive in any definite direction, one shall find, between these numbers, relations which are almost constant." (Siméon-Denis Poisson, "Poisson’s Law of Large Numbers", 1837)

"Without the aid of the calculus of probability you run a great risk of being mistaken about the necessity of that conclusion. However, the calculus leaves nothing vague here and in addition provides necessary rules for determining the chance of the change of the causes indicated by comparing the observed facts at different times." (Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", 1837)

"That which can affect our senses in any manner whatever, is termed matter." (Siméon-Denis Poisson, "A Treatise of Mechanics", 1842)

"Life is good for only two things, discovering mathematics and teaching mathematics." (Simeon-Denis Poisson) [in Mathematical Magazine, Volume 64, Number 1, February 1991]

"That which can affect our senses in any manner whatever, is termed matter." (Siméon-Denis Poisson) 

"The engineer should receive a complete mathematical education, but for what should it serve him? To see the different aspects of things and to see them quickly; he has no time to hunt mice." (Siméon-Denis Poisson)

References: Siméon-Denis Poisson, "Researches into the Probabilities of Judgements in Criminal and Civil Cases", "Règles générales des probabilités", 1837 [source] 

Peter Borwein - Collected Quotes

"Arguments for the Riemann hypothesis often include its widespread ramifications and appeals to mathematical beauty; however, we also have a large corpus of hard facts. With the advent of powerful computational tools over the last century, mathematicians have increasingly turned to computational evidence to support conjectures, and the Riemann hypothesis is no exception." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"In one of the largest calculations done to date, it was checked that the first ten trillion of these zeros lie on the correct line. So there are ten trillion pieces of evidence indicating that the Riemann hypothesis is true and not a single piece of evidence indicating that it is false. A physicist might be overwhelmingly pleased with this much evidence in favour of the hypothesis, but to some mathematicians this is hardly evidence at all. However, it is interesting ancillary information." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"Number-theoretic equivalences of the Riemann hypothesis provide a natural method of explaining the hypothesis to nonmathematicians without appealing to complex analysis. While it is unlikely that any of these equivalences will lead directly to a solution, they provide a sense of how intricately the Riemann zeta function is tied to the primes"  (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"One of the current ideas regarding the Riemann hypothesis is that the zeros of the zeta function can be interpreted as eigenvalues of certain matrices. This line of thinking is attractive and is potentially a good way to attack the hypothesis, since it gives a possible connection to physical phenomena. [...] Empirical results indicate that the zeros of the Riemann zeta function are indeed distributed like the eigenvalues of certain matrix ensembles, in particular the Gaussian unitary ensemble. This suggests that random matrix theory might provide an avenue for the proof of the Riemann hypothesis." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"So the prime number theorem is a relatively weak statement of the fact that an integer has equal probability of having an odd number or an even number of distinct prime factors." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"Solving any of the great unsolved problems in mathematics is akin to the first ascent of Everest. It is a formidable achievement, but after the conquest there is sometimes nowhere to go but down. Some of the great problems have proven to be isolated mountain peaks, disconnected from their neighbors. The Riemann hypothesis is quite different in this regard. There is a large body of mathematical speculation that becomes fact if the Riemann hypothesis is solved. We know many statements of the form “if the Riemann hypothesis, then the following interesting mathematical statement”, and this is rather different from the solution of problems such as the Fermat problem." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"The first and easies tproofs [of the prime number theorem] are analytic and exploit the rich connections between number theory and complex analysis. It has resisted trivialization, and no really easy proof is known. This is especially true for the so-called elementary proofs, which use little or no complex analysis, just considerable ingenuity and dexterity. The primes arise sporadically and, apparently, relatively randomly, at least in thes ense that there is no easy way to find a large prime number with no obvious congruences. So even the amount of structure implied by the prime number theorem is initially surprising." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

"Why is the Riemann hypothesis so important? Why is it the problem that many mathematicians would sell their souls to solve? There are a number of great old unsolved problems in mathematics, but none of them has quite the stature of the Riemann hypothesis. This stature can be attributed to a variety of causes ranging from mathematical to cultural. As with the other old great unsolved problems, the Riemann hypothesis is clearly very difficult. It has resisted solution for 150 years and has been attempted by many of the greatest minds in mathematics." (Peter Borwein et al, "The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike", 2007)

Steve McKillup - Collected Quotes

"A correlation between two variables means they vary together. A positive correlation means that high values of one variable are associated with high values of the other, while a negative correlation means that high values of one variable are associated with low values of the other." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Accuracy is the closeness of a measured value to the true value. Precision is the ‘spread’ or variability of repeated measures of the same value." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Correlation is an exploratory technique used to examine whether the values of two variables are significantly related, meaning whether the values of both variables change together in a consistent way. (For example, an increase in one may be accompanied by a decrease in the other.) There is no expectation that the value of one variable can be predicted from the other, or that there is any causal relationship between them." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Designing a well-controlled, appropriately replicated and realistic experiment has been described by some researchers as an ‘art’. It is not, but there are often several different ways to test the same hypothesis, and hence several different experiments that could be done. Consequently, it is difficult to set a guide to designing experiments beyond an awareness of the general principles." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Even an apparently well-designed mensurative or manipulative experiment may still suffer from a lack of realism." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"First, if you already know that the population from which your sample has been taken is normally distributed (perhaps you have data for a variable that has been studied before), you can assume the distribution of sample means from this population will also be normally distributed. Second, the central limit theorem […] states that the distribution of the means of samples of about 25 or more taken from any population will be approximately normal, provided the population is not grossly non-normal (e.g. a population that is bimodal). Therefore, provided your sample size is sufficiently large you can usually do a parametric test. Finally, you can examine your sample. Although there are statistical tests for normality, many statisticians have cautioned that these tests often indicate the sample is significantly non normal even when a t-test will still give reliable results." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Graphs may reveal patterns in data sets that are not obvious from looking at lists or calculating descriptive statistics. Graphs can also provide an easily understood visual summary of a set of results." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Inaccurate and imprecise measurements or a poor or unrealistic sampling design can result in the generation of inappropriate hypotheses. Measurement errors or a poor experimental design can give a false or misleading outcome that may result in the incorrect retention or rejection of an hypothesis." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"It has often been said, ‘There is no such thing as a perfect experiment.’ One inherent problem is that, as a design gets better and better, the cost in time and equipment also increases, but the ability to actually do the experiment decreases. An absolutely perfect design may be impossible to carry out. Therefore, every researcher must choose a design that is ‘good enough’ but still practical. There are no rules for this – the decision on design is in the hands of the researcher, and will be eventually judged by their colleagues who examine any report from the work." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"It is important to realise that Type 1 error can only occur when the null hypothesis applies. There is absolutely no risk if the null hypothesis is false. Unfortunately, you are most unlikely to know if the null hypothesis applies or not - if you did know, you would not be doing an experiment to test it! If the null hypothesis applies, the risk of Type 1 error is the same as the probability level you have chosen" (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Linear correlation analysis assumes that the data are random representatives taken from the larger population of values for each variable, which are normally distributed and have been measured on a ratio, interval or ordinal scale. A scatter plot of these variables will have what is called a bivariate normal distribution. If the data are not normally distributed, or the relationship does not appear to be linear, they may be able to be analysed by nonparametric tests for correlation [...]" (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Many statistics texts do not mention this and students often ask, ‘What if you get a probability of exactly 0.05?’ Here the result would be considered not significant, since significance has been defined as a probability of less than 0.05 (<0.05). Some texts define a significant result as one where the probability is less than or equal to 0.05 ( 0.05). In practice this will make very little difference, but since Fisher proposed the ‘less than 0.05’ definition, which is also used by most scientific publications, it will be used here." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"No hypothesis or theory can ever be proven - one day there may be evidence that rejects it and leads to a different explanation (which can include all the successful predictions of the previous hypothesis).Consequently we can only falsify or disprove hypotheses and theories – we can never ever prove them." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"One of the nastiest pitfalls is appearing to have a replicated manipulative experimental design, which really is not replicated." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"One way of generating hypotheses is to collect data and look for patterns. Often, however, it is difficult to see any pattern from a set of data, which may just be a list of numbers. Graphs and descriptive statistics are very useful for summarising and displaying data in ways that may reveal patterns." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Parametric tests are designed for analyzing data from a known distribution, and the majority assume a normally distributed population. Although parametric tests are quite robust to departures from normality, and major ones can often be reduced by transformation, there are some cases where the population is so grossly non-normal that parametric testing is unwise. In these cases a powerful analysis can often still be done by using a non-parametric test." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Sample statistics like the mean, variance, standard deviation, and especially the standard error of the mean are estimates of population statistics that can be used to predict the range within which 95% of the means of a particular sample size will occur. Knowing this, you can use a parametric test to estimate the probability that a sample mean is the same as an expected value, or the probability that the means of two samples are from the same population." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Statistical tests are just a way of working out the probability of obtaining the observed, or an even more extreme, difference among samples (or between an observed and expected value) if a specific hypothesis (usually the null of no difference) is true. Once the probability is known, the experimenter can make a decision about the difference, using criteria that are uniformly used and understood." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"The essential features of the ‘hypothetico-deductive’ view of scientific method are that a person observes or samples the natural world and uses all the information available to make an intuitive, logical guess, called an hypothesis, about how the system functions. The person has no way of knowing if their hypothesis is correct - it may or may not apply. Predictions made from the hypothesis are tested, either by further sampling or by doing experiments. If the results are consistent with the predictions then the hypothesis is retained. If they are not, it is rejected, and a new hypothesis formulated." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"The unavoidable problem with using probability to help you make a decision is that there is always a chance of making a wrong decision and you have no way of telling when you have done this." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"The use of a t-test makes three assumptions. The first is that the data are normally distributed. The second is that each sample has been taken at random from its respective population and the third is that for an independent sample test, the variances are the same. It has, however, been shown that t-tests are actually very ‘robust’ – that is, they will still generate statistics that approximate the t distribution and give realistic probabilities even when the data show considerable departure from normality and when sample variances are dissimilar." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Unfortunately, the only way to estimate the appropriate minimum sample size needed in an experiment is to know, or have good estimates of, the effect size and standard deviation of the population(s). Often the only way to estimate these is to do a pilot experiment with a sample. For most tests there are formulae that use these (sample) statistics to give the appropriate sized sample for a desired power." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"When expected frequencies are small, the calculated chi-square statistic is inaccurate and tends to be too large, therefore indicating a lower than appropriate probability, which increases the risk of Type 1 error. It used to be recommended that no expected frequency in a goodness of fit test should be less than five, but this has been relaxed somewhat in the light of more recent research, and it is now recommended that no more than 20% of expected frequencies should be less than five." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Whenever you make a decision based on the probability of a result, there is a risk of either a Type 1 or a Type 2 error. There is only a risk of Type 1 error when the null hypothesis applies, and the risk is the chosen probability level. There is only a risk of Type 2 error when the null hypothesis is false." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

"Without compromising the risk of Type 1 error, the only way a researcher can reduce the risk of Type 2 error to an acceptable level and therefore ensure sufficient power is to increase their sample size. Every researcher has to ask themselves the question, ‘What sample size do I need to ensure the risk of Type 2 error is low and therefore power is high?’ This is an important question because samples are usually costly to take, so there is no point in increasing sample size past the point where power reaches an acceptable level." (Steve McKillup, "Statistics Explained: An Introductory Guide for Life Scientists", 2005)

David R Cox - Collected Quotes

"Exact truth of a null hypothesis is very unlikely except in a genuine uniformity trial." (David R Cox, "Some problems connected with statistical inference", Annals of Mathematical Statistics 29, 1958) 

"Assumptions that we make, such as those concerning the form of the population sampled, are always untrue." (David R Cox, "Some problems connected with statistical inference", Annals of Mathematical Statistics 29, 1958) 

"Overemphasis on tests of significance at the expense especially of interval estimation has long been condemned." (David R Cox, "The role of significance tests", Scandanavian Journal of Statistics 4, 1977) 

"There are considerable dangers in overemphasizing the role of significance tests in the interpretation of data."  (David R Cox, "The role of significance tests", Scandanavian Journal of Statistics 4, 1977) 

"In any particular application, graphical or other informal analysis may show that consistency or inconsistency with H0 is so clear cut that explicit calculation of p is unnecessary." (David R Cox, "The role of significance tests", Scandanavian Journal of Statistics 4, 1977) 

"The central point is that statistical significance is quite different from scientific significance and that therefore estimation [...] of the magnitude of effects is in general essential regardless of whether statistically significant departure from the null hypothesis is achieved." (David R Cox, "The role of significance tests", Scandanavian Journal of Statistics 4, 1977) 

"At a simpler level, some elementary but important suggestions for the clarity of graphs are as follows: (i) the axes should be clearly labelled with the names of the variables and the units of measurement; (ii) scale breaks should be used for false origins; (iii) comparison of related diagrams should be made easy, for example by using identical scales of measurement and placing diagrams side by side; (iv) scales should be arranged so that systematic and approximately linear relations are plotted at roughly 45° to the x-axis; (v) legends should make diagrams as nearly self-explanatory, i.e. independent of the text, as is feasible; (vi) interpretation should not be prejudiced by the technique of presentation, for example by superimposing thick smooth curves on scatter diagrams of points faintly reproduced." (David R Cox,"Some Remarks on the Role in Statistics of Graphical Methods", Applied Statistics 27 (1), 1978)

"Most graphs used in the analysis of data consist of points arising in effect from distinct individuals, although there are certainly other possibilities, such as the use of lines dual to points. In many cases of exploratory analysis, however, the display of supplementary information attached to some or all of the points will be crucial for successful interpretation. The primary co-ordinate axes should, of course, be chosen to express the main dependence explicitly, if not initially certainly in the final presentation of conclusions." (David R Cox,"Some Remarks on the Role in Statistics of Graphical Methods", Applied Statistics 27 (1), 1978)

"So far as is feasible, diagrams should be planned so that (a) departures from "standard" conditions should be revealed as departures from linearity, or departures from totally random scatter, or as departures of contours from circular form; (b) different points should have approximately independent errors; (c) points should have approximately equal errors, preferably known and indicated, or, if equal errors cannot be achieved, major differences in the precision of individual points should be indicated, at least roughly; (d) individual points should have clearcut interpretation; (e) variables plotted should have clearcut physical interpretation; (f) any non-linear transformations applied should not accentuate uninteresting ranges; (g) any reasonable invariance should be exploited." (David R Cox,"Some Remarks on the Role in Statistics of Graphical Methods", Applied Statistics 27 (1), 1978)

"There are two general decisions to be made when displaying supplementary information, the first concerning the amount of such information and the second the precise technique to be used. The amount of supplementary information that it is sensible to show depends on the number of points. The possibility of showing such information only for relatively extreme points and possibly for a sample of the more central points should be considered when the number of points is large; thus in a probability plot of contrasts from a large factorial experiment it may be enough to label only the more extreme values." (David R Cox,"Some Remarks on the Role in Statistics of Graphical Methods", Applied Statistics 27 (1), 1978)

"It is very bad practice to summarise an important investigation solely by a value of P." (David R Cox, "Statistical significance tests", British Journal of Clinical Pharmacology 14, 1982) 

"The criterion for publication should be the achievement of reasonable precision and not whether a significant effect has been found." (David R Cox, "Statistical significance tests", British Journal of Clinical Pharmacology 14, 1982) 

"The continued very extensive use of significance tests is alarming." (David R Cox, "Some general aspects of the theory of statistics", International Statistical Review 54, 1986) 

"It has been widely felt, probably for thirty years and more, that significance tests are overemphasized and often misused and that more emphasis should be put on estimation and prediction. While such a shift of emphasis does seem to be occurring, for example in medical statistics, the continued very extensive use of significance tests is on the one hand alarming and on the other evidence that they are aimed, even if imperfectly, at some widely felt need." (David R Cox, "Some general aspects of the theory of statistics", International Statistical Review 54, 1986) 

"Most real life statistical problems have one or more nonstandard features. There are no routine statistical questions; only questionable statistical routines." (David R Cox)

Michael J Moroney

"A statistical analysis, properly conducted, is a delicate dissection of uncertainties, a surgery of suppositions." (Michael J Moroney, "Facts from Figures", 1951)

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

"If you are young, then I say: Learn something about statistics as soon as you can. Don’t dismiss it through ignorance or because it calls for thought. [...] If you are older and already crowned with the laurels of success, see to it that those under your wing who look to you for advice are encouraged to look into this subject. In this way you will show that your arteries are not yet hardened, and you will be able to reap the benefits without doing overmuch work yourself. Whoever you are, if your work calls for the interpretation of data, you may be able to do without statistics, but you won’t do as well." (Michael J Moroney, "Facts from Figures", 1951)

"Statistics is not the easiest subject to teach, and there are those to whom anything savoring of mathematics is regarded as for ever anathema." (Michael J Moroney, "Facts from Figures", 1951)

"The statistician’s job is to draw general conclusions from fragmentary data. Too often the data supplied to him for analysis are not only fragmentary but positively incoherent, so that he can do next to nothing with them. Even the most kindly statistician swears heartily under his breath whenever this happens." (Michael J Moroney, "Facts from Figures", 1951)

"There is more than a germ of truth in the suggestion that, in all society where statisticians thrive, liberty and individuality are likely to be emasculated." (Michael J Moroney, "Facts from Figures", 1951)

Richard Levins - Collected Quotes

"A mathematical model is neither an hypothesis nor a theory. Unlike the scientific hypothesis, a model is not verifiable directly by experiment. For all models are both true and false. Almost any plausible proposed relation among aspects of nature is likely to be true in the sense that it occurs (although rarely and slightly). Yet all models leave out a lot and are in that sense false, incomplete, inadequate. The validation of a model is not that it is 'true' but that it generates good testable hypotheses relevant to important problems. A model may be discarded in favor of a more powerful one, but it usually is simply outgrown when the live issues are not any longer those for which it was designed." (Richard Levins, "The Strategy of Model Building in Population Biology", American Scientist 54(4), 1966)

"For population genetics, a population is specified by the frequencies of genotypes without reference to the age distribution, physiological state as a reflection of past history, or population density. A single population or species is treated at a time, and evolution is usually assumed to occur in a constant environment. Population ecology, on the other hand, recognizes multispecies systems, describes populations in terms of their age distributions, physiological states, and densities. The environment is allowed to vary but the species are treated as genetically homogeneous, so that evolution is ignored." (Richard Levins, "The Strategy of Model Building in Population Biology", American Scientist 54(4), 1966)

"It is of course desirable to work with manageable models which maximize generality, realism, and precision toward the overlapping but not identical goals of understanding, predicting, and modifying nature. But this cannot be done. Therefore, several alternative strategies have evolved: (1) Sacrifice generality to realism and precision. (2) Sacrifice realism to generality and precision. (3) Sacrifice precision to realism and generality." (Richard Levins, "The strategy of model building in population biology", American Scientist Vol. 54 (4), 1966) 

"The multiplicity of models is imposed by the contradictory demands of a complex, heterogeneous nature and a mind that can only cope with few variables at a time; by the contradictory desiderata of generality, realism, and precision; by the need to understand and also to control; even by the opposing esthetic standards which emphasize the stark simplicity and power of a general theorem as against the richness and the diversity of living nature. These conflicts are irreconcilable. Therefore, the alternative approaches even of contending schools are part of a larger mixed strategy. But the conflict is about method, not nature, for the individual models, while they are essential for understanding reality, should not be confused with that reality itself." (Richard Levins, "The Strategy of Model Building in Population Biology", American Scientist 54(4), 1966)

"The validation of a model is not that it is 'true' but that it generates good testable hypotheses relevant to important problems." (Richard Levins, "The Strategy of Model Building in Population Biology", American Scientist 54(4), 1966)

"[…] truth is the intersection of independent lies." (Richard Levins, "The Strategy of Model Building in Population Biology", 1966)

"Unlike the theory, models are restricted by technical considerations to a few components at a time, even in systems which are complex. Thus a satisfactory theory is usually a cluster of models. These models are related to each other in several ways : as coordinate alternative models for the same set of phenomena, they jointly produce robust theorems; as complementary models they can cope with different aspects of the same problem and give complementary as well as overlapping results; as hierarchically arranged 'nested' models, each provides an interpretation of the sufficient parameters of the next higher level where they are taken as given." (Richard Levins, "The Strategy of Model Building in Population Biology", American Scientist 54(4), 1966)

"All evolutionary theories, whether of physical, biological, or social phenomena, are theories of change. The present state of a system is seen as different from its past states, and its future states are predicted to again differ from the present. But the simple assertion that past, present, and future differ from one another is not in itself an evolutionary world view." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

"Parts and wholes evolve in consequence of their relationship, and the relationship itself evolves. These are the properties of things that we call dialectical: that one thing cannot exist without the other, that one acquires its properties from its relation to the other, that the properties of both evolve as a consequence of their interpenetration." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

"The concept of adaptation implies that there is a preexistent form, problem, or ideal to which organisms are fitted by a dynamical process. The process is adaptation and the end result is the state of being adapted. Thus a key may be adapted to fit a lock by cutting and filing it, or a part made for one model of a machine may be used in a different model by using an adaptor to alter its shape. There cannot be adaptation with out the ideal model according to which the adaptation is taking place. Thus the very notion of adaptation inevitably carried over into modern biology the theological view of a preformed physical world to which organisms were fitted." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

"The large-scale computer models of systems ecology do not fit under the heading of holism at all. Rather they are forms of large-scale reductionism: the objects of study are the naively given 'parts' -abundances or biomasses of populations. No new objects of study arise at the community level. The research is usually conducted on a single system - a lake, forest, or prairie - and the results are measurements of and projections for that lake, forest, or prairie, with no attempts to find the properties of lakes, forests, or prairies in general. Such modeling requires vast amounts of data for its simulations, and much of the scientific effort goes into problems of estimation." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

"The organism cannot be regarded as simply the passive object of autonomous internal and external forces; it is also the subject of its own evolution." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

"Things are similar: this makes science possible. Things are different: this makes science necessary. At various times in the history of science important advances have been made either by abstracting away differences to reveal similarity or by emphasizing the richness of variation within a seeming uniformity. But either choice by itself is ultimately misleading. The general does not completely contain the particular as cases, but the empiricist refusal to group, generalize, and abstract reduces science to collecting - if not specimens, then examples." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

"We believe that science, in all its sense, is a social process that both causes and is caused by social organisation." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

"We can hardly have a serious discussion of a science without abstraction. What makes science materialist is that the process of abstraction is explicit and recognized as historically contingent within the science. Abstraction becomes destructive when the abstract is reified and when the historical process of abstraction is forgotten, so that the abstract descriptions are taken for descriptions of the actual objects. The level of abstraction appropriate in a given science at a given time is a historical issue." (Richard Levins & Richard C Lewontin, "The Dialectical Biologist", 1985)

Steven Pinker - Collected Quotes

"Free will is an idealization of human beings that makes the ethics game playable. Euclidean geometry requires idealizations like infinite straight lines and perfect circles, and its deductions are sound and useful even though the world does not really have infinite straight lines or perfect circles. The world is close enough to the idealization that the theorems can usefully be applied. Similarly, ethical theory requires idealizations like free, sentient, rational, equivalent agents whose behavior is uncaused, and its conclusions can be sound and useful even though the world, as seen by science, does not really have uncaused events. As long as there is no outright coercion or gross malfunction of reasoning, the world is close enough to the idealization of free will that moral theory can meaningfully be applied to it." (Steven Pinker, "How the Mind Works", 1997)

"Suppose the reasoning centers of the brain can get their hands on the mechanisms that plop shapes into the array and that read their locations out of it. Those reasoning demons can exploit the geometry of the array as a surrogate for keeping certain logical constraints in mind. Wealth, like location on a line, is transitive: if A is richer than B, and B is richer than C, then A is richer than C. By using location in an image to symbolize wealth, the thinker takes advantage of the transitivity of location built into the array, and does not have to enter it into a chain of deductive steps. The problem becomes a matter of plop down and look up. It is a fine example of how the form of a mental representation determines what is easy or hard to think." (Steven Pinker, "How the Mind Works", 1997)

"Visual thinking is often driven more strongly by the conceptual knowledge we use to organize our images than by the contents of the images themselves. Chess masters are known for their remarkable memory for the pieces on a chessboard. But it's not because people with photographic memories become chess masters. The masters are no better than beginners when remembering a board of randomly arranged pieces. Their memory captures meaningful relations among the pieces, such as threats and defenses, not just their distribution in space."(Steven Pinker, "How the Mind Works", 1997)

"Equality is not the empirical claim that all groups of humans are interchangeable; it is the moral principle that individuals should not be judged or constrained by the average properties of their group." (Steven Pinker, The Blank Slate: The Modern Denial of Human Nature", 2002)

"Nature is a hanging judge," goes an old saying. Many tragedies come from our physical and cognitive makeup. Our bodies are extraordinarily improbable arrangements of matter, with many ways for things to go wrong and only a few ways for things to go right. We are certain to die, and smart enough to know it. Our minds are adapted to a world that no longer exists, prone to misunderstandings correctable only by arduous education, and condemned to perplexity about the deepest questions we can ascertain." (Steven Pinker, The Blank Slate: The Modern Denial of Human Nature", 2002)

"Much can be gained be contrasting a theory with its alternatives, even ones that look too extreme to be true. You can really understand something when you know what it is not." (Steven Pinker, "The Stuff of Thought: Language as a Window into Human Nature", 2005)

"Semantics is about the relation of words to thoughts, but it also about the relation of words to other human concerns. Semantics is about the relation of words to reality—the way that speakers commit themselves to a shared understanding of the truth, and the way their thoughts are anchored to things and situations in the world." (Steven Pinker, "The Stuff of Thought: Language as a Window into Human Nature", 2005)

"The better you know something, the less you remember about how hard it was to learn. The curse of knowledge is the single best explanation I know of why good people write bad prose." (Steven Pinker, "The Sense of Style: The Thinking Person's Guide to Writing in the 21st Century", 2014)

"The Second Law of Thermodynamics states that in an isolated system (one that is not taking in energy), entropy never decreases. (The First Law is that energy is conserved; the Third, that a temperature of absolute zero is unreachable.) Closed systems inexorably become less structured, less organized, less able to accomplish interesting and useful outcomes, until they slide into an equilibrium of gray, tepid, homogeneous monotony and stay there." (Steven Pinker, "The Second Law of Thermodynamics", 2017)

"Our greatest enemies are ultimately not our political adversaries but entropy, evolution (in the form of pestilence and the flaws in human nature), and most of all ignorance - a shortfall of knowledge of how best to solve our problems." (Steven Pinker, "Enlightenment Now: The Case for Reason, Science, Humanism, and Progress", 2018)

"Cognitive psychology has shown that the mind best understands facts when they are woven into a conceptual fabric, such as a narrative, mental map, or intuitive theory. Disconnected facts in the mind are like unlinked pages on the Web: They might as well not exist." (Steven Pinker)

Terry Gannon - Collected Quotes

"In modern mathematics there is a strong tendency towards formulations of concepts that minimise the number and significance of arbitrary choices. This crispness tends to emphasise the naturality of the construction or definition, at the expense sometimes of accessibility. Our mathematics is more conceptual today – more beautiful perhaps – but the cost of less explicitness is the compartmentalism that curses our discipline." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Like moonlight itself, Monstrous Moonshine is an indirect phenomenon. Just as in the theory of moonlight one must introduce the sun, so in the theory of Moonshine one must go well beyond the Monster. Much as a book discussing moonlight may include paragraphs on sunsets or comet tails, so do we discuss fusion rings, Galois actions and knot invariants." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine concerns the occurrence of modular forms in algebra and physics, and care is taken to avoid analytic complications as much as possible. But spaces here are unavoidably infinite-dimensional, and through this arise subtle but significant points of contact with analysis." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is interested in the correlation functions of a class of extremely symmetrical and well-behaved quantum field theories called rational conformal field theories - these theories are so special that their correlation functions can be computed exactly and perturbation is not required." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Moonshine is profoundly connected with physics (namely conformal field theory and string theory). String theory proposes that the elementary particles (electrons, photons, quarks, etc.) are vibrational modes on a string of length about 10^−33 cm. These strings can interact only by splitting apart or joining together – as they evolve through time, these (classical) strings will trace out a surface called the world-sheet. Quantum field theory tells us that the quantum quantities of interest (amplitudes) can be perturbatively computed as weighted averages taken over spaces of these world-sheets. Conformally equivalent world-sheets should be identified, so we are led to interpret amplitudes as certain integrals over moduli spaces of surfaces. This approach to string theory leads to a conformally invariant quantum field theory on two-dimensional space-time, called conformal field theory (CFT). The various modular forms and functions arising in Moonshine appear as integrands in some of these genus-1 (‘1-loop’) amplitudes: hence their modularity is manifest." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"Physics reduces Moonshine to a duality between two different pictures of quantum field theory: the Hamiltonian one, which concretely gives us from representation theory the graded vector spaces, and another, due to Feynman, which manifestly gives us modularity. In particular, physics tells us that this modularity is a topological effect, and the group SL2(Z) directly arises in its familiar role as the modular group of the torus." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

"The appeal of Monstrous Moonshine lies in its mysteriousness: it unexpectedly associates various special modular functions with the Monster, even though modular functions and elements of Mare conceptually incommensurable. Now, ‘understanding’ something means to embed it naturally into a broader context. Why is the sky blue? Because of the way light scatters in gases. Why does light scatter in gases the way it does? Because of Maxwell’s equations. In order to understand Monstrous Moonshine, to resolve the mystery, we should search for similar phenomena, and fit them all into the same story." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)

Jacques Ozanam - Collected Quotes

"Although the Mathematicks according to its Etymology, signifies only Discipline, yet it merits the Name of Science better than any other, because its Principles are self-evident, and independent on any sensible Experience, and its Propositions demonstrated beyond all possible Doubt or Opposition. Youth were anciently instructed herein before Philosophy, on which Account Aristotle called it the Science of Children. This was taught them not only to raise and excite their Genius, but also as a fit preparative to the Study of Nature; and it was upon this Account that the Divine Plato inscribed on his School... that none wholly ignorant of Geometry should be admitted there." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"By Science is understood a Knowledge acquired by, or founded on clear and self evident Principles, whence it follows that the Mathematicks may truly be stiled such." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"Mathematicks therefore is a Science which teaches or contemplates whatever is capable of Measure or Number as such. When it relates to Number, it is called Arithmetick; but when to measure, as Length, Breadth, Depth, Degrees of Velocity in Motion, Intenseness or Remissness of Sounds, Augmentation or Diminution of Quality, etc. it is called Geometry." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"The Essential Parts of the Simple or Pure Mathematicks are Arithmetick and Geometry, which mutually assist one another, and are independent on any other Sciences, except perhaps on Artificial Logick: But doubtless Natural Logick may be sufficient to a Man of Sense. The other parts are chiefly Physical Subjects explained by the Principles of Arithmetics or Geometry." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702)

"The Usefulness of the Mathematicks in General, and of some Parts of them in Particular, in the common Affairs of Humane Life, has rendered some competent Knowledge of them very necessary to a great Part of Mankind, and very convenient to all the Rest that are any way conversant beyond the Limits of their own particular callings." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"There are two general Methods made use of in the Mathematicks, viz. Synthesis and Analysis, which we shall explain, after having acquainted the Reader, that the Method we make use of to resolve a Mathematical Problem, is called Zetetick; and that that Method which determines when, and by what way, and how many different ways a Problem may be resolved, is called Poristick. But in treating of Methods, we will first premise, that in general, a Method is the Art of disposing a Train of Arguments or Consequences in a right Order, either to discover the Truth of a Theorem, which we would find out, or to demonstrate it to others, when found." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"To be perfectly ignorant in all the Terms of them is only tolerable in those, who think their Tongues of as little Use to them, as generally their Understandings are. Those whom Necessity has obliged to get their Bread by Manual Industry, where some Degree of Art is required to go along with it, and who have had some Insight into these Studies, have very often found Advantages from them sufficient to reward the Pains they were at in acquiring them. And whatever may have been imputed (how justly I'm not now to determine) to some other Studies, under the Notion of Insignificancy and Loss of Time ; yet these, I believe, never caused Repentance in any, except it was for their Remissness in the Prosecution of them." (Jacques Ozanam, "A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms", 1702) 

"Arithmetic and geometry, according to Plato, are the two wings of the mathematician. The object indeed of all mathematical questions, is to determine the ratios of numbers, or of magnitudes ; and it may even be said, to continue the comparison of the ancient philosopher, that arithmetic is the mathematician's right wing; for it is an incontestable truth, that geometrical determinations would, for the most part, present nothing satisfactory to the mind, if the ratios thus determined could not be reduced to numerical ratios. This justifies the common practice, which we shall here follow, of beginning with arithmetic." (Jacques Ozanam et al, "Recreations in Mathematics and Natural Philosophy", 1803)

John Haigh - Collected Quotes

"All this, though, is to miss the point of gambling, which is to accept the imbalance of chance in general yet deny it for the here and now. Individually we know, with absolute certainty, that 'the way things hap pen' and what actually happens to us are as different as sociology and poetry." (John Haigh," Taking Chances: Winning With Probability", 1999)

"As so often happens in mathematics, a convenient re-statement of a problem brings us suddenly up against the deepest questions of knowledge." (John Haigh," Taking Chances: Winning With Probability", 1999)

"But despite their frequently good intuition, many people go wrong in two places in particular. The first is in appreciating the real differences in magnitude that arise with rare events. If a chance is expressed as 'one in a thousand' or 'one in a million', the only message registered maybe that the chance is remote and yet one figure is a thousand times bigger than the other. Another area is in using partial information […]" (John Haigh," Taking Chances: Winning With Probability", 1999)

"It is the same with the numbers generated by roulette: the smoothness of probability in the long term allows any amount of local lumpiness on which to exercise our obsession with pattern. As the sequence of data lengthens, the relative proportions of odd or even, red or black, do indeed approach closer and closer to the 50-50 ratio predicted by probability, but the absolute discrepancy between one and the other will increase." (John Haigh," Taking Chances: Winning With Probability", 1999)

"Normal is safe; normal is central; normal is unexceptional. Yet it also means the pattern from which all others are drawn, the standard against which we measure the healthy specimen. In its simplest statistical form, normality is represented by the mean (often called the 'average') of a group of measurements." (John Haigh," Taking Chances: Winning With Probability", 1999)

"Probability therefore is a kind of corrective lens, allowing us, through an understanding of the nature of Chance, to refine our conclusions and approximate, if not achieve, the perfection of Design." (John Haigh," Taking Chances: Winning With Probability", 1999)

"The psychology of gambling includes both a conviction that the unusual must happen and a refusal to believe in it when it does. We are caught by the confusing nature of the long run; just as the imperturbable ocean seen from space will actually combine hurricanes and dead calms, so the same action, repeated over time, can show wide deviations from its normal expected results - deviations that do not themselves break the laws of probability. In fact, they have probabilities of their own." (John Haigh," Taking Chances: Winning With Probability", 1999)

"These so-called stochastic processes show up everywhere randomness is applied to the output of another random function. They provide, for instance, a method for describing the chance component of financial markets: not every value of the Dow is possible every day; the range of chance fluctuation centers on the opening price. Similarly, shuffling takes the output of the previous shuffle as its input. So, if you’re handed a deck in a given order, how much shuffling does it need to be truly random?" (John Haigh," Taking Chances: Winning With Probability", 1999)

"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)

"We search for certainty and call what we find destiny. Everything is possible, yet only one thing happens - we live and die between these two poles, under the rule of probability. We prefer, though, to call it Chance: an old familiar embodied in gods and demons, harnessed in charms and rituals. We remind one another of fortune’s fickleness, each secretly believing himself exempt. I am master of my fate; you are dicing with danger; he is living in a fool’s paradise." (John Haigh," Taking Chances: Winning With Probability", 1999)

"Winning and losing is not simply a pastime; it is the model science uses to explore the universe. Flipping a coin or rolling a die is really asking a question: success or failure can be defined as getting a yes or no. So the distribution of probabilities in a game of chance is the same as that in any repeated test - even though the result of any one test is unpredictable." (John Haigh," Taking Chances: Winning With Probability", 1999)

Scott L Zeger - Collected Quotes

"Longitudinal data sets are comprised of repeated observations of an outcome and a set of covariates for each of many subjects. One objective of statistical analysis is to describe the marginal expectation of the outcome variable as a function of the covariates while accounting for the correlation among the repeated observations for a given subject." (Scott L Zeger & Kung-Yee Liang, "Longitudinal Data Analysis for Discrete and Continuous Outcomes", Biometrics Vol. 42(1), 1986)

"Longitudinal data sets in which the outcome variable cannot be transformed to be Gaussian are more difficult to analyze for two reasons. First, simple models for the conditional expectation of the outcome do not imply equally simple models for the marginal expectation, as is the case for Gaussian data. Hence, the analyst must choose to model either the marginal or conditional expectation. Second, likelihood analyses often lead to estimators of the regression coefficients which are consistent only when the time dependence is correctly specified." (Scott L Zeger & Kung-Yee Liang, "Longitudinal Data Analysis for Discrete and Continuous Outcomes", Biometrics Vol. 42(1), 1986)

"Statistical models are sometimes misunderstood in epidemiology. Statistical models for data are never true. The question whether a model is true is irrelevant. A more appropriate question is whether we obtain the correct scientific conclusion if we pretend that the process under study behaves according to a particular statistical model." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"Statistical models for data are never true. The question whether a model is true is irrelevant. A more appropriate question is whether we obtain the correct scientific conclusion if we pretend that the process under study behaves according to a particular statistical model." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"Statistical reasoning is based upon two simple precepts: (1) that natural processes can usefully be described by stochastic models and (2) that by studying apparently haphazard collections of autonomous individuals, one can discover, at a higher level, systematic patterns of potential scientific import." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"The rise of statistical reasoning was a key step in the birth of many empirical sciences, especially epidemiology. The ability to focus on the aggregate behavior amidst apparently chaotic variation across autonomous individuals has dramatically increased our understanding of disease processes that affect the health of the public. Simple statistical models based upon the laws of probability provide the language for this population perspective." (Scott Zeger, "Statistical reasoning in epidemiology", American Journal of Epidemiology, 1991)

"Longitudinal data comprise repeated observations over time on each of many individuals. Longitudinal data are in contrast to cross-sectional data where only a single response is available for each person. The statistical analysis of longitudinal data presents special opportunities and challenges because the repeated outcomes for one individual tend to be correlated with one another." (Scott L Zeger & Kung‐Yee Liang, "An overview of methods for the analysis of longitudinal data", Statistics in medicine vol. 11, 1992)

"We have two objectives for statistical models of longitudinal data: (1) to adopt the conventional regression tools, which relate the response variables to the explanatory variables; and (2) to account for the within subject correlation." (Scott L Zeger & Kung‐Yee Liang, "An overview of methods for the analysis of longitudinal data", Statistics in medicine vol. 11, 1992)

Richard Bach - Collected Quotes

"In the path of our happiness shall we find the learning for which we have chosen this lifetime."  (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

"Learning is finding out what you already know. Doing is demonstrating that you know it. Teaching is reminding others that they know just as well as you. You are all learners, doers, teachers." (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

"There is no such thing as a problem without a gift for you in its hands. You seek problems because you need their gifts." (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

"You teach best what you most need to learn." (Richard Bach, "Illusions: The Adventures of a Reluctant Messiah", 1977)

"The only things that matter are those made of truth and joy, and not of tin and glass." (Richard Bach, :There’s No Such Place as Far Away", 1979)

"There are no mistakes. The events we bring upon ourselves, no matter how unpleasant, are necessary in order to learn what we need to learn; whatever steps we take, they’re necessary to reach the places we’ve chosen to go." (Richard Bach, "The Bridge across Forever", 1984)

"An easy life doesn’t teach us anything. In the end it’s the learning that matters: what we’ve learned and how we’ve grown." (Richard Bach, "One", 1988)

"In every language, from Arabic to Zulu to calligraphy to shorthand to math to music to art to wrought stone, everything from the Unified Field Theory to a curse to a sixpenny nail to an orbiting satellite, anything expressed is a net around some idea." (Richard Bach, "One", 1988)

"No matter how qualified or deserving we are, we will never reach a better life until we can imagine it for ourselves and allow ourselves to have it." (Richard Bach, "One", 1988)

"No one can solve problems for someone whose problem is that they don’t want problems solved." (Richard Bach, "One", 1988)

"The exciting thing about ideas is putting them to work. The moment we try them on our own, launch them away from shore, they switch from what-if to become daring plunges down white rivers, as dangerous and as exhilarating." (Richard Bach, "One", 1988)

"The only way to avoid all frightening choices is to leave society and become a hermit, and that is a frightening choice." (Richard Bach, "One", 1988)

"There is no such thing as a problem without a gift. We seek problems because we need their gifts." (Richard Bach)

Douglas Adams - Collected Quotes

"Far out in the uncharted backwaters of the unfashionable end of the Western spiral arm of the Galaxy lies a small unregarded yellow sun. Orbiting this at a distance of roughly ninety million miles is an utterly insignificant little blue-green planet whose ape-descended life forms are so amazingly primitive that they still think digital watches are a pretty neat idea." (Douglas Adams, "The Hitch-Hiker’s Guide to the Galaxy", [radio series episode] 1978)

"The chances of finding out what really is going on are so absurdly remote that the only thing to do is to say hang the sense of it and just keep yourself occupied."  (Douglas Adams, "The Hitch-Hiker’s Guide to the Galaxy", [radio series episode] 1978)

"There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarrely inexplicable. There is another theory which states that this has already happened." (Douglas Adams, "The Hitch-Hiker’s Guide to the Galaxy", [radio series episode]1978)

"You begin to suspect that if there’s any real truth it’s that the entire multidimensional infinity of the Universe is almost certainly being run by a bunch of maniacs." (Douglas Adams, "The Hitch-Hiker’s Guide to the Galaxy", [radio series episode]1978)

"The main reason he had had such a wild and successful life was that he never really understood the significance of anything he did." (Douglas Adams, "The Hitchhiker’s Guide to Galaxy", 1979)

"The whole fabric of the space-time continuum is not merely curved, it is in fact totally bent." (Douglas Adams, "The Restaurant at the End of the Universe", 1980)

"One of the interesting things about space [...] is how dull it is." (Douglas Adams, "Life, the Universe, and Everything", 1982)

"Their minds sang with the ecstatic knowledge that either what they were doing was completely and utterly and totally impossible or that physics had a lot of catching up to do." (Douglas Adams, "So Long, and Thanks for All the Fish", 1985)

"The complexities of cause and effect defy analysis." (Douglas Adams, "Dirk Gently's Holistic Detective Agency", 1987)

"The impossible often has a kind of integrity to it which the merely improbable lacks." (Douglas Adams, "The Long Dark Tea-Time of the Soul", 1988)

"Words used carelessly, as if they did not matter in any serious way, often allowed otherwise well-guarded truths to seep through." (Douglas Adams, "The Long Dark Tea-Time of the Soul", 1988)

"Assumptions are the things you don’t know you’re making." (Douglas Adams, "Last Chance to See", 1990)

"Human beings, who are almost unique in having the ability to learn from the experience of others, are also remarkable for their apparent disinclination to do so." (Douglas Adams, "Last Chance to See", 1990)

"It does highlight the irony that everything you go to see is changed by the very action of going to see it, which is the sort of problem which physicists have been wrestling with for most of this century." (Douglas Adams, "Last Chance to See", 1990)

"The great thing about being the only species that makes a distinction between right and wrong is that we can make up the rules for ourselves as we go along." (Douglas Adams, "Last Chance to See", 1990)

"A common mistake that people make when trying to design something completely foolproof was to underestimate the ingenuity of complete fools." (Douglas Adams, "Mostly Harmless", 1992)

"The major difference between a thing that might go wrong and a thing that cannot possibly go wrong is that when a thing that cannot possibly go wrong goes wrong it usually turns out to be impossible to get at or repair." (Douglas Adams, "Mostly Harmless", 1992)

Stanislaw Lem - Collected Quotes

"Insanity, gentlemen, is not a catchall for every human action that involves motives we don’t understand. Insanity has its own structure, its own internal logic." (Stanislaw Lem, "The Investigation", 1959)

"So-called common sense relies on programmed nonperception, concealment, or ridicule of everything that doesn’t fit into the conventional nineteenth century vision of a world that can be explained down to the last detail." (Stanislaw Lem, "The Investigation", 1959)

"What if the world isn’t scattered around us like a jigsaw puzzle - what if it’s like a soup with all kinds of things floating around in it, and from time to time some of them get stuck together by chance to make some kind of whole? What if everything that exists is fragmentary, incomplete, aborted, events with ends but no beginnings, events that only have middles, things that have fronts or rears but not both, with us constantly making categories, seeking out, and reconstructing, until we think we can see total love, total betrayal and defeat, although in reality we are all no more than haphazard fractions. [...] Using religion and philosophy as the cement, we perpetually collect and assemble all the garbage comprised by statistics in order to make sense out of things, to make everything respond in one unified voice like a bell chiming to our glory. But it’s only soup..." (Stanislaw Lem, "The Investigation", 1959)

"Are we not, in the end, a clamorous prelude to the final silence, a marriage bed to engender dust, a universe for microbes, microbes that strive to circumnavigate us? We are as unfathomable, as inscrutable as That which brought us into being, and we choke on our own enigma..." (Stanislaw Lem, "Memoirs Found in a Bathtub", 1961

"How do you expect to communicate with the ocean, when you can’t even understand one another?" (Stanislaw Lem, "Solaris", 1961)

"Man does not create gods, in spite of appearances. The times, the age, impose them on him." (Stanislaw Lem, "Solaris", 1961)

"The more complex a civilization, the more vital to its existence is the maintenance of the flow of information; hence the more vulnerable it becomes to any disturbance in that flow." (Stanislaw Lem, "Memoirs Found in a Bathtub", 1961)

"We take off into the cosmos, ready for anything: for solitude, for hardship, for exhaustion, death. Modesty forbids us to say so, but there are times when we think pretty well of ourselves. And yet, if we examine it more closely, our enthusiasm turns out to be all sham. We don’t want to conquer the cosmos, we simply want to extend the boundaries of Earth to the frontiers of cosmos. [...] We have no need of other worlds. We need mirrors."  (Stanislaw Lem, "Solaris", 1961)

"Where there are no men, there cannot be motives accessible to men." (Stanislaw Lem, "Solaris", 1961)

"Once there lived a certain engineer-cosmogonist who lit stars to dispel the dark." (Stanislaw Lem, ‘"Uranium Earpieces", 1965),

"Science explains the world, but only Art can reconcile us to it." (Stanislaw Lem, "King Globares and the Sages", 1965)

"These dwarfs amass knowledge as others do treasure; for this reason they are called Hoarders of the Absolute. Their wisdom lies in the fact that they collect knowledge but never use it." (Stanislaw Lem, "How Erg the Self-Inducing Slew a Paleface", 1965)

"For years astrophysicists have been racking their brains over the reason for the great difference in the amounts of cosmic dust in various galaxies. The answer, I think, is quite simple: the higher a civilization is, the more dust and refuse it produces. This is a problem more for janitors than for astrophysicists." (Stanislaw Lem, ‘"Let Us Save the Universe (An Open Letter from Ijon Tichy, Space Traveller", 1966)

"By squandering nuclear energy, polluting asteroids and planets, ravaging the Preserve, and leaving litter everywhere we go, we shall ruin outer space and turn it into one big dump. It is high time we came to our senses and enforced the laws. Convinced that every minute of delay is dangerous, I sound the alarm: Let us save the Universe." (Stanislaw Lem, ‘"Let Us Save the Universe (An Open Letter from Ijon Tichy, Space Traveller", 1966)

"Every intelligent creature was curious - and curiosity prompted it to act when something incomprehensible took place."(Stanislaw Lem, "The Hunt", 1968)

"Is there anything more contemptible than Nature? The scientists, the philosophers have always tried to understand Nature, while the thing to do is to destroy it!" (Stanislaw Lem, "The Sanitorium of Dr. Vliperdius", 1971)