On Numbers: .05 level

"The results of t shows that P is between .02 and .05. The result must be judged significant, though barely so [...] we find... t=1.844 [with 13 df, P = 0.088]. The difference between the regression coefficients, though relatively large, cannot be regarded as significant." (Ronald A Fisher, "Statistical Methods for Research Workers", 1925) 

"The value for which P=0.05, or 1 in 20, is 1.96 or nearly 2; it is convenient to take this point as a limit in judging whether a deviation ought to be considered significant or not. [...] If P is between .1 and .9 there is certainly no reason to suspect the hypothesis tested. If it is below .02 it is strongly indicated that the hypothesis fails to account for the whole of the facts. Belief in the hypothesis as an accurate representation of the population sampled is confronted by the logical disjunction: Either the hypothesis is untrue, or the value of χ2 has attained by chance an exceptionally high value. The actual value of P obtainable from the table by interpolation indicates the strength of the evidence against the hypothesis. A value of χ2 exceeding the 5 per cent. point is seldom to be disregarded." (Ronald A Fisher, "Statistical Methods for Research Workers", 1925) 

"The attempts that have been made to explain the cogency of tests of significance in scientific research, by reference to hypothetical frequencies of possible statements, based on them, being right or wrong, thus seem to miss the essential nature of such tests. A man who 'rejects' a hypothesis provisionally, as a matter of habitual practice, when the significance is at the 1% level or higher, will certainly be mistaken in not more than 1% of such decisions. For when the hypothesis is correct he will be mistaken in just 1% of these cases, and when it is incorrect he will never be mistaken in rejection. This inequality statement can therefore be made. However, the calculation is absurdly academic, for in fact no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and his ideas. Further, the calculation is based solely on a hypothesis, which, in the light of the evidence, is often not believed to be true at all, so that the actual probability of erroneous decision, supposing such a phrase to have any meaning, may be much less than the frequency specifying the level of significance." (Ronald A Fisher, "Statistical Methods and Scientific Inference", 1956)

"[...] blind adherence to the .05 level denies any consideration of alternative strategies, and it is a serious impediment to the interpretation of data." (James K Skipper Jr. et al, "The sacredness of .05: A note concerning the uses of statistical levels of significance in social science", The American Sociologist 2, 1967)

"The current obsession with .05 [...] has the consequence of differentiating significant research findings and those best forgotten, published studies from unpublished ones, and renewal of grants from termination. It would not be difficult to document the joy experienced by a social scientist when his F ratio or t value yields significance at .05, nor his horror when the table reads 'only' .10 or .06. One comes to internalize the difference between .05 and .06 as 'right' vs. 'wrong', 'creditable' vs. 'embarrassing', 'success' vs. 'failure'." (James K Skipper Jr. et al, "The sacredness of .05: A note concerning the uses of statistical levels of significance in social science", The American Sociologist 2, 1967)

"Rejection of a true null hypothesis at the 0.05 level will occur only one in 20 times. The overwhelming majority of these false rejections will be based on test statistics close to the borderline value. If the null hypothesis is false, the inter-ocular traumatic test ['hit between the eyes'] will often suffice to reject it; calculation will serve only to verify clear intuition." (Ward Edwards et al,Bayesian Statistical Inference for Psychological Research", 1992)

"[...] they [confidence limits] are rarely to be found in the literature. I suspect that the main reason they are not reported is that they are so embarrassingly large!" (Jacob Cohen,The earth is round" (p<.05)", American Psychologist 49, 1994)

"After four decades of severe criticism, the ritual of null hypothesis significance testing—mechanical dichotomous decisions around a sacred .05 criterion - still persist. This article reviews the problems with this practice [...] 'What’s wrong with [null hypothesis significance testing]? Well, among many other things, it does not tell us what we want to know, and we so much want to know what we want to know that, out of desperation, we nevertheless believe that it does!" (Jacob Cohen,The earth is round" (p<.05)", American Psychologist 49, 1994)

"It’s a commonplace among statisticians that a chi-squared test (and, really, any p-value) can be viewed as a crude measure of sample size: When sample size is small, it’s very difficult to get a rejection" (that is, a p-value below 0.05), whereas when sample size is huge, just about anything will bag you a rejection. With large n, a smaller signal can be found amid the noise. In general: small n, unlikely to get small p-values. Large n, likely to find something. Huge n, almost certain to find lots of small p-values." (Andrew Gelman,The sample size is huge, so a p-value of 0.007 is not that impressive", 2009)

"There is a growing realization that reported 'statistically significant' claims in statistical publications are routinely mistaken. Researchers typically express the confidence in their data in terms of p-value: the probability that a perceived result is actually the result of random variation. The value of p" (for 'probability') is a way of measuring the extent to which a data set provides evidence against a so-called null hypothesis. By convention, a p- value below 0.05 is considered a meaningful refutation of the null hypothesis; however, such conclusions are less solid than they appear." (Andrew Gelman & Eric Loken,The Statistical Crisis in Science", American Scientist Vol. 102(6), 2014)

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] 

On Numbers: Large Numbers II

"A good description of the data summarizes the systematic variation and leaves residuals that look structureless. That is, the residuals exhibit no patterns and have no exceptionally large values, or outliers. Any structure present in the residuals indicates an inadequate fit. Looking at the residuals laid out in an overlay helps to spot patterns and outliers and to associate them with their source in the data." (Christopher H Schrnid, "Value Splitting: Taking the Data Apart", 1991)

"Skewness is a measure of symmetry. For example, it's zero for the bell-shaped normal curve, which is perfectly symmetric about its mean. Kurtosis is a measure of the peakedness, or fat-tailedness, of a distribution. Thus, it measures the likelihood of extreme values." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Data that are skewed toward large values occur commonly. Any set of positive measurements is a candidate. Nature just works like that. In fact, if data consisting of positive numbers range over several powers of ten, it is almost a guarantee that they will be skewed. Skewness creates many problems. There are visualization problems. A large fraction of the data are squashed into small regions of graphs, and visual assessment of the data degrades. There are characterization problems. Skewed distributions tend to be more complicated than symmetric ones; for example, there is no unique notion of location and the median and mean measure different aspects of the distribution. There are problems in carrying out probabilistic methods. The distribution of skewed data is not well approximated by the normal, so the many probabilistic methods based on an assumption of a normal distribution cannot be applied." (William S Cleveland, "Visualizing Data", 1993)

"The logarithm is one of many transformations that we can apply to univariate measurements. The square root is another. Transformation is a critical tool for visualization or for any other mode of data analysis because it can substantially simplify the structure of a set of data. For example, transformation can remove skewness toward large values, and it can remove monotone increasing spread. And often, it is the logarithm that achieves this removal." (William S Cleveland, "Visualizing Data", 1993)

"Factoring big numbers is a strange kind of mathematics that closely resembles the experimental sciences, where nature has the last and definitive word. […] as with the experimental sciences, both rigorous and heuristic analyses can be valuable in understanding the subject and moving it forward. And, as with the experimental sciences, there is sometimes a tension between pure and applied practitioners." (Carl B Pomerance, "A Tale of Two Sieves", The Notices of the American Mathematical Society 43, 1996)

"Clearly, the mean is greatly influenced by extreme values, but it can be appropriate for many situations where extreme values do not arise. To avoid misuse, it is essential to know which summary measure best reflects the data and to use it carefully. Understanding the situation is necessary for making the right choice. Know the subject!" (Herbert F Spirer et al, "Misused Statistics" 2nd Ed, 1998)

"Big numbers warn us that the problem is a common one, compelling our attention, concern, and action. The media like to report statistics because numbers seem to be 'hard facts' - little nuggets of indisputable truth. [...] One common innumerate error involves not distinguishing among large numbers. [...] Because many people have trouble appreciating the differences among big numbers, they tend to uncritically accept social statistics (which often, of course, feature big numbers)." (Joel Best, "Damned Lies and Statistics: Untangling Numbers from the Media, Politicians, and Activists", 2001)

"Use a logarithmic scale when it is important to understand percent change or multiplicative factors. […] Showing data on a logarithmic scale can cure skewness toward large values." (Naomi B Robbins, "Creating More effective Graphs", 2005) 

"Outliers or influential data points can be defined as data values that are extreme or atypical on either the independent (X variables) or dependent (Y variables) variables or both. Outliers can occur as a result of observation errors, data entry errors, instrument errors based on layout or instructions, or actual extreme values from self-report data. Because outliers affect the mean, the standard deviation, and correlation coefficient values, they must be explained, deleted, or accommodated by using robust statistics." (Randall E Schumacker & Richard G Lomax, "A Beginner’s Guide to Structural Equation Modeling" 3rd Ed., 2010)

"Comparisons are the lifeblood of empirical studies. We can’t determine if a medicine, treatment, policy, or strategy is effective unless we compare it to some alternative. But watch out for superficial comparisons: comparisons of percentage changes in big numbers and small numbers, comparisons of things that have nothing in common except that they increase over time, comparisons of irrelevant data. All of these are like comparing apples to prunes." (Gary Smith, "Standard Deviations", 2014)

"It is not enough to give a single summary for a distribution - we need to have an idea of the spread, sometimes known as the variability. [...] The range is a natural choice, but is clearly very sensitive to extreme values [...] In contrast the inter-quartile range (IQR) is unaffected by extremes. This is the distance between the 25th and 75th percentiles of the data and so contains the ‘central half’ of the numbers [...] Finally the standard deviation is a widely used measure of spread. It is the most technically complex measure, but is only really appropriate for well-behaved symmetric data since it is also unduly influenced by outlying values." (David Spiegelhalter, "The Art of Statistics: Learning from Data", 2019)

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On Numbers: Binary Numbers

"[The information of a message can] be defined as the 'minimum number of binary decisions which enable the receiver to construct the message, on the basis of the data already available to him.' These data comprise both the convention regarding the symbols and the language used, and the knowledge available at the moment when the message started." (Dennis Gabor, "Optical transmission" in Information Theory : Papers Read at a Symposium on Information Theory, 1952)

"[...] there can be such a thing as a simple probabilistic system. For example, consider the tossing of a penny. Here is a perfectly simple system, but one which is notoriously unpredictable. It maybe described in terms of a binary decision process, with a built-in even probability between the two possible outcomes." (Stafford Beer, "Cybernetics and Management", 1959)

"Bivalence trades accuracy for simplicity. Binary outcomes of yes and no, white and black, true and false simplify math and computer processing. You can work with strings of 0s and 1s more easily than you can work with fractions. But bivalence requires some force fitting and rounding off [...] Bivalence holds at cube corners. Multivalence holds everywhere else." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"Fuzziness has a formal name in science: multivalence. The opposite of fuzziness is bivalence or two-valuedness, two ways to answer each question, true or false, 1 or 0. Fuzziness means multivalence. It means three or more options, perhaps an infinite spectrum of options, instead of just two extremes. It means analog instead of binary, infinite shades of gray between black and white." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"Somewhere in the process wishful thinking seems to take over. Scientists start to believe they do math when they do science. This holds to greatest degree in an advanced science like physics or at the theoretical frontier of any science where the claims come as math claims. The first victim is truth. What was inaccurate or fuzzy truth all along gets bumped up a letter grade to the all-or-none status of binary logic. Most scientists draw the line at giving up the tentative status of science. They will concede that it can all go otherwise in the next experiment. But most have crossed the bivalent line by this point and believe that in the next experiment a statement or hypothesis or theory may jump from TRUE to FALSE, from 1 to 0." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"The binary logic of modern computers often falls short when describing the vagueness of the real world. Fuzzy logic offers more graceful alternatives." (Bart Kosko & Satoru Isaka, "Fuzzy Logic,” Scientific American Vol. 269, 1993)

"A dictionary definition of the word ‘complex’ is: ‘consisting of interconnected or interwoven parts’ […] Loosely speaking, the complexity of a system is the amount of information needed in order to describe it. The complexity depends on the level of detail required in the description. A more formal definition can be understood in a simple way. If we have a system that could have many possible states, but we would like to specify which state it is actually in, then the number of binary digits (bits) we need to specify this particular state is related to the number of states that are possible." (Yaneer Bar-Yamm, "Dynamics of Complexity", 1997)

"A bit involves both probability and an experiment that decides a binary or yes-no question. Consider flipping a coin. One bit of in-formation is what we learn from the flip of a fair coin. With an unfair or biased coin the odds are other than even because either heads or tails is more likely to appear after the flip. We learn less from flipping the biased coin because there is less surprise in the outcome on average. Shannon's bit-based concept of entropy is just the average information of the experiment. What we gain in information from the coin flip we lose in uncertainty or entropy." (Bart Kosko, "Noise", 2006)

"Why should anyone know or care about binary numbers? One reason is that working with numbers in an unfamiliar base is an example of quantitative reasoning that might even improve understanding of how numbers work in good old base ten. Beyond that, it’s also important because the number of bits is usually related in some way to how much space, time or complexity is involved. And fundamentally, computers are worth understanding, and binary is central to their operation." (Brian W Kernighan, "Understanding the Digital World", 2017)

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On Numbers: Natural Numbers

"These Exponents they call Logarithms, which are Artificial Numbers, so answering to the Natural Numbers, as that the addition and Subtraction of these, answers to the Multiplication and Division of the Natural Numbers. By this means, (the Tables being once made) the Work of Multiplication and Division is performed by Addition and Subtraction; and consequently that of Squaring and Cubing, by Duplication and Triplication; and that of Extracting the Square and Cubic Root, by Bisection and Trisection; and the like in the higher Powers." (John Wallis, "Of Logarithms, Their Invention and Use", 1685)

"These primitive propositions […] suffice to deduce all the properties of the numbers that we shall meet in the sequel. There is, however, an infinity of systems which satisfy the five primitive propositions. [...] All systems which satisfy the five primitive propositions are in one-to-one correspondence with the natural numbers. The natural numbers are what one obtains by abstraction from all these systems; in other words, the natural numbers are the system which has all the properties and only those properties listed in the five primitive propositions." (Giuseppe Peano, "I fondamenti dell’aritmetica nel Formulario del 1898" ["The Principles of Arithmetic, presented by a new method"], 1889)

"If one intends to base arithmetic on the theory of natural numbers as finite cardinals, one has to deal mainly with the definition of finite set; for the cardinal is, according to its nature, a property of a set, and any proposition about finite cardinals can always be expressed as a proposition about finite sets. In the following I will try to deduce the most important property of natural numbers, namely the principle of complete induction, from a definition of finite set which is as simple as possible, at the same time showing that the different definitions [of finite set] given so far are equivalent to the one given here." (Ernst Zermelo,  "Ueber die Grundlagen der Arithmetik", Atti del IV Congresso Internazionale dei Matematici, 1908)

"It would not be an exaggeration to say that all of mathematics derives from the concept of infinity. In mathematics, as a rule, we are not interested in individual objects (numbers, geometric figures), but in whole classes of such objects: all natural numbers, all triangles, and so on. But such a collection consists of an infinite number of individual objects." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"[…] mathematicians and philosophers have always been interested in the concept of infinity. This interest arose at the very moment when it became clear that each natural number has a successor, i.e., that the number sequence is infinite. However, even the first attempts to cope with infinity lead to numerous paradoxes." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

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

"For the advancing army of physics, battling for many a decade with heat and sound, fields and particles, gravitation and spacetime geometry, the cavalry of mathematics, galloping out ahead, provided what it thought to be the rationale for the natural number system. Encounter with the quantum has taught us, however, that we acquire our knowledge in bits; that the continuum is forever beyond our reach. Yet for daily work the concept of the continuum has been and will continue to be as indispensable for physics as it is for mathematics." (John A Wheeler, "Hermann Weyl and the Unity of Knowledge", American Scientist Vol. 74, 1986)

"When all the mathematical smoke clears away, Godel's message is that mankind will never know the final secret of the universe by rational thought alone. It's impossible for human beings to ever formulate a complete description of the natural numbers. There will always be arithmetic truths that escape our ability to fence them in using the tools, tricks and subterfuges of rational analysis." (John L Casti, "Reality Rules: Picturing the world in mathematics" Vol. II, 1992)

"[…] i is not a real number-not ordered anywhere relative to the real numbers! In other words, it does not even have the central property of ‘numbers’, indicating a magnitude that can be linearly compared to all other magnitudes. You can see why i has been called imaginary. It has almost none of the properties of the small natural numbers-not subitizability, not groupability, and not even relative magnitude. If i is to be a number, it is a number only by virtue of closure and the laws of arithmetic." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"The word 'discrete' means separate or distinct. Mathematicians view it as the opposite of 'continuous'. Whereas, in calculus, it is continuous functions of a real variable that are important, such functions are of relatively little interest in discrete mathematics. Instead of the real numbers, it is the natural numbers 1, 2, 3, ... that play a fundamental role, and it is functions with domain the natural numbers that are often studied." (Edgar G Goodaire & Michael M Parmenter, "Discrete Mathematics with Graph Theory" 2nd Ed., 2002)

"The so-called ‘imaginary numbers’ were successfully used long before they were properly defined as elements of a concrete field extension of the real numbers. The axiomatic description of a theory generally appears only in its mature stages, after many of its properties have been informally explored. Perhaps the longest duration between the usage and formalization of a notion is that of the natural numbers." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"But we also have to know that every model has its limitations. The model of natural numbers and their sums is very successful to determine the number of objects in the union of two different groups of well-distinguished objects. But as a mathematical model, the arithmetic of numbers is not generally true but only validated and confirmed for certain well-controlled situations. […] If a model makes valid predictions in many concrete cases, if it already has been applied and tested successfully in many situations, we have some right to trust in that model. By now, we believe in the model 'natural numbers and their arithmetic' and in its predictions without having to check it every time. We do not expect that the result might be wrong; hence the verification step is not needed any longer for validating the model. If the model had a flaw, it would have been eliminated already in the past." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"Most mathematicians are not particularly worried by the fact that there are natural numbers so huge that they cannot be conceptualized exactly. Typically, when applying numbers to reality, approximate quantities are sufficient, and extremely large numbers would rarely be needed. In theory, the natural numbers are just a sequence whose structure is axiomatically described by the Peano axioms. As a mathematician, one typically does not care about the practical realizability of particular numbers. That every number has a unique successor is simply true by assumption; it needs no practical verification." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"The branch of philosophy of mathematics that would not accept objects or expressions that nobody can construct in any practical sense is called ultrafinitism. According to this view, not even the concept of natural numbers would be accepted without restrictions, and, of course, an ultrafinitist would refuse to talk about infinity. To most mathematicians, this view would be too extreme. Reducing mathematics to finite and not-too-large objects would restrict mathematics and its usefulness in an intolerable way." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"When we extend the system of natural numbers and counting to embrace infinite cardinals, the larger system need not have all of the properties of the smaller one. However, familiarity with the smaller system leads us to expect certain properties, and we can become confused when the pieces don’t seem to fit. Insecurity arose when the square of a complex number violated the real number principle that all squares are positive. This was resolved when we realised that the complex numbers cannot be ordered in the same way as their subset of reals." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"God made the natural numbers. all else is the work of man." (Leopold Kronecker)

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

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Charles Livingston - Collected Quotes

"Cautions about combining groups: apples and oranges. In computing an average, be careful about combining groups in which the average for each group is of more interest than the overall average. […] Avoid combining distinct quantities in a single average." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"Central tendency is the formal expression for the notion of where data is centered, best understood by most readers as 'average'. There is no one way of measuring where data are centered, and different measures provide different insights." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"Concluding that the population is becoming more centralized by observing behavior at the extremes is called the 'Regression to the Mean' Fallacy. […] When looking for a change in a population, do not look only at the extremes; there you will always find a motion to the mean. Look at the entire population." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"Data often arrive in raw form, as long lists of numbers. In this case your job is to summarize the data in a way that captures its essence and conveys its meaning. This can be done numerically, with measures such as the average and standard deviation, or graphically. At other times you find data already in summarized form; in this case you must understand what the summary is telling, and what it is not telling, and then interpret the information for your readers or viewers." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"If a hypothesis test points to rejection of the alternative hypothesis, it might not indicate that the null hypothesis is correct or that the alternative hypothesis is false." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"Limit a sentence to no more than three numerical values. If you've got more important quantities to report, break those up into other sentences. More importantly, however, make sure that each number is an important piece of information. Which are the important numbers that truly advance the story?" (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"Numbers are often useful in stories because they record a recent change in some amount, or because they are being compared with other numbers. Percentages, ratios and proportions are often better than raw numbers in establishing a context." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"Probability is sometimes called the language of statistics. […] The probability of an event occurring might be described as the likelihood of it happening. […] In a formal sense the word "probability" is used only when an event or experiment is repeatable and the long term likelihood of a certain outcome can be determined." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"Roughly stated, the standard deviation gives the average of the differences between the numbers on the list and the mean of that list. If data are very spread out, the standard deviation will be large. If the data are concentrated near the mean, the standard deviation will be small." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"The basic idea of going from an estimate to an inference is simple. Drawing the conclusion with confidence, and measuring the level of confidence, is where the hard work of professional statistics comes in." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"The central limit theorem […] states that regardless of the shape of the curve of the original population, if you repeatedly randomly sample a large segment of your group of interest and take the average result, the set of averages will follow a normal curve." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"The dual meaning of the word significant brings into focus the distinction between drawing a mathematical inference and practical inference from statistical results." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

"The percentage is one of the best (mathematical) friends a journalist can have, because it quickly puts numbers into context. And it's a context that the vast majority of readers and viewers can comprehend immediately." (Charles Livingston & Paul Voakes, "Working with Numbers and Statistics: A handbook for journalists", 2005)

Gerd Gigerenzer - Collected Quotes

"A heuristic is ecologically rational to the degree that it is adapted to the structure of an environment. Thus, simple heuristics and environmental structure can both work hand in hand to provide a realistic alternative to the ideal of optimization, whether unbounded or constrained." (Gerd Gigerenzer & Peter M Todd, "Fast and Frugal Heuristics: The Adaptive Toolbox" [in "Simple Heuristics That Make Us Smart"], 1999)

"Fast and frugal heuristics employ a minimum of time, knowledge, and computation to make adaptive choices in real environments. They can be used to solve problems of sequential search through objects or options, as in satisficing. They can also be used to make choices between simultaneously available objects, where the search for information (in the form of cues, features, consequences, etc.) about the possible options must be limited, rather than the search for the options themselves. Fast and frugal heuristics limit their search of objects or information using easily computable stopping rules, and they make their choices with easily computable decision rules." (Gerd Gigerenzer & Peter M Todd, "Fast and Frugal Heuristics: The Adaptive Toolbox" [in "Simple Heuristics That Make Us Smart"], 1999)

"At present, high school education in some countries covers only little, if any, statistical thinking. Algebra, geometry, and calculus teach thinking in a world of certainty - not in the real world, which is uncertain. [...] Furthermore, in the medical and social sciences, data analysis is typically taught as a set of statistical rituals rather than a set of methods for statistical thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"External representations are not just passive inputs to an active mind. They can do part of the reasoning or calculation, making relevant properties of the same information more accessible." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Good numeric representation is a key to effective thinking that is not limited to understanding risks. Natural languages show the traces of various attempts at finding a proper representation of numbers. [...] The key role of representation in thinking is often downplayed because of an ideal of rationality that dictates that whenever two statements are mathematically or logically the same, representing them in different forms should not matter. Evidence that it does matter is regarded as a sign of human irrationality. This view ignores the fact that finding a good representation is an indispensable part of problem solving and that playing with different representations is a tool of creative thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Ignorance of relevant risks and miscommunication of those risks are two aspects of innumeracy. A third aspect of innumeracy concerns the problem of drawing incorrect inferences from statistics. This third type of innumeracy occurs when inferences go wrong because they are clouded by certain risk representations. Such clouded thinking becomes possible only once the risks have been communicated." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"In my view, the problem of innumeracy is not essentially 'inside' our minds as some have argued, allegedly because the innate architecture of our minds has not evolved to deal with uncertainties. Instead, I suggest that innumeracy can be traced to external representations of uncertainties that do not match our mind’s design - just as the breakdown of color constancy can be traced to artificial illumination. This argument applies to the two kinds of innumeracy that involve numbers: miscommunication of risks and clouded thinking. The treatment for these ills is to restore the external representation of uncertainties to a form that the human mind is adapted to." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Information needs representation. The idea that it is possible to communicate information in a 'pure' form is fiction. Successful risk communication requires intuitively clear representations. Playing with representations can help us not only to understand numbers (describe phenomena) but also to draw conclusions from numbers (make inferences). There is no single best representation, because what is needed always depends on the minds that are doing the communicating." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Natural frequencies facilitate inferences made on the basis of numerical information. The representation does part of the reasoning, taking care of the multiplication the mind would have to perform if given probabilities. In this sense, insight can come from outside the mind." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Overcoming innumeracy is like completing a three-step program to statistical literacy. The first step is to defeat the illusion of certainty. The second step is to learn about the actual risks of relevant events and actions. The third step is to communicate the risks in an understandable way and to draw inferences without falling prey to clouded thinking. The general point is this: Innumeracy does not simply reside in our minds but in the representations of risk that we choose." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Statistical innumeracy is the inability to think with numbers that represent uncertainties. Ignorance of risk, miscommunication of risk, and clouded thinking are forms of innumeracy. Like illiteracy, innumeracy is curable. Innumeracy is not simply a mental defect 'inside' an unfortunate mind, but is in part produced by inadequate 'outside' representations of numbers. Innumeracy can be cured from the outside." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"The creation of certainty seems to be a fundamental tendency of human minds. The perception of simple visual objects reflects this tendency. At an unconscious level, our perceptual systems automatically transform uncertainty into certainty, as depth ambiguities and depth illusions illustrate." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002) 

"The key role of representation in thinking is often downplayed because of an ideal of rationality that dictates that whenever two statements are mathematically or logically the same, representing them in different forms should not matter. Evidence that it does matter is regarded as a sign of human irrationality. This view ignores the fact that finding a good representation is an indispensable part of problem solving and that playing with different representations is a tool of creative thinking." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"The possibility of translating uncertainties into risks is much more restricted in the propensity view. Propensities are properties of an object, such as the physical symmetry of a die. If a die is constructed to be perfectly symmetrical, then the probability of rolling a six is 1 in 6. The reference to a physical design, mechanism, or trait that determines the risk of an event is the essence of the propensity interpretation of probability. Note how propensity differs from the subjective interpretation: It is not sufficient that someone’s subjective probabilities about the outcomes of a die roll are coherent, that is, that they satisfy the laws of probability. What matters is the die’s design. If the design is not known, there are no probabilities." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"When does an uncertainty qualify as a risk? The answer depends on one’s interpretation of probability, of which there are three major versions: degree of belief, propensity, and frequency. Degrees of belief are sometimes called subjective probabilities. Of the three interpretations of probability, the subjective interpretation is most liberal about expressing uncertainties as quantitative probabilities, that is, risks. Subjective probabilities can be assigned even to unique or novel events." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"When natural frequencies are transformed into conditional probabilities, the base rate information is taken out (this is called normalization). The benefit of this normalization is that the resulting values fall within the uniform range of 0 and 1. The cost, however, is that when drawing inferences from probabilities (as opposed to natural frequencies), one has to put the base rates back in by multiplying the conditional probabilities by their respective base rates." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"Why does representing information in terms of natural frequencies rather than probabilities or percentages foster insight? For two reasons. First, computational simplicity: The representation does part of the computation. And second, evolutionary and developmental primacy: Our minds are adapted to natural frequencies." (Gerd Gigerenzer, "Calculated Risks: How to know when numbers deceive you", 2002)

"A heuristic is defined as a simple rule that exploits both evolved abilities to act fast and struc- tures of the environment to act accurately and frugally. The complexity and uncertainty of an environment cannot be determined independently of the actor. What matters is the degree of complexity and uncertainty encountered by the decision maker." (Christoph Engel & Gerd Gigerenzer, "Law and Heuristics: An interdisciplinary venture" [in "Heuristics and the Law", 2006)

"Heuristics are needed in situations where the world does not permit optimization. For many real-world problems (as opposed to optimization-tuned textbook problems), optimal solutions are unknown because the problems are computationally intractable or poorly defined." (Christoph Engel & Gerd Gigerenzer, "Law and Heuristics: An interdisciplinary venture" [in "Heuristics and the Law", 2006)

"[...] a gut feeling is not caprice or a sixth sense. An executive may be buried under a mountain of information, some of which is contradictory, some of questionable reliability, and some that makes one wonder why it was sent in the first place. But despite the excess of data, there is no algorithm to calculate the best decision. In this situation, an experienced executive may have a feeling of what the best action is - a gut feeling. By definition, the reasons behind this feeling are unconscious." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"A rule of thumb, or heuristic, enables us to make a decision fast, without much searching for information, but nevertheless with high accuracy. [...] A heuristic can be safer and more accurate than a calculation, and the same heuristic can underlie both conscious and unconscious decisions." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

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

"Teaching statistical thinking means giving people tools for problem solving in the real world. It should not be taught as pure mathematics. Instead of mechanically solving a dozen problems with the help of a particular formula, children and adolescents should be asked to find solutions to real-life problems. That’s what teaches them how to solve problems, and also shows that there may be more than one good answer in the first place. Equally important is encouraging curiosity, such as asking for answers to questions by doing experiments." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"The taming of chance created mathematical probability. [...] Probability is not one of a kind; it was born with three faces: frequency, physical design, and degrees of belief. [...] in the first of its identities, probability is about counting. [...] Second, probability is about constructing. For example, if a die is constructed to be perfectly symmetrical, then the probability of rolling a six is one in six. You don’t have to count. [...] Probabilities by design are called propensities. Historically, games of chance were the prototype for propensity. These risks are known because people crafted, not counted, them. [...] Third, probability is about degrees of belief. A degree of belief can be based on anything from experience to personal impression." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"There is a mathematical theory that tells us why and when simple is better. It is called the bias-variance dilemma. [...] How far we go in simplifying depends on three features. First, the more uncertainty, the more we should make it simple. The less uncertainty, the more complex it should be. [...] Second, the more alternatives, the more we should simplify; the fewer, the more complex it can be. The reason is that complex methods need to estimate risk factors, and more alternatives mean that more factors need to be estimated, which leads to more estimation errors being made. [...] Finally, the more past data there are, the more beneficial for the complex methods." (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"To calculate a risk is one thing, to communicate it is another. Risk communication is an important skill for laypeople and experts alike. Because it’s rarely taught, misinterpreting numbers is the rule rather than the exception. Each of the three kinds of probability - relative frequency, design, or degree of belief - can be framed in either a confusing or a transparent way. So far we have seen two communication tools for reporting risks: Use frequencies instead of single-event probabilities. Use absolute instead of relative risks."  (Gerd Gigerenzer, "Risk Savvy: How to make good decisions", 2014)

"A heuristic is a strategy that ignores part of the information, with the goal of making decisions more quickly, frugally, and/or accurately than more complex methods." (Gerd Gigerenzer et al, "Simply Rational: Decision Making in the Real World", 2015)

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

Laws II: The Law of Large Numbers

"[…] probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; law of large numbers […]" (Jacob Bernoulli, "Ars Conjectandi" ["The Art of Conjecturing"], 1713)

"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)

"It is a common fallacy to believe that the law of large numbers acts as a force endowed with memory seeking to return to the original state, and many wrong conclusions have been drawn from this assumption." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)

“We know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?” (Albert Claude, [Nobel Prize Lecture], 1974)

"The law of truly large numbers states: With a large enough sample, any outrageous thing is likely to happen." (Frederick Mosteller, Methods for Studying Coincidences Journal of the American Statistical Association, Volume 84, 1989)

"All the law [of large numbers] tells us is that the average of a large number of throws will be more likely than the average of a small number of throws to differ from the true average by less than some stated amount. And there will always be a possibility that the observed result will differ from the true average by a larger amount than the specified bound." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"Jacob Bernoulli's theorem for calculating probabilities a posteriori is known as the Law of Large Numbers. Contrary to the popular view, this law does not provide a method for validating observed facts, which are only an incomplete representation of the whole truth. Nor does it say that an increasing number of observations will increase the probability that what you see is what you are going to get. The law is not a design for improving the quality of empirical tests […]." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"The Law of Large Numbers does not tell you that the average of your throws will approach 50% as you increase the number of throws; simple mathematics can tell you that, sparing you the tedious business of tossing the coin over and over. Rather, the law states that increasing the number of throws will correspondingly increase the probability that the ratio of heads thrown to total throws will vary from 50% by less than some stated amount, no matter how small. The word 'vary' is what matters. The search is not for the true mean of 50% but for the probability that the error between the observed average and the true average will be less than, say, 2% - in other words, that increasing the number of throws will increase the probability that the observed average will fall within 2% of the true average." (Peter L Bernstein, "Against the Gods: The Remarkable Story of Risk", 1996)

"The law of small numbers is not really a law. It is a sarcastic name describing the misguided attempt to apply the law of large numbers when the numbers aren't large." (Leonard Mlodinow, "The Drunkard’s Walk: How Randomness Rules Our Lives", 2008)

"The law of large numbers is a law of mathematical statistics. It states that when random samples are sufficiently large they match the population extremely closely. […] The 'law' of small numbers is a widespread human misconception that even small samples match the population closely." (Geoff Cumming, "Understanding the New Statistics", 2012)

David Wells - Collected Quotes

"All the [mathematical] models that we have considered […] have been rough and ready. They have all been obviously crude approximations, and no one supposes that they are anything more. This does not mean that they are useless - far from it - but it does mean that the answers they give to practical questions are also approximations. There is a pragmatic payoff here between the use of simple models which give good-enough answers which are good value for money, and the use of much more sophisticated models which are more powerful, but also more complex to use, perhaps requiring more advanced mathematics and the use of computers." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"By practice and experience, much of it vicarious, through studying the achievements of others, we develop the strength to tackle novel and unfamiliar situations." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Chess players recognize and applaud good play, from a single smart move to a brilliant combination to an entire game which is considered a masterpiece. Mathematicians recognize and applaud good mathematics, from clever tricks to brilliant proofs, and from beautiful conceptions to grand and deep ideas which advance our understanding of mathematics as a whole. It takes imagination and insight to discover the best moves, at chess or mathematics, and the more difficult the position, the harder they are to find. Chess players learn by experience to recognize types of positions and situations and to know what kind of moves are likely to be successful; they exploit brilliant local tactics as well as deep stategical ideas. So do mathematicians. Neither games nor mathematics play themselves - they both need a player with understanding, good ideas, judgement and discrimination to play them. To develop these essential attributes, the player must explore the game by playing it, thinking about it and analysing it. For the chess player and the mathematician, this process is scientific: you test ideas, experiment with new possibilities, develop the ones that work and discard the ones that fail. This is how chess players and mathematicians develop their tactical and strategical understanding; it is how they give meaning to chess and mathematics." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Do mathematicians know exactly what they are talking about, or don't they? This is the first bridge that mathematicians have to cross in their search for certainty, and it has to be admitted that they cannot cross it with complete confidence, so their search for absolute certainty is already compromised. They can certainly rely on the long history of their most basic ideas, and the universal agreement that these concepts 'work'. However, this is another way of saying that, if a concept is new and untried, then even mathematicians should be wary of it. They may well discover with experience that their intuitive expectations of it are false. Indeed, this is one of the great values of experimental mathematics - experiments provide the mathematician with data against which mathematicians' natural expectations can be continually tested." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Graph theory is typical of much modern mathematics. Its subject matter is not traditional, and it is not a development from traditional theories. Its applications are not traditional either. […] Graph theory is not concerned with continuous quantities. It often involves counting, but in integers, not measuring using fractions. Graph theory is an example of discrete mathematics. Graphs are put together in pieces, in chunks, rather like Meccano or Lego, or a jigsaw puzzle." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"However mathematics starts, whether it is in counting and measuring in everyday life, or in puzzles and riddles, or in scientific queries about projectiles, floating bodies, levers and balances, or magnetic lines of force, it eventually becomes detached from its roots and develops a life of its own. It becomes more powerful, because it can be applied not just to the situations in which it originated but to all other comparable situations. It also becomes more abstract, and more game-like." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Ironically, mathematicians often infer all sorts of properties about objects which they only suspect, or hope, actually exist. If their suspicions turn out to be unfounded, then they seem to end up by knowing rather a lot about something which does not exist and which might seem, therefore, not to have any properties at all." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"It is typical that there is more than one way of looking at a geometrical figure, just as there are many ways of looking at lines of algebra. Perception, 'seeing', is an essential feature of mathematics. This is obvious when we are looking for patterns - how can you possibly 'spot' a pattern if you cannot in some sense 'see' it? But it is just as true when the mathematician is looking for hidden connections, or studying a position in a mathematical game, searching for a tactical sequence, or trying to 'see' the possibilities clearly. Superficially, it might seem that it is only geometry (and related fields of mathematics) that depends on perception, but this is not so. Perception is everywhere in mathematics." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematical models are continually invoking ideas of infinitely smooth surfaces, weightless strings, weightless beams, perfectly spherical balls, projectiles flying through airless space, gases which are perfectly compressible and liquids which are perfectly incompressible, and so on. The purpose of such simplifications is, in theory, to understand the world better despite the oversimplification, which you hope either will not matter or will be corrected when you construct a second (better) model." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematicians get a different kind of pleasure from the illumination of solving a problem, when what was once mysterious and obscure is made plain. Revealing the hidden connections in a situation is delightful - like reaching the top of a mountain after a hard climb, and seeing the landscape spread out before you. All of a sudden, everything is clear! If the result is extremely simple, so much the better . To start with confusing complexity and transform it into revealing simplicity is a marvellous reward for hard work. It really does give the mathematician a 'kick'!" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematics-as-science naturally starts with mysterious phenomena to be explained, and leads (if you are successful) to powerful and harmonious patterns. Mathematics-as-a-game not only starts with simple objects and rules, but involves all the attractions of games like chess: neat tactics, deep strategy, beautiful combinations, elegant and surprising ideas. Mathematics-as-perception displays the beauty and mystery of art in parallel with the delight of illumination, and the satisfaction of feeling that now you understand." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mathematics depends on the ability to 'see', literally and abstractly, in novel and unexpected ways, which is another way of saying that it depends on the brilliance and subtlety of the human brain which just happens to be wonderfully adapted to this very purpose. [...] It is one of the marvels of mathematics that everything can be seen in different ways from different points of view, sometimes literally, sometimes metaphorically speaking." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Mystery is found as much in mathematics as in detective stories. Indeed, the mathematician could well be described as a detective, brilliantly exploiting a few initial clues to solve the problem and reveal its innermost secrets. An especially mathematical mystery is that you can often search for some mathematical object, and actually know a lot about it, if it exists, only to discover that in fact it does not exist at all - you knew a lot about something which cannot be." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Playing the game of mathematics is much harder than investigating scientifically! To jot down some numbers, a few differences, and spot a pattern is child's play compared to playing the game of algebra." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Puzzle composers share another feature with mathematicians. They know that, generally speaking, the simpler a puzzle is to express, the more attractive it is likely to be found: similarly, simplicity is for both a desirable feature of the solution. Especially satisfying solutions are often described as 'elegant', a word that - no surprise here - is also used by scientists, engineers and designers, indeed by anyone with a problem to solve. However, simplicity is by no means the only reward of success. Far from it! Mathematicians (and scientists and others) can reasonably expect two further returns: they are (in no particular order) firstly the power to do things, and secondly the perception of connections which were never before suspected, leading in turn to the insight and illumination that mathematicians expect from their best arguments." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995) 

"'Technique' is a term used equally by chess players and mathematicians to describe sequences of moves which are standard, familiar and unoriginal. Once upon a time, the particular technique was an invention, a new discovery, but no longer. The precise sequence of moves required may never have been played before in the history of the world, yet no new ideas, no originality and no imagination are demanded, at least of the experienced player. (To the learner, of course, the most mundane sequences will appear novel and require original thought.)" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"The easiest way to discover mathematical facts and theorems is to treat mathematical objects just as if they were objects in the real world, and to make observations and do experiments, by drawing and measuring on geometrical figures, and making calculations with numbers (or in algebra) […]" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"[…] the most important techniques are the final results of some of the most significant breakthroughs in the history of mathematics, beautifully simplified and explained for the benefit of players everywhere. As such, technique is used by pure mathematicians and applied mathematicians alike, but in rather different ways. Pure mathematicians use familiar techniques on the way to discovering or proving new results; applied mathematicians use techniques to model phenomena in the real world […]" (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"There is a difficulty, however, which has to be borne in mind. As long as you are behaving like a scientist, you can never be quite certain that your mathematical results are true. Therefore, scientific practice, although it is an essential tool of the mathematician, is also a source of danger. The mathematician who is fast asleep will not see the danger lurking, and will fall straight into the traps and snares that await. The mathematician who is wide awake, however, will gain the double benefit from scientific experiment and observation, of discovering all sorts of supposed facts, and gaining some idea why they might be true." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"There is no sharp dividing line between scientific theories and models, and mathematics is used similarly in both. The important thing is to possess a delicate judgement of the accuracy of your model or theory. An apparently crude model can often be surprisingly effective, in which case its plain dress should not mislead. In contrast, some apparently very good models can be hiding dangerous weaknesses." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"This process, of answering deep questions and so discovering what you really meant by what you've been talking about all the time, is very common in mathematics. It is also very important and very scientific. It has to be scientific, because you cannot prove that one definition is superior to another by logic: you can only make judgements that this definition is preferable to that. You could say that, until mathematicians have examined very deeply and scientifically the objects they talk about so glibly, then they do not really know what they are talking about." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

"Yet, Einstein's theories are also not the last word: quantum theory and relativity are inconsistent, and Einstein himself, proclaiming that 'God does not play dice!', rejected the basic reliance of quantum theory on chance events, and looked forward to a theory which would be deterministic. Recent experiments suggest that this view of Einstein's conflicts with his other deeply held beliefs about the nature of the physical universe. Certain it is that somewhere, beyond physicists' current horizons, are even more powerful theories of how the world is." (David Wells, "You Are a Mathematician: A wise and witty introduction to the joy of numbers", 1995)

Peter M Higgins - Collected Quotes

"A little ingenuity is involved, but once a couple of tricks are learnt, it is not hard to show many sets of numbers are countable, which is the term we use to mean that the set can be listed in the same fashion as the counting numbers. Otherwise a set is called uncountable." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Algebraic symbols carry a universality of interpretation that allows them to be manipulated in a way that words cannot. Indeed, this was the key breakthrough that allowed mathematics to flourish in a way that was not possible until the advent of algebra. All higher mathematics relies on constant use of algebraic manipulation and would be impossible without it." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"For mathematics to be applicable in any sense at all we need to be able to do something with it. In practice this nearly always means developing forms of calculation, and this imperative channels its practitioners into algebraic manipulations of one form or another and ultimately into producing numbers. To the modern mind, this might seem natural and inevitable." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"If we refused to use complex numbers out of stubbornness disguised as some kind of bogus philosophical objection, a solution to a whole range of important problems would remain forever out of reach.[...] The plane of the complex numbers is the natural arena of discourse for much if not most of mathematics." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Nonetheless, some hesitation persisted. After all, the very word imaginary betrays ambivalence, and suggests that in our heart of hearts we do not believe these numbers exist. On the other hand, by calling every number representable by a decimal expansion real, we are making the psychological distinction more stark. Indeed the adjective imaginary is a somewhat unfortunate one - although an intriguing name, some students’ perceptions are so colored by the word that they consequently fail to come to grips with the idea." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Perhaps the greatest legacy of the solution of the cubic was the arrival, without invitation, of the imaginary number i into the world of mathematics." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Since they represent so natural a sequence, it is almost irresistible to search for patterns among the primes. There are however no genuinely useful formulas for prime numbers. That is to say there is no rule that allows you to generate all prime numbers or even to calculate a sequence that consists entirely of different primes." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The beauty of the complex plane is that we may finally carry out all our mathematical work in a single number arena. However, although there may be no pressing mathematical difficulty that is driving us further, we can ask the question whether or not it is possible to go beyond the complex plane into some larger realm of number." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The importance of continued fractions in approximation of irrationals by rationals is that the so called convergents of the fraction, which are the rational approximations of the original number that result from truncating the representation at some point and working out the corresponding rational number, are the best approximation possible in the sense that any better approximation will have a larger denominator than that of the convergents." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The rationals therefore also form a countable set, as do the euclidean numbers, and indeed if we consider the set of all numbers that arise from the rationals through taking roots of any order, the collection produced is still countable. We can even go beyond this: the collection of all algebraic numbers, which are those that are solutions of ordinary polynomial equations∗ form a collection that can, in principle, be arrayed in an infinite list: that is to say it is possible, with a little more crafty argument, to describe a systematic listing that sweeps them all out." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The simple algebraic numbers, like √2, seem closest in nature to the rationals, while we might expect that non-algebraic numbers, the transcedentals, to live apart and not to have close rational neighbors. Surprisingly, the opposite is true. On the one hand, it can be proved that any irrational number that can be well-approximated by rationals (in a sense that can be made precise) must be transcendental. Indeed this affords one of the standard techniques for showing that a number is transcendental." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"[...] the use of complex numbers reveals a connection between the exponential, or power function and the seemingly unrelated trigonometric functions. Without passing through the portal offered by the square root of minus one, the connection may be glimpsed, but not understood. The so-called hyperbolic functions arise from taking what are known as the even and odd parts of the exponential function."  (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"[...] transcendental numbers, those numbers that lie beyond those that arise through euclidean geometry and ordinary algebraic equations. [...] The transcendentals are the numbers that fill the huge void between the more familiar algebraic numbers and the collection of all decimal expansions: to use an astronomical comparison, the transcendentals are the dark matter of the number world." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Transcendental numbers then are numerous but exceedingly slippery. As a rule of thumb, a number that arises in mathematics is almost always transcendental unless it is obvious that it is not. However, showing that a particular number is transcendental can be exceedingly difficult. Number theory throws up endless problems of this kind where everyone feels sure what the answer must be but at the same time no-one has any real idea how it could ever by proved." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

On Numbers: Transcedentals

"Since one could directly derive the expansion in series of algebraic functions according to the powers of an increment, the derivatives, and the integral, one not only held that it was possible to assume the existence of such a series, derivative, and integral for all functions in general, but one never even had the idea that herein lay an assertion, whether it now be an axiom or a theorem - so self-evident did the transfer of the properties of algebraic functions to transcendental ones seem in the light of the geometrical view of curves representing functions. And examples in which purely analytic functions displayed singularities that were clearly different from those of algebraic functions remained entirely unnoticed." (Hermann Hankel, 1870)

"As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious." (George B Mathews, "Theory of Numbers", 1892)

"The proof that π is a transcendental number will forever mark an epoch in mathematical science. It gives the final answer to the problem of squaring the circle and settles this vexed question once for all. This problem requires to derive the number π by a finite number of elementary geometrical processes, i.e. with the use of the ruler and compasses alone. As a straight line and a circle, or two circles, have only two intersections, these processes, or any finite combination of them, can be expressed algebraically in a comparatively simple form, so that a solution of the problem of squaring the circle would mean that π can be expressed as the root of an algebraic equation of a comparatively simple kind, viz. one that is solvable by square roots." (Felix Klein, "Lectures on Mathematics", 1911)

"The number was first studied in respect of its rationality or irrationality, and it was shown to be really irrational. When the discovery was made of the fundamental distinction between algebraic and transcendental numbers, i. e. between those numbers which can be, and those numbers which cannot be, roots of an algebraical equation with rational coefficients, the question arose to which of these categories the number π belongs. It was finally established by a method which involved the use of some of the most modern of analytical investigation that the number π was transcendental. When this result was combined with the results of a critical investigation of the possibilities of a Euclidean determination, the inferences could be made that the number π, being transcendental, does not admit of a construction either by a Euclidean determination, or even by a determination in which the use of other algebraic curves besides the straight line and the circle are permitted." (Ernest W Hobson, "Squaring the Circle", 1913) 

"The simple algebraic numbers, like √2, seem closest in nature to the rationals, while we might expect that non-algebraic numbers, the transcedentals, to live apart and not to have close rational neighbors. Surprisingly, the opposite is true. On the one hand, it can be proved that any irrational number that can be well-approximated by rationals (in a sense that can be made precise) must be transcendental. Indeed this affords one of the standard techniques for showing that a number is transcendental." (Peter M. Higgins, "Number Story: From Counting to Cryptography", 2008)

"[...] transcendental numbers, those numbers that lie beyond those that arise through euclidean geometry and ordinary algebraic equations. [...] The transcendentals are the numbers that fill the huge void between the more familiar algebraic numbers and the collection of all decimal expansions: to use an astronomical comparison, the transcendentals are the dark matter of the number world." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Transcendental numbers then are numerous but exceedingly slippery. As a rule of thumb, a number that arises in mathematics is almost always transcendental unless it is obvious that it is not. However, showing that a particular number is transcendental can be exceedingly difficult. Number theory throws up endless problems of this kind where everyone feels sure what the answer must be but at the same time no-one has any real idea how it could ever by proved." (Peter M. Higgins, "Number Story: From Counting to Cryptography", 2008)

"It turns out π is different. Not only is it incapable of being expressed as a fraction, but in fact π fails to satisfy any algebraic relationship whatsoever. What does π do? It doesn’t do anything. It is what it is. Numbers like this are called transcendental (Latin for 'climbing beyond'). Transcendental numbers - and there are lots of them - are simply beyond the power of algebra to describe." (Paul Lockhart, "Measurement", 2012)

"A transcendental number is defined as a number that isn’t the solution of any polynomial equation with integer constants times the x’s." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

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Alexander Humez - Collected Quotes

"A sequence can be finite or infinite. One way of defining a finite sequence is to list every one of its terms. But to define an infinite sequence, the terms must be capable of being generated, in order, by some procedure with a predictable result for each place in the string." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"An infinite set is any set from which you can remove some members without reducing its size. In fact, we can start with an infinite set, remove an infinite set, and still have an infinite set left behind […]." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Finite-state machines, fundamental to the automatic translation and interpretation of languages used by computer programmers, can accept a limited set of inputs, and as the name implies, allow only a limited number of states. […] When it receives an input item (a coin for the vending machine, a pitch for the baseball count), the finite-state machine chooses a subsequent state based on both the input and the current state." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"In mathematics, a function is a sort of converter: It converts a value to some other value. […] the mathematical definition of a continuous function requires that any small change in the input will result in a small change to the output." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Like the noble gases (helium, neon, argon, krypton, xenon, and radon), primes exist in splendid isolation; conversely, any composite number is the product of a unique set of prime factors." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"'Randomness', too, is in the eye of the beholder." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"The four-color map theorem is an assertion about graph theory, which is the study of discrete points and the lines that connect them; each point is called a vertex and each line is called an edge." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Why is it so important to find primes, or to show that a certain integer is one? A very practical application in cryptography rests on the fact that since it is extremely hard to factor very large numbers, a two-hundred-digit number that was the product of two primes could govern text encoding: It would be virtually impossible to guess what the two numbers were if you didn't know them in advance, and out of the question (save perhaps on a state-of-the-art supercomputer) to go at it by trial and error." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Zero is the only number that is neither positive nor negative. As such, it represents a quantity: If three is the name we give to the number of items in a trilogy, a trinity, or a triad, zero is our name for the number of items in an empty, or null set, i.e., one having no members. This is not the same as saying the set doesn't exist; in fact, we can and do make valid assertions about null sets […]" (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Zero is where it all begins, the clean slate. We speak of zero-sum games (in which anyone who wins anything does so only at the equal expense of the losers), zero hour (the time at which a military operation begins), ground zero (the impact point of a bomb, particularly a nuclear one), to zero in on something (getting it precisely in the cross hairs), zero degrees of temperature-which, depending on the scale you use, can be the freezing point of water (Centigrade), fortified wine (Fahrenheit), or the universe (Kelvin); the last, a bit chillier than - 2730 C or - 459' F, is aptly called absolute zero." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

On Numbers: Irrationals

"It is told that those who first brought out the irrationals from concealment into the open perished in a shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantaneously destroyed and shall remain forever exposed to the play of the eternal waves." (Proclus Lycaeus, cca 5th century)

"For this evil I have found a remedy and obtained a method, by which without experimentation the roots of such binomials can be extracted, imaginaries being no hindrance, and not only in the case of cubics but also in higher equations. This invention rests upon a certain peculiarity which I will explain later. Now I will add certain rules derived from the consideration of irrationals (although no mention is made of irrationals), by which a rational root can easily be extracted from them." (Gottfried W Leibniz, cca. 1675)

"From the irrationals are born the impossible or imaginary quantities whose nature is very strange but whose usefulness is not to be despised." (Gottfried W Leibniz, "Specimen novum analyses pro Scientia infinity circa summas et quadraturas", 1700)

"It is probable that the number π is not even contained among the algebraical irrationalities, i.e., that it cannot be a root of an algebraical equation with a finite number of terms, whose coefficients are rational. But it seems to be very difficult to prove this strictly." (Adrien-Marie Legendre, "Elements de geometrie", 1794)

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

"Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this." (Richard Dedekind, "On Continuity and Irrational Numbers", 1872)

"Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers)." (Henry B Fine, "The Number-System of Algebra", 1890)

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

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

"The number was first studied in respect of its rationality or irrationality, and it was shown to be really irrational. When the discovery was made of the fundamental distinction between algebraic and transcendental numbers, i. e. between those numbers which can be, and those numbers which cannot be, roots of an algebraical equation with rational coefficients, the question arose to which of these categories the number π belongs. It was finally established by a method which involved the use of some of the most modern of analytical investigation that the number π was transcendental. When this result was combined with the results of a critical investigation of the possibilities of a Euclidean determination, the inferences could be made that the number π, being transcendental, does not admit of a construction either by a Euclidean determination, or even by a determination in which the use of other algebraic curves besides the straight line and the circle are permitted." (Ernest W Hobson, "Squaring the Circle", 1913)

"In particular, in introducing new numbers, mathematics is only obliged to give definitions of them, by which such a definiteness and, circumstances permitting, such a relation to the older numbers are conferred upon them that in given cases they can definitely be distinguished from one another. As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics. Here I perceive the reason why one has to regard the rational, irrational, and complex numbers as being just as thoroughly existent as the finite positive integers." (Georg Cantor, "Contributions to the Founding of the Theory of Transfinite Numbers", 1915)

"Complex numbers, though capable of a geometrical interpretation, are not demanded by geometry in the same imperative way in which irrationals are demanded. A 'complex' number means a number involving the square root of a negative number, whether integral, fractional, or real. Since the square of a negative number is positive, a number whose square is to be negative has to be a new sort of number." (Bertrand Russell," Introduction to Mathematical Philosophy", 1919)

"What does it matter whether π is rational or irrational? A mathematician faced with [this] question is in much the same position as a composer of music being questioned by someone with no ear for music. Why do you select some sets of notes and have them repeated by musicians, and reject others as worthless? It is difficult to answer except to say that there are harmonies in these things which we find that we can enjoy. It is true of course that some mathematics is useful. [...] But the so-called pure mathematicians do not do mathematics for such [practical applications]. It can be of no practical used to know that π is irrational, but if we can know it would surely be intolerable not to know. (Edward C Titchmarsh, "Mathematics for the General Reader", 1948)

"It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands." (Philip J Davis, "The Mathematics of Matrices", 1965)

"In contrast to the irrational numbers, whose discovery arose from a mundane problem in geometry, the first transcendental numbers were created specifically for the purpose of demonstrating that such numbers exist; in a sense they were 'artificial' numbers. But once this goal was achieved, attention turned to some more commonplace numbers, specifically π and e." (Eli Maor, "e: The Story of a Number", 1994)

"The discovery of complex numbers was the last in a sequence of discoveries that gradually filled in the set of all numbers, starting with the positive integers (finger counting) and then expanding to include the positive rationals and irrational reals, negatives, and then finally the complex." (Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998)

"Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Mathematical language is littered with pejorative and mystical terms - such as irrational, imaginary, surd, transcendental - that were once used to ridicule supposedly impossible objects. And these are just terms applied to numbers. Geometry also has many concepts that seem impossible to most people, such as the fourth dimension, finite universes, and curved space - yet geometers (and physicists) cannot do without them. Thus there is no doubt that  mathematics flirts with the impossible, and seems to make progress by doing so." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"The words 'imaginary' and 'complex' again demonstrate how difficult it is to make a major change in conceptual systems - a difficulty that we already encountered with negative numbers, fractions, zero, and irrational numbers. The word 'imaginary' tells us that these numbers are unreal from the perspective of someone grounded in the real number system." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)

"Clearly e is different from child-safe numbers such as four or 10, which wouldn’t dream of inducing sudden loss of cranial integrity. But this wantonness isn’t peculiar to e. In fact, the number line is chock full of numbers, like e, whose decimal representations are effectively infinite. They’re called irrational numbers." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"Ever since the discovery of irrational numbers fractured the Greek belief that all numbers were proportions, mathematicians have sorted numbers into categories and hunted for numbers that defied existing categories." (David Perkins, "φ, π, e & i", 2017)

"It just so happens that π can be characterised precisely without any reference to decimals, because it is simply the ratio of any circle’s circumference to its diameter. Likewise can be characterised as the positive number which squares to 2. However, most irrational numbers can’t be characterised in this way." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Since it’s impossible to express an irrational number such as π as a fraction, the quest for a fraction equal to π could never be successful. Ancient mathematicians didn’t know that, however. As noted above, it wasn’t until the eighteenth century that the irrationality of π was demonstrated. Their labors weren’t in vain, though. While enthusiastically pursuing their fundamentally doomed enterprise, they developed a lot of interesting mathematics as well as impressively accurate approximations of π." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)

"The reason we need irrational numbers in the first place is to fill in the 'gaps' that are doomed to be in between all the rational numbers." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

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