On Logarithms - From Fiction to Science-Fiction

"Consequently, these laws of nature need only be discovered, and then man will no longer be answerable for his actions, and his life will become extremely easy. Needless to say, all human actions will then be calculated according to these laws, mathematically, like a table of logarithms, up to 108,000, and entered into a calendar; or, better still, some well-meaning publications will appear, like the present-day encyclopedic dictionaries, in which everything will be so precisely calculated and designated that there will no longer be any actions or adventures in the world." (Fyodor Dostoevsky, "Notes from Underground", 1864)

"Every fact is a logarithm; one added term ramifies it until it is thoroughly transformed. In the general aspect of things, the great lines of creation take shape and arrange themselves into groups; beneath lies the unfathomable." (Victor Hugo, "The Toilers of the Sea", 1866)

"What logarithms are to mathematics that mathematics are to the other sciences." (Friederich von Hardenberg [Novalis], "Schriften", 1901)

"Those terrible logarithms, when I happened to open a table of them, made my head swim, with their columns of figures; actual fright, not unmixed with respect, overwhelmed me on the very threshold of that arithmetical cave." (Jean-Henri Fabre, "The Life of the Fly", 1913)

"She thinks of the Heat Death of the Universe. A logarithmic of those late summer days, endless as the Irish serpent twisting through jewelled manuscripts forever, tail in mouth, the heat pressing, bloating, doing violence." (Pamela Zoline, "The Heat Death of the Universe", 1967)

"You could probably prove, by judicious use of logarithms and congruent triangles, that real life is a lot more like soap opera than most people will admit." (Molly Ivins, The Progressive, 1988)

Phillip I Good - Collected Quotes

 "A major problem with many studies is that the population of interest is not adequately defined before the sample is drawn. Don’t make this mistake. A second major source of error is that the sample proves to have been drawn from a different population than was originally envisioned." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"A permutation test based on the original observations is appropriate only if one can assume that under the null hypothesis the observations are identically distributed in each of the populations from which the samples are drawn. If we cannot make this assumption, we will need to transform the observations, throwing away some of the information about them so that the distributions of the transformed observations are identical." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"A well-formulated hypothesis will be both quantifiable and testable - that is, involve measurable quantities or refer to items that may be assigned to mutually exclusive categories. [...] When the objective of our investigations is to arrive at some sort of conclusion, then we need to have not only a hypothesis in mind, but also one or more potential alternative hypotheses." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"Before we initiate data collection, we must have a firm idea of what we will measure. A second fundamental principle is also applicable to both experiments and surveys: Collect exact values whenever possible. Worry about grouping them in interval or discrete categories later. […] You can always group your results (and modify your groupings) after a study is completed. If after-the-fact grouping is a possibility, your design should state how the grouping will be determined; otherwise there will be the suspicion that you chose the grouping to obtain desired results." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"Estimation methods should be impartial. Decisions should not depend on the accidental and quite irrelevant labeling of the samples. Nor should decisions depend on the units in which the measurements are made." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"Every statistical procedure relies on certain assumptions for correctness. Errors in testing hypotheses come about either because the assumptions underlying the chosen test are not satisfied or because the chosen test is less powerful than other competing procedures."(Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"[…] finding at least one cluster of events in time or in space has a greater probability than finding no clusters at all (equally spaced events)." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"Graphical illustrations should be simple and pleasing to the eye, but the presentation must remain scientific. In other words, we want to avoid those graphical features that are purely decorative while keeping a critical eye open for opportunities to enhance the scientific inference we expect from the reader. A good graphical design should maximize the proportion of the ink used for communicating scientific information in the overall display." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"If the sample is not representative of the population because the sample is small or biased, not selected at random, or its constituents are not independent of one another, then the bootstrap will fail. […] For a given size sample, bootstrap estimates of percentiles in the tails will always be less accurate than estimates of more centrally located percentiles. Similarly, bootstrap interval estimates for the variance of a distribution will always be less accurate than estimates of central location such as the mean or median because the variance depends strongly upon extreme values in the population." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"More important than comparing the means of populations can be determining why the variances are different." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"Most statistical procedures rely on two fundamental assumptions: that the observations are independent of one another and that they are identically distributed. If your methods of collection fail to honor these assumptions, then your analysis must fail also." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"Never assign probabilities to the true state of nature, but only to the validity of your own predictions." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"The greatest error associated with the use of statistical procedures is to make the assumption that one single statistical methodology can suffice for all applications. […] But one methodology can never be better than another, nor can estimation replace hypothesis testing or vice versa. Every methodology has a proper domain of application and another set of applications for which it fails. Every methodology has its drawbacks and its advantages, its assumptions, and its sources of error." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"The sources of error in applying statistical procedures are legion and include all of the following: (•) Using the same set of data both to formulate hypotheses and to test them. (•) Taking samples from the wrong population or failing to specify the population(s) about which inferences are to be made in advance. (•) Failing to draw random, representative samples. (•) Measuring the wrong variables or failing to measure what you’d hoped to measure. (•) Using inappropriate or inefficient statistical methods. (•) Failing to validate models. But perhaps the most serious source of error lies in letting statistical procedures make decisions for you." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"The vast majority of errors in estimation stem from a failure to measure what one wanted to measure or what one thought one was measuring. Misleading definitions, inaccurate measurements, errors in recording and transcription, and confounding variables plague results. To forestall such errors, review your data collection protocols and procedure manuals before you begin, run several preliminary trials, record potential confounding variables, monitor data collection, and review the data as they are collected." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"The vast majority of errors in Statistics - and not incidentally, in most human endeavors - arise from a reluctance (or even an inability) to plan. Some demon (or demonic manager) seems to be urging us to cross the street before we’ve had the opportunity to look both ways. Even on those rare occasions when we do design an experiment, we seem more obsessed with the mechanics than with the concepts that underlie it." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"Use statistics as a guide to decision making rather than a mandate." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"When we assert that for a given population a percentage of samples will have a specific composition, this also is a deduction. But when we make an inductive generalization about a population based upon our analysis of a sample, we are on shakier ground. It is one thing to assert that if an observation comes from a normal distribution with mean zero, the probability is one-half that it is positive. It is quite another if, on observing that half the observations in the sample are positive, we assert that half of all the possible observations that might be drawn from that population will be positive also." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

"While a null hypothesis can facilitate statistical inquiry - an exact permutation test is impossible without it - it is never mandated. In any event, virtually any quantifiable hypothesis can be converted into null form. There is no excuse and no need to be content with a meaningless null. […] We must specify our alternatives before we commence an analysis, preferably at the same time we design our study." (Phillip I Good & James W Hardin, "Common Errors in Statistics (and How to Avoid Them)", 2003)

Robert Hooke - Collected Quotes

"Accounting figures are a blend of facts and arbitrary procedures that are designed to facilitate the recording and communication of business transactions. Their usefulness in the decision process is sometimes grossly overestimated." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"All of us learn by experience. Except for pure deductive processes, everything we learn is from someone's experience. All experience is a sample from an immense range of possible experience that no one individual can ever take in. It behooves us to know what parts of the information we get from samples can be trusted and what cannot." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Being experimental, however, doesn't necessarily make a scientific study entirely credible. One weakness of experimental work is that it can be out of touch with reality when its controls are so rigid that conclusions are valid only in the experimental situation and don't carryover into the real world." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Correlation analysis is a useful tool for uncovering a tenuous relationship, but it doesn't necessarily provide any real understanding of the relationship, and it certainly doesn't provide any evidence that the relationship is one of cause and effect. People who don't understand correlation tend to credit it with being a more fundamental approach than it is." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Experiments usually are looking for 'signals' of truth, and the search is always ham pered by 'noise' of one kind or another. In judging someone else's experimental results it's important to find out whether they represent a true signal or whether they are just so much noise." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

 "First and foremost an experiment should have a goal, and the goal should be something worth achieving, especially if the experimenter is working on someone else's (for example, the taxpayers') money. 'Worth achieving' implies more than just beneficial; it also should mean that the experiment is the most beneficial thing we can think of doing. Obviously we can't predict accurately the value of an experiment (this may not even be possible after we see how it turns out), but we should feel obliged to make as intelligent a choice as we can. Such a choice is sometimes labeled a 'value judgment'." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"In general a small-scale test or experiment will not detect a small effect, or small differences among various products." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Mistakes arising from retrospective data analysis led to the idea of experimentation, and experience with experimentation led to the idea of controlled experiments and then to the proper design of experiments for efficiency and credibility. When someone is pushing a conclusion at you, it's a good idea to ask where it came from - was there an experiment, and if so, was it controlled and was it relevant?" (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"One important way of developing our powers of discrimination between good and bad statistical studies is to learn about the differences between backward-looking (retrospective or historical) data and data obtained through carefully planned and controlled (forward-looking) experiments." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Only a 0 correlation is uninteresting, and in practice 0 correlations do not occur. When you stuff a bunch of numbers into the correlation formula, the chance of getting exactly 0, even if no correlation is truly present, is about the same as the chance of a tossed coin ending up on edge instead of heads or tails." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Randomization is usually a cheap and harmless way of improving the effectiveness of experimentation with very little extra effort." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Science usually amounts to a lot more than blind trial and error. Good statistics consists of much more than just significance tests; there are more sophisticated tools available for the analysis of results, such as confidence statements, multiple comparisons, and Bayesian analysis, to drop a few names. However, not all scientists are good statisticians, or want to be, and not all people who are called scientists by the media deserve to be so described." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Statistical reasoning is such a fundamental part of experimental science that the study of principles of data analysis has become a vital part of the scientist's education. Furthermore, […] the existence of a lot of data does not necessarily mean that any useful information is there ready to be extracted." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"The idea of statistical significance is valuable because it often keeps us from announcing results that later turn out to be nonresults. A significant result tells us that enough cases were observed to provide reasonable assurance of a real effect. It does not necessarily mean, though, that the effect is big enough to be important." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"Today's scientific investigations are so complicated that even experts in related fields may not understand them well. But there is a logic in the planning of experiments and in the analysis of their results that all intelligent people can grasp, and this logic is a great help in determining when to believe what we hear and read and when to be skeptical. This logic has a great deal to do with statistics, which is why statisticians have a unique interest in the scientific method, and why some knowledge of statistics can so often be brought to bear in distinguishing good arguments from bad ones." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

"When a real situation involves chance we have to use probability mathematics to understand it quantitatively. Direct mathematical solutions sometimes exist […] but most real systems are too complicated for direct solutions. In these cases the computer, once taught to generate random numbers, can use simulation to get useful answers to otherwise impossible problems." (Robert Hooke, "How to Tell the Liars from the Statisticians", 1983)

Darell Huff - Collected Quotes

"Another thing to watch out for is a conclusion in which a correlation has been inferred to continue beyond the data with which it has been demonstrated." (Darell Huff, "How to Lie with Statistics", 1954)

"Extrapolations are useful, particularly in the form of soothsaying called forecasting trends. But in looking at the figures or the charts made from them, it is necessary to remember one thing constantly: The trend to now may be a fact, but the future trend represents no more than an educated guess. Implicit in it is 'everything else being equal' and 'present trends continuing'. And somehow everything else refuses to remain equal." (Darell Huff, "How to Lie with Statistics", 1954)

"If you can't prove what you want to prove, demonstrate something else and pretend that they are the something. In the daze that follows the collision of statistics with the human mind, hardly anybody will notice the difference." (Darell Huff, "How to Lie with Statistics", 1954)

"Keep in mind that a correlation may be real and based on real cause and effect -and still be almost worthless in determining action in any single case." (Darell Huff, "How to Lie with Statistics", 1954) 

"Only when there is a substantial number of trials involved is the law of averages a useful description or prediction." (Darell Huff, "How to Lie with Statistics", 1954)

"Percentages offer a fertile field for confusion. And like the ever-impressive decimal they can lend an aura of precision to the inexact. […] Any percentage figure based on a small number of cases is likely to be misleading. It is more informative to give the figure itself. And when the percentage is carried out to decimal places, you begin to run the scale from the silly to the fraudulent." (Darell Huff, "How to Lie with Statistics", 1954)

"Place little faith in an average or a graph or a trend when those important figures are missing."  (Darell Huff, "How to Lie with Statistics", 1954)

"Sometimes the big ado is made about a difference that is mathematically real and demonstrable but so tiny as to have no importance. This is in defiance of the fine old saying that a difference is a difference only if it makes a difference." (Darell Huff, "How to Lie with Statistics", 1954)

"The fact is that, despite its mathematical base, statistics is as much an art as it is a science. A great many manipulations and even distortions are possible within the bounds of propriety. Often the statistician must choose among methods, a subjective process, and find the one that he will use to represent the facts." (Darell Huff, "How to Lie with Statistics", 1954)

"The purely random sample is the only kind that can be examined with entire confidence by means of statistical theory, but there is one thing wrong with it. It is so difficult and expensive to obtain for many uses that sheer cost eliminates it." (Darell Huff, "How to Lie with Statistics", 1954)

"The secret language of statistics, so appealing in a fact-minded culture, is employed to sensationalize, inflate, confuse, and oversimplify. Statistical methods and statistical terms are necessary in reporting the mass data of social and economic trends, business conditions, 'opinion' polls, the census. But without writers who use the words with honesty and understanding and readers who know what they mean, the result can only be semantic nonsense." (Darell Huff, "How to Lie with Statistics", 1954)

"There are often many ways of expressing any figure. […] The method is to choose the one that sounds best for the purpose at hand and trust that few who read it will recognize how imperfectly it reflects the situation." (Darell Huff, "How to Lie with Statistics", 1954)

"To be worth much, a report based on sampling must use a representative sample, which is one from which every source of bias has been removed." (Darell Huff, "How to Lie with Statistics", 1954)

"When numbers in tabular form are taboo and words will not do the work well as is often the case. There is one answer left: Draw a picture. About the simplest kind of statistical picture or graph, is the line variety. It is very useful for showing trends, something practically everybody is interested in showing or knowing about or spotting or deploring or forecasting." (Darell Huff, "How to Lie with Statistics", 1954)

"When you are told that something is an average you still don't know very much about it unless you can find out which of the common kinds of average it is-mean, median, or mode. [...] The different averages come out close together when you deal with data, such as those having to do with many human characteristics, that have the grace to fall close to what is called the normal distribution. If you draw a curve to represent it you get something shaped like a bell, and mean, median, and mode fall at the same point." (Darell Huff, "How to Lie with Statistics", 1954)

"When you find somebody - usually an interested party - making a fuss about a correlation, look first of all to see if it is not one of this type, produced by the stream of events, the trend of the times." (Darell Huff, "How to Lie with Statistics", 1954)

On Logarithms (2000 - )

"A logarithm is a mapping that allows you to multiply by adding." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"As an abstract mathematical function, log maps every positive real number onto a corresponding real number, and maps every product of positive real numbers onto a sum of real numbers. Of course, there can be no table for such a mapping, because it would be infinitely long. But abstractly, such a mapping can be characterized as outlined here. These constraints completely and uniquely determine every possible value of the mapping. But the constraints do not in provide an algorithm for computing such mappings for al1 the real numbers. Approximations to values for real numbers can be made to any degree of accuracy required by doing arithmetic operations on rational numbers." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"The mathematics of physics resides in physical phenomena themselves - there are ellipses in the elliptical orbits of the planets, fractals in the fractal shapes of leaves and branches, logarithms in the logarithmic spirals of snails. This means that 'the books of nature is written in mathematics', which implies that the language of mathematics is the language of nature and that only those who lznow mathematics can truly understand nature." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"Information entropy has its own special interpretation and is defined as the degree of unexpectedness in a message. The more unexpected words or phrases, the higher the entropy. It may be calculated with the regular binary logarithm on the number of existing alternatives in a given repertoire. A repertoire of 16 alternatives therefore gives a maximum entropy of 4 bits. Maximum entropy presupposes that all probabilities are equal and independent of each other. Minimum entropy exists when only one possibility is expected to be chosen. When uncertainty, variety or entropy decreases it is thus reasonable to speak of a corresponding increase in information." (Lars Skyttner, "General Systems Theory: Ideas and Applications", 2001)

"If the intensity of the material world is plotted along the horizontal axis, and the response of the human mind is on the vertical, the relation between the two is represented by the logarithmic curve. Could this rule provide a clue to the relationship between the objective measure of information, and our subjective perception of it?" (Hans Christian von Baeyer, "Information, The New Language of Science", 2003)

"The revelation that the graph appears to climb so smoothly, even though the primes themselves are so unpredictable, is one of the most miraculous in mathematics and represents one of the high points in the story of the primes. On the back page of his book of logarithms, Gauss recorded the discovery of his formula for the number of primes up to N in terms of the logarithm function. Yet despite the importance of the discovery, Gauss told no one what he had found. The most the world heard of his revelation were the cryptic words, 'You have no idea how much poetry there is in a table of logarithms.'" (Marcus du Sautoy, "The Music of the Primes", 2003)

"Mathematics is sometimes described as the science which generates eternal notions and concepts for the scientific method: derivatives‚ continuity‚ powers‚ logarithms are examples. The notions of chaos‚ fractals and strange attractors are not yet mathematical notions in that sense‚ because their final definitions are not yet agreed upon." (Heinz-Otto Peitgen et al, "Chaos and Fractals: New Frontiers of Science", 2004)

"In a time of great mathematical ignorance, John Napier made an outstanding contribution through his discovery of the logarithm. Not only did this discovery provide an algorithm that simplified arithmetical computation, but it also presented a transcendental function that has fascinated mathematicians for centuries." (Tucker McElroy, "A to Z of Mathematicians", 2005)

"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)

"[…] the symmetry group of the infinite logarithmic spiral is an infinite group, with one element for each real number . Two such transformations compose by adding the corresponding angles, so this group is isomorphic to the real numbers under addition." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

On Logarithms (1900-1949)

"The invention of logarithms and the calculation of the earlier tables form a very striking episode in the history of exact science, and, with the exception of the Principia of Newton, there is no mathematical work published in the country which has produced such important consequences, or to which so much interest attaches as to Napier's Descriptio." (James W L Glaisher, "Logarithms", Encyclopedia Britannica 9th Ed., 1914)

"The invention of logarithms came on the world as a bolt from the blue. No previous work had led up to it, nothing had foreshadowed it or heralded its arrival. It stands isolated, breaking in upon human thought abruptly without borrowing from the work of other intellects or following known lines of mathematical thought. It reminds me of those islands in the ocean which rise up suddenly from great depths and which stand solitary with deep water close around all their shores. In such cases we may believe that some cataclysm has thrust them up suddenly with earth-rending force. But can it be so with human thought?" (Lord John F Moulton, "The Invention of Logarithms, Its Genesis and Growth", [address in "The Napier Tercentenary"] 1914)

"To summarize - with the ordinary arithmetical scale, fluctuations in large factors are very noticeable, while relatively greater fluctuations in smaller factors are barely apparent. The logarithmic scale permits the graphic representation of changes in every quantity without respect to the magnitude of the quantity itself. At the same time, the logarithmic scale shows the actual value by reference to the numbers in the vertical scale. By indicating both absolute and relative values and changes, the logarithmic scale combines the advantages of both the natural and the percentage scale without the disadvantages of either." (Willard C Brinton, "Graphic Methods for Presenting Facts", 1919)

"With the ordinary scale, fluctuations in large factors are very noticeable, while relatively greater fluctuations in smaller factors are barely apparent. The semi-logarithmic scale permits the graphic representation of changes in every quantity on the same basis, without respect to the magnitude of the quantity itself. At the same time, it shows the actual value by reference to the numbers in the scale column. By indicating both absolute and relative value and changes to one scale, it combines the advantages of both the natural and percentage scale, without the disadvantages of either." (Allan C Haskell, "How to Make and Use Graphic Charts", 1919)

"The definition of e is usually, in imitation of the French models, placed at the very beginning of the great text books of analysis, and entirely unmotivated, whereby the really valuable element is missed, the one which mediates the understanding, namely, an explanation of why precisely this remarkable limit is used as base and why the resulting logarithms are called natural." (Felix Klein, "Elementary Mathematics from an Advanced Standpoint", 1924)

"The piano keyboard is really a rather inaccurate table of logarithms, a fact which I believe is equally ignored in the teaching of mathematics and of music." (John B S Haldane, "Possible Worlds and Other Essays", 1928)

"Mathematics, indeed, is the very example of brevity, whether it be in the shorthand rule of the circle, c = πd, or in that fruitful formula of analysis, e^iπ = -1, - a formula which fuses together four of the most important concepts of the science - the logarithmic base, the transcendental ratio π, and the imaginary and negative units." (David E Smith, "The Poetry of Mathematics", The Mathematics Teacher, 1926)

On Complex Numbers XXI (Euler’s Formula III)

"Mathematics, indeed, is the very example of brevity, whether it be in the shorthand rule of the circle, c = πd, or in that fruitful formula of analysis, e^iπ = -1, — a formula which fuses together four of the most important concepts of the science — the logarithmic base, the transcendental ratio π, and the imaginary and negative units." (David E Smith, "The Poetry of Mathematics", The Mathematics Teacher, 1926)

"Other questions must be answered as well. Why should e^πi equal, of all things, -1? e^πi has an imaginary number in it; wouldn't you therefore expect the result to be imaginary, not real? e is about differentiation, about change, and π is about circles. What do the ideas involved in change and in circles have to do with the answer? e and n are both transcendental numbers - numbers that are not roots of any algebraic equation. If you operate on one transcendental number with another and then operate on the result with an imaginary number, why should you get a simple integer like -1?" (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)

"What a wealth of insight Euler’s formula reveals and what delicacy and precision of reasoning it exhibits. It provides a definition of complex exponentiation: It is a definition of complex exponentiation, but the definition proceeds in the most natural way, like a trained singer’s breath. It closes the complex circle once again by guaranteeing that in taking complex numbers to complex powers the mathematician always returns with complex numbers. It justifies the method of infinite series and sums. And it exposes that profound and unsuspected connection between exponential and trigonometric functions; with Euler’s formula the very distinction between trigonometric and exponential functions acquires the shimmer of a desert illusion." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)

"I think e^iπ+1=0 is beautiful because it is true even in the face of enormous potential constraint. The equality is precise; the left-hand side is not 'almost' or 'pretty near' or 'just about' zero, but exactly zero. That five numbers, each with vastly different origins, and each with roles in mathematics that cannot be exaggerated, should be connected by such a simple relationship, is just stunning. It is beautiful. And unlike the physics or chemistry or engineering of today, which will almost surely appear archaic to technicians of the far future, Euler's formula will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)

"Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm [ln(-1)]; secondly that a negative quantity has no square root [√-1]; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter [π]." (Augustus De Morgan)

"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence." (Keith Devlin)

On Complex Numbers XXII

"I have finally discovered the true solution: in the same way that to one sine there correspond an infinite number of different angles I have found that it is the same with logarithms, and each number has an infinity of different logarithms, all of them imaginary unless the number is real and positive; there is only one logarithm which is real, and we regard it as its unique logarithm." (Leonhard Euler, [letter to Cramer] 1746)

"If we then compare the position in which we stand with respect to divergent series, with that in which we stood a few years ago with respect to impossible quantities [that is, complex numbers], we shall find a perfect similarity […] It became notorious that such use [of complex numbers] generally led to true results, with now and then an apparent exception. […] But at last came the complete explanation of the impossible quantity, showing that all the difficulty had arisen from too great limitation of definitions." (Augustus de Morgan, Penny Cyclopaedia, cca. 1833-1843)

"The set of complex numbers is another example of a field. It is handy because every polynomial in one variable with integer coefficients can be factored into linear factors if we use complex numbers. Equivalently, every such polynomial has a complex root. This gives us a standard place to keep track of the solutions to polynomial equations." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"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 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)

"Another reason for our ambivalence about the complex numbers is that they feel less real than real numbers. [...] We can directly relate the real numbers to quantities such as time, mass, length, temperature, and so on (though for this usage, we never need the infinite precision of the real number system), so it feels as though they have an independent existence that we observe. But we do not run into the complex numbers in that way. Rather, we play what feels like a sort of game - imagine what would happen if -1 did have a square root." (Timothy Gowers, "Is Mathematics Discovered or Invented?",  ["The Best Writing of Mathematics: 2012"] 2012)

"All of this could have been said using notation that kept √-1 instead of the new representative i, which has the same virtual meaning. But i isolates the concept of rotation from the perception of root extraction, offering the mind a distinction between an algebraic result and an extension of the idea of number." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"It may come as a surprise that the symbol i (even though it is just an abbreviation of the word 'imaginary') has a marked advantage over √-1. In reading mathematics, the difference between a + b√-1 and a + bi is the difference between eating a strawberry while holding your nose, missing the luscious taste, and eating a strawberry while breathing normally." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)

"Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm [ln(-1)]; secondly that a negative quantity has no square root [√-1]; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter [π]." (Augustus De Morgan) [attributed]

On Logarithms ( - 1899)

"Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers. [...] I began therefore to consider in my mind by what certain  and ready art I might remove those hindrances." (John Napier, "Mirifici logarithmorum canonis descriptio", 1614)

"They seem to have been called logarithms by their illustrious inventor because they exhibit to us numbers which always preserve the same ratio to one another." (Henry Briggs, "Arithmetica Logarithmica", 1624)

"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)

"I have finally discovered the true solution: in the same way that to one sine there correspond an infinite number of different angles I have found that it is the same with logarithms, and each number has an infinity of different logarithms, all of them imaginary unless the number is real and positive; there is only one logarithm which is real, and we regard it as its unique logarithm." (Leonhard Euler, [letter to Cramer] 1746)

"The science of calculation [...] is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not to be indulged in by one who is to have a profession to follow for his subsistence." (Thomas Jefferson, [letter to William G Munford] 1799)

"A Logarithmic Table is a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space, by a very easy calculation. It is deservedly called very small, because it does not exceed in size a table of sines; very easy, because by it all multiplications, divisions, and the more difficult extractions of roots are avoided; for by only a very few most easy additions, subtractions, and divisions by two, it measures quite generally all figures and motions."  (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"And if any number of equals to a first sine be multiplied together producing a second, just so many equals to the Logarithm of the first added together produce the Logarithm of the second." (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"Any desired geometrical mean between two sines has for its Logarithm the corresponding arithmetical mean between the Logarithms of the sines." (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"To decrease geometrically is this, that in equal times, first the whole quantity then each of its successive remainders is diminished, always by a like proportional part." (John Napier, "The Construction of the Wonderful Canon of Logarithms", 1889)

"Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skillful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures."(Johann F Herbart, 1890)

"The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms." (Florian Cajori, "A History of Mathematics", 1894)

"Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm [ln(-1)]; secondly that a negative quantity has no square root [√-1]; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter [π]." (Augustus De Morgan) [attributed]

"[logarithms] by shortening the labours doubled the life of the astronomer." (Pierre Simon Laplace, [in Howard Eves'  "Mathematical Circles", 1969])

"As nature puts forth its wonders, most of us are oblivious to the massive calculations and mathematical work needed to explain something that i5 very routine to nature. For example, the Orb spider's web is a simple, but elegant natural creation. When this beautiful structure is analyzed, the mathematical ideas that appear in the web are indeed complicated and surprising - radii, chords, parallel segments, triangles, congruent corresponding angles, the logarithmic spiral, the catenary curve and the transcendental number e. Yet even with all our mathematical forces at work - including chaos and complexity theories - many natural phenomena, such as earthquake and weather predictions, still elude precise mathematical description. The profound study of nature is the most fertile source of mathematical discoveries." (Joseph Fourier)

On Logarithms (1950 - 1999)

"Just as entropy is a measure of disorganization, the information carried by a set of messages is a measure of organization. In fact, it is possible to interpret the information carried by a message as essentially the negative of its entropy, and the negative logarithm of its probability. That is, the more probable the message, the less information it gives. Clichés, for example, are less illuminating than great poems." (Norbert Wiener, "The Human Use of Human Beings", 1950)

"The efforts of computer engineers have already produced a mechanical Briggs (who spent his lifetime computing logarithms) and a mechanical Barlow (whose famous Tables were a life’s work), but no one has ever conceived of a mechanical Napier (for he invented logarithms)." (Bertram V Bowden, "Faster than Thought", 1953)

"In form, the ratio chart differs from the arithmetic chart in that the vertical scale is not divided into equal spaces to represent equal amounts, but is divided logarithmically to represent percentages of gain or loss. On the arithmetic chart equal vertical distances represent equal amounts of change; on the ratio chart equal vertical distances represent equal percentages of change." (Walter E Weld, "How to Chart; Facts from Figures with Graphs", 1959)

So we now have to talk about what we mean by disorder and what we mean by order.[...] Suppose we divide the space into little volume elements. If we have black and white molecules, how many ways could we distribute them among the volume elements so that white is on one side and black is on the other? On the other hand, how many ways could we distribute them with no restriction on which goes where? Clearly, there are many more ways to arrange them in the latter case. We measure 'disorder' by the number of ways that the insides can be arranged, so that from the outside it looks the same. The logarithm of that number of ways is the entropy. The number of ways in the separated case is less, so the entropy is less, or the 'disorder' is less." (Richard P Feynman et al, "Feynman Lectures on Physics" Vol. 1, 1963)

"Since logarithms are clearly part of pure mathematics it may well be surprising to learn that they have been until now the subject of an embarrassing controversy in which whatever side is taken contradictions appear that seem completely impossible to resolve. Meanwhile if truth is to be universal there can be no doubt that these contradictions, [...], however unresolved they seem can only be apparent. [...] I will bring out fully all the contradictions involved so that it may be seen how difficult it is to discover truth and to guard against inconsistency even when two great men are working on the problem." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)

"Logging skewed variables also helps to reveal the patterns in the data. […] the rescaling of the variables by taking logarithms reduces the nonlinearity in the relationship and removes much of the clutter resulting from the skewed distributions on both variables; in short, the transformation helps clarify the relationship between the two variables. It also […] leads to a theoretically meaningful regression coefficient." (Edward R Tufte, "Data Analysis for Politics and Policy", 1974)

"It is common for positive data to be skewed to the right: some values bunch together at the low end of the scale and others trail off to the high end with increasing gaps between the values as they get higher. Such data can cause severe resolution problems on graphs, and the common remedy is to take logarithms. Indeed, it is the frequent success of this remedy that partly accounts for the large use of logarithms in graphical data display." (William S Cleveland, "The Elements of Graphing Data", 1985) 

"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)

"Great inventions generally fall into one of two categories: some are the product of a single person's creative mind, descending on the world suddenly like a bolt out of the blue; others - by far the larger group - are the end product of a long evolution of ideas that have fermented in many minds over decades, if not centuries. The invention of logarithms belongs to the first group, that of the calculus to the second." (Eli Maor, "e: The Story of a Number", 1994)

"If you want to show the growth of numbers which tend to grow by percentages, plot them on a logarithmic vertical scale. When plotted against a logarithmic vertical axis, equal percentage changes take up equal distances on the vertical axis. Thus, a constant annual percentage rate of change will plot as a straight line. The vertical scale on a logarithmic chart does not start at zero, as it shows the ratio of values (in this case, land values), and dividing by zero is impossible." (Herbert F Spirer et al, "Misused Statistics" 2nd Ed, 1998)

Out of Context: On Fractals (Definitions)

 "A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales [...]" (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)

"A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)

"In the mind's eye, a fractal is a way of seeing infinity." (James Gleick, "Chaos: Making a New Science, A Geometry of Nature", 1987)

"Fractals are geometric shapes that are equally complex in their details as in their overall form. That is, if a piece of a fractal is suitably magnified to become of the same size as the whole, it should look like the whole, either exactly, or perhaps after a slight limited deformation." (Benoît B Mandelbrot, "Fractals and an Art for the Sake of Science", 1989)

"Fractals are patterns which occur on many levels." (Györgi Ligeti, [interview] 1999)

"Mathematical fractals are generated by repeating the same simple steps at ever decreasing scales. In this way an apparently complex shape, containing endless detail, can be generated by the repeated application of a simple algorithm." (F David Peat, "From Certainty to Uncertainty", 2002)

[fractal:] "A fragmented geometric shape that can be split up into secondary pieces, each of which is approximately a smaller replica of the whole, the phenomenon commonly known as self similarity." (Khondekar et al, "Soft Computing Based Statistical Time Series Analysis, Characterization of Chaos Theory, and Theory of Fractals", 2013)

"Fractals are generally self-similar (each section looks at all) and are not subordinated to a specific scale. They are used especially in the digital modeling of irregular patterns and structures in nature." (Mauro Chiarella, "Folds and Refolds: Space Generation, Shapes, and Complex Components", 2016)

"A fractal is a mathematical set or concrete object that is irregular or fragmented at all scales […]" (Benoît B Mandelbrot)

On Truth: Scientific Truth (1900-)

"The man who discovers a new scientific truth has previously had to smash to atoms almost everything he had learnt, and arrives at the new truth with hands blood stained from the slaughter of a thousand platitudes." (Jose Ortega y Gasset, "The Revolt of the Masses", 1930)

"The belief in science has replaced in large measure, the belief in God. Even where religion was regarded as compatible with science, it was modified by the mentality of the believer in scientific truth." (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

"Uncertainty is introduced, however, by the impossibility of making generalizations, most of the time, which happens to all members of a class. Even scientific truth is a matter of probability and the degree of probability stops somewhere short of certainty." (Wayne C Minnick,"The Art of Persuasion", 1957)

"The only solid piece of scientific truth about which I feel totally confident is that we are profoundly ignorant about nature." (Lewis Thomas, "The Medusa and the Snail: More Notes of a Biology Watcher", 1979)

"Modern philosophy of science has gone far beyond the naive belief that science reveals the truth. Even if it could, we would have no means of proving it. Certainty seems unattainable. All scientific statements remain open to doubt. […] We cannot reach the absolute at least as far as science is concerned; we have to content ourselves with the relative." (Rolf Sattler, "Biophilosophy", 1986)

"Science doesn't purvey absolute truth. Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match. And this works, not just for the ordinary aspects of science, but for all of life. I should think people would want to know that what they know is truly what the universe is like, or at least as close as they can get to it." (Isaac Asimov, [Interview by Bill Moyers] 1988)

"The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific ‘truth’." (Richard Feynman, "Six Easy Pieces", 1994)

"What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?" (Raymond S Nickerson, "Mathematical Reasoning:  Patterns, Problems, Conjectures, and Proofs", 2010)

On Truth: Scientific Truth (-1899)

"The foundations of chemical philosophy are observation, experiment, and analogy. By observation, facts are distinctly and minutely impressed on the mind. By analogy, similar facts are connected. By experiment, new facts are discovered; and, in the progression of knowledge, observation, guided by analogy, lends to experiment, and analogy confirmed by experiment, becomes scientific truth." (Sir Humphry Davy, "Elements of Chemical Philosophy" Vol. 4, 1812)

"Scientific truth is marvellous, but moral truth is divine; and whoever breathes its air and walks by its light has found the lost paradise." (Horace Mann, "A Few Thoughts for a Young Man", Monthly Literary Miscellany, 1851)

"A mere inference or theory must give way to a truth revealed; but a scientific truth must be maintained, however contradictory it may appear to the most cherished doctrines of religion." (David Brewster, "More Worlds Than One: The Creed of the Philosopher and the Hope of the Christian", 1856)

"Science has fulfilled her function when she has ascertained and enunciated truth."  (Thomas H Huxley, "Man's Place in Nature.", 1863)

"[...] the time has come when scientific truth must cease to be the property of the few, when it must be woven into the common life of the world; for we have reached the point where the results of science touch the very problem of existence, and all men listen for the solving of that mystery." (Jean L R Agassiz, "Methods of Study in Natural History", 1863)

"[...] Scientific truth should be presented in different forms, and should be regarded as equally scientific whether it appears in the robust form and the vivid colouring of a physical illustration, or in the tenuity and paleness of a symbolic expression." (James C Maxwell, [address] 1870)

"It sounds paradoxical to say the attainment of scientific truth has been effected, to a great extent, by the help of scientific errors." (Thomas H Huxley, "The Progress of Science", 1887)

"The scientific spirit is of more value than its products, and irrationally held truths may be more harmful than reasoned errors." (Thomas H Huxley, "Darwiniana", 1893–94)

"By observation, facts are distinctly and minutely impressed in the mind; by analogy, similar facts are connected ; by experiment, new facts are discovered ; and, in the progression of knowledge, observation, guided by analogy, leads to experiment, and analogy, confirmed by experiment, becomes scientific truth." (Sir Humphry Davy)

Horace Mann - Collected Quotes

"Scientific truth is marvellous, but moral truth is divine; and whoever breathes its air and walks by its light has found the lost paradise." (Horace Mann, "A Few Thoughts for a Young Man", Monthly Literary Miscellany, 1851)

"Astronomy is one of the sublimest fields of human investigation. The mind that grasps its facts and principles receives something of the enlargement and grandeur belonging to the science itself. It is a quickener of devotion." (Horace Mann, "Thoughts Selected from the Writings of Horace Mann", 1872)

"Education is an organic necessity of a human being." (Horace Mann, "Thoughts Selected from the Writings of Horace Mann", 1872)

"If ever there can be a cause worthy to be upheld by all toil or sacrifice that the human heart can endure, it is the cause of education." (Horace Mann, "Thoughts Selected from the Writings of Horace Mann", 1872)

"The teacher who is attempting to teach without inspiring the pupil to learn is hammering on cold iron." (Horace Mann, "Thoughts Selected from the Writings of Horace Mann", 1872)

“A human being is not, in any proper sense, a human being till he is educated.” (Horace Mann)

"Education, then, beyond all other devices of human origin, is the great equalizer of the conditions of men–the balance-wheel of the social machinery." (Horace Mann)

On Truth: Universal Truth

"So poetry is something more philosophical and more worthy of serious attention than history, for while poetry is concerned with universal truth, history treats of particular facts [...]" (Aristotle, "Poetics", cca. 350 BC)

"The most useful truths are always universal, and unconnected with accidents and customs." (Samuel Johnson, The Idler, 1767)

"When an induction, based on observations, is made, it is not intended that it shall be accepted as a universal truth, but it is advanced as a hypothesis for further study. Additional observations are then made and the results compared with the results expected from the hypothesis. If there is more deviation between the experimental results and the computed results than can be expected from the inaccuracies of observation and measurement, the scientist discards the' hypothesis and tries to formulate another." (Mayme I Logsdon, "A Mathematician Explains", 1935)

"Scientific truth is universal, because it is only discovered by the human brain and not made by it, as art is." (Konrad Lorenz, "On Aggression", 1963)

"While the equations represent the discernment of eternal and universal truths, however, the manner in which they are written is strictly, provincially human. That is what makes them so much like poems, wonderfully artful attempts to make infinite realities comprehensible to finite beings." (Michael Guillen, "Five Equations That Changed the World", 1995)

"Every truth - if it really is truth - presents itself as universal, even if it is not the whole truth. If something is true, then it must be true for all people and at all times." (Pope John Paul II, "Encyclical Fides et Ratio", 1998)

"The passion and beauty and joy of science is that we humans have invented a process to understand the universe in a way that is true for everyone. We are finding universal truths." (Bill Nye, 2000)

On Truth: Absolute Truth I

"The existence of infinite sets, at least with non-actual members, is something which I now regard as sufficiently proved and defended; as also, that the set of all absolute truths is an infinite set." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"The great truths with which it [mathematics] deals, are clothed with austere grandeur, far above all purposes of immediate convenience or profit. It is in them that our limited understandings approach nearest to the conception of that   absolute and infinite, towards which in most other things they aspire in vain. In the pure mathematics we contemplate absolute truths, which existed in the divine mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven." (Edward Everett, "Orations and Speeches" Vol. 8, 1870)

"Words are but symbols for the relations of things to one another and to us; nowhere do they touch upon absolute truth." (Friedrich Nietzsche, "Philosophy in the Tragic Age of the Greeks", 1873)

"In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1874)

"In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of infinitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible." (William S Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1887)

"It may be impossible for human intelligence to comprehend absolute truth, but it is possible to observe Nature with an unbiased mind and to bear truthful testimony of things seen." (Sir Richard A Gregory, "Discovery, Or, The Spirit and Service of Science", 1916)

"The axioms and provable theorems (i.e. the formulas that arise in this alternating game [namely formal deduction and the adjunction of new axioms]) are images of the thoughts that make up the usual procedure of traditional mathematics; but they are not themselves the truths in the absolute sense. Rather, the absolute truths are the insights (Einsichten) that my proof theory furnishes into the provability and the consistency of these formal systems." (David Hilbert; "Die logischen Grundlagen der Mathematik." Mathematische Annalen 88 (1), 1923)

 "Science makes no pretension to eternal truth or absolute truth; some of its rivals do." (Eric T Bell, "Mathematics: Queen and Servant of Science", 1951)

"The picture of scientific method drafted by modern philosophy is very different from traditional conceptions. Gone is the ideal of a universe whose course follows strict rules, a predetermined cosmos that unwinds itself like an unwinding clock. Gone is the ideal of the scientist who knows the absolute truth. The happenings of nature are like rolling dice rather than like revolving stars; they are controlled by probability laws, not by causality, and the scientist resembles a gambler more than a prophet. He can tell you only his best posits - he never knows beforehand whether they will come true. He is a better gambler, though, than the man at the green table, because his statistical methods are superior. And his goal is staked higher - the goal of foretelling the rolling dice of the cosmos. If he is asked why he follows his methods, with what title he makes his predictions, he cannot answer that he has an irrefutable knowledge of the future; he can only lay his best bets. But he can prove that they are best bets, that making them is the best he can do - and if a man does his best, what else can you ask of him?" (Hans Reichenbach, "The Rise of Scientific Philosophy", 1951)

On Truth: Absolute Truth II

"We can never achieve absolute truth but we can live hopefully by a system of calculated probabilities. The law of probability gives to natural and human sciences - to human experience as a whole - the unity of life we seek." (Agnes E Meyer, "Education for a New Morality", 1957)

"It will never be possible by pure reason to arrive at some absolute truth." (Werner K Heisenberg, "Physics and Philosophy: The revolution in modern science", 1958)

"Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"All views are only probable, and a doctrine of probability which is not bound to a truth dissolves into thin air. In order to describe the probable, you must have a firm hold on the true. Therefore, before there can be any truth whatsoever, there must be absolute truth." (Jean-Paul Sartre, "The Philosophy of Existentialism", 1965)

"The laws of nature 'discovered' by science are merely mathematical or mechanical models that describe how nature behaves, not why, nor what nature 'actually' is. Science strives to find representations that accurately describe nature, not absolute truths. This fact distinguishes science from religion." (George O Abell, "Exploration of the Universe", 1969)

"Science, since people must do it, is a socially embedded activity. It progresses by hunch, vision, and intuition. Much of its change through time does not record a closer approach to absolute truth, but the alteration of cultural contexts that influence it so strongly. Facts are not pure and unsullied bits of information; culture also influences what we see and how we see it. Theories, moreover, are not inexorable inductions from facts. The most creative theories are often imaginative visions imposed upon facts; the source of imagination is also strongly cultural." (Stephen J Gould, "The Mismeasure of Man", 1980)

"Science does not promise absolute truth, nor does it consider that such a thing necessarily exists. Science does not even promise that everything in the Universe is amenable to the scientific process."(Isaac Asimov, "'X' Stands for Unknown", 1984)

"Science doesn't purvey absolute truth. Science is a mechanism. It's a way of trying to improve your knowledge of nature. It's a system for testing your thoughts against the universe and seeing whether they match. And this works, not just for the ordinary aspects of science, but for all of life. I should think people would want to know that what they know is truly what the universe is like, or at least as close as they can get to it." (Isaac Asimov, [Interview by Bill Moyers] 1988)

"The absolutist view of mathematical knowledge is that it consists of certain and unchallengeable truths. According to this view, mathematical knowledge is made up of absolute truths, and represents the unique realm of certain knowledge, apart from logic and statements true by virtue of the meanings of terms […]" (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"Mathematicians are the ultimate scientists, discovering absolute truths not just about this physical universe but about any possible universe." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!" (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)

"The danger arises when a culture takes its own story as the absolute truth, and seeks to impose this truth on others as the yardstick of all knowledge and belief." (F David Peat, "From Certainty to Uncertainty", 2002)

"'There is an old debate', Erdos liked to say, 'about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?' Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws, which will be revealed in the coming chapters. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

On Truth: Mathematical Truth (-1899)

"And thus many are ignorant of mathematical truths, not out of any imperfection of their faculties, or uncertainty in the things themselves, but for want of application in acquiring, examining, and by due ways comparing those ideas." (John Locke, "An Essay Concerning Human Understanding", 1689)

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

"The mathematics are the friends to religion, inasmuch as they charm the passions, restrain the impetuosity of the imagination, and purge the mind from error and prejudice. Vice is error, confusion and false reasoning; and all truth is more or less opposite to it. Besides, mathematical truth may serve for a pleasant entertainment for those hours which young men are apt to throw away upon their vices; the delightfulness of them being such as to make solitude not only easy but desirable." (John Arbuthnot, "An Essay on the Usefulness of Mathematical Learning", 1701)

"He that gives a portion of his time and talent to the investigation of mathematical truth will come to all other questions with a decided advantage over his opponents."  (Charles C Colton, "Lacon", 1820)

"Each mathematician for himself, and not anyone for any other, not even all for one, must tread that more than royal road which leads to the palace and sanctuary of mathematical truth.” (Sir William R Hamilton, “Report of the Fifth Meeting of the British Association for the Advancement of Science”, [Address] 1835)

"The peculiar character of mathematical truth is that it is necessarily and inevitably true; and one of the most important lessons which we learn from our mathematical studies is a knowledge that there are such truths." (William Whewell, "Principles of English University Education", 1838)

"What is exact about mathematics but exactness? And is not this a consequence of the inner sense of truth?" (Johann Wolfgang von Goethe, "Sprüche in Prosa", 1840)

"Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparisons and measurements. But it was found practicable, and being practicable was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which was obvious to the most careless observer, and which compose the first principles and ultimate premises of the science." (John S Mill, "A System of Logic, Ratiocinative and Inductive", 1843)

"He that gives a portion of his time and talent to the investigation of mathematical truth will come to all other questions with a decided advantage over his opponents."  (Colton, Charles Caleb, "Lacon; or Many Things in a Few Words", 1849)

"Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but half comprehends the meaning. Experience sees that the assertions are true, but she sees not how profound and absolute is their truth. She unhesitatingly assents to the laws which geometry delivers, but she does not pretend to see the origin of their obligation. She is always ready to acknowledge the sway of pure scientific principles as a matter of fact, but she does not dream of offering her opinion on their authority as a matter of right; still less can she justly claim to herself the source of that authority." (William Whewell, "The Philosophy of the Inductive Sciences", 1858)

"The peculiarity of the evidence of mathematical truths is that all the argument is on one side." (John S Mill, "On Liberty", 1859)

"It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all." (James J Sylvester, "On certain inequalities related to prime numbers", Nature Vol. 38, 1888)

"Mathematics connect themselves on the one side with common life and physical science; on the other side with philosophy in regard to our notions of space and time, and in the questions which have arisen as to the universality and necessity of the truths of mathematics and the foundation of our knowledge of them." (Arthur Cayley, 1888)

"Geometry, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions." (Augustus de Morgan, "On the Study and Difficulties of Mathematics", 1898)

On Truth: Mathematical Truth (2000-)

"Mathematical truth is like any other truth. A statement is true if our embodied understanding of the statement accords with our embodied understanding of the subject matter and the situation at hand. Truth, including mathematical truth, is thus dependent on embodied human cognition." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"Mathematics is an objective feature of the universe; mathematical objects are real; mathematical truth is universal, absolute, and certain." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)

"Mathematics is not placid, static and eternal. […] Most mathematicians are happy to make use of those axioms in their proofs, although others do not, exploring instead so-called intuitionist logic or constructivist mathematics. Mathematics is not a single monolithic structure of absolute truth!" (Gregory J Chaitin, "A century of controversy over the foundations of mathematics", 2000)

"That a proof must be convincing is part of the anthropology of mathematics, providing the key to understanding mathematics as a human activity. We invoke the logic of mathematics when we demand that every informal proof be capable of being formalized within the confines of a definite formal system. Finally, the epistemology of mathematics comes into play with the requirement that a proof be surveyable. We can't really say that we have created a genuine piece of knowledge unless it can be examined and verified by others; there are no private truths in mathematics." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"Traditionally, mathematical truths have been considered to be a priori truths, either in the sense that they are truths that would be true in any possible universe, or in the sense that they are truths whose validity is independent of our sensory impressions." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)

"While mathematical truth is the aim of inquiry, some falsehoods seem to realize this aim better than others; some truths better realize the aim than other truths and perhaps even some falsehoods realize the aim better than some truths do. The dichotomy of the class of propositions into truths and falsehoods should thus be supplemented with a more fine-grained ordering - one which classifies propositions according to their closeness to the truth, their degree of truth-likeness or verisimilitude. The problem of truth-likeness is to give an adequate account of the concept and to explore its logical properties and its applications to epistemology and methodology." (Graham Oddie, "Truth-likeness", Stanford Encyclopedia of Philosophy, 2001)

"In essence, mathematicians wanted to prove two things: 1.Mathematics is consistent: Mathematics contains no internal contradictions. There are no slips of reason or ambiguities. No matter from what direction we approach the edifice of mathematics, it will always display the same rigor and truth. 2.Mathematics is complete: No mathematical truths are left hanging. Nothing needs adding to the system. Mathematicians can prove every theorem with total rigor so that nothing is excluded from the overall system." (F David Peat, "From Certainty to Uncertainty", 2002)

"'There is an old debate', Erdos liked to say, 'about whether you create mathematics or just discover it. In other words, are the truths already there, even if we don't yet know them?' Erdos had a clear answer to this question: Mathematical truths are there among the list of absolute truths, and we just rediscover them. Random graph theory, so elegant and simple, seemed to him to belong to the eternal truths. Yet today we know that random networks played little role in assembling our universe. Instead, nature resorted to a few fundamental laws, which will be revealed in the coming chapters. Erdos himself created mathematical truths and an alternative view of our world by developing random graph theory." (Albert-László Barabási, "Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life", 2002)

"Where we find certainty and truth in mathematics we also find beauty. Great mathematics is characterized by its aesthetics. Mathematicians delight in the elegance, economy of means, and logical inevitability of proof. It is as if the great mathematical truths can be no other way. This light of logic is also reflected back to us in the underlying structures of the physical world through the mathematics of theoretical physics." (F David Peat, "From Certainty to Uncertainty", 2002)

"Mathematical truth is not totally objective. If a mathematical statement is false, there will be no proofs, but if it is true, there will be an endless variety of proofs, not just one! Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact. [...] each proof will emphasize different aspects of the problem, each proof will lead in a different direction. Each one will have different corollaries, different generalizations. [...] the world of mathematical truth has infinite complexity […]" (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)

"We do not discover mathematical truths; we remember them from our passages through this world outside our own." (Ivar Ekeland, "The Best of All Possible Worlds", 2006)

"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others." (William Byers, "How Mathematicians Think", 2007)

"There is an absolute nature to truth in mathematics, which is unmatched in any other branch of knowledge. A theorem, once proven, requires independent checking but not repetition or independent derivation to be accepted as correct. […] Truth in mathematics is totally dependent on pure thought, with no component of data to be added. This is unique. Associated with truth in mathematics is an absolute certainty in its validity" (James Glimm, "Reflections and Prospectives", 2009)

"Truth in mathematics is totally dependent on pure thought, with no component of data to be added. This is unique. Associated with truth in mathematics is an absolute certainty in its validity. Why does this matter, and why does it go beyond a cultural oddity of our profession? The answer is that mathematics is deeply embedded in the reasoning used within many branches of knowledge. That reasoning often involves conjectures, assumptions, intuition. But whatever aspect has been reduced to mathematics has an absolute validity. As in other subjects search for truth, the mathematical components embedded in their search are like the boulders in the stream, providing a solid footing on which to cross from one side to the other." (James Glimm, "Reflections and Prospectives", 2009)

"What is the basis of this interest in beauty? Is it the same in both mathematics and science? Is it rational, in either case, to expect or demand that the products of the discipline satisfy such a criterion? Is there an underlying assumption that the proper business of mathematics and science is to discover what can be discovered about reality and that truth - mathematical and physical - when seen as clearly as possible, must be beautiful? If the demand for beauty stems from some such assumption, is the assumption itself an article of blind faith? If such an assumption is not its basis, what is?" (Raymond S Nickerson, "Mathematical Reasoning:  Patterns, Problems, Conjectures, and Proofs", 2010)

"Despite its deductive nature, mathematics yields its truths much like any other intellectual pursuit: someone asks a question or poses a challenge, others react or propose solutions, and gradually the edges of the debate are framed and a vocabulary is built." (David Perkins, "Calculus and Its Origins", 2012)

"There are thousands of apparent mathematical truths out there that we humans have discovered and believe to be true but have so far been unable to prove. They are called conjectures. A conjecture is simply a statement about mathematical reality that you believe to be true [..]" (Paul Lockhart, "Measurement", 2012)

On Truth: Mathematical Truth (1975-1999)

"Mathematics has been trivialized, derived from indubitable, trivial axioms in which only absolutely clear trivial terms figure, and from which truth pours down in clear channels." (Imre Lakatos, "Mathematics, Science and Epistemology", 1980)

"Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or ‘mathematics envy’." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

"In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)

"In brief, the way we do mathematics is human, very much so. But mathematicians have no doubt that there is a mathematical reality beyond our puny existence. We discover mathematical truth, we do not create it. We ask ourselves what seems to be a natural question and start working on it, and not uncommonly we find the solution (or someone else does). And we know that the answer could not have been different." (David Ruelle, "Chance and Chaos", 1991)

"Mathematical truth ultimately depends on an irreducible set of assumptions, which are adopted without demonstration. But to qualify as true knowledge, the assumptions require a warrant for their assertion. There is no valid warrant for mathematical knowledge other than demonstration or proof. Therefore the assumptions are beliefs, not knowledge, and remain open to challenge, and thus to doubt." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"The philosophy of mathematics is the branch of philosophy whose task is to reflect on, and account for the nature of mathematics. [...] The role of the philosophy of mathematics is to provide a systematic and absolutely secure foundation for mathematical knowledge, that is for mathematical truth." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"The absolutist view of mathematical knowledge is that it consists of certain and unchallengeable truths. According to this view, mathematical knowledge is made up of absolute truths, and represents the unique realm of certain knowledge, apart from logic and statements true by virtue of the meanings of terms […]" (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"There is one qualitative aspect of reality that sticks out from all others in both profundity and mystery. It is the consistent success of mathematics as a description of the workings of reality and the ability of the human mind to discover and invent mathematical truths." (John D Barrow, "Theories of Everything: The quest for ultimate explanation. New", 1991)

"This absolutist view of mathematical knowledge is based on two types of assumptions: those of mathematics, concerning the assumption of axioms and definitions, and those of logic concerning the assumption of axioms, rules of inference and the formal language and its syntax. These are local or micro-assumptions. There is also the possibility of global or macro-assumptions, such as whether logical deduction suffices to establish all mathematical truths." (Paul Ernest, "The Philosophy of Mathematics Education", 1991)

"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)

"Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God." (Malba Tahan & Patricia R Baquero,"The Man Who Counted", 1993)

"Mathematical beauty and mathematical truth share the fundamental property of objectivity, that of being inescapably context-dependent. Mathematical beauty and mathematical truth, like any other objective characteristics of mathematics, are subject to the laws of the real world, on a par with the laws of physics." (Gian-Carlo Rota, "The Phenomenology of Mathematical Beauty", Synthese, 111(2), 1997)

"Mathematical logic deals not with the truth but only with the game of truth." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)

"It is often the case in mathematics that the definition of truth is assumed to be clear-cut, unambiguous, and unproblematic. While this is often justifiable as a simplifying assumption, the fact is that it is incorrect and that the meaning of the concept of truth in mathematics has changed significantly over time." (Paul Ernest, "Social Constructivism as a Philosophy of Mathematics", 1998)

"Mathematical truth is found to exceed the proving of theorems and to elude total capture in the confining meshes of any logical net." (John Polkinghorne, "Belief in God in an Age of Science", 1998)

"The reason why a 'crude', experimental approach is not adequate for determining mathematical truth lies in the nature of what mathematics is and is intended to be. Though its roots lie in the physical world, mathematics is a precise and idealized discipline. The 'points', 'lines', 'planes', and other ideal constructs of mathematics have no exact counterpart in reality. What the mathematician does is to take a totally abstract, idealized view of the world, and reason with his abstractions in an entirely precise and rigorous fashion." (Keith Devlin, "Mathematics: The New Golden Age", 1998)

"Whatever the ins and outs of poetry, one thing is clear: the manner of expression - notation - is fundamental. It is the same with mathematics - not in the aesthetic sense that the beauty of mathematics is tied up with how it is expressed - but in the sense that mathematical truths are revealed, exploited and developed by various notational innovations." (James R Brown, "Philosophy of Mathematics", 1999)

On Truth: Mathematical Truth (1900-1974)

 "The mathematician, carried along on his flood of symbols, dealing apparently with purely formal truths, may still reach results of endless importance for our description of the physical universe." (Karl Pearson, “The Grammar of Science”, 1900)

"The motive for the study of mathematics is insight into the nature of the universe. Stars and strata, heat and electricity, the laws and processes of becoming and being, incorporate mathematical truths. If language imitates the voice of the Creator, revealing His heart, mathematics discloses His intellect, repeating the story of how things came into being. And the value of mathematics, appealing as it does to our energy and to our honor, to our desire to know the truth and thereby to live as of right in the household of God, is that it establishes us in larger and larger certainties. As literature develops emotion, understanding, and sympathy, so mathematics develops observation, imagination, and reason." (William E Chancellor,"A Theory of Motives, Ideals and Values in Education" 1907)

"[…] because mathematics contains truth, it extends its validity to the whole domain of art and the creatures of the constructive imagination." (James B Shaw, "Lectures on the Philosophy of Mathematics", 1918)

"Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional." (Charles P Steinmetz, 1923)

"Conventionalism as geometrical and mathematical truths are created by our choices, not dictated by or imposed on us by scientific theory. The idea that geometrical truth is truth we create by the understanding of certain conventions in the discovery of non-Euclidean geometries." (Clifford Singer, "Engineering a Visual Field", 1955)

"Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth. That is, mathematics determines what conclusions will follow logically from given premises. The conjunction of mathematics and philosophy, or of mathematics and science is frequently of great service in suggesting new problems and points of view." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

"Mathematics is a body of knowledge, but it contains no truths." (Morris Kline, “Mathematics in Western Culture”, 1964)

"Now a mathematician has a matchless advantage over general scientists, historians, politicians, and exponents of other professions: He can be wrong. A fortiori, he can also be right. [...] A mistake made by a mathematician, even a great one, is not a 'difference of a point of view' or 'another interpretation of the data' or a 'dictate of a conflicting ideology', it is a mistake. The greatest of all mathematicians, those who have discovered the greatest quantities of mathematical truths, are also those who have published the greatest numbers of lacunary proofs, insufficiently qualified assertions, and flat mistakes." (Clifford Truesdell, "Late Baroque Mechanics to Success, Conjecture, Error, and Failure in Newton's Principia" [in "Essays in the History of Mechanics"], 1968)

Out of Context: On Bifurcation (Definitions)

"[…] bifurcations - the abrupt changes that can take place in the behavior, and often in the complexity, of a system when the value of a constant is altered slightly." (Edward N Lorenz, "The Essence of Chaos", 1993)

"A bifurcation is an event that occurs in the evolution of a dynamic system in which the characteristic behavior of the system is transformed." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"The concept of bifurcation, present in the context of non-linear dynamic systems and theory of chaos, refers to the transition between two dynamic modalities qualitatively distinct; both of them are exhibited by the same dynamic system, and the transition (bifurcation) is promoted by the change in value of a relevant numeric parameter of such system." (Emilio Del-Moral-Hernandez, "Chaotic Neural Networks", Encyclopedia of Artificial Intelligence, 2009)

"In mathematical models, a bifurcation occurs when a small change made to a parameter value of a system causes a sudden qualitative or topological change in its behavior." (Dmitriy Laschov & Michael Margaliot, "Mathematical Modeling of the λ Switch: A Fuzzy Logic Approach", 2010)

"In dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour. Generally, at a bifurcation, the local stability properties of equilibria, periodic orbits or other invariant sets changes." (Gregory Faye, "An introduction to bifurcation theory",  2011)

"Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations can occur in both continuous systems (described by ODEs, DDEs, or PDEs) and discrete systems (described by maps)." (Tianshou Zhou, "Bifurcation", 2013)

"The qualitative structure of the flow can change as parameters are varied. In particular, fixed points can be created or destroyed, or their stability can change. These qualitative changes in the dynamics are called bifurcations, and the parameter values at which they occur are called bifurcation points." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)

"[…] what exactly do we mean by a bifurcation? The usual definition involves the concept of 'topological equivalence': if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)