♾️On Complex Numbers II

“I have obtained these values by a singular analogy based on the passages from the real to the imaginary, passages that can be considered as a means of discovery.” (Pierre-Simon Laplace)

“I did not understand how such a quantity could be real, when imaginary or impossible numbers were used to express it.” (Gottfried W Leibniz) 

“But it is just that the Roots of Equations should be often impossible (complex), lest they should exhibit the cases of Problems that are impossible as if they were possible." (Isaac Newton, “Universal Mathematic” 2nd Ed., 1728)

"Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers." (Carl F Gauss, cca. 1831)

“[…] such numbers, which by their natures are impossible, are ordinarily called imaginary or fanciful numbers, because they exist only in the imagination.”  (Leohnard Euler, 1732)

“We completely repudiate the symbol √-1, abandoning it without regret because we do not know what this alleged symbolism signifies nor what meaning to give to it.” (Augustin-Louis Cauchy, 1847)

“Analysis […] would lose immensely in beauty and balance and would be forced to add very hampering restrictions to truths which would hold generally otherwise, if […] imaginary quantities were to be neglected.” (Garrett Birkhoff, 1973)

"It is a curious fact that the first introduction of the imaginaries occurred in the theory of cubic equations, in the case where it was clear that real solutions existed though in an unrecognizable form, and not in the theory of quadratic equations, where our present textbooks introduce them." (Dirk J Struik, “A Concise History of Mathematics” Vol. I, 1948)

"We have shown the symbol √-1 to be void of meaning, or rather self-contradictory and absurd. Nevertheless, by means of such symbols, a part of algebra is established which is of great utility. It depends upon the fact, which must be verified by experience, that the common rules of algebra may be applied to these expressions without leading to any false results." (Augustus De Morgan)

"The word ‘imaginary’ is the great algebraical calamity, but it is too well established for mathematicians to eradicate. It should never have been used. Books on elementary algebra give a simple interpretation of imaginary numbers in terms of rotations. […] Although the interpretation by means of rotations proves nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed ‘imaginaries’." (Philip E B Jourdain, "The Nature of Mathematics" in [James R Newman, “The World of Mathematics” Vol. I, 1956])

See also:
5 Books 10 Quotes: Complex Numbers V
Complex Numbers IV
Complex Numbers III
Complex Numbers I

🎓On Education: On Learning (Aphorisms)

"For the things we have to learn before we can do, we learn by doing." (Aristotle, "Nicomachean Ethics", Book II, 349 BC)

"A little learning is a dangerous thing." (Alexander Pope)

"A few moments to learn, a lifetime to master." (proverb)

"Poor is the pupil who does not surpass his master." (Leonardo da Vinci)

"Much learning does not teach understanding." (Heraclitus, "Fragments", 6th c. BC)

"The learning of many things does not teach intelligence […]." (Pythagoras of Samos)

"Much learning does not teach a man to have intelligence." (Heraclitus of Ephesus)

"Curiosity is the wick in the candle of learning." (William A Ward)

"Learning is a treasure which accompanies its owner everywhere." (proverb)

"Learning is its own exceeding great reward." (William Hazlitt, "The Plain Speaker", 1826)

"What we learn with pleasure we never forget." (Louis Mercier)

"Whatever is good to know is difficult to learn." (Greek proverb)

"We learn to walk by stumbling." (Bulgarian proverb)

"He who is afraid to ask is ashamed of learning." (Danish proverb)

"It takes ten pounds of common sense to carry one pound of learning." (Persian proverb)

"He who has imagination without learning has wings but no feet." (Joseph Joubert)

"Learning is not attained by chance. It must be sought for with ardor and attended to with diligence." (Abigail Adams)

"Learning hath gained most by those books by which the printers have lost." (Thomas Fuller)

"[…] education is not something which the teacher does, but that it is a natural process which develops spontaneously in the human being." (Maria Montessori)

♟️On Games: Chess and Mathematics I

"A chess problem is genuine mathematics, but it is in some way ‘trivial’ mathematics. However, ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. The best mathematics is serious as well as beautiful –‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"We could compare mathematics so formalized to a game of chess in which the symbols correspond to the chessmen; the formulae, to definite positions of the men on the board; the axioms, to the initial positions of the chessmen; the directions for drawing conclusions, to the rules of movement; a proof, to a series of moves which leads from the initial position to a definite configuration of the men." (Friedrich Waismann & Karl Menger, "Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics", 1951)

"It [mathematics] is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst." (Norbert Wiener, "Ex-prodigy: My Childhood and Youth", 1953)

"The advantage is that mathematics is a field in which one’s blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one’s best moments that count and not one’s worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician’s reputation." (Norbert Wiener, "Ex-Prodigy: My Childhood and Youth", 1953)

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

"Geometry, whatever others may think, is the study of different shapes, many of them very beautiful, having harmony, grace and symmetry. […] Most of us, if we can play chess at all, are content to play it on a board with wooden chess pieces; but there are some who play the game blindfolded and without touching the board. It might be a fair analogy to say that abstract geometry is like blindfold chess - it is a game played without concrete objects." (Edward Kasner & James R Newman, "New Names for Old", 1956)

"In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics." (Stanislaw M Ulam, "Adventures of a Mathematician", 1976)

"[…] mathematics can never prove anything. No mathematics has any content. All any mathematics can do is – sometimes – turn out to be useful in describing some aspects of our so-called ‘physical universe’. That is a bonus; most forms of mathematics are as meaning-free as chess." (Robert A Heinlein, "The Number of the Beast", 1980)

"[…] mathematics is not best learned passively; you don’t sop it up like a romance novel. You’ve got to go out to it, aggressive, and alert, like a chess master pursuing checkmate." (Robert Kanigel, "The Man Who Knew Infinity: A Life of the Genius Ramanujan", 1991)

"Mathematics is not the study of an ideal, preexisting nontemporal reality. Neither is it a chess-like game with made-up symbols and formulas. Rather, it is the part of human studies which is capable of achieving a science-like consensus, capable of establishing reproducible results. The existence of the subject called mathematics is a fact, not a question. This fact means no more and no less than the existence of modes of reasoning and argument about ideas which are compelling an conclusive, ‘noncontroversial when once understood’." (Philip J Davis & Rueben Hersh, "The Mathematical Experience", 1995)

⏳On History of Mathematics (-1899)

"[…] we are far from having exhausted all the applications of analysis to geometry, and instead of believing that we have approached the end where these sciences must stop because they  have reached the limit of the forces of the human spirit, we ought to avow rather we are only at the first steps of an immense career. These new [practical] applications, independently of the utility which they may have in themselves, are necessary to the progress of analysis in general; they give birth to questions which one would not think to propose; they demand that one create new methods. Technical processes are the children of need; one can say the same for the methods of the most abstract sciences. But we owe the latter to the needs of a more noble kind, the need to discover the new truths or to know better the laws of nature." (Nicolas de Condorcet, 1781)

"It would be difficult and rash to analyze the chances which the future offers to the advancement of mathematics; in almost all its branches one is blocked by insurmountable difficulties; perfection of detail seems to be the only thing which remains to be done. All of these difficulties appear to announce that the power of our analysis is practically exhausted." (Jean B J Delambre, "Rapport historique sur le progres des sciences mathematiques depuis 1789 et leur etat actuel, 1808)

 “In most sciences, one generation tears down what another has built and what one has established another undoes. In mathematics alone, each generation adds a new story to the old structure” (Hermann Hankel, “Die Entwicklung der Mathematik in den letzten Jahrhunderten, 1884) 

"The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks. The subsequent history may be divided into three periods [...] The first period is that [...] under Greek influence [...] the second is that of [...] the middle ages and the renaissance [...] the third is that of modern mathematics [...]" (W W Rouse Ball, "A Short Account of the History of Mathematics", 1888)

"I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history." (James W L Glaisher, [opening address] 1890)

“The history of mathematics is important […] as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development.” (Florian Cajori, “A History of Mathematics”, 1893)

 "The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store." (Florian Cajori, "A History of Mathematics", 1893)

 "The whole history of the development of mathematics has been a history of the destruction of old definitions, old hobbies, old idols." (David E Smith, American Mathematical Monthly, Vol. 1, No 1, 1894)

📝Misquoted: Herbert G Wells on Mathematical Literacy

"Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write."

The above quote on statistical literacy is often attributed to Herbert G Wells though it belongs to the statistician Samuel S Wilks, who in a 1951 presidential address was paraphrasing Wells:

"Perhaps H. G. Wells was right when he said ‘statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write’!" [4]

The original quote comes from “Mankind in the Making”, first published in 1903 (and not in 1911 as Wikipedia states):

"The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world-wide States that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write." [1]

Even if Wells mentions averages, maxima and minima, tools of statistics, the text refers to mathematical analysis and not statistics. Wilk’s paraphrasing makes sense in nowadays contexts, and seems somehow natural, even if statistical literacy is more about understanding and (critically) evaluating statements that involve rates and percentages.

Another paraphrasing of the same quote and probably closer to the essence of statistical literacy can be found in George A Lundberg paper published in 1940, however without giving credit to Wells:

"The time is perhaps at hand when it will be recognized that for intelligent living in modern society it is as necessary to be able to think in averages, percentages, and deviations as it is to be able to read and write." [2]

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References:
[1] “Mankind in the Making”, by Herbert G Wells, 1903 [Source]
[2] “Statistics in Modern Social Thought”, by George A Lundberg [in “Contemporary Social Theory”, Ed. by H. E. Barnes, H. Becker & F. Becker, 1940] [Source]
[3] “The H. G. Wells Quote on Statistics: A Question of Accuracy”, by James W Tankard Jr., Historia Mathematics 6, 1979 [Source]
[4] “Undergraduate Statistical Education”, by  Samuel S Wilks, Journal of the American Statistical Association, Vol. 46, 1951 [Source]

On (Scientific) Bias I

“It is a capital mistake to theorize before you have all the evidence. It biases the judgment.” (Sir Arthur Conan Doyle, “A Study in Scarlet”, 1887)

 “The eye sees only what the mind is prepared to comprehend.” (Robertson Davies, “Tempest-Tost”, 1951)

“Men judge things according to the disposition of their minds, and had rather imagine things than understand them.” (Baruch Spinoza, “Ethics”, Book I) “Science is a way of trying not to fool yourself. The first principle is that you must not fool yourself, and you are the easiest person to fool.” (Richard Feynman) [attributed to]

“But our ways of learning about the world are strongly influenced by the social preconceptions and biased modes of thinking that each scientist must apply to any problem. The stereotype of a fully rational and objective ‘scientific method’, with individual scientists as logical (and interchangeable) robots, is self-serving mythology.” (Stephen Jay Gould, “This View of Life. In the Mind of the Beholder”, “Natural History”, Vol. 103, No. 2, 1994)

“Numbers have undoubted powers to beguile and benumb, but critics must probe behind numbers to the character of arguments and the biases that motivate them.” (Stephen Jay Gould, “An Urchin in the Storm: Essays About Books and Ideas”, 1987)

“The classification of facts, the recognition of their sequence and relative significance is the function of science, and the habit of forming a judgment upon these facts unbiased by personal feeling is characteristic of what may be termed the scientific frame of mind.” (Karl Pearson, “The Grammar of Science”, 1892)

“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)

“A scientist has to be neutral in his search for the truth, but he cannot be neutral as to the use of that truth when found. If you know more than other people, you have more responsibility, rather than less.” (Charles P Snow) [attributed to]

“Knowing that one may be subject to bias is one thing; being able to correct it is another.” (Jon Elster, “Explaining Social Behavior: More Nuts and Bolts for the Social Sciences”, 2007)

“Science is the search for truth, that is the effort to understand the world: it involves the rejection of bias, of dogma, of revelation, but not the rejection of morality.” (Linus C Pauling)

📊On Statistics: Statistical Thinking

"To a very striking degree our culture has become a Statistical culture. Even a person who may never have heard of an index number is affected [...] by [...] of those index numbers which describe the cost of living. It is impossible to understand Psychology, Sociology, Economics, Finance or a Physical Science without some general idea of the meaning of an average, of variation, of concomitance, of sampling, of how to interpret charts and tables." (Carrol D Wright, 1887)

"I define statistical thinking as thought processes, which recognize that variation is all around us and present in everything we do, all work is a series of interconnected processes, and identifying, characterizing, quantifying, controlling, and reducing variation provide opportunities for improvement." (Ron Snee, “Statistical Thinking and Its Contribution to Total Quality”, The American Statistician, Vol. 44, No. 2 1990) [Link]

“Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” (Samuel S Wilks, 1951) [paraphrasing Herbert G Wells]

“Statistical thinking is a general, fundamental, and independent mode of reasoning about data, variation, and chance.” (David S Moore, 1998)

“Statistics at its best provides methodology for dealing empirically with complicated and uncertain information, in a way that is both useful and scientifically valid” (John M Chambers, 1993)

“It is all too easy to notice the statistical sea that supports our thoughts and actions. If that sea loses its buoyancy, it may take a long time to regain the lost support.” (William Kruskal, “Coordination Today: A Disaster or a Disgrace”, The American Statistician, Vol. 37, No. 3, 1983)

“[…] statistical thinking, though powerful, is never as easy or automatic as simply plugging numbers into formulas. In order to use statistical methods appropriately, you need to understand their logic, not just the computing rules.” (Ann E Watkins et al, “Statistics in Action: Understanding a World of Data”, 2007)

"Numbers already rule your world. And you must not be in the dark about this fact. See how some applied scientists use statistical thinking to make our lives better. You will be amazed how you can use numbers to make everyday decisions in your own life." (Kaiser Fung, "Numbers Rule the World", 2010)

"The issue of group differences is fundamental to statistical thinking. The heart of this matter concerns which groups should be aggregated and which shouldn’t." (Kaiser Fung, "Numbers Rule the World", 2010)

"What is so unconventional about the statistical way of thinking? First, statisticians do not care much for the popular concept of the statistical average; instead, they fixate on any deviation from the average. They worry about how large these variations are, how frequently they occur, and why they exist. [...] Second, variability does not need to be explained by reasonable causes, despite our natural desire for a rational explanation of everything; statisticians are frequently just as happy to pore over patterns of correlation. [...] Third, statisticians are constantly looking out for missed nuances: a statistical average for all groups may well hide vital differences that exist between these groups. Ignoring group differences when they are present frequently portends inequitable treatment. [...] Fourth, decisions based on statistics can be calibrated to strike a balance between two types of errors. Predictably, decision makers have an incentive to focus exclusively on minimizing any mistake that could bring about public humiliation, but statisticians point out that because of this bias, their decisions will aggravate other errors, which are unnoticed but serious. [...] Finally, statisticians follow a specific protocol known as statistical testing when deciding whether the evidence fits the crime, so to speak. Unlike some of us, they don’t believe in miracles. In other words, if the most unusual coincidence must be contrived to explain the inexplicable, they prefer leaving the crime unsolved." (Kaiser Fung, "Numbers Rule the World", 2010) 

📊On Statistics: On Prediction I

“No matter how solidly founded a prediction may appear to us, we are never absolutely sure that experiment will not contradict it, if we undertake to verify it . […] It is far better to foresee even without certainty than not to foresee at all.” (Henri Poincaré, “The Foundations of Science”, 1913)

“[…] the statistical prediction of the future from the past cannot be generally valid, because whatever is future to any given past, is in tum past for some future. That is, whoever continually revises his judgment of the probability of a statistical generalization by its successively observed verifications and failures, cannot fail to make more successful predictions than if he should disregard the past in his anticipation of the future. This might be called the ‘Principle of statistical accumulation’.” (Clarence I Lewis, “Mind and the World-Order: Outline of a Theory of Knowledge”, 1929)

“Factual science may collect statistics, and make charts. But its predictions are, as has been well said, but past history reversed.” (John Dewey, “Art as Experience”, 1934)

“To say that observations of the past are certain, whereas predictions are merely probable, is not the ultimate answer to the question of induction; it is only a sort of intermediate answer, which is incomplete unless a theory of probability is developed that explains what we should mean by ‘probable’ and on what ground we can assert probabilities.” (Hans Reichenbach, “The Rise of Scientific Philosophy”, 1951)

“Predictions, prophecies, and perhaps even guidance – those who suggested this title to me must have hoped for such-even though occasional indulgences in such actions by statisticians has undoubtedly contributed to the characterization of a statistician as a man who draws straight lines from insufficient data to foregone conclusions!” (John W Tukey, “Where do We Go From Here?”, Journal of the American Statistical Association Vol. 55 (289), 1960) [Link]

“Can there be laws of chance? The answer, it would seem should be negative, since chance is in fact defined as the characteristic of the phenomena which follow no law, phenomena whose causes are too complex to permit prediction.” (Félix E Borel, “Probabilities and Life”, 1962)

“[…] “All predictions are statistical, but some predictions have such a high probability that one tends to regard them as certain.” (Marshall J Walker, “The Nature of Scientific Thought”, 1963)

“The moment you forecast you know you’re going to be wrong, you just don’t know when and in which direction.” (Edgar R Fiedler, 1977)

“The aim of every science is foresight (prevoyance). For the laws of established observation of phenomena are generally employed to foresee their succession. All men, however little advanced make true predictions, which are always based on the same principle, the knowledge of the future from the past.” (Auguste Compte)

“The only useful function of a statistician is to make predictions, and thus to provide a basis for action.” (William E Deming)

📊On Statistics: Statistics and Truth

“A statistical estimate may be good or bad, accurate or the reverse; but in almost all cases it is likely to be more accurate than a casual observer’s impression, and the nature of things can only be disproved by statistical methods.” (Sir Arthur L Bowley, “Elements of Statistics”, 1901)

“The statistics themselves prove nothing; nor are they at any time a substitute for logical thinking. There are […] many simple but not always obvious snags in the data to contend with. Variations in even the simplest of figures may conceal a compound of influences which have to be taken into account before any conclusions are drawn from the data.” (Alfred R Ilersic, “Statistics”, 1959)

“[…] in the statistical world you can multiply ignorance by a constant and get truth.” (Raymond F Jones, “The Non-Statistical Man”, 1964)

“There are two kinds of statistics, the kind you look up and the kind you make up.” (Rex Todhunter Stout, “Death of a Doxy”, 1966)

“When we can’t prove our point through the use of sound reasoning, we fall back upon statistical ‘mumbo jumbo’ to confuse and demoralize our opponents. (Audrey Haber & Richard P. Runyon, “General Statistics”, 1973)

“No matter how much reverence is paid to anything purporting to be ‘statistics’, the term has no meaning unless the source, relevance, and truth are all checked.” (Tom Burnam, “The Dictionary of Misinformation”, 1975)

“Do not trust any statistics you did not fake yourself.” (Winston Churchill)

“I can prove anything by statistics except the truth.” (George Canning)

“In earlier times, they had no statistics, and so they had to fall back on lies.” (Stephen Leacock)

“it is easy to lie with statistics, but easier to lie without them” (Frederick Mosteller)

📊On Statistics: Statistics Fallacies I

“Always expect to find at least one error when you proofread your own statistics. If you don’t, you are probably making the same mistake twice.” (Cheryl Russell)

“Do not put faith in what statistics say until you have carefully considered what they do not say.” (William W Watt)

"Facts are stubborn things, but statistics are pliable." (Mark Twain)

"I can prove anything by statistics except the truth." (George Canning)

“If the statistics are boring, you've got the wrong numbers.” (Edward Tufte)

"It is easy to lie with statistics. It is hard to tell the truth without it." (Andrejs Dunkels)

“Torture numbers, and they'll confess to anything.” (Gregg Easterbrook)