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 Chess I: 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)

Beyond the History of Mathematics I

"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)

"The history of mathematics is the mirror of civilization." (Lancelot Hogben, "Mathematics for the Million", 1917)

"[…] a history of mathematics is largely a history of discoveries which no longer exist as separate items, but are merged into some more modern generalization, these discoveries have not been forgotten or made valueless. They are not dead, but transmuted." (John W N Sullivan, "The History of Mathematics in Europe", 1925)

"In the history of mathematics, the ‘how’ always preceded the ‘why’, the technique of the subject preceded its philosophy." (Tobias Dantzig, "Number: The Language of Science", 1930)

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the […] development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes." (Julian L Coolidge, "A History of Geometrical Methods", 1940)

"The study of the history of mathematics shows clearly enough that after each period of research and extension there follows a period of review and synthesis during which more general methods are evolved and the foundation of mathematics consolidated." (Gustave Choquet, "What is Modern Mathematics", 1963)

"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

"Mathematics is a vast adventure of ideas; its history reflects some of the noblest thoughts of countless generations." (Dirk J Struik, "A Concise History of Mathematics", 1967)

"Under the present dominance of formalism, one is tempted to paraphrase Kant: the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the of mathematics, has become empty." (Imre Lakatos, "Proofs and Refutations: The Logic of Mathematical Discovery", 1976)

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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]