Mathematical Circles Squared: A Third Collection of Mathematical Stories and Anecdotes

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It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention. Brown, George Spencer - To arrive at the simplest truth, as Newton knew and practiced, requires years of contemplation. Not activity Not reasoning. Not calculating. Not busy behaviour of any kind. Not reading. Not talking. Not making an effort. Not thinking. Simply bearing in mind what it is one needs to know. And yet those with the courage to tread this path to real discovery are not only offered practically no guidance on how to do so, they are actively discouraged and have to set abut it in secret, pretending meanwhile to be diligently engaged in the frantic diversions and to conform with the deadening personal opinions which are continually being thrust upon them.

The Laws of Form. Browne, Sir Thomas God is like a skilful Geometrician. Religio Medici I , Browne, Sir Thomas All things began in Order, so shall they end, and so shall they begin again, according to the Ordainer of Order, and the mystical mathematicks of the City of Heaven. Hydriotaphia, Urn-burial and the Garden of Cyrus , Browne, Sir Thomas Buck, Pearl S. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships.

I , Burke, Edmund The age of chivalry is gone. That of sophisters, economists and calculators has succeeded. Reflections on the Revolution in France. Butler, Bishop To us probability is the very guide of life. Preface to Analogy. Butler, Samuel - There can be no doubt about faith and not reason being the ultima ratio.

Even Euclid, who has laid himself as little open to the charge of credulity as any writer who ever lived, cannot get beyond this. He has no demonstrable first premise. He requires postulates and axioms which transcend demonstration, and without which he can do nothing.

His superstructure indeed is demonstration, but his ground his faith. Nor again can he get further than telling a man he is a fool if he persists in differing from him. He says "which is absurd," and declines to discuss the matter further. Faith and authority, therefore, prove to be as necessary for him as for anyone else.

The Way of All Flesh. Byron When Newton saw an apple fall, he found A mode of proving that the earth turnd round In a most natural whirl, called gravitation; And thus is the sole mortal who could grapple Since Adam, with a fall or with an apple. Cardano, Girolamo - To throw in a fair game at Hazards only three-spots, when something great is at stake, or some business is the hazard, is a natural occurrence and deserves to be so deemed; and even when they come up the same way for a second time if the throw be repeated.

If the third and fourth plays are the same, surely there is occasion for suspicion on the part of a prudent man. De Vita Propria Liber. Carlyle, Thomas - It is a mathematical fact that the casting of this pebble from my hand alters the centre of gravity of the universe. Sartor Resartus III. Carlyle, Thomas Teaching school is but another word for sure and not very slow destruction.

Carlyle, Thomas A witty statesman said, you might prove anything by figures. Carroll, Lewis What I tell you three times is true. The Hunting of the Snark. Alice in Wonderland. Carroll, Lewis "Can you do addition? Carroll, Lewis "Alice laughed: "There's no use trying," she said; "one can't believe impossible things. Why, sometimes I've believed as many as six impossible things before breakfast. Carroll, Lewis "Then you should say what you mean," the March Hare went on.

Carroll, Lewis "It's very good jam," said the Queen. Carroll, Lewis "When I use a word," Humpty Dumpty said, in a rather scornful tone, "it means just what I choose it to mean - neither more nor less. C'est le vide. Voyage au bout de la nuit. Paris: Gallimard. Carmichael, R. A thing is obvious mathematically after you see it.

Rose ed. Cauchy, Augustin-Louis - Men pass away, but their deeds abide. Cayley, Arthur As for everything else, so for a mathematical theory: beauty can be perceived but not explained. Cayley, Arthur Projective geometry is all geometry. Chebyshev To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls. Chekov, Anton - There is no national science just as there is no national multiplication table; what is national is no longer science. Ponomarev Mysli o nauke Kishinev , Chesterton, G. Mathematicians go mad, and cashiers; but creative artists very seldom.

I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not in imagination. Orthodoxy ch. The Man who was Orthodox. It is that they can't see the problem. Christie, Agatha "I think you're begging the question," said Haydock, "and I can see looming ahead one of those terrible exercises in probability where six men have white hats and six men have black hats and you have to work it out by mathematics how likely it is that the hats will get mixed up and in what proportion.

If you start thinking about things like that, you would go round the bend. Let me assure you of that! Toronto: Bantam Books, Christie, Agatha I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours I found it quite enthralling. An Autobiography.

Churchill, [Sir] Winston Spencer It is a good thing from an uneducated man to read books of quotations. Roving Commission in My Early Life. Depth beyond depth was revealed to me - the Byss and Abyss. I saw - as one might see the transit of Venus or even the Lord Mayor's Show - a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go.

Churchman, C. The measure of our intellectual capacity is the capacity to feel less and less satisfied with our answers to better and better problems. Littlewood A Mathematician's Miscellany. Methuen and Co. Cocteau The composer opens the cage door for arithmetic, the draftsman gives geometry its freedom. Coleridge, Samuel Taylor The Theory of Life. Quoted by T. Huxley in Fortnightly Review , Vol. II, N. Conrad, Joseph Don't talk to me of your Archimedes' lever.

He was an absentminded person with a mathematical imagination. Mathematics commands all my respect, but I have no use for engines. Give me the right word and the right accent and I will move the world. Preface to A Personal Record. Coolidge, Julian Lowell - [Upon proving that the best betting strategy for "Gambler's Ruin" was to bet all on the first trial. I have only proved that a man who does anything else is an even bigger fool. Copernicus, Nicholaus Mathematics is written for mathematicians. De Revolutionibus. Crick, Francis Harry Compton - In my experience most mathematicians are intellectually lazy and especially dislike reading experimental papers.

What Mad Pursuit. London: Weidenfeld and Nicolson, Crowe, Michael Revolutions never occur in mathematics. Historia Mathematica. D'Alembert, Jean Le Rond Thus metaphysics and mathematics are, among all the sciences that belong to reason, those in which imagination has the greatest role. I beg pardon of those delicate spirits who are detractors of mathematics for saying this The imagination in a mathematician who creates makes no less difference than in a poet who invents Of all the great men of antiquity, Archimedes may be the one who most deserves to be placed beside Homer.

Discours Preliminaire de L'Encyclopedie , Tome 1, Dantzig The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it.

Then there is no end of surprise and delight. Dantzig Neither in the subjective nor in the objective world can we find a criterion for the reality of the number concept, because the first contains no such concept, and the second contains nothing that is free from the concept. How then can we arrive at a criterion? Not by evidence, for the dice of evidence are loaded. Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiplied necessity that we call mathematics. How then shall mathematical concepts be judged?

They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.

Darwin, Charles Every new body of discovery is mathematical in form, because there is no other guidance we can have. Darwin, Charles Mathematics seems to endow one with something like a new sense. Davis, Philip J. The numbers are a catalyst that can help turn raving madmen into polite humans. One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.

Number , Scientific American , , Sept. Dehn, Max Mathematics is the only instructional material that can be presented in an entirely undogmatic way. De Morgan, Augustus [When asked about his age. De Morgan, Augustus It is easier to square the circle than to get round a mathematician. De Morgan, Augustus Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no symbols.

Transactions Cambridge Philosophical Society , vol. X, , p. The first was never to accept anything as true if I had not evident knowledge of its being so; that is, carefully to avoid precipitancy and prejudice, and to embrace in my judgment only what presented itself to my mind so clearly and distinctly that I had no occasion to doubt it. The second, to divide each problem I examined into as many parts as was feasible, and as was requisite for its better solution.

The third, to direct my thoughts in an orderly way; beginning with the simplest objects, those most apt to be known, and ascending little by little, in steps as it were, to the knowledge of the most complex; and establishing an order in thought even when the objects had no natural priority one to another.

And the last, to make throughout such complete enumerations and such general surveys that I might be sure of leaving nothing out. Simmons Calculus Gems. La Geometrie. With me everything turns into mathematics. The main thing is to use it well. De Sua, F. Quantum mechanics for example would be a religion under this definition. But mathematics would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified.

Diophantus [His epitaph. And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. After consoling his grief by this science of numbers for four years, he reached the end of his life.

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The country should express its gratitude in some substantial way. Archived from the original on 19 May Clay was cool and dignified; Lincoln was cordial and hearty. He refused entirely to give us the desired aid. Afterward, in speaking of this incident, President Lincoln said that the lady, as a representative of her class in Alexandria, reminded him of the story of the young man who had an aged father and mother owning considerable property. From the ground breaking and life saving to the wacky and implausible, Dr Karl Kruszelnicki reveals some of the best moments in science. University of St Andrews.

Dirac, Paul Adrien Maurice I think that there is a moral to this story, namely that it is more important to have beauty in one's equations that to have them fit experiment. If Schroedinger had been more confident of his work, he could have published it some months earlier, and he could have published a more accurate equation.

It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory.

Scientific American , May Dirac, Paul Adrien Maurice Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. Dirac, Paul Adrien Maurice In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite. Disraeli, Benjamin There are three kinds of lies: lies, damned lies, and statistics.

Mark Twain. Donatus, Aelius 4th Century Pereant qui ante nos nostra dixerunt. Jerome, his pupil]. Doyle, Sir Arthur Conan Detection is, or ought to be, an exact sciences and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism, which produces much the same effect as if you worked a love story or an elopement into the fifth proposition of Euclid.

The Sign of Four. Doyle, Sir Arthur Conan When you have eliminated the impossible, what ever remains, however improbable must be the truth. A study in Scarlet Doyle, Sir Arthur Conan It is a capital mistake to theorize before one has data. Scandal in Bohemia. Dryden, John Mere poets are sottish as mere drunkards are, who live in a continual mist, without seeing or judging anything clearly.

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A man should be learned in several sciences, and should have a reasonable, philosophical and in some measure a mathematical head, to be a complete and excellent poet. Notes and Observations on The Empress of Morocco. Gauss replied, when asked how soon he expected to reach certain mathematical conclusions, that he had them long ago, all he was worrying about was how to reach them! In Mechanisms of Discovery in I. Gordon and S. Sorkin eds.

Dunsany, Lord Logic, like whiskey, loses its beneficial effect when taken in too large quantities. Now the sole reason why painters of this sort are not aware of their own error is that they have not learnt Geometry, without which no one can either be or become an absolute artist; but the blame for this should be laid upon their masters, who are themselves ignorant of this art.

The Art of Measurement. J Heidrich ed. Course in the Art of Measurement. Dyson, Freeman I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce. Missed Opportunities , Gibbs Lecture? Dyson, Freeman For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created.

Mathematics in the Physical Sciences. Dyson, Freeman The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design. Interview with Donald J. Albers, The College Mathematics Journal , vol 25, no.

Eddington, Sir Arthur Proof is the idol before whom the pure mathematician tortures himself. Eddington, Sir Arthur We used to think that if we knew one, we knew two, because one and one are two. Eddington, Sir Arthur We have found a strange footprint on the shores of the unknown.

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We have devised profound theories, one after another, to account for its origins. At last, we have succeeded in reconstructing the creature that made the footprint. And lo! It is our own. Space, Time and Gravitation. Eddington, Sir Arthur It is impossible to trap modern physics into predicting anything with perfect determinism because it deals with probabilities from the outset.

Eddington, Sir Arthur I believe there are 15,,,,,,,,,,,,,,,,,,,,,,,,,, protons in the universe and the same number of electrons. The Philosophy of Physical Science. Cambridge, Eddington, Sir Arthur To the pure geometer the radius of curvature is an incidental characteristic - like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidental to the grin.

Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the "cat without a grin" and the "grin without a cat" are equally set aside as purely mathematical phantasies. The Expanding Universe. Eddington, Sir Arthur Human life is proverbially uncertain; few things are more certain than the solvency of a life-insurance company. Edwards, Jonathon When I am violently beset with temptations, or cannot rid myself of evil thoughts, [I resolve] to do some Arithmetic, or Geometry, or some other study, which necessarily engages all my thoughts, and unavoidably keeps them from wandering.

Mallon A Book of One's Own. Egrafov, M. If you ask mathematicians what they do, yo always get the same answer. They think. They think about difficult and unusual problems. They do not think about ordinary problems: they just write down the answers. Mathematics Magazine , v. Eigen, Manfred - A theory has only the alternative of being right or wrong.

A model has a third possibility: it may be right, but irrelevant. Jagdish Mehra ed. The Physicist's Conception of Nature , Einstein, Albert [During a lecture:]This has been done elegantly by Minkowski; but chalk is cheaper than grey matter, and we will do it as it comes. Einstein, Albert Everything should be made as simple as possible, but not simpler.

Reader's Digest. Einstein, Albert I don't believe in mathematics. Quoted by Carl Seelig. Albert Einstein. Einstein, Albert Imagination is more important than knowledge. On Science. Einstein, Albert The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. What I Believe. Einstein, Albert The bitter and the sweet come from the outside, the hard from within, from one's own efforts.

Out of My Later Years. Einstein, Albert Common sense is the collection of prejudices acquired by age eighteen. Bell Mathematics, Queen and Servant of the Sciences. Einstein, Albert God does not care about our mathematical difficulties.

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He integrates empirically. Infeld Quest , Einstein, Albert How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality? Einstein, Albert [About Newton] Nature to him was an open book, whose letters he could read without effort. Einstein, Albert As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

Einstein, Albert What is this frog and mouse battle among the mathematicians? Brouwer vs. Hilbert] In H. Einstein, Albert Raffiniert ist der Herr Gott, aber boshaft ist er nicht. God is subtle, but he is not malicious. Inscribed in Fine Hall, Princeton University. Einstein, Albert Nature hides her secrets because of her essential loftiness, but not by means of ruse. Einstein, Albert The human mind has first to construct forms, independently, before we can find them in things. Einstein, Albert Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.

Schilpp ed. Albert Einstein, Philosopher-Scientist , Evanston, Einstein, Albert Do not worry about your difficulties in mathematics, I assure you that mine are greater. Einstein, Albert The truth of a theory is in your mind, not in your eyes. Einstein, Albert These thoughts did not come in any verbal formulation. I rarely think in words at all. A thought comes, and I may try to express it in words afterward. Einstein, Albert A human being is a part of the whole, called by us "Universe," a part limited in time and space.

He experiences himself, his thoughts and feelings as something separated from the resta kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty.

Nobody is able to achieve this completely, but the striving for such achievement is in itself a part of the liberation and a foundation for inner security. Einstein, Albert The world needs heroes and it's better they be harmless men like me than villains like Hitler. Einstein, Albert It is nothing short of a miracle that modern methods of instruction have not yet entirely strangled the holy curiousity of inquiry. Einstein, Albert Everything that is really great and inspiring is created by the individual who can labor in freedom.

Einstein, Albert The search for truth is more precious than its possession. The American Mathematical Monthly v. Einstein, Albert If my theory of relativity is proven successful, Germany will claim me as a German and France will declare that I am a citizen of the world.

Address at the Sorbonne, Paris. Einstein, Albert We come now to the question: what is a priori certain or necessary, respectively in geometry doctrine of space or its foundations? Formerly we thought everything; nowadays we think nothing. Already the distance-concept is logically arbitrary; there need be no things that correspond to it, even approximately. Einstein, Albert Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.

The Evolution of Physics. Einstein, Albert Science without religion is lame; religion without science is blind. Reader's Digest , Nov. Ellis, Havelock The mathematician has reached the highest rung on the ladder of human thought. The Dance of Life. Ellis, Havelock It is here [in mathematics] that the artist has the fullest scope of his imagination. Erath, V. God is a child; and when he began to play, he cultivated mathematics. It is the most godly of man's games. Das blinde Spiel. Euler, Leonhard - If a nonnegative quantity was so small that it is smaller than any given one, then it certainly could not be anything but zero.

To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be. These supposed mysteries have rendered the calculus of the infinitely small quite suspect to many people. Those doubts that remain we shall thoroughly remove in the following pages, where we shall explain this calculus. Euler, Leonhard Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.

Euler, Leonhard [upon losing the use of his right eye] Now I will have less distraction. Everett, Edward 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. Quoted by E. Bell in The Queen of the Sciences , Baltimore, Eves, Howard W. A formal manipulator in mathematics often experiences the discomforting feeling that his pencil surpasses him in intelligence.

An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity. Mathematics may be likened to a large rock whose interior composition we wish to examine.

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The older mathematicians appear as persevering stone cutters slowly attempting to demolish the rock from the outside with hammer and chisel. The later mathematicians resemble expert miners who seek vulnerable veins, drill into these strategic places, and then blast the rock apart with well placed internal charges.

One is hard pressed to think of universal customs that man has successfully established on earth. There is one, however, of which he can boast the universal adoption of the Hindu-Arabic numerals to record numbers. In this we perhaps have man's unique worldwide victory of an idea. Ewing, John If the entire Mandelbrot set were placed on an ordinary sheet of paper, the tiny sections of boundary we examine would not fill the width of a hydrogen atom. Physicists think about such tiny objects; only mathematicians have microscopes fine enough to actually observe them.

John Smith who has applied for a position in your department. I should start by saying that I cannot recommend him too highly. In fact, there is no other student with whom I can adequately compare him, and I am sure that the amount of mathematics he knows will surprise you. His dissertation is the sort of work you don't expect to see these days. It definitely demonstrates his complete capabilities.

In closing, let me say that you will be fortunate if you can get him to work for you. Sincerely, A. Visor Prof. Feynman, Richard Philips - We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover up all the tracks, to not worry about the blind alleys or describe how you had the wrong idea first, and so on.

So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work.

Mathematics: Rhyme and Reason

Nobel Lecture, Finkel, Benjamin Franklin The solution of problems is one of the lowest forms of mathematical research, It is the ladder by which the mind ascends into higher fields of original research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem. The American Mathematical Monthly , no.

Fisher, Irving The effort of the economist is to "see," to picture the interplay of economic elements. The more clearly cut these elements appear in his vision, the better; the more elements he can grasp and hold in his mind at once, the better. The economic world is a misty region. The first explorers used unaided vision. Mathematics is the lantern by which what before was dimly visible now looms up in firm, bold outlines. The old phantasmagoria disappear. We see better. We also see further. Transactions of Conn.

Academy , Fisher, Ronald Aylmer - Natural selection is a mechanism for generating an exceedingly high degree of improbability. Fisher, Ronald Aylmer To call in the statistician after the experiment is done may be no more than asking hm to perform a postmortem examination: he may be able to say what the experiment died of. Indian Statistical Congress, Sankhya, ca Flaubert, Gustave Since you are now studying geometry and trigonometry, I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon.

It is the month of May. How old is the captain? Fontenelle, Bernard Le Bovier Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another. Quoted in V. Fontenelle, Bernard Le Bovier A work of morality, politics, criticism will be more elegant, other things being equal, if it is shaped by the hand of geometry. Fontenelle, Bernard Le Bovier Leibniz never married; he had considered it at the age of fifty; but the person he had in mind asked for time to reflect.

This gave Leibniz time to reflect, too, and so he never married. Eloge de le Leibniz. Frankland, W. Whereas at the outset geometry is reported to have concerned herself with the measurement of muddy land, she now handles celestial as well as terrestrial problems: she has extended her domain to the furthest bounds of space. Hodder and Stoughton, The Story of Euclid. Frayn, Michael For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock.

And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems. Frederick the Great To your care and recommendation am I indebted for having replaced a half-blind mathematician with a mathematician with both eyes, which will especially please the anatomical members of my Academy. Euler had vacated the post. Frege, Gottlob - A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished.

I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press. In Scientific American , May , p Galbraith, John Kenneth There can be no question, however, that prolonged commitment to mathematical exercises in economics can be damaging. It leads to the atrophy of judgement and intuition Economics, Peace, and Laughter. Galilei, Galileo - [The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written.

It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Opere Il Saggiatore p. Galilei, Galileo - Measure what is measurable, and make measurable what is not so. Quoted in H. Galilei, Galileo - And who can doubt that it will lead to the worst disorders when minds created free by God are compelled to submit slavishly to an outside will?

When we are told to deny our senses and subject them to the whim of others? When people devoid of whatsoever competence are made judges over experts and are granted authority to treat them as they please? These are the novelties which are apt to bring about the ruin of commonwealths and the subversion of the state.

Galois, Evariste Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing difficulties. Galton, [Sir] Francis Whenever you can, count. Galton, Sir Francis [Statistics are] the only tools by which an opening can be cut through the formidable thicket of difficulties that bars the path of those who pursue the Science of Man.

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Galton, Sir Francis I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway.

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It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along. Gardner, Martin Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals -- the flotsam and jetsam of historical currents. The men who radically altered history, the great scientists and mathematicians, are seldom mentioned, if at all.

Gardner, Martin Mathematics is not only real, but it is the only reality. That is that entire universe is made of matter, obviously. And matter is made of particles. It's made of electrons and neutrons and protons. So the entire universe is made out of particles. Now what are the particles made out of?

They're not made out of anything. The only thing you can say about the reality of an electron is to cite its mathematical properties. So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure. Gauss, Karl Friedrich I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.

Gauss, Karl Friedrich If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries. Gauss, Karl Friedrich There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science. Gauss, Karl Friedrich You know that I write slowly.

This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length. Gauss, Karl Friedrich We must admit with humility that, while number is purely a product of our minds, space has a reality outside our minds, so that we cannot completely prescribe its properties a priori. Letter to Bessel, Gauss, Karl Friedrich I have had my results for a long time: but I do not yet know how I am to arrive at them. Arber The Mind and the Eye Gauss, Karl Friedrich [His second motto:] Thou, nature, art my goddess; to thy laws my services are bound Shakespeare King Lear.

Gauss, Karl Friedrich [attributed to him by H. Foreword of H. Gauss, Karl Friedrich It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again; the never-satisfied man is so strange if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another.

I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others. Letter to Bolyai, Gauss, Karl Friedrich Finally, two days ago, I succeeded - not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.

Gauss, Karl Friedrich A great part of its [higher arithmetic] theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.

Gauss, Karl Friedrich I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect Koenderink Solid Shape , Cambridge Mass. Gay, John Lest men suspect your tale untrue, Keep probability in view. Gibbs, Josiah Willard - One of the principal objects of theoretical research in my department of knowledge is to find the point of view from which the subject appears in its greatest simplicity.

Gilbert, W. The Pirates of Penzance. Act 1. Glaisher, J. The mathematician requires tact and good taste at every step of his work, and he has to learn to trust to his own instinct to distinguish between what is really worthy of his efforts and what is not. Glanvill, Joseph And for mathematical science, he that doubts their certainty hath need of a dose of hellebore. Goedel, Kurt I don't believe in natural science. Addison Wesley, Goethe It has been said that figures rule the world.

But I am sure that figures show us whether it is being ruled well or badly. Eckermann, Conversations with Goethe. Goethe Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight.

Goodman, Nicholas P. There are no deep theorems -- only theorems that we have not understood very well. The Mathematical Intelligencer , vol. Gordon, P This is not mathematics, it is theology. Scientific American October Mathematicians have long since regarded it as demeaning to work on problems related to elementary geometry in two or three dimensions, in spite of the fact that it it precisely this sort of mathematics which is of practical value.

Handbook of Applicable Mathematics. Hadamard, Jacques The shortest path between two truths in the real domain passes through the complex domain. Quoted in The Mathematical Intelligencer , v. Hadmard, Jacques Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle. Haldane, John Burdon Sanderson In scientific thought we adopt the simplest theory which will explain all the facts under consideration and enable us to predict new facts of the same kind.

The catch in this criterion lies in the world "simplest. The physicist reverses this judgment, and his statement is certainly the more fruitful of the two, so far as prediction is concerned. It is, however, a statement about something very unfamiliar to the plainman, namely, the rate of change of a rate of change. Possible Worlds , Haldane, John Burdon Sanderson A time will however come as I believe when physiology will invade and destroy mathematical physics, as the latter has destroyed geometry. Halmos, Paul R. Mathematics is not a deductive science -- that's a cliche.

When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. I remember one occasion when I tried to add a little seasoning to a review, but I wasn't allowed to. The paper was by Dorothy Maharam, and it was a perfectly sound contribution to abstract measure theory.

The domains of the underlying measures were not sets but elements of more general Boolean algebras, and their range consisted not of positive numbers but of certain abstract equivalence classes. My proposed first sentence was: "The author discusses valueless measures in pointless spaces. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case. The joy of suddenly learning a former secret and the joy of suddenly discovering a hitherto unknown truth are the same to me -- both have the flash of enlightenment, the almost incredibly enhanced vision, and the ecstasy and euphoria of released tension.

Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess. Hamilton, [Sir] William Rowan Who would not rather have the fame of Archimedes than that of his conqueror Marcellus?

Hamilton, Sir William Rowan I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have recourse to x, y, z, etc.. In a letter from Tait to Cayley. Hamilton, Sir William Rowan On earth there is nothing great but man; in man there is nothing great but mind.

Lectures on Metaphysics. Hamming, Richard W. Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane. Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.

Hardy, Godfrey H. I had ridden in taxi cab number and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics. Exposition, criticism, appreciation, is work for second-rate minds. Albers, The College Mathematics Journal, vol.

Beauty is the first test: there is no permanent place in this world for ugly mathematics. Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances may suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized no doubt rightly as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.

Hardy, Thomas Far from the Madding Crowd. Harish-Chandra I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naivete, unburdened by conventional wisdom, can sometimes be a positive asset. Harris, Sydney J. The real danger is not that computers will begin to think like men, but that men will begin to think like computers.

Hawking, Stephen Williams God not only plays dice. He also sometimes throws the dice where they cannot be seen. Heath, Sir Thomas [The works of Archimedes] are without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader. What opportunities do we provide for children to gather data on living and nonliving aspects of their environments?

What opportunities do we give children to explore, hypothesize, take risks, and engage in symbolic and dramatic play with confidence? What opportunities do we give children to experiment with word, language, number, and shape patterns?

How do we assist children to use pattern making and pattern continuation for problem solving and investigation? What opportunities do we give children to explore their local environment and record what they see using visual means? How do we assist children to gather information, ask questions, seek clarification, and consider possibilities about their own lives?

How do we encourage children to generate a range of ideas and to use the processes of play, reflection, and investigation to find answers to problems? In what ways do we provide opportunities for children to reflect upon their mathematical pattern making? How do we encourage children to participate in group discussions and brainstorms around the properties of shapes?

In what ways do we provide opportunities for children to use their imagination to generate interesting shapes or patterns? How do we encourage children to demonstrate an understanding that symbols are a powerful means of communication? How do we encourage children to use different communication strategies to describe shapes and their properties? What opportunities do we provide for children to play with shapes and communicate their findings in a variety of ways?

What opportunities do we provide for children to explore the ideas and concepts of data representation? What opportunities do we provide for each child to accept new challenges, make new discoveries, and celebrate effort and achievement? What opportunities do we provide for children to predict and manage change in their daily routines and record the patterns of their lives? How do we encourage children to engage in a variety of active and quiet activities in order to experience a balance?

What opportunities do we provide for children to make discoveries that are new to them about shape and space? What opportunities do we provide for children to predict and manage change in their daily routines? In what ways do we assist children to represent varied physical activities and games through patterns and symbols? In what ways do we assist children to engage in a variety of physical activities and games that use geometric ideas? How do we encourage children to collect, analyze, and represent data about their physical activity?

As an illustration of the power of the numeracy matrix, consider the column headed by the powerful mathematical idea Argumentation that. Providing such justification, while clearly important as children develop their mathematics, is also important in many other areas of learning and certainly contributes in numerous ways to the developmental learning outcomes. We would firstly need to make sure that children felt safe in talking up about their solutions and those of others.

We want them to say what they think but in ways that will not hurt anyone. That will depend a lot on the atmosphere in the group, but it will also need the kids to know the maths that they are talking about. The contribution of each powerful mathematical idea to the developmental learning outcome Children are intellectually inquisitive also provides an example of the power of the matrix.

While few of the early childhood educators involved in SNI would have argued against mathematics contributing to this DLO, none was able to articulate how that might occur in a learning area such as mathematics with its perceived underlying and constraining structure. Through their use of the numeracy matrix, the educators are now able to see how each of the powerful ideas contributes to the DLO. One of them was able to suggest that the work with the numeracy matrix had helped them see how the DLOs were the capstones to all that they were trying to do in all learning areas:. When I thought about shapes and geometry, I thought all that was needed was for the children to know the names of some regular shapes.

It was really not something I thought they would be inquisitive about. By using the matrix, I can see that they can develop their inquisitiveness by asking lots of questions about lots of different shapes in their environment—not just triangles and circles—and can investigate why things are the way they are. This will take them into asking about how things are used, where they come from, whether some shapes are better than others for a particular job, and why some shapes look better than others. It is exciting for the children—and for me! The approach to assessment known as Learning Stories has been pioneered by Carr They include relationships, dispositions, and an interpretation by someone who knows the child well.

Educators use their evaluation of the learning story to plan for future, ongoing learning. In South Australia, learning stories have been used by preschool educators for some time, especially in the area of literacy learning. However, they tended not to be used in the area of mathematics, partly because the preschool educators did not have sufficient confidence in their ability to link what they were observing with mathematical learning outcomes. The introduction of the numeracy matrix has given this confidence to the group of educators working with the authors and has produced some outstanding results.

Two mathematical learning stories illustrate this point:. Luke was playing outside on the lawn with the portable padded climbing shapes. He decided that he would like to design his own shape and began moving the pieces to form the climbing path he wanted. He experimented over and over, rearranging the pieces in as many ways as he could think of, trying it out each time. Evaluation Luke! How creative you are. You have shown your ability to plan and design and build, as well as your awareness of shape and size spatial concepts. You were able to concentrate on the task in hand for a long time, and you were very involved in what you were doing.

What Next? Perhaps we could draw some designs next time. Luke might enjoy helping to set up an obstacle course in the climbing area! The most obvious is the link between the DLO Children develop trust and confidence and the powerful mathematical idea Spatial and Geometric Thinking. One important aspect of this encouragement is assisting the children to recognize the mathematical ideas that they are experiencing in their lives. To do this, the children need to be introduced explicitly to the mathematical vocabulary needed to describe these ideas.


In short, children need sufficient language to allow them to understand their peers and their teachers as explanations are presented, and to allow them to give their own explanations. Zac was at the making table and had already made quite a few phones. I went over to have a look and asked how many phones he had made. Zac counted them. Placing them into a pile as he counted, he reached the total of 6. I suggested to Zac that maybe he could make a telephone for everyone at kindy. How many will you need to make?

Zac set about counting all the children at kindy and reached He shrugged his shoulders. What comes next? Zac counted on 7, 8, 9, 10, 11, 12, 13, 14, 15 as I held up my fingers for each number. How many fingers do I have held up? Zac counted them in his head moving his lips giving the answer of 9.

Zac found 9 more boxes and set about making them into phones. This process took some time. Zac thought this was a good idea. I left the roll on the table and told Zac which was the next name tag he would need to get by looking at the roll. This seemed like a much better idea so I explained to Zac that where there was a tick that was the name to write.

Zac decided to put something under the name he was writing so he knew where he was up to. He completed this successfully, and then it was time to give out the phones to everyone. Zac gave this a go the next time he gave one of the children a phone. Evaluation Zac has an understanding of numbers and could work out what number came after 14; he was also able to count on from 6. With support, Zac was learning how to work out how many boxes he needed. Zac is interested in writing and copying print as seen in the photo ; he produces writing for a particular purpose and understands the reason behind labeling things with names.

When Zac is focused, his determination and motivation is at a heightened level to complete the task. This learning experience also shows Zac's willingness to take advice given by staff to support him in his learning and put this into practice, which gave him a great sense of achievement. In the How Many Phones learning story, the most obvious link to the numeracy matrix is through the powerful mathematical idea Number Sense and Mental Computation.

The purpose of this paper was to introduce the numeracy matrix, which has been developed as part of the Southern Numeracy Initiative in South Australia, and to celebrate the work of the early childhood educators who have been involved in its development. Anecdotal evidence from the participants in the Southern Numeracy Initiative suggest that the use of the numeracy matrix and the thinking behind it have had positive effects on the pedagogical practices of the early childhood educators involved.

The learning stories assessment methodologies have allowed the preschool educators to meet their reporting obligations while at the same time remaining true to their early childhood philosophies. These results suggest that the SNI preschool project will lead to improved practices and, consequently, improved learning outcomes for the children who are fortunate enough to be taught by this enthusiastic group of educators.

The numeracy matrix has helped me to rethink the way I am teaching and the way children are learning at my center. Using the matrix and being able to develop one inquiry question and explore it in depth has illustrated to me how and when to assess children in a variety of ways.

We wish to acknowledge the assistance and expertise of Nicole Hentschke in this project. Position paper on early childhood mathematics. Carr, Margaret. Assessment in early childhood settings: Learning stories. London: Paul Chapman. Tracking the development of learning dispositions. Assessment in Education, 9 1 , Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design. Mahwah, NJ: Erlbaum.

Department of Education, Training and Employment. South Australian curriculum, standards and accountability framework. A good start to numeracy: Effective numeracy strategies from research and practice in early childhood. Melbourne: Australian Council for Educational Research. Greenes, Carole; Ginsburg, Herbert P. Big math for little kids. Early Childhood Research Quarterly, 19 1 , Adding it up: Helping children learn mathematics. Involvement of children and teacher style: Insights from an international study on experimental education.

Leuven, Belgium: Leuven University Press. Mental computation in the middle grades: The importance of thinking strategies. Mathematics Teaching in the Middle School, 2 5 , Early childhood mathematics: Promoting good beginnings.

Mathematical circles squared; a third collection of mathematical stories and anecdotes

Washington, DC: Author. National Council of Teachers of Mathematics. Principles and standards for school mathematics. Reston, VA: Author. In Lyn D. English Ed. In Helen L. Vincent Eds. Melbourne: University of Melbourne. What did you do in maths today? Australian Journal of Early Childhood, 30 3 , Numeracy in the early years: Project Good Start. Bob Perry is currently associate professor of education at the Murray School of Education, Charles Sturt University in Albury, Australia, where he teaches mathematics education and research methods subjects.

Bob has worked in tertiary institutions in Australia and overseas since His research agenda includes early childhood mathematics education, transition to school, education of indigenous children, and community capacity building. Bob Perry Charles Sturt University bperry csu. Since , Sue has been involved in early childhood teacher education and early childhood education research.

Her research agenda is focused on the transition to school and the expectations, experiences and perceptions of all involved. She has published widely, both nationally and internationally in the area of transition to school and early childhood mathematics education. Sue Dockett Charles Sturt University sdockett csu. Elspeth Harley is policy and program officer, early years, in the South Australian Department of Education and Children's Services with a focus on the preschool years. She has taught in preschools, the first years of school, and university early childhood programs, and has been project leader for numerous practitioner research projects, including the preschool component of the Southern Numeracy Initiative.

Elspeth is trained in drama and has a special interest in play in the early years and the therapeutic aspects of play. Elspeth saugov. Issue Archive The aims of SNI included the following: to develop and implement successful teaching and learning practices to improve numeracy, and to challenge teachers to explore their beliefs and understandings about how children develop their understanding of mathematics and how this effort can be supported through the teaching program.

These DLOs are Children develop trust and confidence. Children develop a positive sense of self and a confident personal and group identity. Children develop a sense of being connected with others and their world. Children are intellectually inquisitive. Children develop a range of thinking skills. Children are effective communicators. Children demonstrate a sense of physical well-being.

Children develop a range of physical competencies. Constructing the Numeracy Matrix Two of the authors of this paper worked with a small group of early childhood educators for two days in and two days in as part of the professional development component of the SNI.