It is a good thing for an uneducated man to read books of quotations. Winston Churchill |

Abbot Hor said to his disciple: Take care that you never bring into this cell the words of another.
Thomas Merton, |

Definitions | Proofs | Axioms | Infinity

Numbers | Combinatorics | Logic and set theory | Algebra | Geometry

Probability and Statistics | Codes and cryptography | Algorithms

Philosophy | Education | History | Mathematicians at work

Research | Mathematics as metaphor | Wordplay

Miscellany | Other sources

This collection is a miscellany in Littlewood’s sense (see his definition under Miscellany below). The quotes are almost all derived from my own reading, though a few are culled from dictionaries or have been sent to me by others. I have organised them under several headings (see above) with a few cross-links. I have also tried to put similar or complementary quotes together, so if you are led to one quote, scan the others around it. (See the pair by Noam Chomsky and Thetis Blacker, towards the end of Miscellany, those by Sir Arthur Quiller-Couch and Robert Louis Stevenson, and those by Hermann Hesse and G. K. Chesterton, both pairs also in Miscellany, those by Dante Alighieri and Nicole Oresme at the start of Probability and Statistics, or the three quotes on mathematics and conjuring or astrology at the end of History, for example.)

Anything you would like to send me will be gratefully received:

`p.j.cameron(at)qmul.ac.uk` or `pjc20(at)st-andrews.ac.uk` .

## Definitions

Definitions from Samuel Johnson’s *Dictionary of the English Language* (1755). Johnson took many of these from John Harris (1667–1719) or Ephraim Chambers (c.1680–1740). I have used the edition by Jack Lynch (Levenger Press 2004).

- mathematicks
- That science which contemplates whatever is capable of being numbered or measured; and it is either pure or mixt: pure considers abstracted quantity, without any relation to matter; mixt is interwoven with physical considerations.
- logick
- The art of reasoning.
- arithmetick
- The science of numbers; the art of computation.
- algebra
- This is a peculiar kind of arithmetick, which takes the quantity sought, whether it be a number or a line, or any other quantity, as if it were granted, and by means of one or more quantities given, proceeds by consequence, till the quantity at first only supposed to be known, or at least some power thereof, is found to be equal to some quantity or quantities which are known, and consequently itself is known.
- geometry
- Originally signifies the art of measuring the earth, or any distances or dimensions on or within it: but it is now used for the science of quantity, extension, or magnitude abstractedly considered, without any regard to matter.
- trigonometry
- Trigonometry is the art of measuring triangles, or of calculating the sides of any triangle sought, and this is plane or spherical.
- number
- The species of quantity by which it is computed how many.

[eight other meanings] - differential method
- is applied to the doctrine of infinitesimals, or infinitely small quantities, called the arithmetick of fluxions; about the invention of which there has been a contest between Leibnitz and Sir Isaac Newton. It consists in descending from whole quantities to their infinitely small differences, and comparing together these infinitely small differences, of what kind soever they be: and from thence it takes the name of the differential calculus, or analysis of infinitesimals.
- calculus
- The stone in the bladder.
- calculator
- A computer; a reckoner.
- computer
- Reckoner; accountant; calculator.
- algorism, algorithm
- Arabick words, which are used to imply the six operations of arithmetick, or the science of numbers.
- mechanicks
- Dr. Wallis defines mechanicks to be the geometry of motion, a mathematical science, which shews the effects of powers, or moving forces, so far as they are applied to engines, and demonstrates the laws of motion.

In Geometry (which is the only science that it hath pleased God hitherto to bestow on mankind) men begin at settling the significations of their words; which . . . they call Definitions

Thomas Hobbes, *Leviathan*

I found, that in Germany they were engaged in a species of political enquiry, to which they had given the name of Statistics; though I apply a different idea to that word, for by Statistics is meant in Germany, an inquiry for the purpose of ascertaining the political strength of a country, or questions respecting matters of state; whereas, the idea I annex to the term, is an inquiry into the state of a country, for the purpose of ascertaining the quantum of happiness enjoyed by its inhabitants, and the means of its future improvement.

John Sinclair, *Statistical Account of Scotland* (1791–1799), quoted in Mel Bayley, “*Hard times* and statistics”, *BSHM Bulletin* 22 (2007), 92–103.

*Complex*, made up of various qualities or ingredients. A beautiful wife, and good woman, is a *complex* idea, containing three distinct ideas, beauty, wisdom, goodness; you might render it still more complex by the addition of high-born, rich, religious—but I must not make my idea too complex.

*Continuity*, uninterrupted connection; the unviolated union of the parts of an animal body. Latin.

From the Glossary by Henry Hunter to his English translation (1795) of Leonhard Euler’s *Letters to a German Princess* (1768); quoted by June Barrow-Green, “Euler as an educator”, *BSHM Bulletin* 25 (2010), 10-22.

The word *Combination* has been used in many different senses, so that to avoid ambiguity I am obliged to say what I mean by it.

Blaise Pascal, *Traité du Triangle Arithmetique*, quoted by A. W. F. Edwards in *Combinatorics Ancient and Modern* (ed. R. J. Wilson and J. Watkins), OUP 2013.

See also Littlewood’s definition of a Miscellany under Miscellany.

## Proofs

A fine collection of quotations about proof in mathematics, assembled by David Lingard, appears in the *BSHM Bulletin* **1** (2004), 63–66. I can’t find them on the Web, but here, as a sample, are two conflicting views from Riemann and Gauss, quoted by Imre Lakatos in *Proofs and Refutations*:

Georg Bernhard Riemann: “If only I had the theorems! Then I should find the proofs easily enough.”

Karl Friedrich Gauss: “I have had my results for a long time, but I do not yet know how to arrive at them.”

A non-symbolic argument or proof can be quite rigorous when given for a particular value of the variable; the conditions for rigor are that the particular value of the variable should be typical, and that a further generalization to any value should be immediate.

R. J. Gillings, *Mathematics in the Time of the Pharaohs*, MIT Press, Cambridge, 1972.

A proof only becomes a proof after the social act of “accepting it as a proof”.

Yu. I. Manin

This is glossed by Alexandre Borovik in *Mathematics under the Microscope* (AMS 2009) as follows:

Manin describes the act of acceptance as a social act: however, the importance of its personal, psychological component can hardly be overestimated.

The [isoperimetric] problem had occurred to the Greek geometers of over 2000 years ago but its solution was not discovered until the 1880s. This solution, which is that the curve must be a circle, was known to the Greeks, but it is one thing to know what the solution of a problem is and quite another to establish this solution in an adequately convincing form or, as we say, to prove it mathematically.

H. G. Eggleston, “The isoperimetric problem”, in *Exploring University Mathematics* (ed. N. J. Hardiman), Pergamon, 1967.

I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions — they are not just repetitions of each other.

Sir Michael Atiyah, interview in *European Mathematical Society Newsletter*, September 2004.

Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof.

Girolamo Cardano, *The book of my life* (transl. Jean Stoner), New York Review Books, New York, 2002.

Probabilitie and sensible prose, may well serue in thinges naturall: and is commendable: In Mathematicall reasoninges, a probable Argument, is nothying regarded: nor yet the testimony of sense, any whit credited: But onely a perfect demonstration, of truthes certaine, necessary, and inuincible: vniuersally and necessaryly concluded: is allowed as sufficient for `an argument exactly and purely Mathematical.’

From John Dee’s *Mathematicall praeface* to Henry Billingsley’s *Elements of geometrie of Euclid of Megara* (1570), quoted by Jennifer M. Rampling in *BSHM Bulletin* 26 (2011), 135-146.

Lefschetz valued independent thinking and originality above everything. He was, in fact, contemptuous of elegant or rigorous proofs of what he considered obvious points. He once dismissed a clever new proof of one of this theorems by saying, “Don’t come to me with your pretty proofs. We don’t bother with that baby stuff around here.” Legend had it that he never wrote a correct proof or stated an incorrect theorem. His first comprehensive treatise on topology, a highly influential book in which he coined the term “algebraic topology,” “hardly contains one completely correct proof. It was rumored that it had been written during one of Lefschetz’ sabbaticals … when his students did not have the opportunity to revise it.”

Sylvia Nasar, *A Beautiful Mind*, Simon & Schuster, 1998.

(Thanks to Robin Whitty for this)

The proof [of the existence of an infinity of prime numbers] is by *reductio ad absurdum*, and *reductio ad absurdum*, which Euclid loved so much, is one of a mathematician’s favourite weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers *the game*.

G. H. Hardy, *A Mathematician’s Apology*, Cambridge, 1940.

Cardinal arithmetic will be quite important for us, so we spend some time on it. Since, however, it tends to be trivial, we shall not need to spend much of this time on *proofs*.

Keith J. Devlin, *Aspects of Constructibility*, Springer, Berlin, 1973.

[Christopher] Hansteen declared that proofs should not be used in elementary teaching before it was necessary because, he argued, it invited students to memorize without understanding. To this, [Michael] Holmboe replied that one either has to prove all or nothing, as half a proof is worse than no proof.

Andreas Christiansen, Bernt Michael Holmboe (1795–1850) and his mathematics textbooks, *BSHM Bulletin* **24** (2009), 105–113.

## Axioms

A choice of axioms is not purely a subjective task. It is usually expected to achieve some definite aim — some specific theorem or theorems are to be derivable from the axioms — and to this extent the problem is exact and objective. But beyond this there are always other important desiderata of a less exact nature: the axioms should not be too numerous, their system is to be as simple and transparent as possible, and each axiom should have an immediate intuitive meaning by which its appropriateness can be judged directly.

J. von Neumann and O. Morgenstern, *Theory of Games and Economic Behavior*, Princeton University Press, 1944.

To illustrate the strong feelings of independence which, as a part of the old traditions, are so characteristic of the English spirit, I should like to tell how Hardy and Littlewood, when they planned and began their far-reaching and intensive team work, still had some misgivings about it because they feared that it might encroach on their personal freedom, so vitally important to them. Therefore, as a safety measure, … they amused themselves by formulating some so-called ‘axioms’ for their mutual collaboration. There were in all four such axioms.

The first of them said that, when one wrote to the other, …, it was completely indifferent whether what they wrote was right or wrong …

The second axiom was to the effect that, when one received a letter from the other, he was under no obligation whatsoever to read it, let alone to answer it …

The third axiom was to the effect that, although it did not really matter if they both thought about the same detail, still, it was preferable that they should not do so.

And, finally, the fourth, and perhaps most important axiom, stated that it was quite indifferent if one of them had not contributed the least bit to the contents of a paper under their common name …

I think one may safely say that seldom — or never — was such an important and harmonious collaboration founded on such apparently negative axioms.

From the collected works of Harald Bohr, quoted by Bela Bollobás in the foreword to *Littlewood’s Miscellany*, Cambridge University Press, 1986.

“Ignorance of axioms”, the Lecturer continued, “is a great drawback in life. It wastes so much time to have to say them over and over again. For instance, take the Axiom, ‘*Nothing is greater than itself*,’ that is, ‘*Nothing can contain itself*.’ How often you hear people say ‘He was so excited, he was quite unable to contain himself.’ Why, *of course* he was unable! The *excitement* had nothing to do with it!”

Lewis Carroll, *Sylvie and Bruno Concluded*.

To solve the problem of what is mathematical truth, Poincaré said, we should first ask ourselves what is the nature of geometric axioms. Are they synthetic *a priori* judgments, as Kant said? That is, do they exist as a fixed part of man’s consciousness, independently of experience and uncreated by experience? Poincaré thought not. They would then impose themselves on us with such force that we couldn’t conceive the contrary proposition, or build upon it a theoretic edifice. There would be no non-Euclidean geometry.

Should we therefore conclude that the axioms of geometry are eternal verities? Poincaré didn’t think that was so either. If they were, they would be subject to continual change and revision as new laboratory data came in. This seemed to be contrary to the whole nature of geometry itself.

Poincaré concluded that the axioms of geometry are *conventions*, our choice among all possible conventions is *guided* by experimental facts, but it remains *free* and is limited only by the necessity of avoiding all contradiction. Thus it is that the postulates can remain rigorously true even though the experimental laws that have determined their adoption are only approximative. The axioms of geometry, in other words, are merely disguised definitions.

Robert M. Pirsig, *Zen and the Art of Motorcycle Maintenance: An Inquiry into Values*, Bodley Head, London, 1974.

## Infinity

In the Middle Ages the problem of infinity was of interest mainly in connection with arguments about whether the set of angels who could sit on the head of a pin was infinite or not.

N. Ya. Vilenkin, *Stories about Sets*, Academic Press, New York, 1968.

[Infinity] is … the staple of the mystic contemplation of reality — “make me one with everything” as the mystic said to the hamburger vendor

John D. Barrow, *The Infinite Book*, Vintage, 2005.

Inside the museums, infinity goes up on trial.

Voices echo, “This is what salvation must be like, after a while.”

Bob Dylan, “Visions of Johanna”

Why had not the Public Prosecutor asked him: “Defendant Rubashov, what about the infinite?” He would not have been able to answer — and there, there lay the real source of his guilt … Could there be a greater?

Arthur Koestler, *Darkness at Noon*, quoted in Tiresias, *Notes from Overground*, Paladin, London, 1984.

So, naturalists observe, a flea

Hath smaller fleas that on him prey

And these have smaller fleas to bite ‘em

And so proceed *ad infinitum*.

Thus every poet, in his kind,

Is bit by him that comes behind.

Jonathan Swift, “On Poetry” (1733) [often misquoted!]

[A bit of background: infinite group theorists say that a group *G* is *virtually P*, where *P* is some group-theoretic property, if *G* has a subgroup of finite index which has property *P*. This enables them to wind up finite group theorists by saying that they study groups which are virtually trivial.]

[If a surface has complexity at most 1, then its mapping class group] is virtually free, so not interesting. It may even be finite.

Cornelia Drutu, talk to London Algebra Colloquium, 4 February 2010.

… the eighteenth century had been rich in speculative theories about the possibility of a “Big Universe”. These included … Kant’s *Universal Natural History of the Heavens* (1755), which first proposed – though without observational evidence – that … the whole cosmos might be in some sense “infinite”, though it is not clear what exactly “infinite” might mean, as hitherto it was a quality possessed only by God and mathematics.

Richard Holmes, *The Age of Reason: How the Romantic Generation Discovered the Beauty and Terror of Science*, HarperPress, London, 2008.

Maxwell’s talk of a large number of systems, instead of a continuous one, was a characteristic way of expression of a physicist who tends to see the infinite as an approximation to the sufficiently large finite. (A mathematician might think exactly the other way around.)

John von Plato, *Creating Modern Probability*, Cambridge, 1994.

Methinks I lied all winter, when I swore

My love was infinite, if spring make it more.

Love’s Growth

[And finally, let me be vain enough to quote myself:]

Counting is a less precise tool for infinite sets than for finite ones. The shepherdess who can count her flock of a hundred sheep will know if the wolf has taken one; but, if she has an infinite flock, she won’t notice until almost all of her sheep have been lost.

Peter J. Cameron, *Combinatorics* (2nd edition), Cambridge University Press, 1996.

## Numbers

Said to have acquired chopper when on the rocks (Guardian series) (3,3,5,4)

Araucaria, clue in Guardian crossword, 8 June 2013

“Nought usually comes at the beginning,” Ralph said.

“Not necessarily,” said Sibyl. “It might come anywhere. Nought isn’t a number at all. It’s the opposite of number.”

Nancy looked up from the cards. “Got you, aunt,” she said. “What about ten? Nought’s a number there — it’s part of ten.”

“Well, if you say that any mathematical arrangement of one and nought really makes ten — ” Sibyl smiled. “Can it possibly be more than a way of representing ten?”

Charles Williams, *The Greater Trumps*, Victor Gollancz, London, 1932.

How many roads must a man walk down before you can call him a man?

How many seas must the white dove sail before she sleeps in the sand?

How many times must the cannonballs fly, before they’re forever banned?

The answer, my friend, is blowin’ in the wind

The answer is blowin’ in the wind.

Bob Dylan, “Blowin’ in the Wind” (1962)

What could be more general than 2, which can represent two galaxies or two pickles, or one galaxy plus one pickle (the mind doth boggle), or just 2 gently bobbing — where? It, like God, is an “I am” and many have thought that it must be a precipitate of ultimate reality.

Alfred W. Crosby, *The Measure of Reality: Quantification and Western Society, 1250–1600*, Cambridge University Press, Cambridge, 1997.

I have often admired the mystical way of Pythagoras, and the secret magic of numbers.

Sir Thomas Browne, *Religio Medici*, I, 12.

There is divinity in odd numbers, either in nativity, chance or death.

Shakespeare, *The Merry Wives of Windsor*, V, 1.

“I count a lot of things that there’s no need to count,” Cameron said. “Just because that’s the way I am. But I count all the things that need to be counted.”

Richard Brautigan, *The Hawkline Monster: A Gothic Western*, Picador, London, 1976.

Them as counts counts moren them as dont count

Russell Hoban, *Riddley Walker*, Jonathan Cape, 1980.

… mathematical knowledge … is, in fact, merely verbal knowledge. “3” means “2+1″, and “4” means “3+1″. Hence it follows (though the proof is long) that “4” means the same as “2+2″. Thus mathematical knowledge ceases to be mysterious.

Bertrand Russell, *History of Western Philosophy*, George Allen and Unwin, London, 1961.

The man who has learned that three plus one are four doesn’t have to go through a proof of that assertion with coins, or dice, or chess pieces, or pencils. He knows it, and that’s that. He cannot conceive a different sum. There are mathematicians who say that three plus one is a *tautology* for four, a *different way of saying* “four” … If three plus one can be two, or fourteen, then reason is madness.

Jorge Luis Borges, “Blue Tigers”, in *Shakespeare’s Memory* (1983), transl. Andrew Hurley, Penguin, 1999.

I know that two and two make four —and should be glad to prove it too if I could — though I must say if by any sort of process I could convert two and two into five it would give me much greater pleasure.

Lord Byron to his wife (parents of Ada Lovelace) in a letter, 1812

Take one from ten, and what remains?

Ten still, if sermons go for gains.

George Herbert, “Charms and Knots”

Pure mathematics … seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but *because it is so*, because mathematical reality is built that way.

G. H. Hardy, *A Mathematician’s Apology*, Cambridge, 1940.

“Can you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”

“I don’t know,” said Alice. “I lost count.”

“She ca’n’t do Addition,” the Red Queen interrupted.

Lewis Carroll, *Through the Looking-Glass, and what Alice found there*, 1875.

We have learned to pass with such facility from cardinal to ordinal number that the two aspects appear to us as one. To determine the plurality of a collection, that is, its cardinal number, we do not bother anymore to find a model collection with which we can match it — we count it. And to the fact that we have learned to identify the two aspects of number is due our progress in mathematics . . . The operations of arithmetic are based on the tacit assumption that we can always pass from any number to its successor, and this is the essence of the ordinal concept.

And so matching by itself is incapable of creating an art of reckoning. Without our ability to arrange things in ordered succession little progress could have been made. Correspondence and succession, the two principles that permeate all mathematics — nay, all realms of exact thought — are woven into the very fabric of our number system.

Tobias Dantzig, *Number: The Language of Science*, Macmillan,

New York, 1930.

What seems universally acceptable is that numbers are inconceivable—practically, experientially, conceptually, semiotically, historically—in the absense of *counting*. Counting, whether with fingers, pebbles, notches, tally marks, abacus beads, or notations on chalkboards, paper, and computer screens, is an activiity involving *signs*. And, as an activity, counting works through—it *is*—significant repetition. How are we—through what discursive apparatus and technology of symbolic persuasion—to immagine the business of repeating the self-same signifying act without end, of iterating *for ever*? Or, which will come to the same, what would it mean to *deny* the possibility of endlessly repeating a signifying act? Is it in fact possible to coherently imagine an activity of iterating that did not—by definition of the very abstracted purity of the repetition that furthers it—go on forever?

Brian Rotman, *Ad Infinitum: The Ghost in Turing’s Machine*, Stanford, 1993.

You see, the chemists have a complicated way of counting: instead of saying “one, two, three, four, five protons”, they say, “hydrogen, helium, lithium, beryllium, boron.”

Richard Feynman, *QED: The Strange Theory of Light and Matter*, Princeton U.P., 1985.

The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic.

G. F. Leibniz, quoted by Oliver Sacks, *The Man who Mistook his Wife for a Hat*, Duckworth, London, 1985.

The Way begets one; one begets two; two begets three; three begets the myriad creatures.

Lao Tse, *Tao Te Ching*.

All of mathematics can be deduced from the sole notion of an integer; here we have a fact universally acknowledged today.

E. Borel, Contribution a l’analyse arithmetique du continu, *Oeuvres* 3, 1439–1485.

Although the idea that we have no bananas is unlikely to be a new one, or one that is hard to grasp, the idea that no bananas, no sheep, no children, no prospects are really all the same, in that they have the same numerosity, is a very abstract one.

Brian Butterworth, *The Mathematical Brain*, Macmillan,

London, 1999.

Behold the One in all things; it is the second that leads you astray.

Kabir, quoted in Aldous Huxley, *The Perennial Philosophy*, 1944.

Huxley adds:

For example, how significant it is that in the Indo-European languages, as Darmsteter has pointed out, the root meaning “two” should connote badness. The Greek prefix dys- (as in dyspepsia) and the Latin dis- (as in dishonorable) are both derived from “duo”. The cognate bis- gives a pejorative sense to such modern French words as *bévue* (“blunder”, literally “two-sight”). Traces of that “second which leads you astray” can be found in “dubious”, “doubt”, and *Zweifel* — for to doubt is to be double-minded. Bunyan has his Mr. Facing-both-ways, and modern American slang its “two-timers”. Obscurely and unconsciously wise, our language confirms the findings of the mystics and proclaims the essential badness of division — a word, incidentally, in which our old enemy “two” makes another decisive appearance.

In the history of the concept of *number* has been *adjective* (*three* cows, *three* monads) and *noun* (*three*, pure and simple), and now …, *number* seems to be more like a *verb* (*to triple*).

Barry Mazur, *Imagining Numbers*, Penguin, London, 2003.

Numbers written on bills within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the universe.

Douglas Adams, *Life, the Universe and Everything*, Pan, London, 1982.

You can take away my chicken or my sheep so I count them, but no man can take away my years so I don’t count the years.

Miruts Yifter (Yifter the Shifter) (attr.), in John Bryant, *The London Marathon*, Arrow Books, 2006.

There are so many things you can do to a number, and as many reasons why you might want to. Add it, subtract it, cube it, expose it to all the contortions of mathematics. You could even, long ago, count things with it. But you couldn’t steal it. But then again, you wouldn’t have wanted to.

Times change, and there is a new mathematics of number. In the new mathematics, the digits themselves, though crucial, no longer convey the number’s value. In this new order, a small four-digit number, though numerically inferior in all respects to an infinite number of numbers above it, may be much more valuable.

A PIN, for example — an access code, the power to get wealth or credit or information — is worth many times more than the 11-digit telephone number of a pizza delivery outlet, even one that claims to use real mozzarella. The new situation is that a number’s true value can no longer be separated from its place in the world.

So the new mathematics, that is, the useful mathematics, has a calculus quite different from its predecessor. It still has ordering functions, and the concept of value but the number line has become a cloud of shifting points, tied up almost unfathomably with the world in which they exist.

David Whiteland, *Book of Pages*, Ringpull Press, 2000.

Augustine took from the Neoplatonists that interest in number, which to the ordinary reader of his works seems an idiosyncrasy. … Plato, influenced by the Pythagoreans, tended towards the end of his life to see the ideal world—reality—as made up of mathematical concepts and symbols, which were therefore the metaphysical constituents of the visible universe. Plotinus adopted the conception, probably from the Neo-Pythagoreans, and Augustine in turn took it from the Neoplatonists. For him number is the intelligible formula which describes the qualities of being and the manner of change, so that all chage throughout the universe, which presents so much philosophical difficulty, can in a sense be ‘controlled’ by numbers, just as an algebraic formula might express an electrical transformation or an engineering stress. Numbers are, in fact, a rationalization of the seminal reason of things. Numbers as used by Augustine had, of course, no scientific or mathematical basis, and it was easy, as Augustine found, to allegorize them, but the rational, or at least the pseudo-rational, foundation for what seems to many to be a strange aberration of a great genius can be seen to be one more legacy from Neoplatonism.

David Knowles, *The Evolution of Medieval Thought*, reprinted by Random House, Toronto, 1962.

(If you are aware of examples in the work of Augustine, please let me know!)

The idea that area and volume would be discrete had come to me in a flash as I was trying to calculate the volume of some quantum geometry, while I was sitting for an hour in a noisy room in a garage waiting for my car to be fixed. The page of my notebook was filled with many messy integrals, but all of a sudden I saw emerge a formula for counting. I had begun to calculate a quantity on the assumption that the result was a real number, but found instead that, in certain units, all the possible answers would be integers. This meant that areas and volumes cannot take any value, but come in multiples of fixed units.

Lee Smolin, *Three Roads to Quantum Gravity*, Weidenfeld & Nicolson, 2000.

An anticipation of the construction of natural numbers?

Our universe … has twenty-five levels, one above the other [footnote: Ordinarily twenty-four levels are counted … The number twenty-five seems to be due to a peculiar mode of counting, adding the total as a separate item to the available number.

*The life and teachings of Naropa* (transl. Herbert V. Guenther), Oxford University Press 1963.

## Combinatorics

Igor Pak has a collection of answers to the question “What is combinatorics?” here.

Journalists say that when a dog bites a man, that is not news, but when a man bites a dog, that is news . . . Thanks to the mathematics of combinatorics, we will never run out of news.

Steven Pinker, *How the Mind Works*, W. W. Norton, 1997.

*Net*. Anything reticulated, or decussated at equal distances, with interstices between the intersections.

Samuel Johnson, *Dictionary of the English Language* (1775)

… nets, grids, and other types of calculus …

Alan Watts, *The Book* (1972)

This method of deduction … is often called “combinatory”. Its usefulness is not exhausted at this stage, but it does even at the outset lead to some valuable conclusions …

John Chadwick, *The Decipherment of Linear B*, CUP, Cambridge, 1958.

These results, which are partly combinatorial and partly real mathematics …

A. Joseph, lecture to the London Mathematical Society, Oxford, 22 February 1997.

At the end of the thirteenth century, Raymond Lully was prepared to solve all arcana by means of an apparatus of concentric, revolving disks of different sizes, divided into sectors with Latin words; John Stuart Mill, at the beginning of the nineteenth, feared that some day the number of musical combinations would be exhausted and there would be no place in the future for indefinite Webers and Mozarts; Kurd Lasswitz, at the end of the nineteenth, toyed with the staggering fantasy of a universal library which would register all the variations of the twenty-odd orthographical symbols, in other words, all that it is given to express in all languages. Lully’s machine, Mill’s fear and Lasswitz’s chaotic library can be the subject of jokes, but they exaggerate a propensity which is common: making metaphysics and the arts into a kind of play with combinations.

Jorge Luis Borges, *Labyrinths*, New Directions, New York, 1964.

… combinatorics, a sort of glorified dice-throwing …

Robert Kanigel, *The Man who Knew Infinity: A Life of The Genius Ramanujan*, Scribner, New York, 1991.

… the branch of topology we now call “graph theory” …

Stuart Hollingdale, *Makers of Mathematics*, Penguin, London, 1989.

Thus you see, most noble Sir, how this type of solution [to the Königsberg bridge problem] bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle …

In the meantime, most noble Sir, you have assigned this question to the geometry of position, but I am ignorant as to what this new discipline involves, and as to which types of problem Leibniz and Wolff expected to see expressed in this way.

Leonhard Euler, Letter to Carl Ehler, mayor of Danzig, 3 April 1736, transl. Adrian Hollis, quoted in R. J. Wilson, “Euler’s combinatorial mathematics”, *BSHM Bulletin* **23** (2008) 13–23.

More and more I’m aware that the permutations are not unlimited.

Russell Hoban, *Turtle Diary*, Jonathan Cape, London, 1975.

We have not begun to understand the relationship between combinatorics and conceptual mathematics.

Jean Dieudonné, *A Panorama of Pure Mathematics: As seen by N. Bourbaki*, Academic Press, New York, 1982.

The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory — and away from the algorithm of differential equations which dominates mathematical physics.

J. von Neumann and O. Morgenstern, *Theory of Games and Economic Behavior*, Princeton University Press, 1944.

The process is directed always towards analysing and separating the material into a collection of discrete counters, with which the detached intellect can make, observe and enjoy a series of abstract, detailed, artificial patterns of words and images (you may be reminded of the New Criticism) …

Elizabeth Sewell, “Lewis Carroll and T. S. Eliot as Nonsense Poets”, in Neville Braybrooke (ed.), *T. S. Eliot: A Symposium for his Seventieth Birthday*, Hart-Davies, London, 1958.

Some of the particulars recommended by Abulafia contributed to the aura of magic surrounding Kabbalah: the best hour for meditative permutations (known as tzeruf) was midnight. The meditator was to light many candles, wear phylacteries and a prayer shawl, and write out the permutations of the alphabet with ever increasing speed. The resulting ecstatic state accompanied by the desire of the soul to leave the body could be so powerful that there was even the possibility of death. At the peak of ecstatic experience there would be a rush of unintelligible language and the kabbalist had to envision a surrounding circle of angels who could help to decipher the divine message. It was the sheer force of the letters themselves which brought forth the meaning, since the only link between the Sephirot of non-verbal Wisdom and verbal Intelligence was through the letters of the alphabet.

Johanna Drucker, *The Alphabetic Labyrinth: The Letters in History and Imagination*, Thames and Hudson, London, 1995.

Lord of sequence and design

Richard G. Jones, “The Earth is the Lord’s”,

*Hymns for Today* 33.

Television? The word is half Latin and half Greek. No good can come of it.

C. P. Scott (attr.)

There is no problem in all mathematics that cannot be solved by direct counting.

Ernst Mach, quoted by A. T. Benjamin, G. M. Levin, K. Mahlburg and J. J. Quinn, Random approaches to Fibonacci identities, Amer. Math. Monthly 107 (2000), p.511.

While [Maynard Smith] believed that a Marxist in science could take a lot of different positions, he saw the need for “some kind of substitute for Hegelian dialectics … some kind of concept that in dynamical systems there are going to be sudden breaks and thresholds and transformations, and so on”. He added that, in his opinion, “today we really do have a mathematics for thinking about complex systems and things which undergo transformations from quantity into quality”. Here he saw Hopf bifurcations and catastrophe theory as really nothing other than a change of quantity into quality in a dialectical sense.

Ullica Segerstrale, *Defenders of the Truth: The Battle for Science in the Sociobiology Debate and Beyond*, Oxford University Press, Oxford, 2000.

**Note:** I assume that “quantity into quality” can be interpreted as “continuous into discrete”.

Still more recently has it been found that the good Bishop Berkeley’s logical jibes against the Newtonian ideas of fluxions and limiting ratios cannot be adequately appeased in the rigorous mathematical conscience, until our apparent continuities are resolved mentally into discrete aggregates which we only partally apprehend. The irresistible impulse to atomize everything thus proves to be not only a disease of the physicist; a deeper origin, in the nature of knowledge itself, is suggested.

J. Larmor, Introduction to Henri Poincaré, *Science and Hypothesis*, Walter Scott Publ., 1905.

[Roger] Lyndon produces elegant mathematics and thinks in terms of broad and deep ideas … I once asked him whether there was a common thread to the diverse work in so many different fields of mathematics, he replied that he felt the problems on which he had worked had all been combinatorial in nature.

K. I. Appel, in *Contributions to Group Theory* (ed. K. I. Appel, J. G. Ratcliffe and P. E. Schupp), AMS, Providence, 1984.

The miserable wasteland of multidimensional space was first brought home to me in one gruesome solo lunch hour in one of MIT’s sandwich shops. “Wholewheat, rye, multigrain, sourdough or bagel? Toasted, one side or two? Both halves toasted, one side or two? Butter, polyunsaturated margarine, cream cheese or hoummus? Pastrami, salami, lox, honey cured ham or Canadian bacon? Aragula, iceberg, romaine, cress or alfalfa? Swiss, American, cheddar, mozzarella, or blue? Tomato, gherkin, cucumber, onion? Wholegrain, French, English or American mustard? Ketchup, piccalilli, tabasco, soy sauce? Here or to go?” … Comparative genomics and structural biochemistry say that the building blocks of living organisms are few, modular, and combinatorial. Proteins comprise a few hundred protein structural domains; nucleic acids are simpler, with a few tens of structural domains and regulatory binding sites for sequence-specific protein domains. The combinations of these elements are vastly larger than any universe we can comprehend, but a large proportion of these should work, when given familiar network architectures similar to those of existing organisms. Why waste time marking out the hyperspace of possibilities when we already know where the good stuff is?

Review by Myles Aston of *Life without Genes: the History and Future of Genomes* by Adrian Woolfson, in the Balliol College Annual Record 2001.

I have to admit that he was not bad at combinatorial analysis — a branch, however, that even then I considered to be dried up.

Stanislaw Lem, *His Master’s Voice* (1968).

Only connect!

E. M. Forster, *Howards End* (1910)

But many, many stories were told; from what could be gathered, all fifty of the mine’s inhabitants had reacted on each other, two by two, as in combinatorial analysis, that is to say, everyone with all the others, and especially every man with all the women, old maids or married, and every woman with all the men. All I had to do was to select two names at random, better if different sex, and ask a third person, “What happened with those two?” and lo and behold, a splendid story was unfolded for me, since everyone knew the story of everyone else.

Primo Levi, *The Periodic Table* (transl. Raymond Rosenthal), Michael Joseph, 1985.

## Logic and set theory

Reasoning and logic are to each other as health is to medicine, or — better — as conduct is to morality. Reasoning refers to a gamut of natural thought processes in the everyday world. Logic is how we ought to think if objective truth is our goal — and the everyday world is very little concerned with objective truth. Logic is the science of the justification of conclusions we have reached by natural reasoning. My point here is that, for such natural reasoning to occur, consciousness is not necessary. The very reason we need logic at all is because most reasoning is not conscious at all.

Julian Jaynes, *The Origin of Consciousness in the Breakdown of the Bicameral Mind*, Houghton Mifflin, New York, 1976.

[John Henry Newman] wished always “to go by reason, not by feeling”, but he insisted that “it is the concrete being that reasons … the whole man moves; paper logic is but the record of it.”

Geoffrey Faber, *Oxford Apostles: A Character Study of the Oxford Movement*, 1933.

Reason and logic are tools for understanding the world. We need a means of understanding ourselves, too. That is what imagination allows. When a child reads of a Nightingale who bleeds her song into a rose for love’s sake, or of a Selfish Giant who puts a wall round life, or of a Fisherman who wants to be rid of his Soul, or of a statue who feels the suffering of the world more keenly than the Mathematics Master who scoffs at his pupils for dreaming about Angels, the child knows at once both the mystery and truth of such stories. We have all at some point in our lives been the overlooked idiot who finds a way to kill the dragon, win the treasure, marry the princess.

Jeanette Winterson, review of Oscar Wilde’s *The Selfish Giant and Other Stories*, Guardian, 19 October 2013.

The object of mathematical rigor is to sanction and legitimate the conquests of intuition, and there never was any other object for it.

Jacques Hadamard, quoted in Badger, Sangwin and Hawkes (eds.), *Teaching Problem-solving in Undergraduate Mathematics*, HE STEM/Coventry University 2012.

What is the nature of mathematical reasoning? Is it really deductive, as is commonly supposed? Careful analysis shows us that it is nothing of the kind; that it participates to some extent in the nature of inductive reasoning, and for that reason it is fruitful. But none the less does it retain its character of absolute rigour; and this is what must first be shown.

Henri Poincaré, *Science and Hypothesis*, Walter Scott Publ., 1905.

Set theory has a dual role in mathematics. In pure mathematics, it is the place where questions about infinity are studied. Although this is a fascinating study of permanent interest, it does not account for the importance of set theory in applied areas. There the importance stems from the fact that set theory provides an incredibly versatile toolbox for building mathematical models of various phenomena.

Jon Barwise and Lawrence Moss, *Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena*, CSLI, Stanford, 1996.

The Naturalist theory of possibility now to be advanced will be called a Combinatorial theory. It traces the very idea of possibility to the idea of the combinations — all the combinations — of given, actual elements.

–oo–

Set theory is important . . . because mathematics can be exhibited as involving nothing but set-theoretical propositions about set-theoretical entities.

–oo–

Mathematics . . . is concerned with a wider domain than that domain which it is the object of the natural sciences to describe and categorize. The natural sciences are concerned with the actual world. Mathematics is concerned with “all possible worlds”.

–oo–

Philosophers have not found it easy to sort out sets . . .

D. M. Armstrong, *A Combinatorial Theory of Possibility*, Cambridge University Press, Cambridge, 1989.

The standard “foundation” for mathematics starts with sets and their elements. It is possible to start differently, by axiomatising not elements of sets but functions between sets. This can be done by using the language of categories and universal constructions.

–oo–

… in one sense a foundation is a security blanket: If you meticulously follow the rules laid down, no paradoxes or contradictions will arise. In reality there is now no guarantee of this sort of security …

–oo–

… the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories — specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms (like functions) form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.

S. Mac Lane, *Mathematics: Form and Function*, Springer, New York, 1986.

Nowadays, one of the most interesting points in mathematics is that, although all categorical reasonings are formally contradictory, we use them and we never make a mistake. Grothendieck provided a partial foundation in terms of universes but a revolution of the foundations similar to what Cauchy and Weierstrass did for analysis is still to arrive. In this respect, he was pragmatic: categories are useful and they give results so we do not have to look at subtle set-theoretic questions if there is no need. Is today the moment to think about these problems? Maybe …

Pierre Cartier, interview in *Newsletter of the European Mathematical Society*, January 2010.

It seems to us however that mathematics has now reached the stage where formalisation within some particular axiomatic set theory is irrelevant, even for foundational studies. It should be possible to specify conditions on a mathematical theory which would suffice for embeddability within ZF (supplemented by additional axioms of infinity of necessary) but which do not otherwise restrict the possible constructions in that theory. Of course the conditions would apply to ZF itself, and to other possible theories that have been proposed as suitable foundations for mathematics (certain theories of categories, etc.), but would not restrict us to any particular theory. This appendix is in fact a cry for a Mathematicians’ Liberation Movement!

…

The situation is analogous to the theory of vector spaces. Once upon a time there were collections of *n*-tuples of numbers, and the interesting theorems were those that remained invariant under linear transformations of these numbers. Now even the initial definitions are invariant, and vector spaces are defined by axioms rather than as particular objects. However, it is proved that every vector space has a base, so that the new theory is much the same as the old. But now no particular base is distinguished, and usually arguments which distinguish particular bases are cumbrous and inelegant compared to arguments directly in terms of the axioms.

We believe that mathematics itself can be foundsd in an invariant way, which would be equivalent to, but would not involve, formalisation within some theory such as ZF. No particular axiomatic theory like ZF would be needed, and indeed attempts to force arbitrary theories into a single formal straitjacket will probably continue to produce unnecessarily cumbrous and inelegant contortions.

John H. Conway, *On Numbers and Games* (2nd edition), A K Peters, 2001.

“Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.”

Lewis Carroll, *Through the Looking-Glass, and what Alice found there*, 1875.

If, like the truth, falsehood had only one face, we should know better where we are, for we should then take the opposite of what a liar said to be the truth. But the opposite of the truth has a hundred thousand shapes and a limitless field.

Montaigne, *Essays*, I, 9.

… all worthwhile philosophical statements express an insight; and the opposite of an insight is not a contradictory sentence, but a muddle …

A. J. P. Kenny, Thesis, quoted in *A path from Rome*, Oxford, 1985.

During a counterpoint class at UCLA, Schoenberg sent everybody to the blackboard. We were to solve a particular problem he had given and to turn around when finished so that he could check on the correctness of the solution. I did as directed. He said, “That’s good. Now find another solution.” I did. He said, “Another.” Again I found one. Again he said, “Another.” And so on. Finally, I said, “There are no more solutions.” He said, “What is the principle underlying all of the solutions?”

John Cage, Four Statements on the Dance, from *Silence: Lectures and Writings*, Calder and Boyars, 1968.

To be is to be the value of a variable.

Willard Van Ormond Quine, “On What There Is”, 1948.

To every thing there is a season, and a time to every purpose under the heaven:

A time to be born, and a time to die; a time to plant, and a time to pluck up that which is planted;

A time to kill, and a time to heal; a time to break down, and a time to build up;

A time to weep, and a time to laugh; a time to mourn, and a time to dance;

A time to cast away stones, and a time to gather stones together; a time to embrace, and a time to refrain from embracing;

A time to get, and a time to lose; a time to keep, and a time to cast away;

A time to rend, and a time to sew; a time to keep silence, and a time to speak;

A time to love, and a time to hate; a time of war, and a time of peace.

Ecclesiastes, Chapter 3

“For a *complete* logical argument”, Arthur began with admirable solemnity, “we need two prim Misses—”

“Of course!” she interrupted. “I remember that word now. And they produce—”

“A Delusion,” said Arthur.

“Ye–es?” she said dubiously. “I don’t seem to remember that so well. But what is the *whole* argument called?”

“A Sillygism.”

Lewis Carroll, *Sylvie and Bruno*.

In Sumatra, someone wishes to receive a doctorate in prophecy. The master seer who administers his exam asks if he will fail or pass. The candidate replies that he will fail …

Jorge Luis Borges, Review of Edward Kasner and James Newman, * Mathematics and the Imagination*, in *Non-Fiction 1922–1986* (ed. Eliot Weinberger), Penguin 2001.

“I believe in two-faced truth, in the Either, the Or and the Holy Both. I believe that if a statement is true then its opposite must be true. (Aristotle: ‘The knowledge of opposites is one.’) …”

Palinurus, *The Unquiet Grave* (1944)

[Garrett] Birkhoff delighted me with the following story. (Quotation is from memory.)

“You know, many years ago — back in the 1930’s — I thought I was interested in the foundations of mathematics. I even wrote a paper or two in the area. We had a brilliant young logician on our faculty who was making a name for himself. One term I happened to lecture in the same classroom, the hour immediately after he did. Each day, before erasing the blackboard, I would take a look to see what he was up to. Well, in September he started out with Theorem 1. Shortly before Christmas he was up to Theorem 747. You know what it was? ‘If *x*≤*y* and *y*≤*z*, then *x*≤*z*‘! At that point, something within me snapped. I said to myself, ‘There are some things in mathematics that you just have to take for granted!’ And I never again had anything to do with the foundations of mathematics.”

Eugene L. Lawler, Old Stories, in *History of Mathematical Programming: A Collection of Personal Reminiscences* (ed. J. K. Lenstra, A. H. G. Rinnooy Kan, and A. Schrijver), CWI, 1991.

(Thanks to Robin Whitty for this)

… the contradictory opposite of a copulative proposition is a disjunctive proposition composed of the contradictory opposites of its parts.

… the contradictory opposite of a disjunctive proposition is a copulative proposition composed of the contradictories of the parts of the disjunctive proposition.

William of Ockham (Occam), *Summa totius logicae*, 14th century (transl. Philotheus Boehner 1955)

**Note:** This is Ockham’s formulation of De Morgan’s Laws, more than five hundred years before De Morgan. It is just as clear in his Latin text.

The following is … a scientific principle of general application:

We are not allowed to affirm a statement to be true or to maintain that a certain thing exists, unless we are forced to do so either by its self-evidence or by revelation or experience or by a logical deduction from either a revealed truth or a proposition verified by observation.

That this is the real meaning of “Ockham’s razor” can be gathered from various texts in Ockham’s writings. It is quite often stated by Ockham in the form: “Plurality is not to be posited without necessity” (*Pluralitas non est ponenda sine necessitate*), and also, though seldom: “What can be explained by the assumption of fewer things is vainly explained by the assumption of more things” (*Frustra fit per plura quod potest fieri per pauciora*). The form usually given, “Entities must not be multiplied without necessity” (*Entia non sunt multiplicanda sine necessitate*), does not seem to have been used by Ockham. What Ockham demands in his maxim is that everyone who makes a statement must have a sufficient reason for its truth … This principle of “sufficient reason” is epistemological or methodological, certainly not an ontological axiom.

*Ockham: Philosophical Writings* (ed. and transl. Philotheus Boehner), new edition Hackett Publ. Co., 1990 (from the Introduction)

Florensky saw a relationship between the naming of “God” and the naming of sets in set theory; both God and sets were made real by their naming. In fact, the “set of all sets” might be God Himself.

–oo–

Borel’s interest in set theory began with what he at first called a “romantic attraction”, although like many such attractions, it would later cool. Borel subsequently excused this early fascination with set theory by observing, “Like many of the young mathematicians, I had been immediately captivated by the Cantorian theory; I don’t regret it in the least, for that is one mental exercise that truly opens up the mind.”

–oo–

In 1938 Lebesgue was given an honorary degree in Lwow, and he was taken to the coffee shop where the famous Polish mathematician Stefan Banach used to work. The waiter handed him a menu with long descriptions in Polish. Lebesgue glanced at it and answered, “Thank you, I only eat well-defined objects.”

Loren Graham and Jean-Michel Kantor, *Naming Infinity*, Belknap Press, Cambridge, MA, 2009.

## Algebra

You also get dramatic advances when you spot that you can leave out part of the *problem*. Algebra, for instance (and hence the whole of computer programming), derives from the realisation that you can leave out all the messy, intractable numbers.

Douglas Adams, *The Salmon of Doubt*, Macmillan, London, 2002.

Whatever you have to do with a structure-endowed entity Σ try to determine its group of automorphisms … You can expect to gain a deep insight into the constitution of Σ in this way.

H. Weyl, *Symmetry*, Princeton U. P. 1952.

Curiously enough, the twelve-tone system has no zero in it. Given a series: 3, 5, 2, 7, 10, 8, 11, 9, 1, 6, 4, 12 and the plan of obtaining its inversion by numbers which when added to the corresponding ones of the original series will give 12, one obtains 9, 7, 10, 5, 2, 4, 1, 3, 11, 6, 8 and 12. For in this system 12 plus 12 equals 12. There is not enough of zero in it.

John Cage, Eric Satie, from *Silence: Lectures and Writings*, Calder and Boyars, 1968.

**Note:** You can read my commentary on this quote in the second edition of my Introduction to Algebra.

There is a very famous joke about Bose’s work in Giridh. Professor Mahalanobis wanted Bose to visit the paddy fields and advise him on sampling problems for the estimation of yield of paddy. Bose did not very much like the idea, and he used to spend most of the time at home working on combinatorial problems using Galois fields. The workers of the ISI used to make a joke about this. Whenever Professor Mahalanobis asked about Bose, his secretary would say that Bose is working in fields, which kept the Professor happy.

Bose memorial session, in Sankhya **54** (1992) (special issue devoted to the memory of Raj Chandra Bose), i–viii.

Langvetningur: … There are special spots here where the All-thought is manifest in the elements themselves, places where fire has become earth, earth become water, water become air, and air become spirit.

Embi: I thought it was the other way round.

The Langvetningur is of the old Teachers’ College and learned algebra from the Danish book by Pedersen: The order of the factors is immaterial.

Halldór Laxness, *Under the Glacier* (1968) (transl. Magnus Magnusson 1972).

A piece of music can be accompanied by words, movement, or dance, or can simply be appreciated on its own. It is the same with groups. They can be seen as groups of symmetries, permutations, or motions, or can simply be studied and admired in their own right.

Mark Ronan, *Symmetry and the Monster: One of the Greatest Quests of Mathematics*, Oxford Univ. Press, 2006.

The mathematics in this book is not hard, but neither is it very interesting, unless you are an algebra nerd.

Review in *The Economist* of *Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life* by Robin Wilson, 5 July 2008

… the mathematician Oswald Veblen … and the physicist James Jeans were discussing the reform of the mathematics curriculum at Princeton University. Jeans argued that they “may as well cut out group theory,” for it “would never be of any use to physics.” Of course, today, group theory is central to many parts of physics, such as quantum mechanics. Fortunately, Jeans’s advice was not taken.

Steven G. Krantz, *Mathematical Apocrypha*, MAA, 2002.

## Geometry

“It’s a fact I’ve continually observed in the witness box,” he said abstractedly, “that nine people out of ten, off their own subject, are incapable of lucidity, whereas on their own subject they can be as direct as a straight line before Einstein …”

–oo–

“… I think in a line — but there is the potentiality of the plane.” This perhaps was what great art was — a momentary apprehension of the plane at a point in the line.

Charles Williams, *Many Dimensions*, 1931.

“One of the virtues of this simple but at the same time complex design”, said Bembel Rudzak, “this design in which we see the continually reciprocating action of unity and multiplicity, is that it suits the apparent action to the mind of the viewer: those who look outward see the outward pre-eminent; those who look inward see the inward.”

–oo–

“There is transitive motion and there is intransitive motion: the motion of a galloping horse is transitive, it passes through our field of vision and continues on to where it is going; the motion in a tile pattern is intransitive, it does not pass, it moves but it stays in our field of vision. It arises from stillness, and I should like to think about the point at which stillness becomes motion. Another thing I should like to think about is the point at which pattern becomes consciousness.”

–oo–

“When a pattern shows itself in tiles or on paper or in your mind and says, ‘This is the mode of my repetition; in this manner I extend myself to infinity’, it has already done so, it has already been infinite from the very first moment of its being; the potentiality and the actuality are one thing. If two and two can be four then they actually are four, you can only perceive it, you have no part in making it happen by writing it down in numbers or telling it out in pebbles.”

Russell Hoban, *Pilgermann*, Pan, London, 1984.

In Plane Geometry that afternoon, I got into an argument with Mr Shull, the teacher, about parallel lines. I say they have to meet. I’m beginning to think everything comes together somewhere.

William Wharton, *Birdy*, Alfred A. Knopf, New York, 1979.

Voyez-vous cet oeuf. C’est avec cela qu’on renverse toutes les écoles de théologie, et tous les temples de la terre.

Denis Diderot, *Le rêve de d’Alembert*

What? Will the line stretch out to the crack of doom?

William Shakespeare, *Macbeth*

An *active* line on a walk moving freely, without goal. A walk for a walk’s sake.

Paul Klee, *Pedagogical Sketchbook* (1925)

See also the Jorge Luis Borges quote about Tlön under Miscellany.

## Probability and Statistics

For John Sinclair’s definition of Statistics, see under Definitions.

Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.

Bertrand Russell, lecture in 1929, quoted in Ruzsa and Székely, *Algebraic Probability Theory* (1988).

For [Luck] your science finds no measuring-rods; . . .

Her permutations never know truce nor pause

Dante Alighieri, *Commedia I: Inferno* (transl. D. L. Sayers), Penguin, London, 1949. (Written early 14th century.)

**Note:** I interpret the first line as saying that there cannot be a theory of probability; and the second as saying that a random permutation has no fixed points. Neither of these assertions is correct, but Dante was prescient in raising the questions!

It is probable that two proposed unknown ratios are incommensurable because if many unknown ratios are proposed it is most probable that any [one] would be incommensurable to any [other].

Nicole Oresme, *De proportionibus proportionum* (1351), quoted by Karl Petersen, *Ergodic Theory* (1983).

No sector of a circle is so small that two such [bodies bodies moving with uniform but incommensurable velocities] could not conjunct in it at some future time, and could not have conjuncted in it sometime [in the past].

Nicole Oresme, *Tractatus de commensurabilitate vel incommensurabilitate motuum cell*, quoted by Karl Petersen, *ibid*.

**Note:** This is Oresme’s argument against astrology. It is based on his clear understanding that (in modern language) rational numbers form a null set, while multiples of an irrational number mod 1 are dense in the unit interval.

The understanding painter, describing a battle in lively colours, lets not his hand fall so low as the single encounters of mercenaries, light sallies of a desperate troop, nor to the exposure of every common man’s life, though it were in the very mouth of the cannon, for they would be base and idle descensions and not worth the labour of a deserving pencil, but neglecting these he strives only to express the valour of chieftains and generals, the arraigning of battles, overthrow of armies, devastation of cities, conflict of nations, and in the end the triumphs of the returning conquerors. So the Statist that desires to look through a Kingdom must not cast his eye upon humble and abject matters, and so be drawn from the greater, but even at the first let him take measure of the highest, which knowledge making him inwardly acquainted with foreign countries thrives so well in him that at last he grows to be a counsellor in his own. Whensoever thereafter he goeth about that Royal Survey and would to himself draw forth the model and platform of the state of any prince, it is needful he lay hold upon these observations following.

*The Politick Survey of a Kingdome* (possibly by Edward Wray in 1631), described in Chris Lewin, “*The Politick Survey of a Kingdome*: the first statistical template?”, *Significance* **7** (2010), 36-39.

In calculating entropy by molecular-theoretic methods, the word “probability” is often used in a sense differing from the way the word is defined in probability theory. In particular, “cases of equal probability” are often hypothetically stipulated when the theoretical methods employed are definite enough to permit a deduction rather than a stipulation.

A. Einstein, “On a heuristic point of view concerning the production and transformation of light” (1905).

A person who has watched her mother die from Huntington’s disease knows she has a fifty per cent chance of contracting it. But that is not right, is it? No individual can have fifty percent of this disease. She either has a one hundred per cent chance or zero chance, and the probability of each is equal. So all that a genetic test does is unpackage the risk and tell her whether her ostensible risk is actually one hundred per cent or is actually zero.

Matt Ridley, *Genome*, 4th Estate, London, 1999.

… statistics is not a subset of mathematics, and calls for skills and judgement that are not exclusively mathematical. On the other hand, there is a large intersection between the two disciplines, statistical theory is serious mathematics, and most of the fundamental advances, even in applied statistics, have been made by mathematicians like R. A. Fisher.

Sir John Kingman, interview in the EMS Newsletter, March 2002.

The TV scientist who mutters sadly “The experiment is a failure: we have failed to achieve what we hoped for,” is suffering mainly from a bad scriptwriter. An experiment is never a failure solely because it fails to achieve predicted results. An experiment is a failure only when it also fails adequately to test the hypothesis in question, when the data it produces don’t prove anything one way or the other.

Robert M. Pirsig, *Zen and the Art of Motorcycle Maintenance: An Inquiry into Values*, Bodley Head, London, 1974.

Once Bose was teaching a class in which Somesh Das Gupta was the only Indian and all of a sudden, Bose said, “Only Hindus can understand Design of Experiments. You see, in Design of Experiments we work with the same structure in different forms: plots viewed as points, blocks viewed as lines, plot in a block as a point incident with a line and so on. The same thing the Hindus do, they worship the same God in different forms.”

Bose memorial session, in Sankhya **54** (1992) (special issue devoted to the memory of Raj Chandra Bose), i–viii.

… the rationalist’s substitute for demonology — statistics.

Stanslaw Lem, *His Master’s Voice* (1968)

“Would you like me to quote you some statistics?”

“Er, well . . .”

“Please, I would like to. They, too, are quite sensationally dull.”

Douglas Adams, *Life, the Universe and Everything*, Pan, London, 1982.

The method of the physical sciences is based upon the induction which leads us to expect the recurrence of a phenomenon when the circumstances which give rise to it are repeated. If all the circumstances could be simultaneously reproduced, this principle could be fearlessly applied; but this never happens; some of the circumstances will always be missing. Are we absolutely certain that they are unimportant? Evidently not! It may be probable, but it cannot be rigorously certain. Hence the importance of the rôle that is played in the physical sciences by the law of probability. The calculus of probabilities is therefore not merely a recreation, or a guide to the baccarat player; and we must thoroughly examine the principles on which it is based.

Henri Poincaré, *Science and Hypothesis*, Walter Scott Publ., 1905.

. . . the plural of anecdote is not data.

Ben Goldacre, *Bad Science*, Fourth Estate, London, 2008.

No aphorism is more frequently repeated in connection with field trials, than that we should ask nature few questions or, ideally, one question, at a time. The writer is convinced that this view is mistaken. Nature, he suggests, will best respond to a logical and carefully thought out questionnaire; indeed, if we ask her a single question, she will often refuse to answer until some other topic has been discussed.

R. A. Fisher, quoted by Roger Payne at DEMA2011.

The idea is sometimes expressed that one could improve observational accuracy by averaging. The dangers invited by such averaging were well known to statisticians. One has to keep separate the error term (variation) in observations and the numerical accuracy of the readings. The last significant digit of a single observation is the last significant digit of the average. Hence this way, too, is barred from leading to unlimited precision.

John von Plato, *Creating Modern Probability*, Cambridge, 1994.

… a decision tree, so many branches spring

from its trunk, so many choices. Statistics

feels like poetry — endless searching,

never-ending uncertainty.

From “Numerical Landscape” by Evelyn Pye, in *Bridges 2013 Poetry Anthology* (ed. Sarah Glaz).

(Thanks to JoAnne Growney’s blog “Intersections”.)

## Codes and cryptography

(An early use of coding and cryptography)

When Lludd told his brother the purpose of his errand Llevelys said that he already knew why Lludd had come. Then they sought some different way to discuss the problem, so that the wind would not carry it off and the Corannyeid learn of their conversation. Llevelys ordered a long horn of bronze to be made, and they spoke through that, but whatever one said to the other came out as hateful and contrary. When Llevelys perceived there was a devil frustrating them and causing trouble he ordered wine to be poured through the horn to wash it out, and the power of the wine drove the devil out.

“Lludd and Llevelys”, from *The Mabinogion* (earlier than 1325).

The code often used in the Arthurian Chancellery was the same which the Laconians had used in ancient Greece, known in their tongue as ‘skitale’. The Ephors had used it in their letters to ambassadors and generals. The method involved a rod of olive-wood about a span and a half in length, around which was obliquely wrapped a bit of skin; on this the message was written, from top to bottom, in such a way that when the skin was unrolled only detached letters appeared, and to read the message the recipient had to roll the skin again around a rod of the same dimensions.

Alvaro Cunqueiro, *Merlin and Company* (transl. Colin Smith), J. M. Dent, 1996.

I have written these words in code, made only for Your eyes.

Please take them, and read them right away!

Hafiz (transl. Thomas Rain Crowe)

## Algorithms

A “good” algorithm should have three properties:

- it should be clear and simple, laying out step by step the procedures to be followed,
- it should emphasize the general character of its applications by pointing out its appropriateness, not to a single problem but to a group of similar problems, and
- it should show clearly the answer obtained after the prescribed set of operations is completed.

George Gheverghese Joseph, *The Crest of the Peacock: Non-European Roots of Mathematics*, Penguin, London 1991.

I wish to God these calculations had been executed by steam. (Charles Babbage)

–oo–

To spare themselves, the mathematicians farmed out the routine work to the “computers” — a common reference to people who performed calculations. (Computers are not the only devices to have human antecedents. Later in the nineteenth century, “typewriter” referred to a person who typed rather than to the machine itself. It was not unheard of for the boss to elope with his typewriter.)

–oo–

Science was not an established profession and offered no secure career path.

–oo–

Babbage … wrote to Herschel saying that … in a few years his completed engine would produce “logarithmic tables as cheap as potatoes”.

–oo–

[Babbage] took the ergonomics of reading tables to extraordinary lengths, producing samples using every permutation of ink colour and backing paper including — bizarrely it seems — green ink on three shades of green paper. He was a mathematician and a logician, and a permutation is a permutation.

–oo–

From the start there has been a curious affinity between mathematics, mind and computing … It is perhaps no accident that Pascal and Leibniz in the seventeenth century, Babbage and George Boole in the nineteenth, and Alan Turing and John von Neumann in the twentieth — seminal figures in the history of computing — were all, among their other accomplishments, mathematicians, possessing a natural affinity for symbol, representation, abstraction and logic.

–oo–

Yet it is by the laborious route of analysis that [the man of genius] must reach truth; but he cannot pursue this unless guided by numbers; for without numbers it is not given to us to raise the veil which envelopes the mysteries of nature. (L. Menabrea, transl. A. Lovelace)

Doron Swade, *The Cogwheel Brain; Charles Babbage and the Quest to Build the First Computer*, Little, Brown, London, 2000.

Elegance is an algorithm.

Iain M. Banks, *The Algebraist*, Orbit, 2004.

(Thanks to Robin Chapman for drawing my attention to this.)

Although no mathematician can ever define properly what the “elegance” or the “beauty” of a mathematical theorem, proof or construction really means, all of them have an indescribable feeling for it; … however, programmers are more concerned about efficiency and other practical problems …

I. Herman, *The Use of Projective Geometry in Computer Graphics*, Lecture Notes in Computer Science **564**, Springer, Berlin, 1992.

## Philosophy

Idealists and metaphysicists not only fall into confusion in their attempts to answer these basic questions but they go so far as to distort mathematics completely, turning it literally inside out. Thus, seeing the extreme abstractness and cogency of mathematical results, the idealist imagines that mathematics issues from pure thought.

In reality, mathematics offers not the slightest support for idealism or metaphysics. We will convince ourselves of this as we attempt, in general outline, to answer the listed questions about the essence of mathematics. For a preliminary clarification of these questions, it is sufficient to examine the foundations of arithmetic and elementary geometry, to which we now turn.

A. D. Aleksandrov, *A general view of mathematics* (English translation AMS 1962)

## Education

The difference between principles and rules is this, that the former are persuasions and the latter are commands. There is great deal of difference between carrying 2 for such and such a reason, and carrying 2 because you *must* carry 2. You see boys that can cover reams of paper with figures, and do it with perfect correctness too; and at the same time, can give you not a single reason for any part of what they have done. Now this is really doing very little. The rule is soon forgotten, and then all is forgotten.

William Cobbett, “From Petersfield to Kensington”, *Rural Rides* (ed. Ian

Dyck), Penguin Classics 2001. (First published 1830)

I am a profound believer that the proper, and therefore the exact study, even of so humble a subject as elementary Arithmetic is a necessary, in fact the necessary, first step towards the culture of mathematics; and although it is not given to everyone to be a mathematician any more than to be an artist, a musician, or a poet, yet just as every normal educated man or woman is rightly expected to have some fairly correct notions on art, on poetry, and on music, so some reasonable, rational and exact knowledge of numbers and their properties is to be regarded, independent of all commercial and technical applications, as an essential part of the culture that all normal educated persons should have acquired.

J. E. A. Steggall, talk to the Mathematical Association, 1914, quoted in Hilary Mason, “J. E. A. Steggall: Teaching mathematics 1880–1933″, *BSHM Bulletin* **1** (2004), 27–38.

… 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*, Scribner, New York, 1991.

Mathematics makes a nice distinction between the usually synonymous terms “elementary” and “simple”, with “elementary” taken to mean that not very much mathematical knowledge is needed to read the work and “simple” to mean that not very much mathematical ability is needed to understand it. In these terms we think the content is often elementary but in places not so very simple. The reader should expect to make use of pen and paper in many places; mathematics is not a spectator sport!

Julian Havel, *Gamma: Exploring Euler’s Constant*, Princeton University Press, Princeton, 2003.

Whoever then has the effrontery to study physics while neglecting mathematics, should know from the start that he will never make his entry through the portals of wisdom.

Thomas Bradwardine

If we had chosen to develop the theory of rings with operators, establishing our present propositions in the presence of operator domains, [the classical Density Theorem] would follow immediately; however, it seems inadvisable to treat rings with operators in an introductory course.

Irving Kaplansky, *Fields and Rings*, University of Chicago Press 1969.

Problem-solving is at the heart of mathematics: we learn to do the subject by solving problems, and we teach it by posing them. However … we increasingly teach our students simply to solve the type of problems which we have already taught them to solve, and we rarely challenge their ingenuity by confronting them with the unfamiliar. “It’s not fair, we haven’t been told how to solve problems like that” is one of the most depressing complaints a teacher is likely to hear.

From a review by Gareth Jones of *The Art of Mathematics* by Béla Bollobás, London Mathematical Society Newsletter, March 2007.

… in some places [the new math] reached down as far as kindergarten, where lucky students might be taught that a set consisting of exactly the two elements *a* and *b* could be symbolized as {*a, b*}. And the expected story ensued: Johnny’s parents ask the kindergarten teacher how he is doing, and she tells them “Yes, he understands sets, but he has trouble writing the curly brackets.”

Saunders Mac Lane, *A Mathematical Autobiography*, A K Peters 2005.

Students themselves are attracted or repelled by different ways of presenting science, and no single approach can be universally successful.

G. Holton and S. G. Brush, *Introduction to Concepts and Theories in Physical Science* (2nd ed.), Addison-Wesley, 1973.

We would not go so far as to say that the passion with which these young Russian students thought day and night about mathematical questions was unique; the same passion could be found in the corridors of mathematics departments in Cambridge, Bonn, Berkeley, Paris, and elsewhere. One characteristic of the Russian approach, however, stood out — the conviction of the best Russian teachers of mathematics that the most fruitful attack on problems was direct and straightforward, without any preliminary, long, heavy readings. In other words, *start from scratch*. By doing so, one got an almost physical feeling of being directly in contact with mathematical objects and experienced the sensual pleasure of having to fight intellectually with one’s bare hands. One of the great mathematicians of the time, Israel Moissevich Gelfand, would tell his young students, “We should study this topic before it has been tainted by handling.”

Loren Graham and Jean-Michel Kantor, *Naming Infinity*, Belknap Press, Cambridge, MA, 2009.

Ill-used persons. who are forced to load their minds with a score of subjects against an examination, who have too much on their hands to indulge themselves in thinking or investigation, who devour premiss and conclusion together with indiscriminate greediness, who hold whole sciences on faith, and commit demonstrations to memory, and who too often, as might be expected, when their period of education is passed, throw up all they have learned in disgust, having gained nothing really by their anxious labours, except perhaps the habit of application.

John Henry Newman, quoted in Badger, Sangwin and Hawkes (eds.), *Teaching Problem-solving in Undergraduate Mathematics* HE STEM/Coventry University 2012.

*Scholar.* Sir, I thank you: but I think I might the better doe it, if you did shew me the working of it.

*Master.* Yea, but you must prove your self to doe some things without my aid, or else you shall not be able to doe any more than you are taught. And that were rather to learn by wrote [rote] (as they call it) than by reason.

Robert Recorde, *Grounde of Artes* (1542), quoted by June Barrow-Green in *BSHM Bulletin* **21** (2006).

Many teachers will say that ‘you cannot express the inexpressible’, and they do not try. But teachers like Yasutani and Maezumi don’t agree, and I feel as they do: *if you perceive deeply enough*, a clear and simple way to express it can be found.

Tetsugen (Bernard Glassman), quoted in Peter Matthiessen, *Nine-headed Dragon River*, Shambhala 1998.

If I give the answer, you immediately forget about the question. If I don’t give you the answer, you will still have questions and you will be thinking about the problem long after.

Eugene A. Geist

See also the comments by Hansteen and Holmboe from Christiansen’s article under Proofs.

## History

Many quotations with historical relevance are scattered in other parts of this document; see the quotes from Dante and Oresme in the Probability and Statistics section, for example.

The mathematical sciences particularly exhibit order, symmetry and limitation; and these are the greatest forms of the beautiful.

Aristotle, Metaphysics, 3-1078b.

For many parts of nature can neither be invented with sufficient subtility nor demonstrated with sufficient perspicuity nor accommodated into use with sufficient dexterity, without the aid and intervening of the Mathematics: of which sort are Perspective, Music, Astronomy, Cosmology, Architecture, Enginery, and divers others.

Francis Bacon

I often say that when you can measure what you are speaking about and express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind.

William Thompson, Lord Kelvin

… the word “theory” … was originally an Orphic word, which Cornford interprets as “passionate sympathetic contemplation” … For Pythagoras, the “passionate sympathetic contemplation” was intellectual, and issued in mathematical knowledge … To those who have reluctantly learnt a little mathematics in school this may seem strange; but to those who have experienced the intoxicating delight of sudden understanding that mathematics gives, from time to time, to those who love it, the Pythagorean view will seem completely natural …

–oo–

It seems to us unwise to have insisted on teaching geometry to the younger Dionysius, tyrant of Syracuse, in order to make him a good king, but from Plato’s point of view it was essential. He was sufficiently Pythagorean to think that without mathematics no true wisdom is possible.

Bertrand Russell, *History of Western Philosophy*, George Allen and Unwin, London, 1961.

It is given to us to live for the most part under the guidance of mathematics … It is impossible to distinguish from other living creatures anyone who does not understand how to quantify.

Cassiodorus, quoted in David Ewing Duncan, *The Calendar*, Fourth Estate, London, 1998.

Though [scholarship] may have appealed to some — men such as Aldhelm of Malmesbury who could even find tackling fractions an exciting challenge — it still went against the (male) barbarian grain.

Henrietta Leyser, *Medieval Women: A Social History of Women in England 450–1500*, Weidenfeld and Nicholson, London, 1995.

Why waste words? Geometry existed before the Creation, is co-eternal with the mind of God, is God himself (what exists in God that is not God himself?): geometry provided God with a model for the Creation and was implanted into man, together with God’s own likeness — and not merely conveyed to his mind through the eyes.

Johannes Kepler, *Harmonice Mundi*, quoted in Arthur Koestler, *The Sleepwalkers*, Hutchinson, London, 1959.

–oo–

Thus God himself

was too kind to remain idle

and began to play the game of signatures

signing his likeness unto the world:

therefore I chance to think

that all nature and the graceful sky are

symbolized in the art of Geometria.

Now, as God the maker play’d

he taught the game to Nature

whom he created in his image;

taught her the selfsame game

which he played to her.

Johannes Kepler, *Tertius Interveniens*, quoted in *ibid*.

… the professors [at Bologna] were kept in absolute and even humiliating subservience to their students. They had to swear obedience to the student rectors and to the student-made statutes, which bore very hardly upon them, e.g. the professor was fined if he began his teaching a minute late or continued a minute longer than the fixed time, and should this happen the students who failed to leave the lecture-room immediately were themselves fined. In addition, the professor was fined if he shirked explaining a difficult passage, or if he failed to get through the syllabus; he was fined if he left the city for a day without the rector’s permission, and if he married, was allowed only one day off for the purpose. The city, for its part, took a hand in controlling the professors, and they were forced to take an oath not to leave Bologna in search of more lucrative or less onerous posts.

David Knowles, *The Evolution of Medieval Thought*, Vintage Books, New York, 1962.

Astrologers … were suspect because of their mathematical calculations. The memory of Roger Bacon had been besmirched by the assumption that mathematics was part of the black art, and it was notorious that the Edwardian reformers had destroyed mathematical manuscripts at Oxford under the delusion that they were conjuring books. “Where a red letter or a mathematical diagram appeared, they were sufficient to entitle the book to be Popish or diabolical.” (This may account for the disappearance at this period of nearly all the works of the fourteenth-century Merton College school of astronomers.)

Keith Thomas, *Religion and the Decline of Magic*, Weidenfeld & Nicholson, London, 1971.

The seventeenth-century antiquarian John Aubrey reported that the Tudor authorities had “burned Mathematical books for Conjuring books”. Mathematics was still popularly associated with the magical Black Arts, the term “calculating” (sometimes corrupted to “calculing”) being synonymous with conjuration.

Benjamin Woolley, *The Queen’s Conjuror: The Life and Magic of Dr Dee*, Flamingo, London, 2002.

The evidence of title-pages … suggests that in seventeenth-century England the most usual meaning of “mathematician” was someone who made [astrological] predictions. That a mathematician was therefore a somewhat ambivalent figure is demonstrated by the rather hyperbolic example of *The ladies champion*, where John Heydon … attacked a series of pamphlets about prostitution. Heydon apparently felt that he had been slandered, and on his title-page he sarcastically gave himself all the bad names he could think of. One of these was “mathematician”.

Benjamin Wardhaugh, “Poor Robin and Merry Andrew: mathematical humour in Restoration England”, *BSHM Bulletin* **22** (2007), 151–159.

## Mathematicians at work

I wanted to represent these [Fuchsian] functions by the quotient of two series; the idea was perfectly conscious and deliberate; the analogy with elliptic functions guided me. I asked myself what properties these series must have if they existed, and succeeded without difficulty in forming the series I have called thetafuchsian.

Just at this time, I left Caen, where I was living, to go on a geologic excursion under the auspices of the School of Mines. The incidents of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go to some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’s sake I verified the result at my leisure.

Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty, that the arithmetic transformations of indefinite ternary quadratic forms were identical with those of non-Euclidean geometry.

…

Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestible.

Henri Poincaré, lecture at Société de Psychologie, Paris; quoted by Jacques Hadamard, *The Psychology of Invention in the Mathematical Field*; Robert M. Pirsig, *Zen and the Art of Motorcycle Maintenance*; William Byers, *How Mathematicians Think*; and no doubt others.

Two quotes from Simon Singh, *Fermat’s Last Theorem*:

Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into furniture, but gradually you learn where each piece of furniture is. Finally after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of — and couldn’t exist without — the many months of stumbling around in the dark that preceded them.

Andrew Wiles

–oo–

At first I supposed that I should be able to overcome the contradiction quite easily, and that probably there was some trivial error in the reasoning. Gradually, however, it became clear that this was not the case … Throughout the latter half of 1901 I supposed the solution would be easy, but by the end of that time I had concluded that it was a big job … I made a practice of wandering about the common room every night from eleven until one, by which time I came to know the three different noises made by nightjars. (Most people only know one.) I was trying hard to solve the contradiction. Every morning I would sit down before a blank sheet of paper. Throughout the day, with a brief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty.

Bertrand Russell (on his attempt to resolve his paradox)

I recently had an odd and vivid experience. I had been struggling for months to prove a result I was pretty sure was true. When I was walking up a Swiss mountain, fully occupied by the effort, a very odd device emerged — so odd that, though it worked, I could not grasp the resulting proof as a whole. But not only so; I had a sense that my subconscious was saying, “Are you *never* going to get it, confound you; try this.”

J. E. Littlewood, “The Mathematician’s Art of Work”, in *Littlewood’s Miscellany* (ed. B. Bollobás), Cambridge University Press, Cambridge, 1986.

Finally, two days ago, I succeeded, not on account of my painful efforts, but by the grace of God. Like a sudden flash of lightning, the riddle happened to be solved. I myself cannot say what was the conducting thread which connected what I previously knew with what made my success possible.

C. F. Gauss, quoted in Jacques Hadamard, *The Psychology of Invention in the Mathematical Field*, Princeton University Press, 1945.

It was at St Leonards, probably because it was a boarding school, that I discovered and developed as a positive habit the powers of what I call “subliminal learning”. We kept very regular hours and were never tired or stressed at school. Lights out at 9p.m., even when in the sixth form. Out of bed at 7a.m. The accepted wisdom is that one should relax before going to bed, emptying the mind of problems. As far as mathematics is concerned I could not agree less. We looked forward to “drawing-room” each evening, but I usually cheated and stole ten minutes back at my desk before bathtime. I would make certain to sort out in my head, as late as possible, what problems needed to be solved the next day and what might be usefully committed to memory. Before falling asleep, I “drew” with my finger any relevant geometrical figure or algebraic equation on the partitioning of the dormitory cubicle that formed a bedside wall. The result would be miraculous. Without fail, on waking in the morning, the details, the logical argument required or the facts that I needed to recall were clearly imprinted in my mind and, because of the clarity, any required solution would often be clearly “written” on the partition. For this to work, it is essential to make sure to wake at least five or ten minutes before the prescribed time for getting out of bed, giving oneself time to go over what has been resolved while asleep. This became honed to a fine art, without my ever telling anyone, and I have used the technique deliberately ever since.

Kathleen Ollerenshaw, *To talk of many things: An autobiography*, Manchester University Press, 2004.

The difference between a good mechanic and a bad one, like the difference between a good mathematician and a bad one, is precisely this ability to *select* the good facts from the bad ones on the basis of quality. He has to *care*!

Robert M. Pirsig, *Zen and the Art of Motorcycle Maintenance: An Inquiry into Values*, Bodley Head, London, 1974.

I would like to briefly share with you a story about my father, which I believe is typical of him. One day when our family was having tea with some friends, he was enthusiastically talking about his work. He said: “I feel like I am somehow moving through outer space. A particular idea leads me to a nearby star on which I decide to land. Upon my arrival I realize that somebody already lives there. Am I disappointed? Of course not. The inhabitant and I are cordially welcoming each other, and we are happy about our common discovery.” This was typical of my father; he was never envious.

Hilda Abelin-Schur (daughter of Issai Schur), in A. Joseph, A. Melnikov and R. Rentschler (eds.), *Studies in Memory of Issai Schur*, Birkhäuser, Boston, 2002, p. xli.

(Thanks to Robin Whitty for this.)

I just move around in the mathematical waters, thinking about things, being curious, interested, talking to people, stirring up ideas; things emerge and I follow them up. Or I see something which connects up with someting else I know aboout, and I try to put them together and things develop. I have practically never started off with any idea of what I’m going to be doing or where it’s going to go.

Sir Michael Atiyah in *Mathematical Intelligencer*, reprinted in the Problem Corner of the European Mathematical Society Newsletter.

If I was asked a question, rather a difficult one by itself, the result immediately proceeded from my sensibility without my knowing at the first moment how I had obtained it; starting from the result, I then sought the way to be followed for this purpose. That intuitive conception which, curiously enough, has never been shaken by an error, has developed more and more as needs increased. Even now, I have often the sensation of somebody beside me whispering the right way to find the desired result; it concerns some ways where few people have entered before me and which I should certainly not have found if I had sought for them by myself.

It often seems to me, especially when I am alone, that I find myself in another world. Ideas of numbers seem to live. Suddenly, questions of any kind rise before my eyes with their answers.

Ferrol (a mental calculator) in a letter to Möbius, quoted

in Hadamard, *The Mathematician’s Mind*.

“But how do you DO research in mathematics?”, people often wonder. Even a distinguished laboratory scientist may be unable to comprehend how research can be possible without test-tubes, pipettes, and bubbling flasks of evil-smelling brews. Well, you scratch your head; you scratch the back of an envelope with a pencil; you scratch a blackboard with a piece of chalk. You lie in the bath and gaze alternately at the ceiling and at your navel. You do the washing-up or go to sleep and you leave your subconscious to do your thinking for you – often astonishingly successfully. And a good old screaming-match with your colleagues can sometimes help too.

Donald Preece, in *Bulletin of the Institute of Combinatorics and its Applications.*

## Research

Research is the curse of our age. “Research” is the first step on the way to expertdom. There is so much research going on nowadays that teachers are becoming scarce. Already in the universities complaints are being made about there being too many research students and research fellowships. And what, you may ask, is all this research for? Goodness knows.

John Betjeman, “Antiquarian Prejudice”, in *First and Last Loves*, John Murray, London, 1952.

The world of research is going berserk

Too much paperwork

Bob Dylan, *Nettie Moore* (2006)

‘I said to him’, said Knowles, ‘ “This here’s a *re*-search laboratory. *Re*-search means *look again*, don’t it? Means they’re looking for something they found once and it got away somehow, and now they got to *re*-search for it! How come they got to build a building like this, with mayonnaise elevators and all, and fill it with all these crazy people? What is it they’re trying to find again? Who lost what?” Yes, yes!’

Kurt Vonnegut, *Cat’s Cradle*, 1963.

Clearly we must explain more forcibly, especially at the highest levels of government, that the primary goal of universities is teaching and research, and that income is a constraint, and not the value to be maximised.

Andrew Graham, *Balliol College Register*, 2005.

## Mathematics as metaphor

Love was even more mathematical than poetry; it was the pure mathematics of the spirit.

Charles Williams, *Descent into Hell*, 1937 (reprinted William B. Eerdman, Grand Rapids, MI, 1983).

He comes out with a few, excited phrases now and then, dense with meaning, pregnant with suggestion, a language all bone and sinew, like a mathematician’s formulae.

Fosco Mariani, *Secret Tibet* (transl. Eric Moshbacher and Guido Waldman), Harvill Press, London, 2000.

Angling may be said to be so like the mathematics, that it can never be fully learnt.

Izaak Walton, *The Compleat Angler* (1653).

And at the time of the rising of the moon with its blackness of light, bowing low, as it were, with folded hands under the guise of closing the blue lotuses, immediately the stars shone forth … like zero dots … scattered in the sky as if on the ink blue skin rug of the Creator who reckoneth the total with a bit of moon for chalk.

Subandhu, *Vâsavadattâ* (6th century), quoted in David Ewing Duncan, *The Calendar*, Fourth Estate, London, 1998.

There is a strong parallel between mountain climbing and mathematics research. When first attempts on a summit are made, the struggle is to find any route. Once on the top, other possible routes up may be discerned and sometimes a safer or shorter route can be chosen for the descent or for subsequent ascents. In mathematics the challenge is finding a proof in the first place. Once found, almost any competent mathematician can usually find an alternative often much better and shorter proof. At least in mountaineering we know that the mountain is there and that, if we can find a way up and reach the summit, we shall triumph. In mathematics we do not always know that there is a result, or if the proposition is only a figment of the imagination, let alone whether a proof can be found.

Kathleen Ollerenshaw, *To talk of many things: An autobiography*, Manchester University Press, 2004.

The spirit of mathematics and the essence of its beauty are remarkably fragile, because mathematics is about ideas and about thought. Mathematics takes place in the mind, and no two minds are the same. After many years of study and work, a mathematician may stumble on a vast and beautiful vista that unifies and simplifies many things that once appeared disparate and complicated. Mathematicians can share a beautiful vista with one another, but there is no camer that can easily capture an image of such a vista to convey it in full to people who have not trudged along many of the same trails.

William P. Thurston, Inauguration handout for the sculpture *The Eigntfold Way* by Helaman Ferguson at MSRI, Berkeley, 1998.

Alongside the liberating relief of the veteran who tells his story, I now felt in the writing a complex, intense, and new pleasure, similar to that I felt as a student when penetrating the solemn order of differential calculus.

Primo Levi, *The Periodic Table*, Michael Joseph 1985.

The geometry of innocent flesh on the bone

Causes Galileo’s math book to get thrown

At Delilah, who’s sitting worthlessly alone

But the tears on her cheeks are from laughter

Bob Dylan, “Tombstone Blues”

See also the first Charles Williams quote, and the William Wharton quote, under Geometry, the Bob Dylan quote under Infinity, and the Bertrand Russell quotes under Miscellany.

## Wordplay

Ive got to work the E qwations and the low cations

Ive got to comb the nations of it.

Russell Hoban, *Riddley Walker*, Jonathan Cape, London, 1980.

In mathematical terms, Lacan is here pointing out that the first homology group of the sphere is trivial, while those of the other surfaces are profound …

Alan Sokal, Trangressing the boundaries: Towards a transformative hermeneutics of quantum gravity, *Social Text* **46/47** (1996), 217–252.

… human life could be defined as a calculus in which zero was irrational.

Jacques Lacan, quoted in Alan Sokal and Jean Bricmont, *Fashionable Nonsense*, 1998.

**Note:** Sokal and Bricmont quote the paragraph from which this nice metaphor is taken: the rest is not up to this standard!

Art has to move you and design does not, unless it’s good design for a bus.

David Hockney, reported in *The Observer* (1988)

The students [in Moscow in the 1920s] were so devoted to their teachers’ studies of set theory that they made fun of mathematicians who worked in other areas, giving their topics comic titles such as “impartial differential equations”, “theory of improbability”, and “different finitenesses”.

Loren Graham and Jean-Michel Kantor, *Naming Infinity*, Belknap Press, Cambridge, MA, 2009.

. . . the cycle has taken us up through forests.

*Zen and the Art of Motorcycle Maintenance: An Inquiry into Values*, Bodley Head, London, 1974.

See also the anecdote about R. C. Bose under Algebra, the John Barrow quote under Infinity, and several quotes from Lewis Carroll throughout this document.

## Miscellany

A Miscellany is a collection without a natual ordering relation.

J. E. Littlewood, *A Mathematician’s Miscellany*, Methuen & Co., London, 1953; reprinted as *Littlewood’s Miscellany* (ed. Béla Bollobás), Cambridge University Press, Cambridge, 1986.

I admire those men who know about mathematics,

Who calculate, and cipher, and compute;

I used to think that logarithms were things that scuttle about in attics

And surds were little flowers with square roots.

At geometry I don’t excel,

My triangles go parallel,

But nobody else is loved so well

By Isobel, as me.

Jake Thackray, *Isobel*

A mathematician is a machine for turning coffee into theorems.

Alfréd Rényi (attr.) [Rényi's colleage Paul Turán is credited with the refinement that "weak coffee is suitable only for lemmas".]

There’s no maths involved. You solve the puzzle with reasoning and logic.

Sudoku instructions in *The Independent*.

In mathematics, deception as to whether real understanding is present or not, is least possible.

Felix Klein, quoted in Judy Green, “How many women mathematicians can you name?”, *Math Horizons*, MAA.

Mathematics is a hard thing to love. It has the unfortunate habit, like a rude dog, of turning its most unfavourable side towards you when you first make contact with it. That unfavourable side is arithmetic, and most people never really get the legacy of that initial encounter off their fingers — which is where counting starts, and for most, where it remains. Perhaps such an unwelcoming exterior serves a useful purpose — it deters casual visitors from stumbling into the awesome discoveries that lie beyond. Mathematics: you can get lost in such a beautiful place; unlike the real world, it has a purity that makes even simulated reality seem grubby; it is devoid of arbitrary construction in the same way the real world used to be, before peole learned to tear natural things down and throw unnatural things up. It is the best wilderness in which to retreat, to fast, to search for enlightenment. It is, paradoxically, about the only place where you can really take refuge from the modern world’s unrelenting barrage of numbers. But only mathematicians, and the various subspecies and half-breeds thereof, go there, which might be another reason why it is so unpopular with almost everyone else.

David Whiteland, *Book of Pages*, Ringpull Press, 2000.

We were to found a University magazine. A pair of little, active brothers—Livingstone by name, great skippers on the foot, great rubbers of the hands, who kept a book-shop over against the University building—had been debauched to play the part of publishers. We four were to be conjuct editors and, what was the main point of the concern, to print our own works; while, by every rule of arithmetic—that flatterer of credulity—the adventure must succeed and bring great profit. Well, well : it was a bright vision.

Robert Louis Stevenson, “A College Magazine”, in *Memories and Portraits* (1887); reprint Richard Drew Publishing, Glasgow, 1990.

Luckily for [my father], I was, as he had been, an enthusiastic learner, eager to sit beside him on the sofa and be shown how one solved simultaneous equations. It is true that mathematics didn’t lead anywhere, neither in his day nor in mine. It had got him a scholarship to Westminster and an exhibition to Trinity College, Cambridge. It got me a scholarship to Harrow (by mistake) and another to Stowe (which was what I really wanted) and then, later, one to Trinity. But after that, for both of us, our enthusiasm burned itself out. The exciting road we had been following had come to an end; almost the only prospect open to the mathematician was to become a mathematics master; and neither of us could have faced that.

Christopher Milne, *The Enchanted Places*, Eyre Methuen.

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 when the last of their radiant host shall have fallen from heaven.

E. Everett, *Orations and speeches on various occasions*, Volume 3 (Boston 1970), p. 514. (Thanks to Clark Kimberling for help with correcting this reference.)

Like the crest of a peacock, like the gem on the head of a snake, so is mathematics at the head of all knowledge.

Vedanga Jyotisa

Wherefore in all great works are Clerks so much desired?

Wherefore are Auditors so well-fed?

What causeth Geometricians so highly to be enhaunsed?

Why are Astronomers so greatly advanced?

Because that by number such things they find,

which else would farre excell mans minde.

Robert Recorde

Mathematical demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick are the *only* truths that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth according as their Subjects are more or less capable of Mathematical Demonstration.

From Christopher Wren’s inaugural lecture, quoted in Robin Wilson’s inaugural lecture, *Newsletter of the European Mathematical Society*, March 2009.

Philosophy is written in this grand book the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.

Galileo Galilei, *The Assayer*, 1622 (transl. Stillman Drake)

The men of experiment are like the ant, they only collect and use; the reasoners resemble spiders, who make cobwebs out of their own substance. But the bee takes the middle course: it gathers its material from the flowers of the garden and field, but transforms and digests it by a power of its own. Not unlike this is the true business of philosophy (science); for it neither relies solely or chiefly on the powers of the mind, nor does it take the matter which it gathers from natural history and mechanical experiments and lay up in the memory whole, as it finds it, but lays it up in the understanding altered and disgested. Therefore, from a closer and purer league between these two faculties, the experimental and the rational (such as has never been made), much may be hoped.

Francis Bacon, *Novum Organum: Aphorisms Concerning the Interpretation of Nature and the Kingdom of Man*, quoted by James Sneyd, lecture in Dunedin, 2003. (I have not verified the original quotation; this version is taken from Jone Johnson Lewis’ *Wisdom Quotes* webpage.)

Strong Reason and good fancy, joyn’d with experience and tryalls, so that we are assured of the good effects of it.

Nicholas Hawksmoor’s gloss on the “Rules of the Ancients” in architecture, quoted by Kerry Downes, *Hawksmoor*, Thames and Hudson, 1970.

The most captivating and imaginative painter to have lived since Giotto would certainly have been Paolo Uccello, if only he had spent as much time on human figures and animals as he spent, and wasted, on the finer points of perspective. Such details may be attractive and ingenious, but anyone who studies them excessively is squandering time and energy, choking his mind with difficult problems, and, often enough, turning a fertile and spontaneous talent into something sterile and laboured. Artists who devote more attention to perspective than to figures develop a dry and angular style and, moreover, they usually end up solitary, eccentric, melancholy, and poor, as indeed did Paolo Uccello himself. He was endowed by nature with a discriminating and subtle mind, but he found pleasure only in exploring certain difficult, or rather impossible, problems of perspective, which, although fanciful and attractivve, hindered him so much when he came to paint figures that the older he grew the worse he did them.

Giorgio Vasari, *The Lives of the Artists* (transl. George Bull), Penguin, 1965.

To start with invention is the mark of the fertile mind … and leads later to the interpretation of experience; to start with the reproduction of experience is the infallible index of a barren invention.

Dorothy L. Sayers, *Cat o’ Mary*, quoted by James Brabazon, *Dorothy L. Sayers: A Biography*, Gollancz, London, 1988.

… the laws of physics and of logic … the number system … the principle of algebraic substitution. These are ghosts. We just believe in them so thoroughly they seem real.

*Zen and the Art of Motorcycle Maintenance: An Inquiry into Values*, Bodley Head, London, 1974.

Birkhoff’s idea of putting numerical values on spatial and temporal patterns, tonal harmonies, etc., was considered decidedly offbeat in its day, though lately information theory and computer analysis lend a certain stylishness to this kind of approach.

Arthur Berger, Introduction to D. W. Prall, *Aesthetic Analysis*, Apollo, New York, 1967.

It was difficult enough being a mathematician, this being the frightening subject of which even educated people knew nothing, and of which they might proudly boast ignorance.

–oo–

The thesis was that “mind” or psychology could properly be described in terms of Turing machines because they both lay on the same level of description of the world, that of discrete logical systems.

Andrew Hodges, *Alan Turing: The Enigma*, Burnett 1983.

There is something breathtaking about the basic laws of crystals. They are in no sense a discovery of the human mind; they just “are” — they exist quite independently of us. The most that man can do is become aware, in a moment of clarity, that they are there, and take cognizance of them.

–oo–

By keenly confronting the enigmas that surround us, and by considering and analyzing the observations I had made, I ended up in the domain of mathematics.

–oo–

I came to the … open gate of mathematics. From here, well-trodden paths lead in every direction, and since then I have often spent time there. Sometimes I think … I have trodden all the paths … and then I suddenly discover a new path and experience fresh delights.

M. C. Escher

One man may have some special knowledge at first-hand about the character of a river or a spring, who otherwise knows only what everyone else knows. Yet to give currency to this shred of information, he will undertake to write on the whole science of physics. From this fault many great troubles spring.

Montaigne, *Essays*, I, 31.

Mathematics never reveals man to the degree, never expresses him in the way, that any other field of human endeavour does: the extent of the negation of man’s corporeal self that mathematics achieves cannot be compared with anything. Whoever is interested in this subject I refer to my articles. Here I will say only that the world injected its patterns into human language at the very inception of that language; mathematics sleeps in every utterance, and can only be discovered, never invented.

Stanislaw Lem, *His Master’s Voice* (1968).

You can see [the meaning of the statement that "Literature is a living art"] most easily and clearly, perhaps, by contrasting Science and Art at their two extremes — say Pure Mathematics and Acting. Science as a rule deals with things, Art with man’s thought and emotion about things. In Pure Mathematics things are rarefied into ideas, numbers, concepts, but still further and further away from the individual man. Two and two make four, and fourpence is not ninepence (or at any rate four is not nine) whether Alcibiades or Cleon keep the tally. In Acting on the other hand almost everything depends on personal interpretation — on the gesture, the walk, the gaze, the tone of a Siddons, the *rusé* smile of a Coquelin, the exquisite, vibrant intonation of a Bernhardt. ‘English Art?’ exclaimed Whistler, ‘there is no such thing! Art is art and mathematics is mathematics.’ Whistler erred. Precisely because Art is Art, and Mathematics is Mathematics and a Science, Art being Art can be English or French; and more than this, must be the personal expression of an Englishman or a Frenchman, as a ‘Constable’ differs from a ‘Corot’ and a ‘Whistler’ from both.

Sir Arthur Quiller-Couch, *On the Art of Writing*, 1915.

The arts, like arithmetic and geometry, turn away their eyes from the gross, coloured and mobile nature at our feet, and regard instead a certain figmentary abtraction. Geometry will tell us of a circle, a thing never seen in nature; asked about a green circle or an iron circle, it lays its hand upon its mouth. So with the arts. Painting, ruefully comparing sunshine and flake-white, gives up truth of colour, as it had already given up relief and movement; and instead of vying with nature, arranges a scheme of harmonious tints. Literature, above all in its most typical mood, the mood of narrative, similarly flees the direct challenge and pursues instead an independent and creative aim … A proposition of geometry does not compete with life; and a proposition of geometry is a fair and luminous parallel for a work of art. Both are reasonable, both untrue to the crude fact; both inhere in nature, neither represents it.

Robert Louis Stevenson, “A humble remonstrance”, in *Memories and Portraits* (1887); reprinted Richard Drew Publishing, Glasgow, 1990.

… mathematics is not just another language … it is a language plus logic. Mathematics is a tool for reasoning.

–oo–

Everybody who reasons carefully about anything is making a contribution … and if you abstract it away and send it to the Department of Mathematics they put it in books …

–oo–

There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time … On the other hand, I think I can safely say that nobody understands quantum mechanics.

Richard Feynman, *The Character of Physical Law*, BBC, 1965.

“In India, I learned a proverb that says, ‘Distrust the calculation seven times over, the mathematician a hundred times.'”

–oo–

“‘Without the help of mathematics,’ the wise man continued, ‘the art could not advance and all the sciences would perish.'”

Malba Tahan, *The Man who Counted* (tr. Leslie Clark and Alastair Reid), Canongate Press, Edinburgh, 1994.

All these people that I used to know, they’re an illusion to me now.

Some are mathematicians, some are carpenters’ wives.

Bob Dylan, “Tangled up in blue”

… the true mathematician and physicist know very well that the realms of the small and the great often obey quite different rules.

Kurt Singer, *Mirror, Sword and Jewel: The Geometry of Japanese Life*, Croom Helm, London, 1973.

Tout ce qu’on invente est vrai, soi-en sure. La poesie est une chose aussi precise que la geometrie.

Gustave Flaubert, letter to Louise Colet

The formalism and rigor of skaldic poetry is in no way inferior to that of modern mathematics, and anyone who is able to weave kjenninger while respecting the rules of alliteration will also be able to derive theorems from one another following the rules of logic. In both cases, creativity is a bonus; it adds meaning and beauty and distinguishes the artist from the laborer.

Ivar Ekeland, *The Broken Dice, and other Mathematical Tales of Chance*, University of Chicago Press, Chicago, 1993.

On sighting mathematicians [poetry] should unhook the algebra from their minds and replace it with poetry; on sighting poets it should unhook poetry from their minds and replace it with algebra.

Brian Patten, “Prosepoem towards a definition of itself”.

The sea squirt, after an active life, settles on the sea floor and, like a professor given tenure, absorbs its brain.

Steve Jones, *Almost like a Whale: the Origin of Species Updated*, Doubleday, 1999.

O Maximin,

You are the mountain and the valley.

Hildegard of Bingen, “Hymn to St Maximin”.

Mathematics may be the only exception in the sciences that leaves no room for skepicism. But, if mathematical results are exact as no empirical law can ever be, philosophers have discovered that they are not absolutely novel — instead, they are tautological.

–oo–

We chose to do this work mathematically, which has the advantage of precision but is not always appreciated by readers. It is perhaps for this reason that anthropologists have not shown much interest in these models, unlike economists, for example, for whom the use of mathematics poses no problem. However, one could reach the same conclusions by using just a bit of common sense.

Luigi Luca Cavalli-Sforza, *Genes, Peoples and Languages*, Allen Lane, London, 2000.

… preliminary accounting, banking and surveying (known as arithmetic, algebra and geometry) …

Alan Watts, *In My Own Way*, Random House, New York, 1972.

The mathematical sciences wield their particular language made of digits and signs, no less subtle than any other.

Jorge Luis Borges, “Verbiage for poems” (1926).

All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA).

Cryptography has generated number theory, algebraic geometry over finite fields, algebra, combinatorics and computers.

Hydrodynamics procreated complex analysis, partial differential equations, Lie groups and algebra theory, cohomology theory and scientific computing.

Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.

The existence of mysterious relations between all these different domains is the most striking and delightful feature of mathematics (having no rational explanation).

V. I. Arnold, Polymathematics: Is mathematics a single science or a set of arts? in *Mathematics: Frontiers and Perspectives* (ed. V. Arnold, M. Atiyah, P. Lax and B. Mazur), American Math. Soc., 1999, pp. 403–416.

–oo–

It is obvious that mathematics needs both sorts of mathematicians [theory-builders and problem-solvers] … It is equally obvious that different branches of mathematics require different aptitudes. In some, such as algebraic number theory, or the cluster of subjects now known simply as Geometry, it seems … to be important for many reasons to build up a considerable expertise and knowledge of the work that other mathematicians are doing, as progress is often the result of clever combinations of a wide range of existing results. Moreover, if one selects a problem, works on it in isolation for a few years and finally solves it, there is a danger, unless the problem is very famous, that it will no longer be regarded as all that significant.

At the other end of the spectrum is, for example, graph theory, where the basic object, a graph, can be immediately comprehended. One will not get anywhere in graph theory by sitting in an armchair and trying to understand graphs better. Neither is it particularly necessary to read much of the literature before tackling a problem: it is of course helpful to be aware of some of the most important techniques, but the interesting problems tend to be open precisely because the established techniques cannot easily be applied.

W. T. Gowers, The two cultures of mathematics, *ibid*. pp. 65–78.

–oo–

Of the properties of mathematics, as a language, the most peculiar one is that by playing formal games with an input mathematical text, one can get an output text which seemingly carries new knowledge. The basic examples are furnished by scientific or technological calculations: general laws plus initial conditions produce predictions, often only after time-consuming and computer-aided work. One can say that the input contains an implicit knowledge which is thereby made explicit. One could try to find a parallel in the humanities by comparing this to hermeneutics: the art of finding hidden meanings of sacred texts. Legal discourse, too, has some common traits with scientific discourse. In the course of history, the modern language of science slowly emerged from these two archaic activities, and it still owes a lot to them, especially in the more descriptive and less mathematicised domains.

Yu. I. Manin, Mathematics as profession and vocation, *ibid*. pp. 153–159.

–oo–

First, I want to quote a definition of what is mathematics due to Davis and Hersh … “The study of mental objects with reproducible properties is called mathematics.” I love this definition because it doesn’t try to limit mathematics to what has been called mathematics in the past but really attempts to say why certain communications are classified as math, others as science, others as art, others as gossip. Thus reproducible properties of the physical world are science whereas reproducible mental objects are math. Art lives on the mental plane (the real painting is not the set of dry pigments on the canvas nor is a symphony the sequence of sound waves that convey it to our ear) but, as the post-modernists insist, is reinterpreted in new contexts by each appreciator. As for gossip, which includes the vast majority of our thoughts, its essence is its relation to a unique local part of time and space.

David Mumford, The dawning of the age of stochasticity,

*ibid*. pp. 197–218.

The book *Dynamic Programming* by Richard Bellman is an important, pioneering work in which a group of problems is collected together at the end of some chapters under the heading “Exercises and Research Problems,” with extremely trivial questions appearing in the midst of deep, unsolved problems. It is rumored that someone once asked Dr. Bellman how to tell the exercises apart from the research problems, and he replied: “If you can solve it, it is an exercise; otherwise it’s a research problem.”

Donald E. Knuth, *The Art of Computer Programming*

I recall a lecture by John Glenn, the first American to go into orbit. When asked what went through his mind while he was crouched in the rocket nose-cone, awaiting blast-off, he replied, “I was thinking that the rocket has 20,000 components, and each was made by the lowest bidder.”

Martin Rees, in *The Times Higher Education Supplement*, 30 May 2003.

To exist (in mathematics), said Henri Poincaré, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.

Mark Kac and Stanislaw M. Ulam, *Mathematics and Logic* (1968)

Among mathematicians, even in those days, the reputation of being a good Glass Bead Game player meant a great deal; it was equivalent to being a very good mathematician.

Hermann Hesse, *The Glass Bead Game* (1943)

Tlön’s geometry is made up of two rather distinct disciplines — visual geometry and tactile geometry. Tactile geometry corresponds to our own, and is subordinate to the visual. Visual geometry is based on the surface, not the point; it has no parallel lines, and it claims that as one’s body moves through space, it modifies the shapes that surround it. The basis of Tlön’s arithmetic is the notion of indefinite numbers; it stresses the importance of the concepts “greater than” and “less than”, which our own mathematicians represent with the symbols > and <. The people of Tlön are taught that the act of counting modifies the amount counted, turning indefinites into definites. The fact that several persons counting the same quantity come to the same result is for the psychologists of Tlön an example of the association of ideas or of memorization.

Jorge Luis Borges, “Tlön, Uqbar, Orbis Tertius”, transl. Andrew Hurley, Penguin, 1999.

To criticize mathematics for its abstraction is to miss the point entirely. *Abstraction is what makes mathematics work.* If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer.

Ian Stewart, *Does God play dice? The mathematics of chaos*, Penguin, London, 1990.

They presume to alter the holy scriptures, to abandon the ancient rule of faith, and to form their opinions according to the subtile precepts of logic. The science of the church is neglected for the study of geometry, and they lose sight of Heaven while they are employed in measuring the earth. Euclid is perpetually in their hands. Aristotle and Theophrastus are the objects of their admiration; and they express an uncommon reverence for the works of Galen. Their errors are derived from the abuse of the arts and sciences of the infidels, and they corrupt the simplicity of the gospel by the refinements of human reason.

Eusebius on the followers of Artemon, quoted by Edward Gibbon, *The History of the Decline and Fall of the Roman Empire* (1776).

… Next

Lie bills and calculations much perplexed,

With steam-boats, frigates, and machinery quaint

Traced over them in blue and yellow paint.

Then comes a range of mathematical

Instruments, for plans nautical and statical;

A heap of rosin, a queer broken glass

With ink in it;—a china cup that was

What it will never be again, I think,—

A thing from which sweet lips were wont to drink

The liquor doctors rail at—and which I

Will quaff in spite of them—and when we die

We’ll toss up who died first of drinking tea

and cry out,—’Heads or tails?’ where’er we be.

Near that a dusty paint-box, some odd hooks,

A half-burnt match, an ivory block, three books,

Where conic sections, spherics, logarithms,

To great Laplace, from Saunderson and Sims,

Lie heaped in their harmonious disarray

Of figures,—disentangle them who may.

Percy Bysshe Shelley, *Letter to Maria Gisborne* (1820).

The question [of whether boots or shoes are better for walking] may be pursued through all its ramfications; and no doubt those who like quantitative thinking could ultimately produce some sort of determination … Where comfort and utility only are concerned, the vulgar processes of comparing, adding and subtracting are quite sufficient to lead to a conclusion.

But quantitative reasoning, though invaluable in politics, is very poor fun. Life would have little flavour without occasional qualitative excursions into the *a priori*.

A. H. Sidgwick, *Walking Essays* (1912), quoted in Ducan Minshull (ed.), *Over the Hills and Far Away*, Vintage, London, 2000.

“You know, Mouse,” [Tabby] said, “a brilliant cat like me should have smart friends; people who can count to more than four.”

“I can count to more than four,” answered Mouse, very offended. “And I can do hard sums, and I know geography and history, and I can knit and …”

Ruth Park, *The Muddle-Headed Wombat at School*, 1962.

… let us recall the well-known statement of a university professor in the Republic of the Massagetes: ‘Not the faculty but His Excellency the General can properly determine the sum of two and two.’

Hermann Hesse, *The Glass Bead Game* (transl. Richard and Clara Winston), Penguin, London, 1972.

It is only the last and wildest kind of courage that can stand on a tower before ten thousand people and tell them that twice two is four.

G. K. Chesterton, *Heretics*

Those arts which are, to be sure, not finite, as geometry and arithmetic, do not suffer adornment; others, contrarily, are rather subject to division and embellishment, such as astronomy and jurisprudence.

Girolamo Cardano, *The book of my life* (transl. Jean Stoner), New York Review Books, New York, 2002.

If anyone is racist or sexist, it is those who claim that women and minorities are unable to deal with traditional mathematics.

Barry Simon, quoted in David Klein, “A quarter century of US ‘math wars’ and political partisanship”, *BSHM Bulletin* **22** (2007).

From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician.

James Jeans, *The Mysterious Universe* (1930).

I used to love mathematics for its own sake, and I still do, because it allows for no hypocrisy and no vagueness, my two *bêtes noires*.

Stendhal, *La Vie d’Henri Brulard* (1890).

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Bertrand Russell, *Mysticism and Logic* (1918).

–oo–

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture.

Bertrand Russell, *Philosophical Essays* (1910).

Outside observers often assume that the more complicated a piece of mathematics is, the more mathematicians admire it. Nothing could be further from the truth. Mathematicians admire elegance and simplicity above all else, and the ultimate goal in solving a problem is to find the method that does the job in the most efficient manner. Though the major accolades are given to the individual who solves a particular problem first, credit (and gratitude) always goes to those who subsequently find a simpler solution.

Keith Devlin, *The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern*, Basic Books, New York, 2008.

No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game … I do not know an instance of a major mathematical advance initiated by a man past fifty.

–oo–

Mathematics is not a contemplative but a creative subject; no one can draw much consolation from it when he has lost the power or the desire to create; and that is apt to happen to a mathematician rather soon. It is a pity, but in that case he does not matter a great deal anyhow, and it would be silly to bother about him.

G. H. Hardy, *A Mathematician’s Apology*, Cambridge, 1940.

In the applied mathematical textbooks current during my undergraduate days, the formal arguments were mostly tolerably sound (though often very irrelevant to current scientific progress). But their scientific philosophy was very often quite unsound and superficial, as witness the statement in one of them: “It is inconceivable that the original laws on which Mechanics is based could be erroneous.”

K. E. Bullen, Applied Mathematics in the Twentieth Century, in J. C. Butcher (ed.), *A Spectrum of Mathematics: Essays Presented to H. G. Forder*, Auckland and Oxford, 1971.

Men sholde nat knowe of Goddes pryvetee

Ye, blessed be alwey, a lewed man

That noght but oonly his believe kan!

So ferde another clerk with astromye,

He walked in the feelds, for to prye

Upon the sterres, what ther sholde bifalle,

Til he was in a marle-pit yfalle.

Geoffrey Chaucer, “The Miller’s Tale”, *The Canterbury Tales*.

Did chemistry theorems exist? No: therefore you had to go further, not be satisfied with the *quia*, go back to the origins, to mathematics and physics. The origins of chemistry were ignoble, or at least equivocal: the dens of the alchemists, their abominable hodgepodge of ideas and language, their confessed interest in gold, their Levantine swindles typical of charlatans and magicians; instead, at the origin of physics lay the strenuous clarity of the West—Archimedes and Euclid.

Primo Levi, *The Periodic Table*, Michael Joseph 1985.

There is no point in saying that one should not doubt or that one should believe. Just to say “I believe” does not mean that you understand and see. When a student works on a mathematical problem, he comes to a stage beyond which he does not know how to proceed, and where he is in doubt or perplexity. As long as he has this doubt, he cannot proceed. If he wants to proceed, he must resolve this doubt. And there are ways of resolving that doubt. Just to say “I believe”, or “I do not doubt”, will certainly not solve the problem. To force oneself to believe and to accept a thing without understanding is political, and not spiritual or intellectual.

Walpola Rahula, *What the Buddha taught*.

We who often glorify our tendency to ignore reason, installing in its place blind faith, valuing it as spiritual, are forever paying for its cost with the obscuration of our mind and destiny.

Rabindranath Tagore

Many teachers will say that ‘you cannot express the inexpressible’, and they do not try. But teachers like Yasutani and Maezumi don’t agree, and I feel as they do: *if you perceive deeply enough*, a clear and simple way to express it can be found.

Tetsugen (Bernard Glassman), quoted in Peter Matthiessen, *Nine-headed Dragon River*, Shambhala 1998.

As a Marxist mathematician, Kol’man insisted that human knowledge finds its origin in the material world, not in the minds of scientists. Marx and Engels had written that mathematics arose in the ancient world when humans found it necessary to quantify material things like olive oil and grain, and to measure land in primitive surveying operations. Thus, for Marxists, mathematics was a science of material relationships. According to Kol’man, while some areas of mathematics may have become very abstract in modern times, the discipline never lost its contact with the exterior world. He maintained that mathematics must be interpreted from the standpoint of philosophical materialism.

Opposed to this view, said Kol’man, was the “idealistic, religious” view that mathematics is merely created by human beings — that it is a product of their minds, without a necessary relationship to the material world. In a 1931 article Kol’man even used technical arguments about Luzin’s treatment of the continuum, saying that Luzin “eliminates all points with rational coordinates, which has even less to do with reality than absolute continuity”. Kol’man reproached Luzin for his “inability to understand the unity of continuous and discrete”. In his denunciation Kol’man accused Luzin of saying that numbers “exist as a function of the mind of the mathematician”. Here Kol’man was using the debate of twenty years earlier and Luzin’s anti-intuitionist inclinations to accuse the latter of absolute idealism — the belief that a person gives a thing existence by thinking about it.

Loren Graham and Jean-Michel Kantor, *Naming Infinity*, Belknap Press, Cambridge, MA, 2009.

As we know,

There are known knowns.

There are things we know we know:

We also know

There are known unknowns.

That is to say

We know there are some things

We do not know.

But there are also unknown unknowns,

The ones we don’t know we don’t know.

Donald H. Rumsfeld, quoted in Hart Seely, *Pieces of Intelligence: The Existential Poetry of Donald H. Rumsfeld*, Simon and Schuster, 2003.

In my own professional work I have touched on a variety of different fields. I’ve done work in mathematical linguistics, for example, without any professional credentials in mathematics; in this subject I am completely self-taught, and not very well taught. But I’ve often been invited by universities to speak on mathematical linguistics at mathematics seminars and colloquia. No one has ever asked me whether I have the appropriate credentials to speak on these subjects; the mathematicians couldn’t care less. What they want to know is what I have to say. No one has ever objected to my right to speak, asking whether I have a doctor’s degree in mathematics, or whether I have taken advanced courses in this subject. That would never have entered their minds. They want to know whether I am right or wrong, whether the subject is interesting or not, whether better approaches are possible — the discussion dealt with the subject, not with my right to discuss it.

But on the other hand, in discussion and debate concerning social issues or American foreign policy, Vietnam or the Middle East, for example, the issue is constantly raised, often with considerable venom. I’ve repeatedly been challenged on grounds of credentials, or asked, what special training do you have that entitles you to speak of these matters. The assumption is that people like me, who are outsiders from a professional viewpoint, are not entitled to speak on such things.

Compare mathematics and the political sciences — it’s quite striking. In mathematics, in physics, people are concerned with what you say, not with your certification. But in order to speak about social reality, you must have the proper credentials, particularly if you depart from the accepted framework of thinking. Generally speaking, it seems fair to say that the richer the intellectual substance of a field, the less there is a concern for credentials, and the greater is the concern for content.

Noam Chomsky, *Language and Responsibility*, Pantheon, New York, 1979; quoted by Alan Sokal and Jean Bricmont, *Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science*, Picador, New York, 1998.

One evening I was in a group of people interested in psychology and religion, where a friend of mine gave a talk on the means of perceiving the qualitative reality of life. She helped illumine the meaning of life for her. I thought her talk excellent, but no sooner had she stopped speaking than she was pecked at from all sides:

“How can you say that poetic experience is ‘real’? It cannot even be demonstrated mathematically, and what is expressible mathematically is the only true reality.”

“How can you possibly say that you were ‘at one’ with Nature when you had no proof that Nature was ‘at one’ with you? It probably felt quite the contrary!”

I was upset because it appeared so obvious to me that what my friend was talking about was not measurable at all by quantitative criteria. She used words which usually express space and time and solid substance, like *inner* and *outer* and *higher* and *lower* and *real* and *unreal*, because there were no other words for expressing what she meant. But she used them symbolically and qualitatively, not with their usual quantitative and spatial connotations; and because of this, her meaning was not apparent to her listeners. They were really quite nasty to her, and when I tried to come to her defence, I was pecked at too. When I said that they had not understood the qualitative reality which my friend had been talking about, they quite justifiably pointed out that I, not understanding mathematics, was ignorant of what they knew was reality.

Thetis Blacker, *A Pilgrimage of Dreams*, Turnstone Books, London, 1973.

I am very glad there are quite a number of people born with a gift and a liking for all of this; like great chessplayers who play sixteen games at once blindfold and die quite soon of epilepsy. Serve them right! I hope the Mathematicians, however, are well rewarded. I promise never to blackleg their profession nor take the bread out of their mouths.

Winston Churchill, *My Early Life*

[This is part of Oliver Sacks' commentary on a patient who had, as the result of a stroke, lost perception on her left side.]

Knowing it intellectually, knowing it inferentially, she has worked out strategies to deal with her imperception. She cannot look left, directly, she cannot turn left, so what she does is to turn right – and right through a circle … if she cannot find something which she knows should be there, she swivels to the right, through a circle, until it comes into view. … If her portions seem too small, she will swivel to the right, keeping her eyes to the right, until the previously missed half now comes into view; she will eat this, or rather half of this, and feel rather less hungry than before. But if she is still hungry, or if she thinks on the matter, and realises that she may have only perceived half of the missing half, she will make a second rotation until the remaining quarter comes into view, and, in turn, bisect this yet again. This usually suffices – after all, she has how eaten seven-eighths of the portion – but she may, if she is feeling particularly hungry or obsessive, make a third turn, and secure another sixteenth of her portion (leaving, of course, the remaining sixteenth, the left sixteenth, on her plate). “It’s absurd,” she says. “I feel like Zeno’s arrow – I never get there …”

Oliver Sacks, *The Man who Mistook his Wife for a Hat*, Picador, 1985.

## Other sources

I keep other quotes, not necessarily mathematical, at the top and bottom of my front page, and also in various other places around the site, for example here. Another quote on the ethos of a university is at the end of this page.

Further sources of quotes include:

- Robin Whitty’s “A Scientist’s Comfort Zone” contains quotes from mathematicians about actually
*doing*research (rather like the Mathematicians at work section above, but more extensive). - Mathematician of the Day includes a quote by a mathematician who was born or died on this day.
- Fifty mathematics quotes can be found here.

You can of course find *many* more with Google …

Some recommended books include:

- J. E. Littlewood,
*Littlewood’s Miscellany*(ed. B. Bollobás), Cambridge University Press, Cambridge, 1986. - Steven G. Krantz,
*Mathematical Apocrypha*, MAA, Washington DC, 2002.

This page is referred to in Sasha Borovik’s book *Mathematics under the Microscope*, which I also recommend highly. (He uses the Myles Aston quote under Combinatorics.)

To doubt the straight line

you first have to know how many points

it has.

from “On the Life of Ptolemy” by Nichita Stanescu (trans. Sean Cotter) in WHEEL WITH A SINGLE SPOKE (Archipelago Books, 2012).

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