Probability and Statistics

For [Luck] your science finds no measuring-rods; . . .
Her permutations never know truce nor pause

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

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).

. . . 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.

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)

 

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 Sigma try to determine its group of automorphisms . . . You can expect to gain a deep insight into the constitution of Sigma 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.

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).

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)

 

Combinatorics

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 he 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.

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.

[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)

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.

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. MacLane, Mathematics: Form and Function, Springer, New York, 1986.

"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.

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.

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.

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

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

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

Numbers

"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.

. . . 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.

"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.

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 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.

 

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

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.

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.

 

 莱布尼茨(Gottfriend Wilhelm Leibniz)是17、18世纪之交德国最重要的数学家、物理学家和哲学家，一个举世罕见的科学天才，和牛顿同为微积分的创建人。他博览群书，涉猎百科，对丰富人类的科学知识宝库做出了不可磨灭的贡献。

生平事迹

    1646年7月1日，莱布尼茨出生于德国东部莱比锡的一个书香之家，父亲弗里德希·莱布尼茨是莱比锡大学的道德哲学教授，母亲凯瑟琳娜·施马克出身于教授家庭，虔信路德新教。

    莱布尼茨的父母亲自做孩子的启蒙教师，耳濡目染使莱布尼茨从小就十分好学，并有很高的天赋，幼年时就对诗歌和历史有着浓厚的兴趣。不幸的是，父亲在他6岁时去世，却给他留下了丰富藏书。

    莱布尼茨的父亲在他年仅六岁时便去世了，给他留下了比金钱更宝贵的丰富的藏书，知书达理的母亲担负起了儿子的幼年教育。莱布尼茨因此得以广泛接触古希腊罗马文化，阅读了许多著名学者的著作，由此而获得了坚实的文化功底和明确的学术目标。

    8岁时，莱布尼茨进入尼古拉学校，学习拉丁文、希腊文、修词学、算术、逻辑、音乐以及《圣经》、路德教义等。

    1661年，15岁的莱布尼茨进入莱比锡大学学习法律，一进校便跟上了大学二年级标准的人文学科的课程，他还抓紧时间学习哲学和科学。1663年5月，他以《论个体原则方面的形而上学争论》一文获学士学位。这期间莱布尼茨还广泛阅读了培根、开普勒、伽利略等人的著作，并对他们的著述进行深入的思考和评价。在听了教授讲授的欧几里得的《几何原本》的课程后，莱布尼茨对数学产生了浓厚的兴趣。

    1664年1月，莱布尼茨完成了论文《论法学之艰难》，获哲学硕士学位。是年2月12日，他母亲不幸去世。18岁的莱布尼茨从此只身一人生活，他—生在思想、性格等方面受母亲影响颇深。

    1665年，莱布尼茨向莱比锡大学提交了博士论文《论身份》，1666年，审查委员会以他太年轻(年仅20岁)而拒绝授予他法学博士学位，黑格尔认为，这可能是由于莱布尼茨哲学见解太多，审查论文的教授们看到他大力研究哲学，心里很不乐意。他对此很气愤，于是毅然离开莱比锡，前往纽伦堡附近的阿尔特多夫大学，并立即向学校提交了早已准备好的那篇博士论文，1667年2月，阿尔特多夫大学授予他法学博士学位，还聘请他为法学教授。

    这一年，莱布尼茨发表了他的第一篇数学论文《论组合的艺术》。这是一篇关于数理逻辑的文章，其基本思想是想把理论的真理性论证归结于一种计算的结果。这篇论文虽不够成熟，但却闪耀着创新的智慧和数学的才华，后来的一系列工作使他成为数理逻辑的创始人。

    1666年莱布尼茨获得法学博士学位后，在纽伦堡加入了一个炼金术士团体，1667年，通过该团体结识了政界人物博因堡男爵约翰·克里斯蒂文，并经男爵推荐给选帝迈因茨，从此莱布尼茨登上了政治舞台，便投身外交界，在美因茨大主教舍恩博恩的手下工作。

    167l～1672年冬季，他受迈因茨选帝侯之托，着手准备制止法国进攻德国的计划。1672年，莱布尼茨作为一名外交官出使巴黎，试图游说法国国王路易十四放弃进攻，却始终未能与法王见上一面，更谈不上完成选帝侯交给他的任务了。这次外交活动以失败而告终，然而在这期间，他深受惠更斯的启发，决心钻研高等数学，并研究了笛卡儿、费尔马、帕斯卡等人的著作，开始创造性的工作。

    1673年1月，为了促使英国与荷兰之间的和解，他前往伦敦进行斡旋未果。他却趁这个机会与英国学术界知名学者建立了联系。他见到了与之通信达三年的英国皇家学会秘书、数学家奥登伯以及物理学家胡克、化学家波义耳等人。1673年3月莱布尼茨回到巴黎，4月即被推荐为英国皇家学会会员。这一时期，他的兴趣越来越明显地表现在数学和自然科学方面。

    1672年10月，迈因茨选帝侯去世，莱布尼茨失去了职位和薪金，而仅是一位家庭教师了。当时，他曾多方谋求外交官的正式职位，或者希望在法国科学院谋一职位，都没有成功。无奈，只好接受汉诺威公爵约翰·弗里德里希的邀请，前往汉诺威。

    1676年10月4日，莱布尼茨离开巴黎，他先在伦敦作了短暂停留。继而前往荷兰，见到了使用显微镜第一次观察了细菌、原生动物和精子的生物学家列文虎克，这些对莱布尼茨以后的哲学思想产生了影响。在海牙，他见到了斯宾诺莎。1677年1月，莱布尼茨抵达汉诺威，担任布伦兹维克公爵府法律顾问兼图书馆馆长，并负责国际通信和充当技术顾问。汉语威成了他的永久居住地。

    在繁忙的公务之余，莱布尼茨广泛地研究哲学和各种科学、技术问题，从事多方面的学术文化和社会政治活动。不久，他就成了宫廷议员，在社会上开始声名显赫，生活也由此而富裕。1682年，莱布尼茨与门克创办了近代科学史上卓有影响的拉丁文科学杂志《学术纪事》(又称《教师学报》)，他的数学、哲学文章大都刊登在该杂志上；这时，他的哲学思想也逐渐走向成熟。

    1679年12月，布伦兹维克公爵约翰·弗里德里却突然去世，其弟奥古斯特继任爵位，莱布尼茨仍保留原职。新公爵夫人苏菲是他的哲学学说的崇拜者，“世界上没有两片完全相同的树叶”这一句名言，就出自他与苏菲的谈话。

    奥古斯特为了实现他在整个德国出人头地的野心，建议莱布尼茨广泛地进行历史研究与调查，写一部有关他们家庭近代历史的著作。1686年他开始了这项工作。在研究了当地有价值的档案材料后，他请求在欧洲作一次广泛的游历。

    1687年11月，莱布尼茨离开汉诺威，于1688年初夏5月抵达维也纳。他除了查找档案外，大量时间用于结识学者和各界名流。在维也纳，他拜见了奥地利皇帝利奥波德一世，为皇帝构画出一系列经济、科学规划，给皇帝留下了深刻印象。他试图在奥地利宫庭中谋一职位，但直到1713年才得到肯定答复，而他请求古奥地利建立一个“世界图书馆”的计划则始终未能实现。随后，他前往威尼斯，然后抵达罗马。在罗马，他被选为罗马科学与数学科学院院士。1690年，莱布尼茨回到了汉诺威。由于撰写布伦兹维克家族历史的功绩，他获得了枢密顾问官职务。

    在1700年世纪转变时期，莱布尼茨热心地从事于科学院的筹划、建设事务。他觉得学者们各自独立地从事研究既浪费了时间又收效不大，因此竭力提倡集中人才研究学术、文化和工程技术，从而更好地安排社会生产，指导国家建设。

    从1695年起，莱布尼茨就一直为在柏林建立科学院四处奔波，到处游说。1698年，他为此亲自前往柏林。1700年，当他第二次访问柏林时，终于得到了弗里德里希一世，特别是其妻子(汉诺威奥古斯特公爵之女)的赞助，建立了柏林科学院，他出任首任院长。1700年2月，他还被选为法国科学院院士。至此，当时全世界的四大科学院：英国皇家学会、法国科学院、罗马科学与数学科学院、柏林科学院都以莱布尼次作为核心成员。

    1713年初，维也纳皇帝授予莱布尼茨帝国顾问的职位，邀请他指导建立科学院。俄国的彼得大帝也在17ll～1716年去欧洲旅行访问时，几次听取了莱布尼茨的建议。莱布尼茨试图使这位雄才大略的皇帝相信，在彼得堡建立一个科学院是很有价值的。彼得大帝对此很感兴趣，1712年他给了莱布尼茨一个有薪水的数学、科学宫廷顾问的职务。1712年左右，他同时被维出纳、布伦兹维克、柏林、彼得堡等王室所雇用。这一时期他一有机会就积极地鼓吹他编写百科全书，建立科学院以及利用技术改造社会的计划。在他去世以后，维也纳科学院、彼得堡科学院先后都建立起来了。据传，他还曾经通过传教士，建议中国清朝的康熙皇帝在北京建立科学院。

    就在莱布尼茨倍受各个宫廷青睐之时，他却已开始走向悲惨的晚年了。1716年11月14日，由于胆结石引起的腹绞痛卧床一周后，莱布尼茨孤寂地离开了人世，终年70岁。

    莱布尼茨一生没有结婚，没有在大学当教授。他平时从不进教堂，因此他有一个绰号 Lovenix，即什么也不信的人。他去世时教士以此为借口，不予理睬，曾雇用过他的宫廷也不过问，无人前来吊唁。弥留之际，陪伴他的只有他所信任的大夫和他的秘书艾克哈特。艾克哈特发出讣告后，法国科学院秘书封登纳尔在科学院例会时向莱布尼茨这位外国会员致了悼词。1793年，汉诺威人为他建立了纪念碑；1883年，在莱比锡的一座教堂附近竖起了他的一座立式雕像；1983年，汉诺威市政府照原样重修了被毁于第二次世界大战中的“莱布尼茨故居”，供人们瞻仰。

　

始创微积分

    17世纪下半叶，欧洲科学技术迅猛发展，由于生产力的提高和社会各方面的迫切需要，经各国科学家的努力与历史的积累，建立在函数与极限概念基础上的微积分理论应运而生了。

    微积分思想，最早可以追溯到希腊由阿基米德等人提出的计算面积和体积的方法。1665年牛顿创始了微积分，莱布尼茨在1673～1676年间也发表了微积分思想的论著。

    以前，微分和积分作为两种数学运算、两类数学问题，是分别的加以研究的。卡瓦列里、巴罗、沃利斯等人得到了一系列求面积(积分)、求切线斜率(导数)的重要结果，但这些结果都是孤立的，不连贯的。

    只有莱布尼茨和牛顿将积分和微分真正沟通起来，明确地找到了两者内在的直接联系：微分和积分是互逆的两种运算。而这是微积分建立的关键所在。只有确立了这一基本关系，才能在此基础上构建系统的微积分学。并从对各种函数的微分和求积公式中，总结出共同的算法程序，使微积分方法普遍化，发展成用符号表示的微积分运算法则。因此，微积分“是牛顿和莱布尼茨大体上完成的，但不是由他们发明的”。

    然而关于微积分创立的优先权，在数学史上曾掀起了一场激烈的争论。实际上，牛顿在微积分方面的研究虽早于莱布尼茨，但莱布尼茨成果的发表则早于牛顿。

    莱布尼茨1684年10月在《教师学报》上发表的论文《一种求极大极小的奇妙类型的计算》，是最早的微积分文献。这篇仅有六页的论文，内容并不丰富，说理也颇含糊，但却有着划时代的意义。

    牛顿在三年后，即1687年出版的《自然哲学的数学原理》的第一版和第二版也写道：“十年前在我和最杰出的几何学家莱布尼茨的通信中，我表明我已经知道确定极大值和极小值的方法、作切线的方法以及类似的方法，但我在交换的信件中隐瞒了这方法，……这位最卓越的科学家在回信中写道，他也发现了一种同样的方法。他并诉述了他的方法，它与我的方法几乎没有什么不同，除了他的措词和符号而外”(但在第三版及以后再版时，这段话被删掉了)。

    因此，后来人们公认牛顿和莱布尼茨是各自独立地创建微积分的。

    牛顿从物理学出发，运用集合方法研究微积分，其应用上更多地结合了运动学，造诣高于莱布尼茨。莱布尼茨则从几何问题出发，运用分析学方法引进微积分概念、得出运算法则，其数学的严密性与系统性是牛顿所不及的。

    莱布尼茨认识到好的数学符号能节省思维劳动，运用符号的技巧是数学成功的关键之一。因此，他所创设的微积分符号远远优于牛顿的符号，这对微积分的发展有极大影响。1713年，莱布尼茨发表了《微积分的历史和起源》一文，总结了自己创立微积分学的思路，说明了自己成就的独立性。

　

高等数学上的众多成就

    莱布尼茨在数学方面的成就是巨大的，他的研究及成果渗透到高等数学的许多领域。他的一系列重要数学理论的提出，为后来的数学理论奠定了基础。

    莱布尼茨曾讨论过负数和复数的性质，得出复数的对数并不存在，共扼复数的和是实数的结论。在后来的研究中，莱布尼茨证明了自己结论是正确的。他还对线性方程组进行研究，对消元法从理论上进行了探讨，并首先引入了行列式的概念，提出行列式的某些理论，此外，莱布尼茨还创立了符号逻辑学的基本概念。

    1673年莱布尼茨特地到巴黎去制造了一个能进行加、减、乘、除及开方运算的计算机。这是继帕斯卡加法机后，计算工具的又一进步。他还系统地阐述了二进制计数法，并把它和中国的八卦联系起来，为计算机的现代发展奠定了坚实的基础。

　

丰硕的物理学成果

    莱布尼茨的物理学成就也是非凡的。1671年，莱布尼茨发表了《物理学新假说》一文，提出了具体运动原理和抽象运动原理，认为运动着的物体，不论多么渺小，它将带着处于完全静止状态的物体的部分一起运动。他还对笛卡儿提出的动量守恒原理进行了认真的探讨，提出了能量守恒原理的雏型，并在《教师学报》上发表了《关于笛卡儿和其他人在自然定律方面的显著错误的简短证明》，提出了运动的量的问题，证明了动量不能作为运动的度量单位，并引入动能概念，第一次认为动能守恒是一个普通的物理原理。

    他又充分地证明了“永动机是不可能”的观点。他也反对牛顿的绝对时空观，认为“没有物质也就没有空见，空间本身不是绝对的实在性”，“空间和物质的区别就象时间和运动的区别一样，可是这些东西虽有区别，却是不可分离的”。这一思想后来引起了马赫、爱因斯坦等人的关注。

    1684年，莱布尼茨在《固体受力的新分析证明》一文中指出，纤维可以延伸，其张力与伸长成正比，因此他提出将胡克定律应用于单根纤维。这一假说后来在材料力学中被称为马里奥特——莱布尼茨理论。

    在光学方面，莱布尼茨也有所建树，他利用微积分中的求极值方法，推导出了折射定律，并尝试用求极值的方法解释光学基本定律。可以说莱布尼茨的物理学研究一直是朝着为物理学建立一个类似欧氏几何公理系统的目标前进的。

多才多艺的莱布尼茨

    莱布尼茨中奋斗的主要目标是寻求一种可以获得知识和创造发明的普遍方法，这种努力导致许多数学的发现。莱布尼茨的多才多艺在历史上很少有人能和他相比，他的研究领域及其成果遍及数学、物理学、力学、逻辑学、生物学、化学、地理学、解剖学、动物学、植物学、气体学、航海学、地质学、语言学、法学、哲学、历史和外交等等。

    l693年，莱布尼茨发表了一篇关于地球起源的文章，后来扩充为《原始地球》一书，提出了地球中火成岩、沉积岩的形成原因。对于地层中的生物化石，他认为这些化石反映了生物物种的不断发展，这种现象的终极原因是自然界的变化，而非偶然的神迹。他的地球成因学说，尤其是他的宇宙进化和地球演化的思想，启发了拉马克、赖尔等人，在一定程度上促进了19世纪地质学理论的进展。

    1677年，他写成《磷发现史》，对磷元素的性质和提取作了论述。他还提出了分离化学制品和使水脱盐的技术。

    在生物学方面，莱布尼茨在1714年发表的《单子论》等著作中，从哲学角度提出了有机论方面的种种观点。他认为存在着介乎于动物、植物之间的生物，水螅虫的发现证明了他的观点。

    在气象学方面。他曾亲自组织人力进行过大气压和天气状况的观察。

    在形式逻辑方面，他区分和研究了理性的真理(必然性命题)、事实的真理(偶然性命题)，并在逻辑学中引入了“充足理由律”，后来被人们认为是一条基本思维定律。

    1696年，莱布尼茨提出了心理学方面的身心平行论，他强调统觉作用，与笛卡儿的交互作用论、斯宾诺莎的一元论构成了当时心理学三大理论。他还提出了“下意识”理论的初步思想。

    1691年，莱布尼茨致信巴本，提出了蒸汽机的基本思想。

    l700年前后，他提出了无液气压原理，完全省掉了液柱，这在气压机发展史上起了重要作用。

    法学是莱布尼茨获得过学位的学科，1667年曾发表了《法学教学新法》，他在法学方面有一系列深刻的思想。

    1677年，莱布尼茨发表《通向一种普通文字》，以后他长时期致力于普遍文字思想的研究，对逻辑学、语言学做出了一定贡献。今天，人们公认他是世界语的先驱。

    作为著名的哲学家，他的哲学主要是“单子论”、“前定和谐”论及自然哲学理论。其学说与其弟子沃尔夫的理论相结合，形成了莱布尼茨—沃尔夫体系，极大地影响了德国哲学的发展，尤其是影响了康德的哲学思想。他开创的德国自然哲学经过沃尔夫、康德、歌德到黑格尔得到了长足的发展。

    在莱布尼茨从事学术研究的生涯中，他发表了大量的学术论文，还有不少文稿生前未发表。在数学方面，格哈特编辑的七卷本《数学全书》是莱布尼茨数学研究较完整的代表性著作。格哈特还编辑过七卷本的《哲学全书》。已出版的各种各样的选集、著作集、书信集多达几十种，从中可以看到莱布尼茨的主要学术成就。今天，还有专门的莱布尼茨研究学术刊物“Leibniz”，可见其在科学史、文化史上的重要地位。

　

中西文化交流之倡导者

    莱布尼茨对中国的科学、文化和哲学思想十分关注，他是最早研究中国文化和中国哲学的德国人。他向耶酥会来华传教士格里马尔迪了解到了许多有关中国的情况，包括养蚕纺织、造纸印染、冶金矿产、天文地理、数学文字等等，并将这些资料编辑成册出版。他认为中西相互之间应建立一种交流认识的新型关系。

    在《中国近况》一书的绪论中，莱布尼茨写道：“全人类最伟大的文化和最发达的文明仿佛今天汇集在我们大陆的两端，即汇集在欧洲和位于地球另一端的东方的欧洲——中国。”“中国这一文明古国与欧洲相比，面积相当，但人口数量则已超过”。“在日常生活以及经验地应付自然的技能方面，我们是不分伯仲的。我们双方各自都具备通过相互交流使对方受益的技能。在思考的缜密和理性的思辩方面，显然我们要略胜一筹”，但“在时间哲学，即在生活与人类实际方面的伦理以及治国学说方面，我们实在是相形见拙了”。

    在这里，莱布尼茨不仅显示出了不带“欧洲中心论”色彩的虚心好学精神，而且为中西文化双向交流描绘了宏伟的蓝图，极力推动这种交流向纵深发展，是东西方人民相互学习，取长补短，共同繁荣进步。

    莱布尼茨为促进中西文化交流做出了毕生的努力，产生了广泛而深远的影响。他的虚心好学、对中国文化平等相待，不含“欧洲中心论”偏见的精神尤为难能可贵，值得后世永远敬仰、效仿。