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From Frank Wikstrom's Mathematical Quotes:

A man whose mind has gone astray should study mathematics.
-- Francis Bacon
Medicine makes people ill, mathematics make them sad and theology makes them sinful.
-- Martin Luther
In mathematics you don't understand things. You just get used to them.
-- Johann von Neumann

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.
-- Johann Wolfgang von Goethe (Maxims and Reflexions, 1829)

Mathematician are a species of Frenchmen: if you say something to them they translate it into their own language and presto! it is something entirely different -- Goethe [Source of quote unknown, quoted in Chapter 3 of Pi in the Sky by John D. Barrow]

From Raven's Memorable Quotes from Alt.Sysadmin.Recovery:

Two of my imaginary friends reproduced once ... with negative results. -- Ben (float)

... More quotes from alt.sysadmin.recovery in Microsoft Forlorn and Computing Quotes

Quotes from the Mathematical Quotation Server.

The cowboys have a way of trussing up a steer or a pugnacious bronco which fixes the brute so that it can neither move nor think. This is the hog-tie, and it is what Euclid did to geometry.
-- Eric Bell, in R Crayshaw-Williams The Search For Truth, p. 191.
The good Christian should beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
-- Saint Augustine, DeGenesi ad Litteram, Book II, xviii, 37 [Note: mathematician = astrologer].
Die ganze Zahl shuf der liebe Gott, alles brige ist Menschenwerk.
(God made the integers, all else is the work of man.)
-- Leopold Kornecker, Jahresberichte der Deutschen Mathematiker Vereinigung
When we ask advice, we are usually looking for an accomplice.
-- Joseph-Louis LaGrange
In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role. In their unhappiness over the events of this world, some immerse themselves in a kind of self-sufficiency in mathematics. (Some have engaged in it for this reason alone.)
-- Stansilaw Ulam, Adventures of a Mathematician
Life is good for only two things, discovering mathematics and teaching mathematics
-- Siméon Poisson
Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
-- Paul R. Halmos, in I Want to be a Mathematician, Washington: MAA Spectrum, 1985.
"Necessity is the mother of invention" is a silly proverb. "Necessity is the mother of futile dodges" is much nearer the truth.
-- Alfred North Whitehead
In the company of friends, writers can discuss their books, economists the state of the economy, lawyers their latest cases, and businessmen their latest acquisitions, but mathematicians cannot discuss their mathematics at all. And the more profound their work, the less understandable it is.
-- Alfred Adler

Q: I'd like some kind of gift suggestion for the "guy who has it all". Any ideas?
A: Give him the set of guys who don't have it all.
-- Exchange on Forum 3000

Tortise: You don't need to give an infinite number of monkeys an infinite amount of time to write Hamlet. A finite number of monkeys and a finite amount of time will do just fine. And like my father used to say, never use an infinite number of monkeys when a finite number will do.
-- Mike Schiraldi, in a sci.math posting

"I think Whitehead and Russell probably win the prize for the most notation-intensive non-machine-generated piece of work that's ever been done. "
-- Stephen Wolfram, www.stephenwolfram.com/publications/talks/mathml/mathml2.html

From Robert Kaplan, The Nothing That Is - A Natural History of Zero:

A seven-year-old of my acquaintance claimed that the last number of all was 23,000. "What about 23,000 and one?" she was asked. After a pause: "Well, I was close."
The syadvada, his English translator explains, is an argument that the world of appearances may or may not be real, or both may and may not be real - or may be indescribable; or may be real and indescribable, or unreal and indescribable; or in the end may be read and unreal and indescribable.
And [Adelard of Bath] brought back with him precious manuscripts, the real treasures of the East: a treatise on alchemy thinly disguised as a text of mixing pigments (though it also contained a recipe for makeing toffee), [...]
[...] a Frenchman, writing in the fifteenth century, expressed the popular view as well: 'Just as the rag doll wanted to be an eagle, the donkey a lion and the monkey a queen, the zero put on airs and pretended to be a digit.'
[...] four people are in a room and seven people leave it. How many must go in before the room is empty? Answer: three.
[Bhaskara] wrote [...] a book he called Lilavati, 'Charming Girl' - perhaps because it was full of problems such as this: "Beautiful and dear delightful girl, whose eyes are like a faun's! If you are skilled in multiplication, tell me, what is 135 times 12?" They don't write math books like that any more.
Don't we need to find even more fundamental truths to derive these last two [distributive and associative] 'laws' from? And if we do, won't they require antecedents, and so down this eternal spiral to where their fires are not quenched? For a truth even deeper that the one which sprang form Robespierre and the revolutionary mod is that the kind of certainty demanded by deductive thought is unattainable because of the nature of deductive thought. To stop the infinite regress we have to say at some point: 'We hold these truths to be self-evident.'
Like the lovers in a story of Isak Dinesen's, [the two rival schools of calculus of Newtown & Leibniz] were two locked caskets, each holding the key to the other.
Unlike the two women shouting from their Edinburgh windows at one another, who would never agree (says Sydney Smith) because they were arguing from different premises, Newton and Leibniz would never agree because their premises were the same [...]
The Sultan Abdul Hamid the Second, perpetrator of the terrible nineteenth-century Armenian massacres, had his censors, they say, remove any reference to H₂O from chemistry books entering his empire, convinced that the symbol stood for 'Hamid the Second is Nothing!'

Quotes from The Penguin Book of Curious and Interesting Mathematics by David Wells:

The Greek philosopher Aristippus was shipwrecked upon a strange shore, when he noticed a geometrical drawing in the sand. "Be of good hope," he said to his companions, "for I see the footprints of men." -- frontspiece to J. F. Scott, A History of Mathematics, Taylor & Francis, 1960
Some people believe that a theorem is proved when a logically correct proof is given; but some people believe a theorem is proved only when the student sees why it is inevitably true. The author tends to belong to this second school of thought. -- Richard Hamming, Coding and Information Theory, Prentice-Hall 1980, p. 155
Nicholas Saunderson (1682-1739) was blinded by smallpox in his twelfth year. Nevertheless, amazing to relate, he was appointed in 1711 to Newton's chair at Cambridge, becoming the fourth Lucasian Professor of Mathematics.
It is difficult to give an idea of the vast extent of moderm mathematics. The word "extent" is not the right one: I mean extent crowded with beautiful details - not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood and flower ... -- Arthur Cayley (1821 - 1895) Presidential Address to the British Association, September 1883 in The Collected Mathematic Papers of Arthur Cayley, CUP, 1896, vol XI, p. 449
When Tennyson wrote "The Vision of Sin", Babbage read it. After doing so, it is daid he wrote the following extraordinary letter to the poet:
"In your otherwise beautiful poem, there is a verse which reads:
'Every moment dies a man,
Every moment one is born.'
"It must be manifest that, were this true, the population of the world would be at a standstill: In truth the rate of birth is slightly in excess of that of death. I would suggest that in the next edition of your poem you have it read:
'Every moment dies a man,
Every moment 1 1/6 is born.'
"Strictly speaking this is not correct. The actual figure is a decimal so long that I cannot get it in the line, bit I believe 1 1/6 will be sufficiently accurate for poetry. I am, etc."
-- from Clifton Fadiman, Fantasia Mathematica, Simon & Schuster, New Yor, 1958, p.293
"I am a mathematician to this extent: I can follow triple integrals if they are done slowly on a large blackboard by a personal friend. -- J. W. McReynolds, 'George's Problem', in Scripta Mathematica, vol 15, 2, June 1949.
Hilbert [...] was perhaps the most absent-minded man who ever lived. He was a great friend of the physicist James Franck. One day when Hilbert was walking in the street he met James Franck and he said, "James, is your wife as mean as mine?" Well, Franck was taken aback by this statement and didn't know quite what to say, and he said, "Well, what has your wife done?" And Hilbert said, "It was only this morning that I discovered quite by accident that my wife does not give me an egg for breakfast. Heaven knows how long this has been going on." -- from The Physicist's Conception of Nature, reprinted in T. Ferris (ed.), The World Treasury of Physics, Astronomy and Mathematics, Little, Brown & Co., 1991, p. 604.
It is hard to communicate understanding because that is something you get by living with a problem for a long time. You study it, perhaps for years, you get the feel of it and it is in your bones. You can't convey that to anyone else. Having studied the problem for five years you may be able to present it in such a way that it would take somebody else less time to get to that point than it took you. But if they haven't struggled with the problem and seen all the pitfalls, then they haven't really understood it. -- Michael Atiyah, in 'An Interview with Michael Atiya', Mathematical Intelligencer, vol 6, no. 1, 1984, p.17.
I have no fault with those who teach grometry. That science is the only one which has not produced sects; it is founded on analysis, and on synthesis and on the calculus; it does not occupy itself with probable truths; moreover it has the same method in every country. -- Frederick the Great (1712-1786), Oeuvres, quoted in A. L. Mackay, The Harvest of a Quiet Eye - a Selection of Scientific Quotations, The Institute of Physics, Bristol, 1977, p. 59.
Mathematicians are like lovers [...] Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consquenece another. -- Fontelle (1657-1757), quoted in A. L. Mackay, The Harvest of a Quiet Eye - a Selection of Scientific Quotations, The Institute of Physics, Bristol, 1977, p. 58.
Mark all mathematical heads which be wholly and only bent on these sciences, how solitary they be themselves, how unfit to be with others, how unapt to serve the world. -- Roger Ascham (1515-1568) in E. G. R. Taylor, Mathematical Practitioners of Tudor and Stuart England, Cambridge University Press, 1954.
It is said that mathematicians are exempt from psychical derangements, but this is not true [...] Codazzi was sub-microcephalic, oxycephalic, alcoholic, sordidly avaricious; to effective insensibility he added vanity so great that while still young he set apart a sum for his own funeral monument, and refused the least help to his starving parents; he admitted no discussion of his judgment even if it only concerned the cut of a coat; and he had taken it into his head that he could compose melodic music with the help of the calculus.
All mathematicians admire the great geometer Bolyai, whose eccentricities were of an insane character; thus he provoked thirteen officials to duels and fought them, and between each duel he played the violin, the only piece of furniture in his house [...]
-- Cesare Lombroso, TheMan of Genius, Walter Scott, 1891, p. 73.
Sylvester was once approached by one of his research students who proposed to use a certain result in his research. Sylvester objected that the claimed theorem could not possibly be true, at which point the student tactfully explained to him that he, Professor Sylvester, had proved it himself, many years previously. -- A. R. Luria, The Mind of a Mnemonist, Johnathon Cape, 1969, p. 131-132.

From The Times [London] Wednesday 3 January 2001 in an article titled Pupils sum up maths teachers as fat nerds by Simon de Bruxelles [http://www.thetimes.co.uk/article/0,,2-61352,00.html]:

MATHEMATICIANS are fat, scruffy and have no friends -- in any language. Youngsters from seven countries, asked to come up with a portrait of the typical mathematician, showed a badly dressed, middle-aged nerd with no social life.
[...]
Most children drew white men with glasses, often with a beard, bald head or weird hair, and shirt pockets filled with pens, who were working at a blackboard or computer. Finnish children had an even more disturbing view of maths teachers: several portrayed them forcing children to do sums at gunpoint.
[...]
He was forced to admit that being a mathematician did little for his social life. "If you are at a party and tell people you're a mathematician, it's the worst turn-off you can imagine," he said.

From Alice's Adventures in Wonderland by Lewis Carroll:

The Mock Turtle went on. 'We had the best of educations ... Reeling and Writhing, of course, to begin with, and then the different branches of Arithmetic: Ambition, Distraction, Uglification, and Derision.'

From Simon Singh, Fermat's Enigma:

Although Shimura had a whimsical streak -- even today he retains host fondness for Zen jokes -- he was far more conservative and conventional than his intellectual partner. Shimura would rise at dawn and immediately get down to work, whereas his colleague would often still be awake at this time, having worked through the night. Visitors to his apartment would often find Taniyama asleep in the middle of the afternoon.
While Shimura was fastidious, Taniyama was sloppy to the point of laziness. Surprisingly this was a trait that Shimura admired: "He was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this and tried in vain to imitate him, but found it quite difficult to make good mistakes."

From Charles Seife, Zero - The Biography of a Dangerous Idea:

In the Egyptian Book of the Dead, when a dead soul is challenged by Aquen, the ferryman who conveys departed spirits across a river in the netherworld, Aqen refuses to allow anyone aboard "who does not know the number of his fingers" The soul must then recite a counting rhyme to tally his fingers, satisying the ferryman. (The Greek ferryman, on the other hand, wanted money, which was stowed under the dead person's tongue.)
The [irational numbers] didn't kill Pythgoras. Beans did. [... Pythagoras] might have gotten away had he not run smack into a bean field. There he stopped. He declared that he would rather be killed than cross the field of beans. His pursuers were more than happy to oblige.
After a baby finishes his 12th month, we all say that the child is one year old; he has finished the first 12 months of life. If the baby turns one when she's already lived a year, isn't the only consistent choice to say that the baby is zero years old before that time? Of course, we say that the child is siz weeks old or nine months old instead -- a clever way of getting around the fact that the baby is zero.
With the introduction of ... the infinitely small and infinitely large, mathematics, usually so strictly ethical, fell from grace. ... The virgin state of absolute validity and irrefutable proof of everything mathematical was gone forever; the realm of controversy was inaugurated, and we have reached the point where most people differentiate and integrate not because they understand what they are doing but from pure faith, because up to now it has always come out right. -- Fridrich Engels, Anti-Duhring.

From John D. Barrow, The Book of Nothing:

Now the sirens have a still more fatal weapon than their song, namely their silence ... someone might possibly have escaped from their singing; but from their silence never. -- Franz Kafka, Parables.
A mathematician may say anything he pleases, but a physicist must be at least partially sane. -- Josiah Willard Gibbs.
"Encyclopædia Britannica full set, no longer needed due to husband knowing everything." -- personal ad, Lancashire Post, quoted in Observer 12 December 1999, p.30.
I have yet to see any problem, however complicated, which you looked at it in the right way, did not become still more complicated -- Poul Anderson.

From Eli Maor, Trigonometric Delights, Chapter 5, Measuring Heaven and Earth:

[Jean Picard] then extended his survey to the French coastline, which resulted in an unwelcome discovery: the west coast of the country had to be shifted 1½° eastward relative to the prime meridian through Paris, causing the monarch Louis XIV to exclaim: "Your journey has cost me a major portion of my realm!" [It was not until 1913 that France recognized the meridian through Greenwich as the prime (zero) meridian, in exchange for England "recognizing" the metric system.]
[Jean Dominic Cassini IV's] huge map, 12 by 12 yards, was published in 182 sheets on a scale of 1:86,400 and showed not only topographical features but also the location of castles, vineyards, and -- this being the time of the French Revolution -- guillotines.
[Captain William Lambton's] men encountered numerous perils: the intense heat of central India, thick vegetation in which tigers were roaming, the ever present threat of malaria, and angry locals who were convinced the surveyors were spying on their wives.

From Morris Kline, Mathematics - The Loss of Certainty:

The mere fact that there can be alternative geometries was in itself a shock. But the greater shock was that one could no longer be sure which geometry was true or whether any one of them was true. [...] Mathematicians were in the position described by Mark Twain: "Man is the religious animal. He's the only one who's got the true religion -- several of them."
-- From Chapter 4: The First Debacle
Negative numbers were not accepted by all the Hindus. Even Bhaskara said, while giving 50 and -5 as solutions of a problem, "The second value is in this case not to be taken, for it is inadequate; people do not approve of negative solutions."
-- From Chapter 5
Bertrand Russell [...] wrote in My Philosophical Development, "Those who taught me the infinitesimal calculus did not know the valid proofs of its fundamental theorems and tried to persuade me to accept the official sophistries as an act of faith."
-- From Chapter 7
[Niels Henrick] Abel wrote [...] "The divergent series are the invention of the devil, and it is a shame to base on them any demonstrations whatsoever.
-- From Chapter 7
The developments in this century bearing on the foundations of mathematics are best summarized in a story. On the banks of the Rhine, a beautiful castle had been standing for centuries. In the cellar of the castle, an intricate network of webbing had been constructed by industrious spiders who lived there. One day a strong wind sprang up and destroyed the web. Frantically, the spiders worked to repair the damage. They thought it was their webbing that was holding up the castle.
-- From Chapter 12 - Disasters
If potential application is the goal, then as the great physical chemist Josiah Willard Gibbs remarked, the pure mathematician can do what he pleases, but the applied mathematician must be at least partially sane.
-- From Chapter 13 - The Isolation of Mathematics
Other contemporary mathematicians are aware of the uncertainties in the foundations [of mathematics] but prefer to take an aloof attitude toward what they characterize as philosophical (as opposed to purely mathematical) questions. They find it hard to believe that there can be any serious concern about the foundations, or at least about their own mathematical activity. They prefer to be suckled in a creed outworn. For these men the unwritten code states: Let us proceed as thought nothing has happened in the last seventy-five years. They speak of proof in some universally accepted sense even though there is no such animal, and write and publish as if the uncertainties were non-existent. What matters for them is new publications, the more the better. If they respect sound foundations at all, it is only on Sundays and on that day they either pray for forgiveness or they desist from writing new papers in order to read what their competitors are doing.
-- From Chapter 15 - The Authority of Nature

From David Ruelle, Chance and Chaos:

What is the origin of the urge, the fascincation that drives physicists, mathematicians, and presumably other scientists as well? Psychoanalysis suggests that it is sexual curiosity. You start by asking where little babies come from, one thing leads to another, and you find yourself preparing nitroglycerine or solving differential equations. This explanation is somewhat irritating, and therefore probably basically correct.

From Wilfrid Hodges, An Editor Recalls Some Hopeless Papers, The Bulletic on Symbolic Logic, Vol 4, No 1, March 1998 [PostScript version available here]:

In English-speaking philosophy (and much European philosophy too) you are taught not to take anything on trust, particularly if it seems obvious and undeniable. You are also taught to criticise anything said by earlier philosophers. Mathematics is not like that; one has to accept some facts as given and not up for arguments. [...] (In the days when I taught philosophy, I remember one student who was told he had failed his course badly. He duly produced a reasoned argument to prove that he hadn't.)
Other authors, less coherently, suggested that Cantor had used the wrong positive integers. He should have allowed integers which have infinite decimal expansions to the left, like the p-adic integers. To these people I usually sent the comment that they were quite right, the set of real numbers does have the same cardinality as the set of natural numbers in their sense of natural numbers; but the phrase 'natural number' already has a meaning, and that meaning is not theirs.

From The Book of Imaginary Beings by Jorge Luis Borges:

[Plato] enriched fantastic zoology with vast spherical animals and cast aspersions on those slow-witted astronomers who failed to understand that the circular course of heavenly bodies was entirely voluntary.
In Alexandria over five hundred years later, Origen, one of the Fathers of the Church, taught that the blessed would come back to life in the form of spheres and would enter rolling into heaven.
[p. 21-22]
More quotes from The Book of Imaginary Beings

From The Number Devil, by Hans Magnus Enzensberger:

Still in a daze the next morning, Robert said to his mother, "Do you know the year I was born? It was 6x1 and 8x10 and 9x100 and 1x1000."
"I don't know what's got into the boy lately," said Robert's mother, shaking her head. "Here," she added, handing him a cup of hot chocolate, "maybe this will help. You say the oddest things."
Robert drank his hot chocolate in silence. There are some things you can't tell your mother, he thought.
[p. 46 -- The Second Night]

From At Home In The Universe - The Search for Laws of Complexity by Stuart Kauffman (Penguin Books, 1995):

Algorithms are a set of procedures to generate the answer to a problem. An example is the algorithm to find the solution to a quadratic equation, which most of us were taught while learning algebra. Not only was I taught the algorithm, but our entire class was invited to tattoo it on our stomachs so that we could solve quadratic equations by rote. [p. 21]

From The Artful Universe by John D. Barrow (Oxford University Press, 1995):

The theoretician's prayer: "Dear Lord, forgive me the sin of arrogance, and Lord, by arrogance I mean the following ..." -- Leon Lederman [source of quote unknown ; appears in Chapter 2, p. 31]
More quotes from "The Artful Universe"

Newsgroups: sci.math
Subject: Re: I'm looking for axioms and proof in math texts
Date: 6 Aug 2003 02:53:12 -0700
From: euclid@softcom.net (prometheus666)

[...]
and being passing familiar with the number line -- I'm sure if I don't
say passing familiar somebody here will say, "You have to know vector
tensor shmelaculus in 15 triad synergies to really understand the
number line. It's not even called that, it's called the real
torticular space." or something to that effect --
[...]

From Pi in the Sky by John Barrow (Oxford University Press, 1992):

Ask a philosopher "What is philosophy?" or a historian "What is history?" and they will have no difficulty in giving an answer. Neither of them, in fact, can pursue his own discipline without knowing what he is searching for. But ask a mathematician "What is mathematics?" and he may justifiably reply that he does not know the answer but that this does not stop him from doing mathematics. -- François Lasserre [Source of quote unknown, quoted at head of Chapter 1]
For most people mathematics lives in classrooms, impenetrable books, income tax returns, and university lecture halls. Mathematicians are the people you don't want to meet at cocktail parties. [Chapter 1, p. 5]
Logic is invincible because in order to combat logic it is necessary to use logic -- Pierre Boutroux [Source of quote unknown, quoted in Chapter 1, p. 15].
[Contrast this with a quote by Steven Weinberg in his book Dreams of a Final Theory, p. 259: "[...] all logical arguments can be defeated by the simple refusal to reason logically"]
[...] it has been suggested that if we were to define a religion to be a system of thought which contains unprovable statements, so it contains an element of faith, the Gödel has taught us that not only is mathematics a religion but it is the only religion able to prove itself to be one. [Chapter 1, p. 19]
When scientists attempt to explain their work to the general public they are urged to simplify the ideas with which they work, to remove unnecessary technical language and generally make contact with the fund of everyday concepts and experience that the average shares with them. Invariably this leads to an attempt to explain esoteric ideas by means of analogies. Thus, to explain how elementary particles of matter interact with one another we might describe them in terms of billiard balls colliding with each other.
[Footnote: One Hungarian physicist once remarked in the course of writing a textbook that, although he would often be referring to the motions and collisions of billiard balls to illustrate the laws of mechanics, he had neither seen nor played this game and his knowledge of it was derived entirely from the study of physics books.]
Near the turn of the [twentieth] century this practice provoked the French mathematician, Henri Poincaré, to criticize it as a very 'English' approach to the study of Nature, typified by the work of Lord Kelvin and his collaborators, who were never content with an abstract mathematical theory of the world but instead sought always to reduce that mathematical abstraction to a picture involving simple mechanical concepts with which they had an intuitive familiarity. Kelvin's preference for simple mechanical pictures of what the equations were saying, in terms of rolling wheels, strings, and pulleys, has come to characterize the popularization of science in the English language. But if we look more closely at what scientists do, it is possible to see their descriptions as the search for analogies that differ from those used as a popularising device only by the degree of sophistication and precision with which they can be endowed. Just as out picture of elementary particles of matter as little billiard balls, or atoms as mini solar systems, breaks down if pushed far enough, so our more sophisticated scientific description in terms of particles, fields or strings may well break down as well if pushed too far.
Mathematics is also seen by many as an analogy. But it is implicitly assumed to be the analogy that never breaks down. Our experience of the world has failed to reveal any physical phenomenon that cannot be described mathematically. That is not to say that there are not things for which such a description is wholly inappropriate or pointless. Rather, there has yet to be found any system in Nature so unusual that it cannot be fitted into one of the strait-jackets that mathematics provides.
This state of affairs leads us to the overwhelming question: Is mathematics just an analogy or is it the real stuff of which the physical realities are but particular reflections?
[Chapter 1, p. 21]
We can hardly imagine a state of mind in which all material objects were regarded as symbols of spiritual truths or episodes in sacred history. Yet, unless we make this effort of imagination, mediaeval art is largely incomprehensible. -- Kenneth Clark [Source of quote unknown, quoted in Chapter 1, p. 23]
To obtain some feeling for what life is like if one has a conception of 'oneness' and 'twoness' but nothing more, one might consider the following vivid account of the world of the Damaras in Southern Africa which is taken from Francis Galton's turn of the century account of his early contacts with these people (it also reveals something of their ingenuity in overcoming their conceptual limitations):
"In practice whatever they may possess in their language, they certainly use no greater number than three. When they wish to express four they take to their fingers, which are to them formidable instruments of calculation as a sliding rule is to an English schoolboy. They puzzle very much after five, because no spare hand remains to grasp and secure the fingers that are required for units. Yet they seldom lose oxen; the way in which they discover the loss of one is not by the number of the herd being diminished, but by the absence of a face they know. When bartering is going on each sheep must be paid for separately. This suppose two stick of tobacco to be the rate of exchange for one sheep, it would sorely puzzle a Damara to take two sheep and give him four sticks. I have done so, and seen a man take two of the sticks apart and take a sight over them at one of the sheep he was about to sell. Having satisfied himself that one was honestly paid for, and finding to his surprise that exactly two sticks remained in hand to settle the account for the other sheep, he would be afflicted with doubts; the transaction seemed to him to come too 'pat' to be correct, and he would refer back to the first couple of sticks, and then his mind got hazy and confused, and wandered from one sheep to the other, and he broke off the transaction until two sticks were placed in his hand and one sheep driven away, and then the other two sticks given him and the second sheep driven away."
[Chapter 2, p. 35-36]
All are lunatics, but he who can analyze his delusion is called a philosopher -- Ambrose Bierce [Source of quote unknown, quoted on p. 36]
English: one/first ; two/second ; three/third ; four/fourth
French: un/premier ; deux/second or deuxième ; trois/troisième ; quatre/quatrième
German: ein/erste ; zwei/ander or zweite ; drei/dritter ; vier/vierte
Italian: uno/primo ; due/secondo ; tre/terzo ; quattro/quarto
In each of these four languages the words for 'one' and 'first' are quite distinct in form and emphasize the distinction between solitariness (one) and priority (being first). In Italian and the more old-fashioned German and French usage of ander and second, there is also a clear difference between the words used for 'two' and 'second', just as there is in English. This reflects the Latin root sense in English, French, and Italian of being second, this is, coming next in line, and this does not necessarily have an immediate association with two quantities. But when we get to three and beyond, there is a clear and simple relationship between the cardinal and ordinal words. Presumably this indicates that the dual aspect of number was appreciated by the time the concepts of 'threeness' and 'fourness' had emerged linguistically, following a period when only words describing 'oneness' and 'twoness' existed with greater quantities described by joining those words together as we described above.
In all the known languages of Indo-European origin, numbers larger than four are never treated as adjectives, changing their form according to the thing they are describing. But, numbers up to and including four are: we say they are 'inflected'. [...] a rather antiquated structure that barely survives in the modern forms of many Indo-European languages. For example, in French we find two words un and une corresponding to the English 'one' and they are used according to the gender of what is being counted. An analogous feature of language that certainly survives in English is the way in which different adjectives are associated with the same quantities of different things. We speak of a pair of shoes, a brace of pheasants, a yoke of oxen, or a couple of people, but we would never speak of a brace of chickens or a couple of shoes. [...]
We have seen that the distinction between cardinal and ordinal aspects of number and the use of inflected adjectives is clear up to the number four but conflated beyond that. [Footnote: In Finnish there are still two kinds of plural, as in classical Greek, Biblical Hebrew and Arabic: one for two things and another for more than two. Also interesting in this respect is the fact that there is no connection between the words for '2' and '½' in the Romance and Slavic languages (nor in Hungarian which is not an Indo-European language) but in all the European languages the words for '3' and '1/3', '4' and '1/4' and so on, are closely related, just as they are in English. This may indicate that the concept of a fraction, or the relation between a number and the concept of a ratio, only emerged after counting beyond 'two'.]
[...]
A curious speculation arises [...] to give special status to the number 8 - the total number of fingers excluding the thumbs - that many known languages originally possessed a base-8 system (which they later replaced by something better), because the word for the number 'nine' appears closely related to the word for 'new' suggesting that nine was a new number added to a traditional system. There are about twenty examples of this link, including Sanskrit, Persian, and the more familiar Latin, where we can see novus = 'new' and novem = 'nine'.
[Chapter 2, p. 37-38]
In Samoa, when elementary schools were first established, the natives developed an absolute craze for arithmetical calculations. They laid aside their weapons and were to be seen going about armed with slate and pencil, setting sums and problems to one another and to European visitors. The Honourable Frederick Walpole declares that his visit to the beautiful island was positively embittered by ceaseless multiplication and division. -- R. Briffault. [Source of quote unknown, quoted in Chapter 2, p. 45]
[...] Nearly as bad was the prejudice that everything had to have a numerological aspect. As a result subjects like medicine, which have little or no need of numbers for diagnosis, introduced them as a display of their philosophical significance.
[Footnote: This state of affairs is not entirely unknown today and its manifestations in some subjects are tellingly documented by the sociologist Stanislav Andreski in his book Social Sciences as Sorcery.]
An amusing example of the power of this approach is the famous occasion on which Leonhard Euler, the great Swiss mathematician who was sometimes tutor to Catherine the Great of Russia during the eighteenth century, decided to bamboozle the Voltarean philosophers at Court in an argument about the existence of God. Calling for a blackboard, he wrote:
(x+y)² = x²+2xy+y²
therefore God exists.
[Presumably in a language other than English -- Fred.]
Unwilling to confess their ignorance of the formula or unable to question its relevance to the question at hand, his opponents accepted his argument with a nod of profound approval.
[Chapter 3, p. 107-108]
Ten years ago a survey of working mathematicians revealed 30 percent of them to be formalists in the Bourbaki mode. One reason for this is the point that Laurent Schwartz makes: most mathematical work is far from its popular image of 'discovery'; rather, it is dominated by the process of refinement so that difficult and complicated proofs are made simpler and shorter until one can claim that their chain of argument is 'obvious' or 'trivial' by which mathematicians mean simply that it appeals to no new type of argument. It is merely cranking though a well-worn set of operations. In order to do this sort of thing it is probably most expedient to act as though one is a formalist even if one might find all the implications of such a cramped perspective rather less attractive if asked to reflect upon it in the armchair over the weekend.
[Chapter 3, p. 133-134]
Truth is stranger than fiction; fiction has to make sense. -- Leo Rosten [Quote in Chapter 3, p. 134]
One is reminded of the story about an astronomer who began a public lecture about stars with the words "Stars are pretty simple things ..." only to hear a voice calling from the back of the room, "You'd look pretty simple too from a distance of a hundred light years!" [Chapter 4, p. 149]
My theology, briefly, is that the universe was dictated but not signed. -- Christopher Morley [Source of quote unknown, quoted in Chapter 4, p. 159]
[...] A certain level of predictability and innate predictive power is required for the successful evolution and survival of living things. However, we must beware of the fact that our brains are altogether too good at finding compressions. It has clearly proved efficacious to overdevelop our pattern recognition capability (presumably because if you see tigers in the bushes when there are none your friends will merely call you paranoiac, whereas if you fail to see tigers in the bushes when there are, then your continued survival must be rather doubtful). As a result we see canals on Mars and all manner of exotic things lurking in inkblots. [...]
[Chapter 4, p. 164]
Poets do not go mad, but chess players do. Mathematicians go mad, and cashiers, but creative artists very seldom. I am not, as will be seen, in any sense attacking logic: I only say that this danger does lie in logic, not in imagination. -- G. K. Chesterton [Source of quote unknown, quoted in Chapter 4, p. 171]
The idea that extraterrestrials might possess an alien way of thinking and reasoning had occurred to Frege in the nineteenth century. His reaction indicates the extent to which mathematicians of that time regarded this possibility as bizarre:
"But what if beings were even found whose laws of thought flatly contradicted ours and therefore frequently led to contrary results even in practice? The psychological logician could only acknowledge the fact and say simple: those laws hold for them, these laws hold for us. I should say: we have a hitherto unknown type of madness."
[Chapter 4, p. 176]
[...] Brouwer denied this 'principle of the excluded middle' as it was known, allowing a third limbo status of 'undecided' to exist for statements whose truth or falsity had not been constructed by following a finite number of deductive steps. This distinction is somewhat reminiscent of the difference between English and Scottish law. In English law the defendant must be found either guilty or not guilty whereas in Scotland there exist not only these two verdicts but a third option of 'not proven'. This last verdict differs from that of 'not guilty' in that it permits the defendant to be retried on the same charge in the future. English law does not permit such a retrial.
[Footnote: The Reader may recall that the essence of the plot in Agatha Christie's first novel The Mysterious Affair at Styles was that a villain who, having committed a murder by some means, deliberately laid false clues to make it appear that the had committed the murder in a different way. He had a cast-iron alibi against having committed the murder in the manner suggested by this trail of false clues, and so if he could be charged and tried for the murder on the basis of the false evidence then the revelation of the alibi at the last moment in court would result in his acquittal with no possibility of him being retried even if the true means of the murder were to be discovered in the future.]
[Chapter 5, p. 186]
The point of philosophy is to start with something so simple as to seem not worth stating, and to end with something so paradoxical that no one will believe it. -- Bertrand Russell [Source of quote unknown, quoted in Chapter 5, p. 188]
Paul Cohen's demonstration of the insolubility of the continuum hypothesis has a remarkable history. Cohen was a brilliant and confident young student who upon entering graduate school at Stanford gathered a group of his fellow students together to ask them if they thought he would become more famous by solving one of Hilbert's remaining unsolved problems or by showing the continuum hypothesis is independent of the axioms of set theory. His colleagues voted for the latter. Whereupon Cohen went off, learnt about the relevant areas of mathematical logic and invented a new proof procedure that enabled him to solve the most difficult unsolved question in mathematics in less than a year. Local mathematicians knew there was only one way to know whether the proof was correct and so soon Cohen was knocking on the door of Gödel's residence in Princeton. He was clearly not the first such called. Gödel opened the door just wide enough for Cohen's proof to pass through, but not so wide as to allow Cohen to follow. But two days later Cohen received an invitation to tea with the Gödels. The proof was correct. The master had given his imprimatur.
[Chapter 6, p. 215]
As a result of Cantor's developments, one could divide the mathematical community into three sorts. There were the finitists, typified by the attitudes of Aristotle or Gauss, who would only speak of potential infinities, not of actual infinities. Then there were the intuitionists like Kronecker and Brouwer who denied that there was any meaningful content to the notion of quantities that are anything but finite. Infinities are just potentialities that can never be actually realised. To manipulate them and include them within the realm of mathematics would be like letting wolves into the sheepfold. Then there were the transfinitists like Cantor himself, who ascribe the same degree of reality to actual completed infinities as they did to finite quantities. In between, there existed a breed of manipulative transfinitists, typified by Hilbert, who felt no compunction or need to ascribe any ontological status to infinities but admitted them as useful ingredients of mathematical formalism whose presence was useful in simplifying and unifying other mathematical theories. "No one," he predicted, "though he speak with the tongue of angels, will keep people from using the principle of the excluded middle."
[Chapter 5, p. 216]
In the West it was the publication in 1967 of a work entitled Foundations of Constructive Analysis by Errett Bishop that rekindled interest in constructive mathematics. [...] Bishop developed the constructive approach in order to keep unverifiable philosophical ideas out of mathematics; in the introduction he makes his famous remark that:
Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer exists, he should know how to find it. If God has mathematics of his own that needs to be done, let him do it himself.
[Chapter 5, p. 220]
The Lord is my thesis adviser; I shall not err.
He arranges for me to be published in the
respectable journals; he teaches me how
to use the reductio argument.
He enshores my validity; he leads me by
the classical logic, for the truth's sake.
Yea, though I walk through the valley of
the existence proofs, I will fear no contradiction;
for he edits my work.
The Axiom of Choice and Zorn's Lemma, they comfort me.
He invites a colloquium on classical analysis,
for my participation, in the absence of the contructivists;
He frequently and approvingly abstracts me
in Mathematical Reviews;
my reputation flourishes internationally.
Surely, honours and grants shall follow me
all the days of my career,
And I shall rise in the ranks of the Department,
to Emeritus.
Amen.
[From a fable by the American mathematician John Hays, quoted in Chapter 5, p. 226]
I knew a mathematician who said, "I do not know was much as God, but I know as much as God did at my age." -- Milton Shulman [Source of quote unknown, quoted at the start of Chapter 6, p. 249]
Plato's philosophy of mathematics grew out of his attempts to understand the relationship between particular things and universal concepts. What we see around us in the world are particular things -- this chair, that chair, big chairs, little chairs and so on. But that quality which they share - let's call it 'chairness' - present a dilemma. It is not itself a chair and unlike all the chairs we know it cannot be located in some place or at some time. But that lack of a place in space and time does not mean that 'chairness' is an imaginary concept. We can locate it in two different pieces of wood and upholstery. These things have something in common. Plato's approach to these universal was to regard them as something real. In some sense they really exist 'out there'. The totality of his reality consisted of all the particular instances of things together with the universals of which they were examples. Thus the particulars which we witness in the world are each imperfect reflections of a perfect exemplar or 'form'.
[...] So, 'somewhere', Plato maintained, there must exist perfect straight lines, perfect circles and triangles, or exactly parallel lines; and these perfect forms would exist even if there were no particular examples of them for us to see. This 'somewhere' was not simply the human mind. Strikingly, Plato maintained that we discover the truths and theorems of mathematics; we do not simply invent them. [...]
The first problem with Plato's picture of reality is to clarify the relationship between universals and particular examples of them. Plato's idea that there exist perfect blueprints of which the particulars are imperfect approximations does not seem very helpful when one gives it a second thought: for, as far as our minds are concerned, the universal blueprint is just another particular. So, whilst we could say that Plato maintained that universals would exist even in the absence of particulars, this statement does not really have any clear meaning. If all the particulars vanished, so would all those mental images of concrete perfect blueprints together with all the blueprints themselves. Aristotle [...] also picked upon this weakness in Plato's doctrine of Ideas, arguing that it leads to an endless regress:
"If all men are alike because they have something in common with Man, the ideal and eternal archetype, how can we explain the fact that one man and Man are alike without assuming another archetype? And will not the same reasoning demand a third, fourth and fifth archetype, and so on into the regress of more and more ideal worlds?"
[...] Plato wants to relate the universal abstract blueprint of a perfect circle to the approximate circles that we see in the real world. But why should we regard the 'approximate' circles, or 'almost parallel lines', or 'nearly triangles' as imperfect examples of perfect blueprints? Why not regard them as perfect exhibits of universals of 'approximate circles', 'almost parallel lines' and 'nearly triangles'? When viewed in this light the distinction between universals and particulars seems to be eroded.
[Chapter 6, p. 253-255]
[...] Even Herman Weyl [...] felt a natural leaning toward the Platonic attitude of regarding mathematical entities as real and transcending the human creative process. He found that his tendency to veer from this dogma towards the more conservative constructive philosophy of the intuitionists felt unnatural and even acted as a restraining influence upon his mathematical creativity because although
"outwardly it does not seem to hamper our daily work, yet I for one confess that it has had a considerable practical influence on my mathematical life. It directed my interests to fields I considered relatively 'safe', and has been a constant drain on the enthusiasm and determination with which I pursued my research work."
Here we see a viewpoint that [...] hints that even if Platonism is not true, it is most effective for the working mathematician to act as if it were true. The French mathematician Emile Borel, writing a century ago, tends a little closer to this attitude:
"Many do, however, have a vague feeling that mathematics exists somewhere, even though, when they think about it, they cannot escape the conclusion that mathematics is exclusively a human creation. Such questions can be asked of many other concepts such as state, moral values, religion, ... we tend to posit existence on all those things which belong to a civilization or culture in that we share them with other people and can exchange thoughts about them. Something becomes objective (as opposed to 'subjective') as soon as we are convinced that it exists in the minds of others in the same form as it does in ours, and that we can think about it and discuss it together. Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence regardless of whether it has another origin."
If we look back to the end of the nineteenth century when Borel was writing then we find some of the most emphatic Platonists. Charles Hermite regarded mathematics as an experimental science:
"I believe that the numbers and functions of analysis are not the arbitrary product of our spirits: I believe that they exist outside us with the same character of necessity as the objects of objective reality; and we find or discover them and study them as do the physicists, chemists and zoologists."
Henri Poincaré highlights Hermite's approach as especially remarkable in that it was not just a philosophy of mathematics, more a psychological attitude. Poincaré had been Hermite's student and was also much taken to careful introspective analysis of his mathematical thought processes and creativity.
Unlike many subscribers to the Platonic philosophy, Hermite did not regard all mathematical discovery as the unveiling of the true Platonic world of mathematical forms; he seems to have drawn a distinction between mathematical discoveries of this pristine sort and others, like Cantor's development of different orders of infinity [...]that he regarded as mere human inventions. Poincaré writes, in 1913, twelve years after Hermite's death, that
"I have never known a more realistic mathematician in the Platonist sense then Hermite ... He accused Cantor of creating objects instead of merely discovering them. Doubtless because of his religious convictions he considered it a kind of impiety to with to penetrate a domain which God alone can encompass [i.e. the infinite], without waiting for Him to reveal its mysteries one by one. He compared the mathematical sciences with the physical sciences. A natural scientist who sought to divine the secret of God, instead of studying experience, would have seemed to him not only presumptuous but also lacking in respect for the divine majesty: the Cantorians seemed to him to want to act in the same way in mathematics. And this is why, a realist in theory, he was an idealist in practice. There is a reality to be known, and it is external to and independent of us; but all we can know if it depends on us, and is no more than a gradual development, a sort of stratification of successive conquests. The rest is real but eternally unknowable."
[...] G. H. Hardy [...] maintained a solid Platonic realism about the nature of mathematics:
"I believe that mathematical reality lies outside us, and that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations' are simply our notes of our observations ... 317 is a prime number, not because we think it 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."
[...]
Nevertheless [...] there are many dissenters to such a confident Platonic view [...] the subtle change of direction in the titles of applied mathematics books which highlight changing attitudes to the status of mathematical descriptions of the world. They emphasize the use of mathematics as a tool for deriving approximate descriptions ('models') of the real thing. There is no implication that the mathematics being presented is the reality.
[Chapter 6, p. 259-262]
[...] At root, the Platonist believes he is exploring and discovering the structure of some other 'world' of truths, yet somehow the Platonic view allows him to feel as though his creativity has free rein. This is at root why Brouwer's philosophy of mathematics was so distasteful. [...] It was not that mathematicians have any deep-seated resentment to the idea of construction as the essence of mathematical reasoning. [...] No, the real objection was that it did not permit all the deductions arrived at by other means to be called 'mathematical truths': it was a restriction upon freedom of thought.
Platonism allows freedom of thought, but only in the sense that it is your fault if you want to think the wrong thoughts. Whereas the formalist is free to create any logical system he chooses, a Platonist like Gödel maintained that only one system of axioms captured the truths that existed in the Platonic world. Although a conjecture like the continuum hypothesis was undecidable from the axioms of standard set theory, Gödel believed it was either true or false in reality and this would be decided by adding appropriate axioms to those we were in the habit of using for set theory. He writes of the undecidability of Cantor's continuum hypothesis:
"Only someone who (like the intuitionist) denies that the concepts and axioms of classical set theory have any meaning ... could be satisfied with such a solution, not someone who believes them to describe some well-determined reality. For in reality Cantor's conjecture must be either true or false, and its undecidability from the axioms as known today can only mean that these axioms do not contain a complete description of reality."
This seems a strange view because there would always exist other undecidable statements no matter what additional axioms are prescribed.
[Chapter 6, p. 263-264]
[...] If the Platonic philosophy of mathematics is not true and mathematics possesses a strong cultural element, or is an approximation to reality that our minds have invented in response to the evolutionary pressures created by our particular terrestrial environment, then extraterrestrial mathematics will not be like ours. [...]
When studying the simultaneous discovery of mathematical concepts one must be careful not to generalize over the whole of mathematics. The motivations for ancient mathematical discoveries are rather different to those which inspire developments in modern times. The most primitive notions of geometry and counting are tied to practical applications in obvious ways and there was clear motivation for their adoption by less developed cultures who wanted to trade and converse with more sophisticated neighbours. But in modern times mathematics need not be tied to practical applications and one cannot argue that notions of pure mathematics are developed in response to social or utilitarian pressures. Modern communications mean that mathematicians and scientists are in effect a single intellectual society that transcends national and cultural borders. [...] it is hard for mathematicians to be independent in the deepest sense. They grow up in the body of mathematical knowledge that is taught to mathematicians of every colour and creed. They attend conferences, or read reports of them, and sense the direction of the subject, they see the great unsolved problems being laid out and conjectures as to their solution being proposed. [...] they have common predecessors, common inherited intuitions and methods. They are fish that swim in the same big pond.
Another intriguing aspect of mathematics that seems to distinguish it from the arts and humanities is the extent to which mathematicians, like scientists, collaborate in their work.
[Chapter 6, p. 266-267]
The mathematician may be compared to a designer of garments, who is utterly oblivious of the creatures whom his garments may fit. To be sure, his art originated in the necessity for clothing such creatures, but this was long ago; to this day a shape will occasionally appear which will fit into the garment as if the garment had been made for it. Then there is no end of surprise and delight! -- Tobias Dantzig [Quotes in Chapter 6, p. 270]

From Words Fail Me by Philip Howard (Hamish Hamilton Ltd, 1980) [Chapter 19, Millennium, p. 109]:

Human kind, especially the English human kind, cannot bear very much mathematics. Plato said that he had hardly ever known a mathematician who was capable of reasoning. The passion for anniversaries, decades, centuries, and all dates with noughts in them is deeply engrained in the human attitude to time And yet it is an irrationally tidy way to measure life, which does not conform to the decimal system. The seventeenth century ought to begin in 1603. The death of Oscar Wilde in 1900 and the Dreyfus case mark the end of an intellectual and moral epoch. [...] But the nineteenth century properly ends in 1914. You cannot wrap men and ideas up in parcels of centuries in order to make literary and historical generalizations about them with the appearance of mathematical exactitude. You should not, but we all do. Since A.D. 1 there have been not nineteen but 1979 complete centuries. The real world is regardless of our systems of reckoning; events and men slip over years with noughts in their dates, with as little shudder as is felt on a liner passing over a tropic or a car crossing a county boundary.

From The Science Show, ABC Radio National (Australia), 1 Nov 2003:

Mark Lythgoe: [...] it was about 4 or 5 years ago - and I read an article by Simon Baron-Cohen. And he'd looked at a thousand students from Cambridge and assessed them on what he called "the autistic spectrum".
Now these were normal, functioning intelligent people that had tried to decide whether they had particular autistic traits or not. And he found that the scientists, as opposed to the arts and humanists, came significantly higher on this autistic spectrum, they had more autistic traits. Then the mathematicians were virtually off the top of the spectrum, they were very autistic.
Robyn Williams: And the engineers.

MP3 audio of quote (143 Kbyte)

When I turned two I was really anxious, because I'd doubled my age in a year. I thought, if this keeps up, by the time I'm six I'll be ninety. -- Steven Wright

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