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Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals.
     Woody Allen, quoted in Howard Eves, 
     Return to Mathematical Circles (1988)


It is quite possible that mathematics was invented in the ancient Middle East to keep track of tax receipts and grain stores. How odd that out of this should come a subtle scientific language that can effectively describe and predict the most arcane aspects of the Universe.
     Isaac Asimov


One of the finest creations of the human mind is mathematics, for not only is it the apotheosis of rational thought but it is also the spine that renders scientific speculation sufficiently rigid to confront experience.  Scientific hypotheses themselves are like jelly; they need the rigidity of mathematical formulation if they are to stand up to experimental verification and fit into the network of concepts that constitute physical science.
     Peter Atkins, Galileo’s Finger:  The Ten Great Ideas of Science (2003)
     “Arithmetic:  The Limits of Reason”


For he who knows not mathematics cannot know any other science; what is more, he cannot discover his own ignorance, or find its proper remedy.
     Roger Bacon


Mathematics is the door and the key to the sciences.
     Roger Bacon, Opus Majur, transl Robert Belle Burke


The greatest shortcoming of the human race is our inability to understand the exponential function.
     Albert A. Bartlett


Aftermath: The horrible headache you have when you've finished the algebra test.
     Tom Batiuk, Funky Winkerbean (comic strip, date unknown)


"Obvious" is the most dangerous word in mathematics.
     Eric Temple Bell


The straightest line between a short distance is two points.
     George Carlin, Brain Droppings (1997)


Recent polls reveal that some people have never been polled. Until recently.
     George Carlin, Brain Droppings (1997)


The Nobel Prize in mathematics was awarded yesterday to a California professor who has discovered a new number. The number is "bleen," which he says belongs between six and seven.
     George Carlin, Napalm & Silly Putty (2001)


'I couldn't afford to learn it.' said the Mock Turtle with a sigh. 'I only took the regular course.'
     'What was that?' inquired Alice.
     'Reeling and Writhing, of course, to begin with,' the Mock Turtle replied; 'and then the different branches of Arithmetic — Ambition, Distraction, Uglification, and Derision.'
     'I never heard of "Uglification,"' Alice ventured to say. 'What is it?'
     The Gryphon lifted up both its paws in surprise. 'What! Never heard of uglifying!' it exclaimed. 'You know what to beautify is, I suppose?'
     'Yes,' said Alice doubtfully: 'it means — to — make — anything — prettier.'
     'Well, then,' the Gryphon went on, 'if you don't know what to uglify is, you are a simpleton.'
          Lewis Carroll, Alice's Adventures in Wonderland (1865)


You can be moved to tears by numbers — provided they are encoded and decoded fast enough.
     Richard Dawkins, River Out of Eden: 
     A Darwinian View of Life (1995)


As far as the properties of mathematics refer to reality, they are not certain: and as far as they are certain, they do not refer to reality.
     Albert Einstein


Statistics: the science of producing unreliable facts from reliable figures.
     Evan Esar


If you took a sample of one million children and counted their digits at birth, you would find that the vast majority of them had twenty in all, while some might have more and some less. If you displayed this digital variation graphically, you would come up with what statisticians call a normal curve. . . . A practicing doctor would call all of the ten-toe-ten-finger kids normal. In fact, they are a statistical mean.
     Anne Fausto-Sterling, in How Things Are: A Science Tool-Kit 
     for the Mind (John Brockman and Katinka Matson, eds.)


A handle is topologically equivalent to a doughnut. (Hence the campus joke, "A topologist is someone who doesn't know the difference between a coffee cup and a doughnut.")
     Timothy Ferris, The Whole Shebang: 
     A State-of-the-Universe(s) Report (1997)


There are 1011 stars in the galaxy. That used to be a huge number. But it's only a hundred billion. It's less than the national deficit! We used to call them astronomical numbers. Now we should call them economical numbers.
     Richard Feynman


If you are interested in the ultimate character of the physical world, or the complete world, and at the present time our only way to understand that is through a mathematical type of reasoning, then I don't think a person can fully appreciate, or in fact can appreciate much of, these particular aspects of the world, the great depth of character of the universality of the laws, the relationships of things, without an understanding of mathematics. I don't know any other way to do it, we don't know any other way to describe it accurately . . . or to see the interrelationships without it. So I don't think a person who hasn't developed some mathematical sense is capable of fully appreciating this aspect of the world — don't misunderstand me, there are many, many aspects of the world that mathematics is unnecessary for, such as love, which are very delightful and wonderful to appreciate and to feel awed and mysterious about; and I don't mean to say that the only thing in the world is physics, but you were talking about physics and if that's what you're talking about, then to not know mathematics is a severe limitation in understanding the world.
     Richard Feynman, "The Pleasure of Finding Things Out" 
     (interview, BBC, Horizon, 1981; shown in US on Nova);
     reprinted in The Pleasure of Finding Things Out: The Best Short 
     Works of Richard P. Feynman (Jeffrey Robbins, ed., 1999)


The Universe is a grand book of philosophy. The book lies continually open to man's gaze, yet none can hope to comprehend it who has not first mastered the language and the characters in which it has been written. This language is mathematics; these characters are triangles, circles, and other geometric figures.
     Galileo Galilei


If the objects of mathematics are in some sense "out there," waiting to be discovered by human minds, permanently embedded in the structure of the cosmos in the way the "form" of a vase is essential to the material vase, then what is surprising about the fact that having extracted mathematics from the universe, of which our brain are a part, we are able to put it back in?
     Martin Gardner, Weird Water & Fuzzy Logic: 
     More Notes of a Fringe Watcher (1996)


I remember one occasion when I tried to add a little seasoning to a review, but I wasn't allowed to. The paper was by Dorothy Maharam, and it was a perfectly sound contribution to abstract measure theory. The domains of the underlying measures were not sets but elements of more general Boolean algebras, and their range consisted not of positive numbers but of certain abstract equivalence classes. My proposed first sentence was: "The author discusses valueless measures in pointless spaces."
     Paul R. Halmos, in I want to be a Mathematician (1985)


317 is a prime, not because we think so, or because our minds are shaped in one way rather than another but because it is so. Mathematical reality is built that way.
     Godfrey Harold Hardy


Beauty is the first test; there is no permanent place in the world for ugly mathematics.
     Godfrey Harold Hardy, A Mathematician's Apology (1941)


I believe that mathematical reality lies outside us, 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.
     Godfrey Harold Hardy, A Mathematician's Apology (1941)


... to characterize the import of pure geometry, we might use the standard form of a movie-disclaimer: No portrayal of the characteristics of geometrical figures or of the spatial properties of relationships of actual bodies is intended, and any similarities between the primitive concepts and their customary geometrical connotations are purely coincidental.
     Carl G. Hempel, "Geometry and Empirical Science" 
     in J. R. Newman (ed.), The World of Mathematics (1956)


Nobody knows why, but the only theories which work are the mathematical ones.
     Michael Holt, Mathematics in Art


Mathematics: A tentative agreement that two and two make four.
     Elbert Hubbard


A table of random numbers, once printed, requires no errata.
     Mark Kak


I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of Science, whatever the matter may be.
     William Thomson, Lord Kelvin, Lecture at the 
     Institution of Civil Engineers (3 May 1883)


Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherialization of common sense.
     William Thomson, Lord Kelvin, in S. P. 
     Thompson, Life of Lord Kelvin (1910)


When asked what it was like to set about proving something, the mathematician likened proving a theorem to seeing the peak of a mountain and trying to climb to the top. One establishes a base camp and begins scaling the mountain's sheer face, encountering obstacles at every turn, often retracing one's steps and struggling every foot of the journey. Finally when the top is reached, one stands examining the peak, taking in the view of the surrounding countryside and then noting the automobile road up the other side!
     Robert J. Kleinhenz


In real life, I assure you, there is no such thing as algebra.
     Fran Lebowitz


... I find that abstract symbols on the blackboard automatically stimulate the organ that secretes eye-glaze juice. If, for instance, I write x = vt (read: ex equals vee times tee), a gasp arises in the lecture hall. It isn't that these brilliant children of parents paying $20,000 tuition per year cannot deal with x = vt. Give them numbers for x and t and ask them to solve for v, and 48 percent would get it right, 15 percent would refuse to answer on advice of counsel, and 5 percent would vote present. (Yes, I know doesn't add up to 100. But I'm an experimenter not a theorist. Besides, dumb mistakes give my class confidence.)
     Leon Lederman, The God Particle: If the Universe is the 
     Answer, What is the Question? (with Dick Teresi, 1993)


A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.
     J. E. Littlewood, A Mathematician's Miscellany (1953)


Medicine makes people ill, mathematics make them sad and theology makes them sinful.
     Martin Luther


Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations.
     Ernst Mach


A talk in mathematics should be one of four things: beautiful, deep, surprising... or short.
     Michel Mendès France, remark, c. 1986


A tendency to drastically underestimate the frequency of coincidence is a prime characteristic of innumerates, who generally accord great significance to correspondences of all sorts while attributing too little significance to quite conclusive but less flashy statistical evidence.
     John Allen Paulos, Innumeracy: Mathematical 
     Illiteracy and its Consequences (1988)


Consider a precise number that is well known to generations of parents and doctors: the normal human body temperature of 98.6° Fahrenheit. Recent investigations involving millions of measurements have revealed that this number is wrong; normal human body temperature is actually 98.2° Fahrenheit. The fault, however, lies not with Dr. Wunderlich's original measurements — they were averaged and sensibly rounded to the nearest degree: 37° Celsius. When this temperature was converted to Fahrenheit, however, the rounding was forgotten, and 98.6 was taken to be accurate to the nearest tenth of a degree. Had the original interval between 36.5° Celsius and 37.5° Celsius been translated, the equivalent Fahrenheit temperatures would have ranged from 97.7° to 99.5°. Apparently, dyscalculia can even cause fevers.
     John Allen Paulos, A Mathematician Reads the Newspaper (1995)


In a math course you learn whether or not there's a solution; in a physics course you find the solution.
     P. A. Piroue


Let no one ignorant of geometry enter my door.


Mathematics consists of proving the most obvious thing in the least obvious way.
     George Polyá, in N. Rose, Mathematical 
     Maxims and Minims (1988)


One of the endearing things about mathematicians is the extent to which they will go to avoid doing any real work.
     Matthew Pordage, in H. Eves, Return 
     to Mathematical Circles (1988)


Pure mathematics consists entirely of such asservations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true. . . . If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
     Bertrand Russell, "Recent Work on the Principles of 
     Mathematics" in International Monthly, vol. 4, p. 84 (1901)


Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of a sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
     Bertrand Russell, The Study of Mathematics (1902)


Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform.
     Bertrand Russell, Principles of Mathematics (1903)


The true spirit of delight, the exaltation, the sense of being more than an, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.
     Bertrand Russell, Mysticism and Logic (1917)


That arithmetic is the basest of all mental activities is proved by the fact that it is the only one that can be accomplished by a machine.
     Arthur Schopenhauer


Numerology is the easiest — and consequently the most dangerous — method for finding patterns. It is easy because anybody can do it, and dangerous for the same reason. The difficulty lies in distinguishing significant numerical patterns from accidental ones. ... The big problem with numerological pattern-seeking is that it generates millions of accidentals for each universal. Nor is it always obvious which is which.
     Ian Stewart, Nature's Numbers: The Unreal 
     Reality of Mathematics (1995)


Each of nature's patterns is a puzzle, nearly always a deep one. Mathematics is brilliant at helping us to solve puzzles. It is a more or less systematic way of digging out the rules and structures that lie behind some observed pattern or regularity, and then using those rules and structures to explain what's going on. Indeed, mathematics has developed alongside our understanding of nature, each reinforcing the other.
     Ian Stewart, Nature's Numbers: The Unreal 
     Reality of Mathematics (1995)


... mathematics is the science of patterns, and nature exploits just about every pattern there is.
     Ian Stewart, Nature's Numbers: The Unreal 
     Reality of Mathematics (1995)


Chaos is overturning our comfortable assumptions about now the world works. It tells us that the universe is far stranger than we think. It casts doubt on many traditional methods of science: merely knowing the laws of nature is no longer enough. On the other hand, it tells us that some things that we thought were just random may actually be consequences of simple laws. Nature's chaos is bound by rules. In the past, science tended to ignore events or phenomena that seemed random, on the grounds that since they had no obvious patterns they could not be governed by simple laws. Not so. There are simple laws right under our noses — laws governing disease epidemics, or heart attacks, or plagues of locusts. If we learn those laws, we may be able to prevent the disasters that follow in their wake.
     Ian Stewart, Nature's Numbers: The Unreal 
     Reality of Mathematics (1995)


A circle is the longest distance to the same point.
     Tom Stoppard


Math was always my bad subject. I couldn't convince my teachers that many of my answers were meant ironically.
     Calvin Trillin


Sometimes, half a dozen figures will reveal, as with a lightning-flash, the importance of a subject which ten thousand labored words, with the same purpose in view, had left at last but dim and uncertain.
     Mark Twain, Life on the Mississippi (1883)


There are no sects in geometry.


There is an astonishing imagination, even in the science of mathematics . . . We repeat, there was far more imagination in the head of Archimedes than in that of Homer.
     Voltaire, Dictionnaire Philosophique (1764)


Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.
     John von Neumann


In Mathematics, you don't understand things; you just get used to them.
     John von Neumann


All science as it grows toward perfection becomes mathematical in its ideas.
     Alfred North Whitehead


I will not go so far as to say that to construct a history of thought without profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. . . But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming — and a little mad.
     Alfred North Whitehead


The study of mathematics is apt to commence in disappointment. ... We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it.
     Alfred North Whitehead, An Introduction to Mathematics (1911)


The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit.
     Alfred North Whitehead, Science and the Modern World (1925)


Factorials were someone's attempt to make math look exciting.
     Steven Wright


I'm kinda tired. I was up all night trying to round off infinity.
     Steven Wright


In Vegas, I got into a long argument with the man at the roulette wheel over what I considered to be an odd number.
     Steven Wright


There are only two kinds of modern mathematics books — those you cannot read beyond the first page and those you cannot read beyond the first sentence.
     C. N. Yang