The job of the artist is always to deepen the mystery.
Francis Bacon,
1273
 
Science is but an image of the truth.
Francis Bacon,
1272
 
Mathematics is the gate and key to the sciences.
Roger Bacon,
16
 
If a man's wit be wandering, let him study the mathematics.
Francis Bacon,
1271
 
I will never be an old man. To me, old age is always 15 years older than I am.
Francis Bacon,
1270
 
Beauty itself is but the sensible image of the Infinite.
Francis Bacon,
1269
 
Simple laws can very well describe complex structures. The miracle is not the complexity of our world, but the simplicity of the equations describing that complexity.
Sander Bais, from The Equations: Icons of Knowledge.
1475
 
What a wealth, what a grandeur of thought may spring from what slight beginnings.
H. J. Baker, quoted in The World of Mathematics, by J.R. Newman.
17
 
Education is indoctrination if you're white; subjugation if you're black.
James Baldwin, quoted in My Soul Looks Back, 'Less I Forget, by Dorothy Winbush Riley.
19
 
The purpose of education...is to create in a person the ability to look at the world for himself, to make his own decisions.
James Baldwin, from "A Talk to Teachers."
18
 
I was an exceedingly shy, withdrawn, and uneasy student. Yet my teachers somehow made me believe that I could learn. And when I could scarcely see for myself any future at all, my teachers told me that the future was mine.
James Baldwin,
768
 
No set of principles can guarantee a recipe for good practice (in teaching).
D. Ball and T. Schroeder, from "Improving Teaching not Standardizing It."
20
 
Mathematicians like to know why. Unfortunately, my experiences with school arithmetic did little to promote this trait.
Thomas Banchoff, from "The Mathematician as a Child and Children as Mathematicians, Teaching Children Mathematics," vol. 6, no. 6, February 2000.
1221
 
Yes, children can be mathematicians, given the opportunity to use their mathematical minds.
Thomas Banchoff, from "The Mathematician as a Child and Children as Mathematicians, Teaching Children Mathematics," vol. 6, no. 6, February 2000.
1223
 
All of us are slaves to the prejudices of our own dimension.
Thomas Banchoff, from the Introduction to the Princeton University Press edition of Flatland.
748
 
[In Flatland] Abbott challenged his readers to imagine trying to understand the nature of phenomena in higher dimensions if all they could see directly were lower-dimensional slices. That is precisely the situation that radiologists face today as they analyze the slices produced by CAT scans or magnetic resonance imaging.
Thomas Banchoff, from the Introduction to the Princeton University Press edition of Flatland.
746
 
Of all the influences on my future mathematical career, the one that made the most difference did not come from a teacher, parent, or friend. It was a comic book!
Thomas Banchoff, from "The Mathematician as a Child and Children as Mathematicians, Teaching Children Mathematics," vol. 6, no. 6, February 2000
1222
 
The slicing technique from Flatland still remains one of the most powerful tools for dealing with aggregates in higher dimensions.
Thomas Banchoff, from the Introduction to the Princeton University Press edition of Flatland.
22
 
Today the major reason for our interest in Flatland is that for the first time we can achieve some of the dreams of our ancestors a century ago and obtain direct visual experience of phenomena in a dimension higher than our own.
Thomas Banchoff, from the Introduction to the Princeton University Press edition of Flatland.
21
 
Looking for patterns trains the mind to search out and discover the similarities that bind seemingly unrelated information together in a whole. A child who expects things to "make sense" looks for the sense in things and from this develops understanding. A child who does not see patterns often does not expect things to make sense and sees all events as discrete, separate, and unrelated.
Mary Baratta-Lorton, quoted in About Teaching Mathematics: A K-8 Resource, by Marliyn Burns.
671
 
It is fairly certain that our space if finite though unbound. Infinite space is simply a scandal to human thought.
Bishop Barnes, quoted in To Infinity and Beyond: A Cultural History of the Infinite by Eli Maor.
1500
 
Thus, although the creation of confusion seems contrary to the role of a teacher, we see that both history and psychology suggest that confusion can be of benefit to students.
Janet Heine Barnett, from "Anomalies and the Development of Mathematical Understanding"
1139
 
The Greek distinction between magnitude and number is evidence of how much resistance a culture can have in the case of accommodation, where significant restructuring of the cognitive schemata is required. In view of the amount of "good" mathematics which was produced in conjunction with this "false" intuition, we see that refraining from correcting certain student errors may not be as detrimental as is typically feared, provided the "correct" intuition is eventually achieved.
Janet Heine Barnett, from "Anomalies and the Development of Mathematical Understanding"
1140
 
I'm not young enough to know everything.
J.M. Barrie,
1699
 
We are the children as well as the mothers of invention.
John Barrow, from Pi in the Sky.
1416
 
Teaching is not a lost art, but the regard for it is a lost tradition.
Jacques Barzun, quoted in The Oxford Book of Quotations, 3rd. edition
444
 
Pedagogy, like language itself, can either liberate or imprison ideas, inspire of suffocate constructive thinking.
Hyman Bass, from "Mathematicians as Educators," in Notices of the American Mathematical Society, Vol. 44, No. 1, January 1997.
24
 
Knowing something for oneself or for communication to an expert colleague is not the same as knowing it for explanation to a student.
Hyman Bass, from "Mathematicians as Educators," in Notices of the American Mathematical Society, Vol. 44, No. 1, January 1997.
23
 
Mathematics is one of the deepest and most powerful expressions of pure human reason, and, at the same time, the most fundamental resource for description and analysis of the experiential world.
Hyman Bass, from his "Statement" as a 1999 candidate for President of the American Mathematical Society.
1189
 
Mathematics is one of the deepest and most powerful expressions of pure human reason, and, at the same time, the most fundamental resource for description and analysis of the experiential world.
Hyman Bass, from his "Statement" as a 1999 candidate for President of the American Mathematical Society.
784
 
A theory is a fantasy constrained by truth.
S. Bastian,
910
 
Most higher education is devoted to affirming the traditions and origins of an existing elite and transmitting them to new members.
Mary Catherine Bateson, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.
25
 
In necessary things, unity;
In doubtful things, liberty;
In all things, charity.
Richard Baxter,
1736
 
This play ["Proof" by David Auburn] is ultimately a love letter to mathematics, and one can only respond to its generosity in kind.
Dave Bayer, Review of Proof, Notices of the American Mathematical Society, vol. 47, no. 9, October, 2000
1141
 
Ever tried. Ever failed. No matter. Try Again. Fail again. Fail better.
Samuel Beckett,
1262
 
Books are not made for furniture, but there is nothing else that so beautifully furnishes a house.
Henry Ward Beecher,
1792
 
Books are not men and yet they stay alive.
Henry Ward Beecher,
1808
 
The whole of mathematics may be interpreted as a battle for supremacy between these two concepts [the continuous and the discrete]. This conflict may be but an echo of the older strife so prominent in early Greek philosophy, the struggle of the One to subdue the Many. But the iimage of a battle is not wholly appropriate, in mathematics at least, as the continuous and the discrete have frequently helped one another to progress.
E.T. Bell, quoted in Calculus: A Liberal Art by W.M. Priestley.
722
 
The mistakes and unresolved difficulties of the past in mathematics have always been the opportunities of its future.
E.T. Bell, quoted in "The Role of Paradoxes in the Evolution of Mathematics", by I. Kleiner and N. Movshovitz-Hadar, American Mathematical Monthly, vol. 101, no. 10, December 1994.
552
 
The average dairy farmer gets up at dawn because he has to go to work in the cow yard. I get up at dawn, too. But it is because I want to find some leaf, hung with dew; or spider web which the dew has made into the most delicate ropes of pearls… I take my camera with me, get down on my knees in the wet grass, and photograph these exquisite bits of nature. Because I do this I can show these lovely things to people who never would have seen them without my help. They will get their daily quart of milk, all right. Other farmers will attend to that. But I think I am giving them something which is just as important
W.A. Bentley, quoted in Snowflake Bentley by Jacqueline Briggs Martin
1190
 
I found that snowflakes were masterpieces of design. No one design was every repeated. When a snowflake melted… just that much beauty was gone, without leaving any record behind.
W.A. Bentley, quoted in Snowflake Bentley by Jacqueline Briggs Martin
1191
 
And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?
George Berkeley,
26
 
Since human beings have never encountered actually infinite collections of things in our material existence, all of our attempts to deal with them must involve projecting our finite experience... Therefore, we must rely on logical reasoning...and then be prepared to accept the consequences of our reasoning, regardless of whether or not they conform to our intuitive feelings.
W. P. Berlinghoff and K. E. Grant, from A Mathematical Sampler: Topics for the Liberal Arts.
27
 
The calculus is the story this [the Western] world first told itself as it became the modern world.
David Berlinski, from A Tour of the Calculus.
732
 
The definition of a limit is essentially his [Cauchy's] creation and is as much of a miracle as those fantastic Swiss clocks of the period in which hundreds of gleaming cogs are made to celebrate not only the time and date but the phases of the moon.
David Berlinski, from A Tour of the Calculus.
733
 
The concepts of mathematics, despite their unfamiliarity, are infinitely accessible. At their deaths, those who have minded mathematics will have known the continuous functions better than the crooked human heart. That so abstract a consideration should in the end be so lucid is a source of wonder.
David Berlinski, from A Tour of the Calculus.
736
 
But now a professional secret must be imparted. The concept of a limit is simple. It is the definition that is complex. The concept involves nothing more obscure than the idea of getting closer and closer to something. It suggests the attempt by one human being to approach another: and the inexpungeable thing in love as in mathematics is that however the distance decreases, it often remains what it always was, which is to say, hopelessly poignant because hopelessly infinite.
David Berlinski, from A Tour of the Calculus.
735
 
The calculus serves to demonstrate with an eerie aptness the extent to which ordinary concepts are not ordinary at all. Simple speed seems a concept on the margins of the infinite, and yet the strangest thing of all, stranger by far than those black holes in space, is the fact that the cat's cradle of words that Cauchy offered the world [as a definition of limit] is sufficient to purge speed of its paradoxes.
David Berlinski, from A Tour of the Calculus.
734
 
We are finite creatures, bound to this place and this time, and helpless before an endless expanse. It is within the calculus that for the first time the infinite is charmed into compliance, its luxuriance subordinated to the harsh concept of a limit.
David Berlinski, from A Tour of the Calculus
1164
 
If the calculus is much like a cathedral, its construction the work of centuries, it remained until the nineteenth century a cathedral suspiciously suspended in midair, the thing simply hanging there, with no one absolutely convinced that one day the gorgeous and elaborate structure would not come crashing down and fracture in a thousand pieces.
David Berlinski, from A Tour of the Calculus
1167
 
In its largest, most architectural aspect, the calculus is a great, even spectacular theory of space and time, a demonstration that in the real numbers there is an instrument adequate to their representation. If science begins in awe as the eye extends itself throughout the cold space, past the girdle of Orion and past the galaxies pinwheeling on their axes, then in the calculus mankind has created an instrument commensurate with its capacity to wonder.
David Berlinski, from A Tour of the Calculus
1166
 
Humped, ancient, and austere, Euclidean geometry is a static theory and thus to some degree a stagnant theory; within its confines, everything remains the same, and from its lucid mirror no form of change is ever shown.
David Berlinski, from A Tour of the Calculus
1165
 
Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minutia
And in narrowest limits no limit in here.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!
Jacques Bernoulli, quoted in To Infinity and Beyond by Eli Maor.
650
 
But just as much as it is easy to find the differential [derivative] of a given quantity, so it is difficult to find the integral of a given differential. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not.
Johann Bernoulli,
28
 
The mind that is not baffled is not employed.
The impeded stream is the one that sings.
Wendell Berry,
1775
 
Whether we and our politicians know it or not, Nature is party to all our deals and decisions, and she has more votes, a longer memory, and a sterner sense of justice than we do.
Wendell Berry,
1709
 
It may be when we no longer know what to do, we have come to our real work, and that when we no longer know which way to go, we have begun our real journey.
Wendell Berry, quoted in Coming to Our Senses by Jon Kabat-Zinn
805
 
Determinism, like the Queen of England, reigns -- but does not govern.
Michael Berry, quoted in Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, by Manfred Schroder.
567
 
For the Future
Planting trees early in spring,
we make a place for birds to sing
in a time to come. How do we know?
They are singing here now.
There is no other guarantee
That singing will ever be.
Wendell Berry,
1204
 
Math is an exceedingly cruel profession. You must notice that if somebody has a bachelor's degree in chemistry, he describes himself as a chemist. But if somebody has been a professor of mathematics for 10 years and you ask him/her, "Are you a mathematician?" he/she may say, "I'm trying to be one!"
L. Bers,
1207
 
The Infinitesimal Calculus, though it cannot wholly dispense with infinite, ... contrives to hide it away before facing the world. Cantor has abandoned this cowardly policy, and has brought the skeleton out of its cupboard... like many skeletons, it was wholly dependent on its cupboard, and vanished in the light of day.
Bertrand Russell, quoted in Using History to Teach Mathematics: An International Perspective, edited by Victor Katz
1138
 
[Richard] Feynman depressed is just a little more cheerful than any other person when he is exuberant.
Hans Bethe, quoted in QED and the Men Who Made It by Silvan S. Schweber.
1408
 
Hurt not the earth, neither the sea, nor the trees.
The Bible, Revelation 7:3.
628
 
Equalizing opportunity through universal higher education subjects the whole population to the intellectual mode natural only to a few. It violates the fundamental egalitarian principle of respect for the differences between people.
Caroline Bird, quoted in The Beacon Book of Quotations by Women, edited by Rosalie Maggio.
29
 
Mathematics is the codified body of all logical thought.
George David Birkhoff,
1659
 
Analysis... would lose immensely in beauty and balance and would be forced to add very hampering restrictions to truths which would hold generally otherwise, if... imaginary quantities were to be neglected.
G. Birkhoff, quoted in "Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)," by Israel Kleiner, Mathematics Teacher, Oct. 1988.
30
 
Mathematics is the codified body of all logical thought.
George David Birkhoff,
1380
 
Everything tries to be round.
Black Elk, quoted in "A Quote a Day Educates" by Monte Zerger, Mathematical Intelligencer, vol. 20, no. 2, Spring 1998.
683
 
Basically, I'm not interested in doing research and I never have been. I'm interested in understanding, which is quite a different thing. And often to understand something you have to work it out yourself because no one else has done it.
David Blackwell,
1561
 
If the doors of perception were cleansed, everything would be seen as it is, infinite.
William Blake, quoted in Listening to Nature by Joseph Cornell.
767
 
To see the world in a grain of sand,
And a heaven in a wild flower;
Hold infinity in the palm of your hand,
And eternity in an hour.
William Blake,
537
 
What is now proved was once only imagined.
William Blake,
1249
 
In reality, no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics. Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding. To understand what they learn, they must enact for themselves verbs that permeate the mathematics curriculum: "examine," "represent," "transform," "solve," "apply," "prove," "communicate." This happens most readily when students work in groups, engage in discussion, make presentations, and in other ways take charge of their own learning.
Mathematical Science Educational Board, from Everybody Counts: A Report to the Nation on the Future of Mathematics Education, National Academy Press, 1989, pp. 58-59
853
 
Somebody came up to me after a talk I had given, and say, "You make mathematics seem like fun." I was inspired to reply, "If it isn't fun, why do it?"
Ralph P. Boas, quoted in Mathematics: A Human Endeavor by Harold R. Jacobs.
32
 
...by a phenomenon that everybody who teaches mathematics has observed: the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.) The average student does not really learn to add fractions in an arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in a calculus class either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on throughout the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it. This is not just because each previous teacher did such a rotten job. It is because there is not time for enough practice on each new topic; and even it there were, it would be insufferably dull.
Ralph P. Boas,
31
 
Also, if examined "objectively," Euclid's work ought to have been any educationist's nightmare. The work presumes to begin from a beginning; that is, it presupposes a certain level of readiness, but makes no other prerequisites. Yet it never offers any "motivations," it has no illuminating "asides," it does not attempt to make anything "intuitive," and it avoids "applications" to a fault. It is so "humorless" in its mathematical purism that, although it is a book about "Elements," it nevertheless does not unbend long enough in its singlemindedness to make the remark, however incidentally, that if a rectangle has a base of 3 inches and a height of 4 inches then it has an area of 12 square inches. Euclid's work never mentions the name of a person; it never makes a statement about, or even an (intended) allusion to, genetic developments of mathematics; it makes no cross references, except once, the exception being in proposition 2 of Book 13, where the text refers to, and repeats the content of, the "first theorem of the tenth book," which, as it happens, is Euclid's "substitute" for the later axiom of Archimedes. Euclid has a fixed pattern for the enunciation of a proposition, and, through the whole length of 13 books, he is never tempted to deviate from it. In short, it is almost impossible to refute an assertion that the Elements is the work of an insufferable pedant and martinet... Euclid's work became one of the all-time best sellers. According to "objective" Pestalozzi criteria, it should have been spurned by students and "progressive" teachers in every generation. But it nevertheless survived intact all the turmoils, ravages, and illiteracies of the dissolving Roman Empire, of the early Dark Ages, of the Crusades, and of the plagues and famines of the later Middle Ages. And, since printing began, Euclid has been printed in as many editions, and in as many languages, as perhaps no other book outside the Bible
Salomon Bochner, from The Role of Mathematics in the Rise of Science
1037
 
A great truth is a statement whose opposite is also a great truth.
Niels Bohr, quoted in Mind Tools: The Five Levels of Mathematical Reality by Rudy Rucker.
527
 
In the Institute of Copenhagen, we used often to comfort ourselves with jokes, among them the old saying of the two kinds of truth. To the one kind belong statements so simple and clear that the opposite assertion obviously could not be defended. The other kind, the so-called "deep truths," are statements in which the opposite also contains deep truth.
Neils Bohr, quoted in From Erso to Gaia by Freeman Dyson (p. 188).
1572
 
Mathematical discoveries, like springtime violets in the woods, have their season which no human can hasten or retard.
John Bolyai, quoted in "Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)," by Israel Kleiner, Mathematics Teacher, Oct. 1988.
33
 
I have discovered such wonderful things that I was amazed... Out of nothing I have created a strange new universe.
Janos Bolyai, quoted in The Magic of Mathematics, by Theoni Pappas.
492
 
[On the use of imaginaries…] It was a wild thought, in the judgment of many; and I too was for a long time of the same opinion. The whole matter seemed to rest on sophistry rather than on truth. Yet I sought so long, until I actually proved this to be the case.
Enrico Bombelli, "Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)" by Israel Kleiner, Mathematics Teacher, October 1988, pp. 583-92.
1703
 
People believe in God because the world is very complicated and they think it is very unlikely that anything as complicated as a flying squirrel or the human eye or a brain could happen by chance. But they should think logically and if they thought logically they would see that they can only ask this question because it has already happened and they exist. And there are billions of planets where there is no life, but there is no one on those planets with brains to notice. And it is like if everyone in the world was tossing coins eventually someone would get 5,698 heads in a row and they would think they were very special. But they wouldn't be because there would be millions of people who didn't get 5,698 heads.
Christopher Boone, character from The Curious Incident of the Dog in the Night-Time by Mark Haddon.
1295
 
The axioms of a group are short and natural, taking less than a line to write down and accounting for the natural notion of the symmetries of things. Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists… It is as if one started to explore a small muddy hole in the ground and found after following difficult narrow passages for several miles that it eventually opened out into a vast cavern whose walls were covered with crystals.
Richard Ewen Borcherds,
1515
 
This is one way to distinguish good science and knowledge from bad: an advanced civilization on Alpha Centauri will have equivalent forms of general relativity, Galois theory, and so on, but it seems unlikely to me that they have anything resembling postmodernism or our religious stories.
Richard Ewen Brocherds,
1516
 
In high school my favorite subject was mathematics. But high school mathematics is not very interesting, so I started to read books about mathematics on my own, such as G.H. Hardy and Edward Wright's The Theory of Numbers.
Richard Ewen Borcherds,
1514
 
'Modernism,' the confluence of historical, artistic, scientific, psychological, and philosophical currents that breached the banks of Western civilization after the turn of the century, has been the focus of countless investigations. Although historians have pondered the influences and effects of this cultural upheaval, particularly in the plastic arts, few have been guileless, or perhaps foolish, enough to suggest any simple reason for society's sharp turn and quick acceptance of a radical new visual paradigm at that particular point in time… There would seem to be little to relate, and nothing to unite, the diverse pantheon of artists and architects mentioned above, except the obvious fact of their birth in the second half of the nineteenth century. There was, however, an international force for change of dramatic potency that never appears in discussions about the roots of modern art… It was the seed pearl of the modern era and it was called kindergarten.
Norman Bosterman, from Inventing Kindergarten
1553
 
Unfortunately, kindergarten for us, and for most of the generations born in this century, was a distortion, a diluted version of what originated as a radical and highly spiritual system of abstract-design activities intended to teach the recognition and appreciation of natural harmony. Kindergarten's universal, perfect, alternative language of geometric form cultivated from children's innate ability to observe, reason, express, and create. Its ultimate aim was to instill in children an understanding of what an earlier generation would have called "the music of the spheres" -the mathematically generated logic underlying the ebb and flow of creation. The explicit, programmatic goal of the early kindergartens was to awaken the sense to what Froebel considered to the God-given structure underlying all growth -animal, vegetable, mineral- in nature. The gifts were intended to be nothing less than a model of universal perfection and the key to recognizing one's place in the natural continuum. Froebel believed that learning the sacred language of geometry in youth would provide a common ground for all people.
Norman Bosterman, from Inventing Kindergarten
1554
 
The triumphant breakthroughs of modern science and mathematics, from relativity theory to the foundations of molecular genetics, have shared the virtues of elegance, economy, clarity, and simplicity, no matter how counterintuitive the discoveries may have been. Why then should mathematics and science be taught in our schools as laden with, and characterized by, the obscure, the complex, the incomprehesible, and the difficult? Here again, one solution lies in the active use of the epistemologically sophisticated linguistic capacities of all learners -- their command of ordinary language.
Leon Botstein, from "Foreward: The Ordinary Experience of Writing", in Writing to Learn Mathematics and Science, edited by Paul Connolly and Teresa Vilardi.
718
 
Major paradoxes provide food for logical thought for decades and sometimes centuries.
Nicholas Bourbaki, quoted in More Joy of Mathematics, by Theoni Pappas.
500
 
When analyzing arbitrary "if / then" rules, many other researchers found that fewer than one-quarter of those tested offer logically correct answers...when analyzing a "social contract" ...70 to 90 percent of volunteers accurately pick out cheaters on a Wasou test.
Bruce Bower, from "Roots of Reason," Science News, Vol 145, January 29, 1994.
34
 
Say we accidentally kill one mouse here. That means all the future families of this one particular mouse are destroyed, right? And all the families of the families of the families of that one mouse! With a stamp of your foot, you annihilate first one, then a dozen, then a thousand, a million, a billion possible mice!...So what? Well, what about the foxes that'll need those mice to survive? For want of ten mice, a fox dies. For want of ten foxes a lion starves. For want of a lion, all manner of insects, vultures, infinite billions of life forms are thrown into chaos and destruction. Eventually it all boils down to this: fifty-nine million years later, a caveman, one of a dozen on the entire world, goes hunting wild boar or saber-toothed tiger for food. But you, friend, have stepped on all the tigers in that region. By stepping on one single mouse. So the caveman starves. And the caveman, please note, is not just any expendable man, no! He is an entire future nation. From his loins would have sprung ten sons. From their loins one hundred sons, and thus onward to a civilization. Destroy this one man, and you destroy a race, a people, an entire history of life. It is comparable to slaying some of Adam's grandchildren. The stomp of your foot, on one mouse, could start an earthquake, the effects of which could shake our earth and destinies down through Time, to their very foundations. With the death of that one caveman, a billion others yet unborn are throttled in the womb. Perhaps Rome never rises on its seven hills. Perhaps Europe is forever a dark forest, and only Asia waxes healthy and teeming. Step on a mouse and you crush the Pyramids. Step on a mouse and you leave your print, like a Grand Canyon, across Eternity. Queen Elizabeth might never be born, Washington might not cross the Delaware, there might never be a United States at all. So be careful. Stay on the Path. Never step off!
Ray Bradbury, through the character Travis in 'A Sound of Thunder'
1581
 
'No, it can't be. Not a little thing like that. No!' Embedded in the mud, glistening green and gold and black, was a butterfly, very beautiful and very dead. 'Not a little thing like that! Not a butterfly!' cried Eckels. It fell to the floor, an exquisite thing, a small thing that could upset balances and knock down a line of small dominoes and then big dominoes and then gigantic dominoes, all down the years across Time. Eckels' mind whirled. It couldn't change things. Killing one butterfly couldn't be that important! Could it?
Ray Bradbury, from 'A Sound of Thunder'
1582
 
Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself.
Claude Bragdon, quoted in "A Quote a Day Educates" by Monte Zerger, Mathematical Intelligencer, vol. 20, no. 2, Spring 1998.
684
 
Make visible what, without you, might perhaps never have been seen.
Robert Bresson,
916
 
The true power of calculus lies in its coupling with infinite processes. Mathematics as we know it and as it has come to shape modern science could never have come into being without a reckless disregard for the dangers of the infinite.
Dave Bressoud, from A Radical Approach to Real Analysis
1145
 
The crisis struck four days before Christmas 1807. The edifice of calculus was shaken to its foundations… The nineteenth century would see ever expanding investigations into the assumptions of the calculus, an inspection and refitting of the structure from the footings to the pinnacle, so thorough a reconstruction that the calculus would be given a new name: analysis. Few of those who witnessed the incident of 1807 would have recognized mathematics as it stood 100 years later.
David Bressoud, from A Radical Approach to Real Analysis.
1562
 
With the development in the mid-1970's of the CAT scan, computer based technologies have revolutionized the field of medical diagnosis.
Entry from Encyclopedia Britanica, From Encyclopedia Britanica, 1995, under "Brain Scanning."
111
 
Statistics are human beings with the tears wiped away.
Paul Brodeur,
603
 
It is not therefore the monopoly of the man who wrote the poem or who made the discovery… The poem or the discovery exists in two moments of vision: the moment of appreciation as much as that of creation; for the appreciator must see the movement, wake to the echo, which was started in the creation of the work… We do not merely nod over someone else's work. We re-enact the creative act, and we ourselves make the discovery again. At bottom, there is no unifying likeness there until we too have seized it, we too have made it for ourselves.
Jacob Bronowski, Science and Human Values, p. 9.
1785
 
The basis for poetry and scientific discovery is the ability to comprehend the unlike in the like and the like in the unlike.
Jacob Bronowski,
35
 
Science is not a mechanism but a human progress, and not a set of findings but a search for them.
Jacob Bronowski, quoted in "Mathematics: an integral part of our culture" by Harald M. Ness, Jr., from Essays in Humanistic Mathematics.
501
 
Mathematicians, like the rest of us, cherish clever ideas; in particular they delight in an ingenious picture. But this appreciation does not overwhelm a prevailing skepticism. After all, a diagram is - at best - just a special case and so can't establish a general theorem. Even worse, it can be downright misleading. Though not universal, the prevailing attitude is that pictures are really no more than heuristic devices; they are psychologically suggestive and pedagogically important - but they prove nothing. I want to oppose this view and to make a case for pictures having a legitimate role to play as evidence and justification - a role well beyond the heuristic. In short, pictures can prove theorems.
James Robert Brown, from Proofs Without Words II
1212
 
Ignorance is not innocence but sin.
Robert Browning,
1043
 
Each human being is both a poet and a mathematician.
Scott Buchanan, from Poetry and Mathematics, p. 7.
1569
 
The aim of the liberal arts is insight, understanding, imagination, and finally the transformation of the student into his own teacher and the teacher of others. The result of liberal education is lifelong learning and teaching.
Scott Buchanan, from Poetry and Mathematics, p. 21.
1570
 
The structures with which mathematics deals are more like lace, the leaves of trees and the play of the light and shadow on a human face than they are like buildings and machines, the least of their representatives.
Scott Buchanan,
36
 
Mathematics suffers much, but most of all from its teachers. As a result of bad pedagogy - and I mean the kind often judged best by administrative pedagogues - the appearance of an algebraic formula, a geometrical figure, or an innocent set of symbols reduces the reader to an unbecoming attitude of hypocritical humility. A great many sometime students of mathematics try to persuade themselves that they haven't mathematical minds, when as a matter of fact they have only had nonmathematical teachers. Mathematics is not what most teachers of mathematics teach. They, with the good intention of conveying what they themselves have only as a skill of manipulation, have unconsciously worked hocus-pocus on their pupils. They have repeated and illustrated opaque formulae, sometimes to the admiration, but almost always to the bewilderment, of their students.
Scott Buchanan, From Poetry and Mathematics.
1614
 
The rediscovery of the liberal arts could be the much-needed beginning of the reconstruction of education in this country.
Scott Buchanan, from Poetry and Mathematics, p. 21.
1571
 
What we are today comes from our thoughts of yesterday, and our present thoughts build our life of tomorrow: Our life is the creation of our mind.
Buddha,
809
 
Mathematics is not a way of hanging numbers on things so that quantitative answers to ordinary questions can be obtained. It is a language that allows one to think about extraordinary questions...(And) getting the picture does not mean writing out the formula or crunching the numbers, it means grasping the mathematical metaphor.
James Bullock, from "Literacy in the Language of Mathematics," American Mathematical Monthly, Oct. 1994.
37
 
The only thing necessary for the triumph of evil is for good (people) to do nothing.
Edmund Burke, from Words I Wish I Wrote by Robert Fulgham
947
 
Seeking patterns is a way of thinking that is essential for making generalizations, seeing relationships, and understanding the logic and order of mathematics. Functions evolve from the investigation of patterns and unify the various aspects of mathematics.
Marilyn Burns, from About Teaching Mathematics.
594
 
Don't go through life, grow through life.
Eric Butterworth,
796
 
Who then will explain the explanation? Who then will explain the explanation?
Lord Byron,
38
 
The power of Thought, the magic of the Mind!
Lord Byron,
1806
 

115 quotes found and displayed.