Mathematics is the gate and key to the sciences.
Roger Bacon,
16
 
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
 
I was an exceedingly shy, withdrawn, and uneasy student.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
 
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
 
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
 
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
 
[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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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 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
 
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
 
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
 
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
 
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
 
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
 
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 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
 
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
 
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
 
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
 
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 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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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 it 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 single-mindedness 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... In short, it is almost impossible to refute an assertion that the Elements is the work of an insufferable pedant and martinet.
S. Bochner, quoted in Calculus: A Liberal Art by W.M. Priestley
1183
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 
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
 

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