The origin and immediate purpose of the introduction of complex magnitudes into mathematics lie in the theory of simple laws of dependence between variable magnitudes expressed by means of operations on magnitudes. If we enlarge the scope of applications of these laws by assigning to the variables they involve complex values, then there appears an otherwise hidden harmony and regularity.
Ebbinghaus, quoted in "Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)," by Israel Kleiner, Mathematics Teacher, Oct. 1988.
96
 
There was just one place where [Einstein’s] theory did not seem to work properly, and that was -- infinity. I think Einstein showed his greatness in the simple and drastic way in which he disposed of difficulties at infinity. He abolished infinity. He slightly altered his equations so as to make space at great distances bend round until it closed up. So that, if in Einstein’s space you keep going right on in one direction, you do not get to infinity; you find yourself back at your starting-point again. Since there was no longer infinity, there could be no difficulties at infinity.
Sir Arthur Eddington, quoted in To Infinity and Beyond by Eli Maor.
658
 
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity by Michael Guillen.
533
 
If we don't stand up for children, then we don't stand for much.
Marian Wright Edelman,
1017
 
There ain't no rules around here! We're trying to accomplish something!
Thomas Edison, quoted in The Fourth, and By Far the Most Recent, 637 Best Things Anybody Ever Said, by Robert Byrne.
95
 
There is great value in disaster. All our mistakes are burned up. Thank God we can start again. [Dec. 10, 1914, the night after a fire destroyed his life work, Edison Industries of West Orange, N.J.]
Thomas Alva Edison,
937
 
The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.
Albert Einstein, quoted in The Magic of Mathematics, by Theoni Pappas.
487
 
Teaching should be such that what is offered is perceived as a valuable gift rather than a hard duty.
Albert Einstein, from the Quotable Einsetin.
498
 
The series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences.
Albert Einstein,
611
 
It is not so very important for a person to learn facts. For that he does not really need a college. He can learn them from books. The value of an education in a liberal arts college is not the learning of many facts but the training of the mind to think something that cannot be learned from textbooks.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
634
 
Humiliation and mental oppression by ignorant and selfish teachers wreak havoc in the youthful mind that can never be undone and often exert a baleful influence in later life.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
635
 
I believe that older people who have scarcely anything to lose ought to be willing to speak out on behalf of those who are young and who are subject to much greater restraint.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
643
 
The state of mind which enables a man to do work of this kind [scientific research]... is akin to that of the religious worshiper or the lover; the daily effort comes from no deliberate intention or program, but straight from the heart.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
637
 
After a certain high level of technical skill is achieved, science and art tend to coalesce in esthetics, plasticity, and form. The greatest scientists are always artists as well.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
638
 
Science is a wonderful thing if one does not have to earn a living at it. One should earn one’s living by work of which one is sure one is capable. Only when we do not have to be accountable to anyone can we find joy in scientific endeavor.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
640
 
People like you and I, though mortal of course, like everyone else, do not grow old no matter how long we live. What I mean is that we never cease to stand like curious children before the great Mystery into which we were born.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
642
 
Teaching should be such that what is offered is perceived as a valuable gift and not as a hard duty.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
636
 
It is the supreme art of the teacher to awaken joy in creative expression and knowledge.
Albert Einstein, quoted in A Teacher's Treasury of Quotations, by Bernard E. Farber.
109
 
In the beginning (if there was such a thing), God created Newton’s laws of motion together with the necessary masses and forces. This is all; everything beyond this follows from the development of appropriate mathematics methods by means of deduction.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
641
 
It is in fact nothing short of a miracle that the modern methods of instruction have not entirely strangled the holy curiosity of inquiry; for this delicate little plant, aside from stimulation, stands mainly in need of freedom; without this it goes to wrack and ruin without fail.
Albert Einstein,
108
 
Science is the attempt to make the chaotic diversity of our sense-experiments correspond to a logically uniform system of thought.
Albert Einstein,
107
 
We never cease to stand like curious children before the great mystery into which we are born.
Albert Einstein, quoted in From Agnesi to Zeno, by Sanderson Smith
106
 
Most of the fundamental ideas of a science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone.
Albert Einstein,
105
 
I have no particular talent. I am only inquisitive.
Albert Einstein, quoted in From Agnesi to Zeno, by Sanderson Smith.
104
 
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Albert Einstein, quoted in A Mathematical Sampler: Topics for Liberal Arts, by W.P. Berlinghoff and K.E. Grant.
103
 
How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?
Albert Einstein,
102
 
Whatever your difficulties in mathematics, I can assure you mine are far greater.
Albert Einstein,
101
 
Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.
Albert Einstein, quoted in The World of Mathematics, by J.R. Newman.
100
 
Imagination is more important than knowledge.
Albert Einstein,
99
 
But there is another reason for the high repute of mathematics, it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.
Albert Einstein,
98
 
Pure mathematics is, in its way, the poetry of logical ideas.
Albert Einstein,
97
 
Great spirits have always encountered violent opposition from mediocre minds.
Albert Einstein, quoted in A Teacher's Treasury of Quotations, by Bernard E. Farber.
110
 
In the middle of difficulty lies opportunity.
Albert Einstein,
1051
 
The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms.
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
639
 
Whoever undertakes to set himself up as a judge of Truth and Knowledge is shipwrecked by the laughter of the gods.
Albert Einstein,
983
 
Logic will get you from A to B. Imagination will take you everywhere.
Albert Einstein,
833
 
The truth of a theory is in your mind, not in your eyes.
Albert Einstein,
862
 
O, Youth: Do you know that yours is not the first generation to yearn for a life full of beauty and freedom? Do you know that all your ancestors have felt the same as you do -- and fell victim to trouble and hatred? Do you know also that your fervent wishes can only find fulfillment if you succeed in attaining love and understanding of people, and animals, and plants, and stars, so that every joy becomes your joy and every pain your pain?
Albert Einstein, quoted in The Quotable Einstein edited by Alice Calaprice.
644
 
We merely want to live in peace with all the world, to trade with them, to commune with them, to learn from their culture as they may learn from ours, so that the products of our toil may be used for our schools and our roads and our churches and not for guns and planes and tanks and ships of war.
Dwight D. Eisenhower,
848
 
Every gun that is made, every warship launched, every rocket fired signifies in the final sense, a theft from those who hunger and are not fed, those who are cold and are not clothed. This world in arms is not spending money alone. It is spending the sweat of its laborers, the genius of its scientists, the hopes of its children. This is not a way of life at all in any true sense. Under the clouds of war, it is humanity hanging on a cross of iron.
Dwight D. Eisenhower, from American Society of Newspaper Editors, 16 April 1953
847
 
Human kind cannot bear much reality.
T.S. Eliot,
866
 
Every great and commanding moment in the annals of the world is the triumph of some enthusiasm.
Ralph Waldo Emerson, quoted in The Teacher"s Quotation Book, edited by Wanda Lincoln and Murray Suid.
629
 
A day is a miniature eternity.
Ralph Waldo Emerson,
871
 
Most of the shadows of this life are caused by standing in our own sunshine
Ralph Waldo Emerson,
933
 
Unless you try to do something beyond what you have already mastered, you will never grow.
Ralph Waldo Emerson,
912
 
We are too civil to books. For a few golden sentences we will turn over and actually read a volume of four or five hundred pages.
Ralph Waldo Emerson,
993
 
If the single man plants himself indomitably on his instincts, and there abides, the huge world will become round to him
Ralph Waldo Emerson,
927
 
When variable magnitudes entered mathematics and when their variability was extended to the infinitely small and infinitely large, then mathematics, usually so very moral, perpetrated the Fall: it ate the apple of knowledge and this opened for it it the road to gigantic achievements but also to delusions. The virgin state of absolute meaningfulness, of irrefutable provability of all things mathematical... belonged to the past. An era of discord had arrived.
Engels, quoted in In Search of Infinity by N.Ya. Vilenkin (translated by Abe Shenitzer).
729
 
If Hilbert could be so wrong about predicting [when the Riemann hypothesis, the transcendence of 2^(root 2), and Fermat’s last theorem would be solved], I think it is even more risky to make predictions now [1993].
Paul Erdos, quoted in Consortium [COMAP Newsletter], No. 71, Fall 1999.
1195
 
If Hilbert could be so wrong about predicting [when the Riemann hypothesis, the transcendence of 2^(root 2), and Fermat’s last theorem would be solved], I think it is even more risky to make predictions now [1993].
Paul Erdos, quoted in Consortium [COMAP Newsletter], No. 71, Fall 1999.
790
 
It will be another million years, at least, before we understand the primes.
Paul Erdos, quoted in "A Quote a Day Educates" by Monte Zerger, Mathematical Intelligencer, vol. 20, no. 2, Spring 1998.
680
 
Another roof, another proof.
Paul Erdos, quoted in "To Prove and Conjecture: Paul Erdos and His Mathematics", by Bela Bollobas, American Mathematical Monthly, vol. 105, no. 3, March 1998.
702
 
Babies can ask questions about primes which grown [wo]men cannot answer.
Paul Erdos, quoted in In Code by Sarah Flannery and David Flannery.
908
 
Every human activity, good or bad, except mathematics, must come to an end.
Paul Erdos, quoted in "To Prove and Conjecture: Paul Erdos and His Mathematics", by Bela Bollobas, American Mathematical Monthly, vol. 105, no. 3, March 1998.
701
 
Determination plus hard work plus concentration equals success, which equals ganas.
Jaime Escalante, quoted in Math Power: How to Help Your Children Love Math, Even if You Don't by Patricia Clark Kenschaft.
1162
 
I never got a pass mark in math... Just imagine - mathematicians now use my prints to illustrate their books. Funny me consorting with all these learned folks, as though I were their long lost brother. I guess they are unaware of the fact that I am ignorant about the whole thing.
M. C. Escher,
112
 
The laws of mathematics are not merely human inventions or creations. The simply ‘are’; they exist quite independently of the human intellect. The most that any man with a keen intellect can do is to find out that they are there and to take cognizance of them.
M.C. Escher, quoted in The Magic of Mathematics, by Theoni Pappas.
488
 
Science and art sometimes can touch one another, like two pieces of the jigsaw puzzle which is our human life, and that contact may be made across the boderline between the two respective domains.
M.C. Escher, quoted in Notebooks, Periodic Drawings, and Related Works of M.C. Escher by Doris Schattschneider.
693
 
Father [M.C. Escher] had difficulty comprehending that the working of his mind was akin to that of a mathematician. He greatly enjoyed the interest in his work by mathematicians and scientists, who readily understood him as he spoke, in his pictures, a common language. Unfortunately, the specialized language of mathematics hid from him the fact that mathematicians were struggling with the same concepts as he was.
George Escher, quoted in Notebooks, Periodic Drawings, and Related Works of M.C. Escher by Doris Schattschneider.
692
 
After all I am happy about the contact and friendship of mathematicians that resulted from it all. They have often given me new ideas, and sometimes there even is an interaction between us. How playful they can be, those learned ladies and gentlemen!
M.C. Escher, quoted in To Infinity and Beyond by Eli Maor.
655
 
By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made, I ended up in the domain of mathematics. Although I am absolutely without training in the exact sciences, I often seem to have more in common with mathematicians than with my fellow-artists.
M.C. Escher, quoted in To Infinity and Beyond by Eli Maor.
654
 
For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.
Leonhard Euler, quoted in Single Variable Calculus, by James Stewart.
113
 
After exponential quantities the circular functions, sine and cosine, should be considered because they arise when imaginary quantities are involved in the exponential.
Leonhard Euler, quoted in Theory of Complex Functions, by Reinhold Remmert.
114
 
As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of the numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards.
Euler,
1217
 
The King [Frederick II] calls me “my Professor”, and I am the happiest man in the world!
Euler, quoted in Analysis by Its History by E. Hairer and G. Wanner.
1109
 
The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such a discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.
Euler,
1218
 
Mighty is geometry; joined with art, resistless.
Euripides,
115
 
If it ain't from the heart then it can't be art;
If you ain't got proof it can't be truth;
If it ain't got legs then it can't run;
If it ain't never started then it can't be done.
Everlast, from "Whitey" on the album "Eat at Whitey's" [Whitey Ford's].
1185
 
There is a distinction between what may be called a problem and what may be considered an exercise. The latter serves to drill a student in some technique or procedure, and requires little if any, original thought... No exercise, then, can always be done with reasonbable dispatch and with a miniumum of creative thinking. In contrast to an exercise, a problem, if it is a good one for its level, should require though on the part of the student.
Howard Eves, quoted in Out of the Mouths of Mathematicians, by Rosemary Schmalz
445
 
It is impossible to overstate the imporance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps... Every new discovery in mathematics, results from an attempt to solve some problem.
Howard Eves, quoted in Out of the Mouths of Mathematicians, by Rosemary Schmalz
446
 
Getting [the] degree meant more to me than an NCAA title, being named All-American or winning an Olympic gold medal.
Patrick Ewing, quoted in My Soul Looks Back, 'Less I Forget, by Dorothy Winbush Riley.
116
 

71 quotes found and displayed.