Quotes & Sayings About Mathematical Problems

The mathematical question is "Why?" It's always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it. So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems. — Paul Lockhart

One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution. — George Polya

Mathematical thinking is not the same as doing mathematics - at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box - a valuable ability in today's world. — Keith Devlin

Yes, I share your concern: how to program well -though a teachable topic- is hardly taught. The situation is similar to that in mathematics, where the explicit curriculum is confined to mathematical results; how to do mathematics is something the student must absorb by osmosis, so to speak. One reason for preferring symbol-manipulating, calculating arguments is that their design is much better teachable than the design of verbal/pictorial arguments. Large-scale introduction of courses on such calculational methodology, however, would encounter unsurmoutable political problems. — Edsger Dijkstra

By the time the average person finishes college, he or she will have taken over 2,600 tests, quizzes, and exams. The right answer approach becomes deeply ingrained in our thinking. This may be fine for some mathematical problems where there is in fact only one right answer. The difficulty is that most of life isn't this way. Life is ambiguous; there are many right answers- all depending on what you're looking for. But if you think there is only one right answer, then you'll stop looking as soon as you find one. — Roger Von Oech

Even with the most stupid video games, kids learn more about learning than they ever did before, because they want to learn codes and moves before other kids figure them out. They're motivated to seek out someone or search the Net for help. A student who makes a video game has to solve mathematical problems to make special effects happen on the screen. — Seymour Papert

If there is a problem you can't solve, then there is an easier problem you can't solve: find it. — George Polya

The mere formulation of a problem is far more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science. — Albert Einstein

Tengo's lectures took on uncommon warmth, and the students found themselves swept up in his eloquence. He taught them how to practically and effectively solve mathematical problems while simultaneously presenting a spectacular display of the romance concealed in the questions it posed. Tengo saw admiration in the eyes of several of his female students, and he realized that he was seducing these seventeen- or eighteen-year-olds through mathematics. His eloquence was a kind of intellectual foreplay. Mathematical functions stroked their backs; theorems sent warm breath into their ears. — Haruki Murakami

Our modern conception of the average person is not a mathematical truth but a human invention, created a century and a half ago by two European scientists to solve the social problems of their era. — Todd Rose

Mathematics is about problems, and problems must be made the focus of a student's mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process - having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work. — Paul Lockhart

The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules ... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms. — Kurt Godel

The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression,more exact,compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write. — H.G.Wells

I could quote enough Nietzsche to bore someone into a coma, solve mathematical problems so beautiful they'd make Pythagoras cry; I could talk so much bullshit the listener didn't know if I was coming or going, but I had gotten to know Rickie well enough to realize that I had no idea how a woman's mind worked. — T.J. Forrester

The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus, whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe. — Joseph Fourier

Ken appeared, was taller than she, wanted her, was acceptable and accepted on all sides; similarly, nagging mathematical problems abruptly crack open. Foxy could find no fault with him, and this challenged her, touched off her stubborn defiant streak. She felt between his handsomeness and intelligence a contradiction that might develop into the convoluted humour of her Jew. Ken looked lika a rich boy and worked like a poor one. From Farmington, he was the only son of a Hartford laywer who never lost a case. Foxy came to imagine his birth as cool and painless, without a tear or outcry. Nothing puzzled him. There were unknowns, but no mysteries. ( ... ) He was better-looking, better-thinking, a better machine. — John Updike

He'd met other prodigies in mathematical competitions. In fact he'd been thoroughly trounced by competitors who probably spent literally all day practising maths problems and who'd never read a science-fiction book and who would burn out completely before puberty and never amount to anything in their future lives because they'd just practised known techniques instead of learning to think creatively. (Harry was something of a sore loser.) — Eliezer Yudkowsky

Mathematics was actually a logical puzzle with endless variations - riddles that could be solved. The trick was not to solve arithmetical problems. Five times five would always be twenty-five. The trick was to understand combinations of the various rules that made it possible to solve any mathematical problem whatsoever. — Stieg Larsson

It is an unfortunate fact that proofs can be very misleading. Proofs exist to establish once and for all, according to very high standards, that certain mathematical statements are irrefutable facts. What is unfortunate about this is that a proof, in spite of the fact that it is perfectly correct, does not in any way have to be enlightening. Thus, mathematicians, and mathematics students, are faced with two problems: the generation of proofs, and the generation of internal enlightenment. To understand a theorem requires enlightenment. If one has enlightenment, one knows in one's soul why a particular theorem must be true. — Herbert S. Gaskill

Before a kid learns how to use a computer that can solve mathematical problems, he or she should know how to do arithmetic without a computer. — Andy Rooney

More people have access to education today than ever before. But I cannot help but feel that the modern educational experience is not preparing us adequately to attend the rich banquet of life. Certainly the young people of today have mastered the use of technology and are capable of solving complex scientific and mathematical problems, but who and what do these serve if they cannot think for themselves? If they have no understanding of the meaning and purpose of their own lives? If they do not know who they are as individuals? — Matthew Kelly

Our teaching of mathematics revolves around a fundamental conflict. Rightly or wrongly, students are required to master a series of mathematical concepts and techniques, and anything that might divert them from doing so is deemed unnecessary. Putting mathematics into its cultural context, explaining what is has done for humanity, telling the story of its historical development, or pointing out the wealth of unsolved problems or even the existence of topics that do not make it into school textbooks leaves less time to prepare for the exam. So most of these things aren't discussed. — Ian Stewart

All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. — James C. Maxwell

The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems
the determination of the diagonal of a square and that of the circumference of a circle
revealed the existence of new mathematical beings for which no place could be found within the rational domain. — David Van Dantzig

Elegance; it lacks the intellect -
he is not a dancer who dances to the beat of love - on the other hand, the heart, can not solve mathematical problems. — Kristian Goldmund Aumann

For most problems found in mathematics textbooks, mathematical reasoning is quite useful. But how often do people find textbook problems in real life? At work or in daily life, factors other than strict reasoning are often more important. Sometimes intuition and instinct provide better guides; sometimes computer simulations are more convenient or more reliable; sometimes rules of thumb or back-of-the-envelope estimates are all that is needed. — Lynn Steen

her on the dorm board. We had only twenty girls in a batch of two hundred. Goodlooking ones were rare; girls don't get selected to IIM for their looks. They get in because they can solve mathematical problems faster than 99.99% of India's population and crack the CAT. Most IIM girls are above shallow things like makeup, fitting clothes, contact lenses, removal of facial hair, body odour and feminine charm. Girls like Ananya, if and when they arrive by freak chance, become instant pin-ups in out testosterone-charged, estrogen-starved campus. — Anonymous

WORK MAKES MEN. A university is not merely a place for making scholars, it is a place for making Christians. A farm is not a place for growing corn, it is a place for growing character, and a man has no character except that which is developed by his life and thought. God's Spirit does the building through the acts which a man performs from day to day. A student who cons out every word in his Latin and Greek instead of consulting a translation finds that honesty is translated into his character. If he works out his mathematical problems thoroughly, he not only becomes a mathematician, but becomes a thorough man. It is by constant and conscientious attention to daily duties that thoroughness and conscientiousness and honorableness are imbedded in our beings. Character is — Henry Drummond

Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems. — George Polya

In fact, I barely missed being number one in France in both schools. In particular I did very well in mathematical problems. — Benoit Mandelbrot

Much of the apparent uniformity of Nature is a uniformity of averages. Our gross senses only take cognizance of the average effect of vast numbers of individual particles and processes; and the regularity of the average might well be compatible with a great degree of lawlessness of the individual. I do not think it is possible to dismiss statistical laws (such as the second law of thermodynamics) as merely mathematical adaptations of the other classes of law to certain practical problems. — Arthur Stanley Eddington

We have a problem and a a problem demands a solution. Problem and solution - these two terms are inseparably connected to each other. It means if a person has a problem there is a certain method - it could be mathematical, algebra - that you can apply to this problem. What you get out of that is the solution. With that the problem is over.

But it is not like that in human life. It is not like that with - now I am going to use the word myself - problems. Human problems, social problems, societal problems will never be solved... It is an illusion that some kind of problem will be solved. — Joseph Weizenbaum

I know that most men - not only those considered clever, but even those who are very clever, and capable of understanding most difficult scientific, mathematical, or philosophic problems - can very seldom discern even the simplest and most obvious truth if it be such as to oblige them to admit the falsity of conclusions they have formed, perhaps with much difficulty - conclusions of which they are proud, which they have taught to others, and on which they have built their lives. — Leo Tolstoy

The tantalizing and compelling pursuit of mathematical problems offers mental absorption, peace of mind amid endless challenges, repose in activity, battle without conflict, refuge from the goading urgency of contingent happenings, and the sort of beauty changeless mountains present to senses tried by the present day kaleidoscope of events. — Morris Kline

The classes of problems which are respectively known and not
known to have good algorithms are of great theoretical interest. [ ... ]
I conjecture that there is no good algorithm for the traveling
salesman problem. My reasons are the same as for any mathematical
conjecture: (1) It is a legitimate mathematical possibility, and
(2) I do not know. — Jack Edmonds

These forays into the real world sharpened his view that scientists needed the widest possible education. He used to say, "How can you design for people if you don't know history and psychology? You can't. Because your mathematical formulas may be perfect, but the people will screw it up. And if that happens, it means you screwed it up." He peppered his lectures with quotations from Plato, Chaka Zulu, Emerson, and Chang-tzu.

But as a professor who was popular with his students - and who advocated general education - Thorne found himself swimming against the tide. The academic world was marching toward ever more specialized knowledge, expressed in ever more dense jargon. In this climate, being liked by your students was a sign of shallowness; and interest in real-world problems was proof of intellectual poverty and a distressing indifference to theory. — Michael Crichton

A philosopher is a mathematician, who might not be good at solving mathematical problems but knows 'which one' to solve and 'why' to solve it... — Victor Ghoshe

Mathematical objects states "only the relationships between mathematically 'undefined objects' and the rules governing operations with them." It doesn't matter what mathematical things are: it's what they do that counts. Thus mathematics hovers uneasily between the real and the not-real; its meaning does not reside in formal abstractions, but neither is it tangible. This may cause problems for philosophers who like tidy categories, but it is the great strength of mathematics - what — Richard Courant

I combat the errors of ages; I meet the violence of mobs; I cope with illegal proceedings from executive authority; I cut the gordian knot of powers, and I solve mathematical problems of universities, with truth-diamond truth; and God is my 'right hand man'. — Joseph Smith Jr.

Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed. They are no exceptions to the rule that God always geometrizes. Their problems of form are in the first instance mathematical problems, their problems of growth are essentially physical problems, and the morphologist is, ipso facto, a student of physical science. — D'Arcy Wentworth Thompson

The life and soul of science is its practical application, and just as the great advances in mathematics have been made through the desire of discovering the solution of problems which were of a highly practical kind in mathematical science, so in physical science many of the greatest advances that have been made from the beginning of the world to the present time have been made in the earnest desire to turn the knowledge of the properties of matter to some purpose useful to mankind. — Lord Kelvin