Quotes About Mathematics

. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories -- specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms (like functions) form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.

Some people think that mathematics is a serious business that must always be cold and dry; but we think mathematics is fun, and we aren't ashamed to admit the fact. Why should a strict boundary line be drawn between work and play? Concrete mathematics is full of appealing patterns; the manipulations are not always easy, but the answers can be astonishingly attractive.

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.

A chess problem is genuine mathematics, but it is in some way "trivial" mathematics. However, ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. The best mathematics is serious as well as beautiful-"important" if you like, but the word is very ambiguous, and "serious" expresses what I mean much better.

No one really understood music unless he was a scientist, her father had declared, and not just a scientist, either, oh, no, only the real ones, the theoreticians, whose language was mathematics. She had not understood mathematics until he had explained to her that it was the symbolic language of relationships. "And relationships," he had told her, "contained the essential meaning of life."

[The error in the teaching of mathematics is that] mathematics is expected either to be immediately attractive to students on its own merits or to be accepted by students solely on the basis of the teacher's assurance that it will be helpful in later life. [And yet,] mathematlcs is the key to understanding and mastering our physical, social and biological worlds.

The mathematics is the odd one, odd because I'm not sure how to measure its effect. It is so fundamental to my outlook on everything and yet I'm not even sure how. It must be because in my formative years it was everything to me, the single place of beauty in my life, and of breathtaking beauty at that. I still believe that pure mathematics is the most creative thing that humanity does, though I am no longer a part of it.

To create a language all of a piece which would be a women's language, that I find quite insane. There does not exist a mathematics which is only a women's mathematics, or a feminine science.

Though the structures and patterns of mathematics reflect the structure of, and resonate in, the human mind every bit as much as do the structures and patterns of music, human beings have developed no mathematical equivalent to a pair of ears. Mathematics can only be "seen" with the "eyes of the mind". It is as if we had no sense of hearing, so that only someone able to sight read music would be able to appreciate its patterns and harmonies.

The proof of Fermat's Last Theorem underscores how stable mathematics is through the centuries - how mathematics is one of humanity's long continuous conversations with itself.

For me, rhythm is a type of divine mathematics in a way. No matter where you're from, we can all understand the mathematics of rhythm. I try to apply this mathematical thinking to my playing.

Mathematics was born and nurtured in a cultural environment. Without the perspective which the cultural background affords, a proper appreciation of the content and state of present-day mathematics is hardly possible.

Philosophically, mathematics is not a part of science. Mathematics studies patterns, science studies nature

Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems-general and specific statements-can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.

I'm sorry to say that the subject I most disliked was mathematics. I have thought about it. I think the reason was that mathematics leaves no room for argument. If you made a mistake, that was all there was to it.

The subject for which I am asking your attention deals with the foundations of mathematics. To understand the development of the opposing theories existing in this field one must first gain a clear understnding of the concept "science"; for it is as a part of science that mathematics originally took its place in human thought.

Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God.

Mathematical economics is old enough to be respectable, but not all economists respect it. It has powerful supporters and impressive testimonials, yet many capable economists deny that mathematics, except as a shorthand or expository device, can be applied to economic reasoning. There have even been rumors that mathematics is used in economics (and in other social sciences) either for the deliberate purpose of mystification or to confer dignity upon common places as French was once used in diplomatic communications.

Probably no branch of mathematics has experienced a more surprising growth than has... topology... Considered as a most specialized and abstract subject in the early 1920's, it is today [1938] an indispensable equipment for the investigation of modern mathematical theories.

Taking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half.

We are servants rather than masters in mathematics.

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

In mathematics, two angles that are said to coincide fit together perfectly. The word "coincidence" does not describe luck or mistakes. It describes that which fits together perfectly.

Nature imitates mathematics.

Mathematics, even in its present and most abstract state, is not detached from life. It is just the ideal handling of the problems of life.

Mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations.

If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither "number" nor "size", but always form. And among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things.

The importance of the "New Mathematics" lies mainly in the fact that it has taught us the difference between the disc and the circle.

Thinking - in particular abstract thinking, which most of us are introduced to through the study of mathematics and literature - helps us learn that we can become problem solvers.

Perhaps the most surprising thing about mathematics is that it is so surprising.

For most people, the major hurdle in grasping modern insights into the nature of the universe is that these developments are usually phrased using mathematics.

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.

There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned.

The more reasonable a student was in mathematics, the more unreasonable she was in the affairs of real life, concerning which fewtrustworthy postulates have yet been ascertained.

One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.

I think the biggest misconception is that everybody has to learn mathematics. That seems to be a complete mistake.

My occupation is an open question. I was once an assistant professor of mathematics. Since then, I have spent time living in the woods of Montana.

Man is full of desires: he loves only those who can satisfy them all. "This man is a good mathematician," someone will say. But I have no concern for mathematics; he would take me for a proposition. "That one is a good soldier." He would take me for a besieged town. I need, that is to say, a decent man who can accommodate himself to all my desires in a general sort of way.

We have these amazing gifts of music and mathematics and painting and Olympic running. I mean, we're the animal that is best of all the animals at long-distance running. Why? It is quite amazing. Superfluous gifts you don't really need to survive.

I would not dare to say that there is a direct relation between mathematics and madness, but there is no doubt that great mathematicians suffer from maniacal characteristics, delirium and symptoms of schizophrenia.

If the proof starts from axioms, distinguishes several cases, and takes thirteen lines in the text book ... it may give the youngsters the impression that mathematics consists in proving the most obvious things in the least obvious way.

Mathematics is a field which has often been compared with chess, but differs from the latter in that it is only one's best moments that count and not one's worst.

When we cannot use the compass of mathematics or the torch of experience...it is certain we cannot take a single step forward.

Things like the financial markets - a proper grounding in mathematics could help the common man. I believe that if people are more familiar with mathematical concepts... it can help deal with modern life, which is increasingly complex.

I finally reached the conclusion that mathematics was the study I was best fitted to follow, though I did not clearly see in what way I should turn the subject to account.

Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional propositions a priori, and it will be your task to find out the false deduction in each case.

On set, the playground for the character, how much it takes varies. Is it like ballet, is it like jazz? The content always lends itself to the form, and it's really not mathematics.

Most of the arts, as painting, sculpture, and music, have emotional appeal to the general public. This is because these arts can be experienced by some one or more of our senses. Such is not true of the art of mathematics; this art can be appreciated only by mathematicians, and to become a mathematician requires a long period of intensive training. The community of mathematicians is similar to an imaginary community of musical composers whose only satisfaction is obtained by the interchange among themselves of the musical scores they compose.