Sayings about mathematics by great mathematicians. Sayings of great people about mathematics

Definition

The first definitions and statements about mathematics can be found in the French philosopher Rene Descartes: “It is necessary to unite under the old, well-known concept of universal mathematics everything that needs to be put in order or measured. And it doesn’t matter what the measurements are taken with, numbers or sounds, stars or figures.”

In the Soviet Union, the statement of A. N. Kolmogorov was considered traditional: “This is a science where the quantitative relationship is closely related to the real form of the surrounding world. But only in an expanded and completely abstract concept.”

Nicolas Bourbaki is a group of scientists from France who have written several books about modern science. The group was created in 1935, statements about mathematics were in the epigraph of the first edition: “The essence of this great science can be called the doctrine of the influence of objects on each other. Some properties of objects may be unknown, but they can be calculated using known, fundamental properties. It's a set of abstract structures."

Hermann Weyl doubted that it was possible to give a clear definition of mathematics at all: “The question of the foundations can be considered open. It is difficult to imagine that over time we will find a definition of mathematics that suits everyone. Since this is rather not a science, but a creative activity, like music or poetry.”

An anecdote about mathematicians and physicists on a train

Three mathematicians and three physicists are going to another city for a conference. ¬Meet in front of the ticket office at the station. The physicists' turn comes first, and they, as expected, buy a ticket per person. Mathematicians buy one ticket for everyone. - How so? - physicists are surprised. - There is a controller on the train, without tickets you will be kicked out! “Don’t worry,” the mathematicians answer, “we have a METHOD.” Before the train departs, physicists are seated in the carriages, and mathematicians are crowded into the toilet. When the inspector knocks on the door, a hand with a ticket sticks out. The controller takes the ticket, and then everyone goes to their destination without any problems. After the conference, the scientists meet again at the station. Physicists, inspired by the example of mathematicians, buy one ticket. Mathematics - none. - What will you show to the controller? - We have a METHOD. On the train, physicists are crowded into one toilet, mathematicians into another. Shortly before departure, one of the mathematicians approaches the toilet where the physicists are hiding. Knocking. A hand with a ticket sticks out. ¬The mathematician takes the ticket and returns to his colleagues. (Moral: You can't use math without understanding it.)

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Quotes about science

Statements about mathematics by great mathematicians and short quotes ask more questions than answer them:

“This is a tool for any scientist, like a scalpel for a surgeon” (N. Abel).
“On earth there is only beauty, the main thing in beauty is form, ideal form is ideal proportions, proportions consist of numbers. Conclusion: beauty is numbers” (A. Augustine).
“The main benefit of mathematics for ordinary people is that it is difficult” (A. Alexandrov).
“This is a science of rigor and clarity. Morally, it can be considered truth, which is clear and does not like fog” (L. Bers).
“Mathematics is an unshakable structure and a true prophecy” (L. Bers).

Catchphrases about mathematics (200 expressions)

One of the most important sciences in the world is mathematics. We live and breathe mathematical formulas, each of us can be decomposed into tiny components, and these pieces can be counted. Even in ancient times they said that “mathematics is the greatest of sciences,” but then scientists had no idea how right they were. Over the centuries, many phrases have been formulated that describe this science. This section contains popular expressions about mathematics.

Geometry is the art of reasoning well from poorly executed drawings.

Infinity! Nothing has moved the human mind so deeply.

Mathematics is to a scientist what a scalpel is to an anatomist.

There is nothing to like except beauty, in beauty there is nothing but forms, in forms there is nothing but proportions, in proportions there is nothing but numbers.

Mathematics is a single symphony of the infinite.

Number underlies all perception of beauty. Only when the sensation of pleasure itself is filled with definite numbers is it capable of approving equal intervals and rejecting irregular ones.

About one of his students: He became a poet - for a mathematician he did not have enough imagination.

The purpose of mathematical rigor is to sanction and legitimize the gains of intuition.

Mathematics is the study of relationships between formulas, devoid of any content.

The good thing about math is that it is difficult.

Mathematics is just a game played according to simple rules and using meaningless notations.

Nowhere, as in mathematics, does the clarity and accuracy of the conclusion allow a person to wriggle out of an answer by talking around the question.

In the huge geometry garden, everyone can choose a bouquet to suit their taste.

Mathematics teaches the accuracy of thought, submission to the logic of proof, the concept of strictly substantiated truth, and all this shapes the personality, perhaps more than music.

We are all convinced that any mathematical problem can be solved. This belief... is a great help for us in our work when we begin to solve a mathematical problem, because we hear within ourselves a constant call: here is the problem, look for a solution. You can find it with the help of pure thinking, for in mathematics there is no ignorabimus (we will not know).

The world around us is a world of geometry.

In my opinion, the real reason why... it was not possible to find an unsolvable mathematical problem is that unsolvable problems do not exist.

Morally, mathematics teaches us to be strict about what is stated as truth, what is put forward as an argument, or what is stated as evidence. Mathematics requires clarity of concepts and statements and does not tolerate fog or unsubstantiated statements.

Where else can one look for reliability and truth, if even mathematical thinking itself misfires?

Cold numbers, outwardly dry formulas of mathematics are full of inner beauty and the heat of the thought concentrated in them.

... Any framework, especially of a national nature, contradicts the spirit of mathematics. Only by absolutely not understanding our science can differences be created between people and races, and the reasons for which this was done are extremely insignificant. Mathematics does not know races... For mathematics, the entire cultural world is a single country.

The uniqueness of geometry, which distinguishes it from other branches of mathematics, and all sciences in general, lies in the inextricable organic combination of living imagination with strict logic. Geometry in its essence is spatial imagination, permeated and organized by strict logic.

Mathematics is not only a science; in its best manifestations it is closer to high art, music, poetry.

The general educational significance of a mathematics course, like any other subject, lies primarily in the general concepts that it gives and which expand a person’s horizons and ways of approaching the phenomena of life. From this point of view, mathematics is important, firstly, for its logic, consistency and accuracy of conclusions. Secondly, the good thing about math is that it is difficult. Her abstract, rigorous reasoning requires great and lengthy mental effort, requiring not so much memory as understanding and consideration.

Modern mathematics is the language of the future.

The importance of mathematics is now continuously increasing. New ideas and methods are born in mathematics. All this expands the scope of its application. Nowadays it is no longer possible to name an area of human activity where mathematics does not play a significant role. It has become an indispensable tool in all natural sciences, technology, and social science. Even lawyers and historians are using mathematical methods.

If geometric axioms affected people's interests, they would be refuted.

An engineer who does not know mathematical methods is not an engineer, but a fitter... An engineer in the full sense of the word is unthinkable without knowledge of mathematics. Nothing can be done without mathematics: a bridge cannot be built, a dam cannot be built, a hydroelectric station cannot be built. Reducing the amount of teaching mathematics is a crime! We need to study it as much as possible, and most importantly, as thoroughly as possible.

The geometry is somewhat reminiscent of wine.

... Any axiomatic processing of mathematical material must be preceded by a concrete, I would say, naive mastery of it.

The key to solving many scientific problems is their successful presentation in the language of mathematics.

I love the saying of the ancient and most famous painter Pamphilus, from whom young people of noble rank began to learn painting. He believed that no painter could paint well without knowing geometry well. Our outlines, which set out the entire art of painting in all absolute perfection, will be easily understood by any geometer, but an ignoramus in geometry will not understand these or any other rules of painting. That's why I say that a painter needs to study geometry.

Life is based on mathematics, all the greatness of art and music is based on mathematics.

And having greatly loved the art of numbers, I thought that without number no philosophical reasoning can be formed, considering it the mother of all wisdom.

Any new discovery is mathematical in form, for there is no other possible guiding thread for us.

Mathematics reveals order, symmetry and certainty, and these are the most important types of beauty.

People who have mastered the great principles of mathematics have one more sense organ than mere mortals.

We enjoy learning about mathematics... It delights us like a lotus flower.

It is not enough to have a good mind, the main thing is to use it well.

Both the teaching of nature and mathematics should be considered only parts of wisdom.

I considered the most reliable truths that I clearly perceived as relating to figures, numbers and other matters belonging to arithmetic, geometry and pure and abstract mathematics in general... Only mathematicians are given the ability to achieve certainty and clarity, for they proceed from what is most easy and simple .

The Pythagoreans were the first to take the mathematical sciences forward; having been brought up on them, the Pythagoreans began to consider them the beginnings of all things.

The field of mathematics includes only those sciences in which either order or measure are considered, and it is not at all important whether these are numbers, figures, stars, sounds or anything else in which this measure is sought...

...In mathematics...not a single proof here refers to the fact that this is better or worse, and in general, nothing like this even comes to anyone’s mind. That is why some sophists, for example Aristippus, treated mathematics with disdain: in other arts, they say, even in crafts, for example, in carpentry and shoemaking, they always refer to the fact that this is better or worse, but mathematical art does not take into account the good and bad.

Infinity is recognizable, but not knowable.

Although the mathematician uses general principles in his own way, the principles of mathematics must be investigated by first philosophy.

All the great achievements of mathematics and the natural sciences... spring from our indefatigable desire to give the world in our minds a more rational form than that which the brute order of our experience has given it.

Having tasted the sweet fruit of mathematics, we become like lotophages, because having used it at least once, we do not want to tear ourselves away from it and it takes possession of us, like a lotus flower.

The most important fact is that all the pictures of nature drawn by science, which can only be in agreement with observational data, are mathematical pictures... We go beyond the limits of mathematical formulas at our own peril and risk.

The saying of Pythagoras should be repeated again and again: There is no doubt that nature remains similar to itself in everything. The situation is like this: there are numbers, thanks to which the harmony of sounds captivates the ear, and these same numbers fill both the eyes and the spirit with wonder. It is easier to find a proof by first acquiring some idea of what we are looking for, than to look for such a proof without any prior knowledge.

When you talk to a mathematician, you may have no idea about mathematics. But at the same time, a sense of humor and awareness of one’s insignificance are absolutely necessary...

A mathematician is someone who knows how to find analogies between statements. The best mathematician is the one who establishes analogies of proofs. The stronger one may notice the analogies of theories. But there are also those who see analogies between analogies.

Mathematics was to me as a child what religion was. Because both religion and mathematics claim that they can explain the whole world.

The main strength of mathematics is that, together with the solution of one specific problem, it creates general techniques and methods that are applicable in many situations that cannot always be foreseen.

There are different ways to start your research. Anyway, the beginning almost always turns out to be a very imperfect, often unsuccessful attempt. There are truths, like countries, the most convenient path to which becomes known only after we have tried all the paths. Someone has to, at risk to themselves, go off the beaten path in order to show others the right path... On the path to the truth, we are almost always doomed to make mistakes.

“Obvious” is the most dangerous word in mathematics.

Mathematics is a tool specially adapted to deal with abstract concepts of any kind, and in this field there is no limit to its power. For this reason, a book about modern physics, unless it is reduced to a mere description of experimental work, must be essentially mathematical.

Mathematics is the queen and handmaiden of the sciences.

It turns out that one of the main features of nature is that the laws of fundamental physics are described by very elegant and powerful mathematical theories. To understand these theories you need to be a high-level mathematician. You may wonder: why is Nature designed this way? The only answer that can be given at the current level of knowledge is that Nature is designed this way. All that remains is to accept it. In other words, God is a very high level mathematician and He used the most advanced mathematics in creating the universe. Our feeble mathematical efforts only allow us to understand the structure of a small piece of the Universe, and as mathematics continues to develop, we hope to understand the structure of the Universe better.

As most mathematicians know from their own experience, much of what one generation of mathematicians considers reliable and satisfactory has a chance of turning into a fine web under the watchful gaze of the next generation.

...We must first of all be guided by considerations of mathematical beauty, without attaching much importance to discrepancies with experience. The discrepancies may well be caused by some secondary effects that will become clearer later.

The definition of mathematics adopted in the 20th century as a science about the infinite should be replaced by another, more correctly reflecting its essence as a science about the relationships between the finite and the infinite.

...It is simply impossible to deceive a person who knows how to observe and analyze. His conclusions will be as infallible as Euclid's theorems.

Don't mathematicians... have their secrets and, moreover, their troubles and contradictions?

Numbers are an indispensable tool of modern civilization, which is used to streamline the scope of its activities.

It has long been noted that geometry is a wonderful logic. But indeed: when the definitions are clear, when the postulates are undeniable, when the properties of figures follow from clear observation with the help of a clear chain of sequential reasoning, and objects are always in the field of vision, then in this way the habit of thinking accurately, consistently and methodically is formed; this habit strengthens and sharpens the mind and becomes necessary during the search for truth in other areas.

Mathematics is less a knowledge than a skill.

Rigor in mathematics means, above all, conscientiousness and clarity.

If a theorem cannot be proven, it becomes an axiom.

I hope that in the future more and more theoretical physicists will acquire a deep knowledge of mathematical principles, and that mathematicians will not limit themselves to the purely aesthetic development of mathematical abstractions.

In mathematics, it is not the formulas that should be remembered, but the processes of thinking.

Apparently, new mathematical discoveries made at the prompting of physics will always be the most important, for nature has paved the way and established the canons that mathematics, which is the language of nature, must follow.

One might ask what these impossible solutions (complex roots) are for. I will answer - for three reasons: for the inviolability of general rules; so that there are no other solutions and because of their usefulness.

Mathematics has its own beauty, just like painting and poetry.

...The relationship between mathematics and painting is especially close, because these types of activities often use the same object of study. Moreover, just as mathematics can be used to analyze painting, the latter in turn (drawing, drawing, in particular) is obviously useful in mathematical research, and not only in geometry. Mathematics and painting in this regard can be seen simply as two different, complementary ways of visualizing the concrete or abstract reality in which we exist.

Mathematics is more than science, it is the language of science.

Mathematical talent is a complex thing. Of course, mathematical ability is a necessary ingredient. But this is not enough. Mathematical ability is more common than is commonly believed. However, mathematics, like science in general, is difficult and to achieve anything worthwhile in it, you need to live it, you need passion.

Mathematics is the sworn enemy of cramming.

In problems in elementary geometry one has to use very ingenious, sometimes subtle techniques, and those who tasted their charm in their youth will never forget them.

It is a well-known truth, obvious at first glance, that mathematics is a human invention.

Mathematics is the shortest path to independent thinking.

... It would be easier to stop the Sun, it would be easier to move the Earth, than to reduce the sum of the angles in a triangle, reduce the parallels to convergence and move the perpendiculars to the straight line to diverge.

If you want to participate in a big life, then fill your head with mathematics while you have the opportunity. She will then provide you with great assistance in all your work.

Mathematics is not there to force hard work on anyone. On the contrary, it exists only for pleasure. For the pleasure of those who like to analyze what they do, or can do, or what they have already done in the hope of doing it even better.

The evaluation of a mathematical theory is determined not only by its correctness. It also depends on the importance of the subject and the scope of application. Beyond this, there must still be room for human free judgment.

Since the time of the Greeks, to say “mathematics” is to say “proof.”

Since ancient times, critical revisions of the foundations of all mathematics as a whole or any of its branches have almost invariably been followed by periods of uncertainty, when contradictions arose that had to be resolved... But for twenty-five centuries mathematicians have been in the habit of correcting their mistakes and seeing this as enrichment, not impoverishment Sciences; this gives them the right to look into the future calmly.

Art is a veiled algebra that takes away life from those who seek to lift its veil.

A person who does not know mathematics is not capable of any other sciences. Moreover, he is not even able to assess the level of his ignorance, and therefore does not seek a cure for it.

We draw wisdom from history, wit from poetry, and insight from mathematics.

Mathematics is the first of all sciences, both useful and necessary for them.

Mathematics is the door and key to the sciences.

Practice is born from a close combination of physics and mathematics.

There is much in nature that cannot be understood deeply enough, proven convincingly enough, or used skillfully and reliably in practice without the help and intervention of mathematics.

Mathematical talents, like musical ones, are often innate, appear early and organically determine the mentality of a given person.

Mathematics helps to understand reality and can be inspired by it, but it does not need reality to confirm or refute any of its proposals.

Physics appears to be mathematics in color, but mathematics is more than black and white physics.

Reality has the final say to test or disprove a scientific theory, but what or who cares about such things in mathematics?

Complex numbers help, because of the other side of the mirror, to cope with the shortcomings of real numbers.

The beauty of mathematics, like the beauty of any thing, is an internal property, it comes from harmony between the different parts of one whole.