“My child doesn’t want to learn at all!” – This is a problem that almost every parent faces. What is the reason? How to get your child interested in learning? How to explain to him how important it is? A single recipe does not exist – every situation requires a unique approach. But some common features can be highlighted.

In general, we can distinguish three main motivators – the reasons that encourage us to do something:
1) The pleasure of the corresponding angles picture
2) Expectation of a reward in the future.
3) Fear of any negative consequences of idleness

These principles, as we can see, work fine for adults as well. We are willing to do boring work for high pay – we are motivated by the expectation of reward. We can repair a leaky pipe, knowing that if we don’t, the house will flood – we are motivated by the fear of trouble in the future. And we don’t need any reason to meet good friends or travel – these things are pleasurable in and of themselves, we are motivated by the pleasure of the process. Also, additional motivation is not too important to us if we really love our work.

The same motivators also govern children’s love or dislike of learning

The school system of grades, verbal encouragement or reprimand from teachers, mostly uses reward and punishment motivators (the proverbial “carrot and stick method”). But as we observe, this system – rewards and punishments – that works so well for adults, works very poorly at school. Why?

The main reason is very simple: grades tend to have little impact on students’ lives. Much less than positions, salary, bonuses and fines! Let’s say Petya got an “A” in math and Vera got a “3”. How would that affect them? Perhaps Peter would be praised and Vera would be scolded. Maybe Peter will even be given some small pocket money. In most cases, this is completely insufficient. Compare, for example, with the prospect of promotion, which will buy a car in six months, or with the prospect of dismissal with a record in his work book – the difference is huge! That is why the system of school rewards and punishments cannot motivate students by itself.

The situation is quite different with the first of the motivators, i.e., enjoyment of the process. If Petya likes to play on the computer, he does not need any more reasons or excuses: he will play as long as he can. If Vera likes to go out with friends, she will not need any fear of reprimand or encouragement in the form of money or gifts – she will just go out with them for hours on end.

There are children who love to learn. Curiosity is one of the traits of our nature; some have it stronger, some weaker. There is also a love of doing a particular thing, a hobby: some like to read, others like animals, someone loves to play soccer – interests, as well as activities, a great many. And everyone has one or another interest – every adult, every child. There are just those who have not found “his” thing – something that can really give pleasure.

A frustrated, upset child, or child with learning difficulties.

There are different ways to find what you love. Of course, the best one is practice. You just need not be afraid to try out different areas. Yes, you did not like volleyball or tennis, but that does not mean that the sport does not suit you in principle – perhaps you will find yourself in golf or fencing. You’re not interested in a high school physics course – that doesn’t mean that a course of lectures by, for example, Feynman can’t awaken in you a passion for that science. But the path of practice, for all its merits, has one important disadvantage: it is too long. To personally try every activity that has been discovered by humankind is an impossible task.

The second way is to share experiences, to communicate with people who have actually done these or those things. This way is also very good, but has a very significant disadvantage: all people are different. For one, the work in direct sales is a horror and torture, which are barely compensated by a salary, but designing structures is a pleasant and easy task; for another, the market analysis will be unbearably difficult and boring, but the work on training will be easy and pleasant. Exchange of experience is always an exchange of personal experience, transmitted through the prism of a given person’s personality.

Finally, to determine the sphere of their interests and find a business that will really give pleasure, you can turn to the services of professionals working in the field of career guidance. Yes, the recommendations received as a result of career guidance will not replace personal experience of the activity. But career guidance testing will help to avoid wasting energy and time. Instead of trying your hand at all the different fields of study and activities, you can focus on a few that really suit you.

In conclusion, I would like to emphasize once again that all people are different. We differ in our interests, in our abilities, in our personalities. But each of us can find a job that will not only maintain a decent standard of living, but will be easy and enjoyable to do. It is only possible to determine what is really “your” job on the basis of personal experience, but it makes a lot of sense to use career guidance services to shorten the search.

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Classical Yoga for Beginners

What should one do if he or she decides to go the classical yoga route? How do you start practicing at home? And is there a classical complex?

The first prerequisite is fa

miliarity with the moral principles of yoga (yama, niyama). Those who take the path of yoga should use them in their life, strive to observe the moral precepts as clearly as possible. However, due to the fact that modern life has a frantic rhythm, one should start mastering classical asanas at the same time. If you’re already practicing at a club, the opportunity to build your home practice will allow you to get firmly on the path of yoga.

Your first set for home practice should include a warm-up. You can only practice by first warming up your body, and then you should include a Padmasana routine to master Padmasana. Your first classes should not be too long, the purpose of yoga is not to tire you out, but to help you become a whole person. Start with 30-40 minutes, but do it regularly. Conclude any practice with a shavasana. Your first set should be simple and safe.

young beauty woman in white lay in yoga asana – fish pose isolated

When your body is ready and you are able to stay in meditative asana for 10 to 15 minutes, you should start learning pranayama. It is important to remember: pranayamas are performed in meditative asanas or in a sitting position, with straight back and crossed legs. Pranayamas should be mastered only under the guidance of a specialist. Incorrect performance of the exercises can lead to very unpleasant consequences. For beginners is recommended full yoga breathing without delays, this is the safest option.

The next step is meditation. There are a huge number of meditative techniques, but here it is also important to remember that meditative practices should be performed under the supervision of a qualified instructor.

To summarize, we would like to remind that the essence of yoga is not the asanas. It doesn’t matter how difficult an asana you can do if, outside the mat, you insult people and malign them. It doesn’t matter how skillfully you do pranayama, what matters is how the living beings breathe around you. Yoga doesn’t start and end on a mat. It starts in our heart and mind and expresses itself through our actions.

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Doing homework with your child? Teacher’s opinion

If you do not do homework in 8th grade language arts with your child – he’s all mixed up, half not done, inevitably rolls to F’s – so think some parents. But what does the teacher think, let’s find out!

I know a teacher who frankly said at a parent-teacher meeting, “I am annoyed by incorrectly done homework. Dear parents, homework is not just for the kids, it’s for you!” That is, she believes that it is the parents’ responsibility to help the children with their homework. This attitude of the teacher leads to the fact that parents make crafts and presentations, write poems and compose stories. Creativity is wonderful. But not when you do it instead of the child.

In my opinion, parents do the assignment for the child for 3 reasons:

Parents are perfectionists, “great at heart.” These moms and dads can’t accept that the child isn’t doing everything as well and accurately as they would like: “Let me show you how to do it!”

Parents just don’t have time. Dad comes home from work, he’s tired, the show or a soccer game is about to start, and his son is there with his homework: “Let me do it faster, and you rewrite it.

Parents are afraid of the teacher and 5th grade math multiple choice questions. Yes, it happens.

Teachers can see perfectly well if the pupil did the work himself or if he had help. I don’t even have to ask about this, I know very well what each pupil is capable of. But I work in a small private school where the number of kids in the class allows me to check on every lesson whether the topic has been learned. The most important principle that should guide the teacher: ask only what you have taught. Because homework – this is a check, how the student understood the material. That’s the main point of homework, and not whether the teacher will praise you, put “5” or “3”. Doing “homework” for a child means to harm him and interfere with the teacher.

How to help your child cope with homework?
My advice: patience. Patience and observation. If a child can cope by himself, does not ask for help, if the teacher is not in touch with you about his homework, then the question of help falls away. Sometimes it is enough to check the very existence of the completed homework. Teachers usually comment on students’ work. Remember such notes in red pen: “No work on mistakes”, “Where is homework 14?”, “Write neater!” and so on. Of course, you have to pay attention to this. But again, patience. Observe and just prompt the child if you notice that he forgot something or did not form his work correctly.

Will I lower the grade if the work in 2nd grade ela is done correctly, but poorly formatted and sloppily written? No, but I will write a reprimand, especially to the child who can write better but does not try. Parents should not be intimidated by the teacher’s notes. This is not a characteristic of the student and his parents. It is a work, a learning process. It can’t work out all at once.

What to do if a child does not succeed in his homework?
This situation, of course, should not be ignored. If you yourself understand the subject, explain it, but try not to show a ready-made solution, but with questions, lead them to it. It is even better to find another similar task. It may well be in the same notebook in the class work. Eliminate all books and manuals with ready-made homework. Let me tell you a secret: there are a lot of mistakes. In formal homework, especially in elementary school, when the foundations of knowledge are laid, they do much more harm than good. If you can’t do anything at all, it’s better to be honest with your teacher and ask him to explain how you should have done the assignment.
To accept the inevitability of homework, you need to understand that this is an important stage in learning. It is a repetition and consolidation of the passed material. One more thing. There are children who absolutely refuse any help. And don’t impose! Maybe you’re just lucky!

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What exercises are there in hatha yoga?

Hatha yoga practices are divided into several groups.

Asanas are certain poses, static or dynamic. Among the most famous are chaturanga dandasana, which resembles the classic push-up; chair, warrior, boat, dog head down or up, and other poses. All can be divided into lying, sitting and standing exercises; bending and flexing, twisting, inverting and balancing. Some are aimed at increasing body endurance and muscle strength, while others are aimed at stretching the muscles.
Pranayamas – breathing exercises, during which you need to take inhalations, exhalations, to perform breath-holding. It is important to maintain maximum concentration on the process. Pranayamas allow you to learn how to keep the energy in the body under control and direct it in a certain direction.
Meditations can be performed during the class, after the exercises, and shavasana is performed at the end of the training. Practitioners assume the corpse pose: lying down on the floor, eyes closed, arms and legs free and away from the body. For maximum relaxation, it is recommended to first strongly stretch the feet and the top of the body in different directions, and then sharply relax. To take your mind off your thoughts, you can focus on the tip of your nose or your breath, the sound of a mantra or another object. This allows you to accumulate the energy received during the practice.
Mudras are special hand postures. They are performed either alone or in combination with other exercises.
Hatha yoga as a philosophical teaching also includes other practices. These include shatkarma (purification) – fasting, nasal washing and so on. Acupuncture massage is used: the impact on the biological active points of the body during classes.

In total in hatha-yoga there are four stages – yamas (five moral foundations, which include non-violence (ahimsa), ascesis (brahmacharya), truthfulness (satya) and others); niyamas – five rules of inner self-development and spiritual development, asanas – static postures, pranayamas – breathing techniques.

Those who do not just perform exercises and meditations, but adhere to the philosophy of hatha yoga in everyday life, eat a certain way. In the menu there is no animal protein, a lot of dairy products, vegetable oils, nuts, plant foods. The most important thing is not to overdo it. After all, one of the basic principles of hatha yoga is the absence of violence, including violence against yourself.

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A woman was awarded the Abel Prize for the first time

The award, which is called the Nobel Prize in Mathematics, will be presented to the American Karen Uhlenbeck

The Norwegian Academy of Sciences has awarded the 2019 Abel Prize to Karen Uhlenbeck, Professor at the University of Texas at Austin, “for her pioneering achievements in geometric PDE theory, gauge invariance theory and integrable systems, and for her fundamental contributions to analysis, geometry and mathematics. physics in general”: It was decided to present this award to a woman for the first time in 17 years of the existence of the award.

Along with the Fields Medal, the Abel Prize is often referred to as the Nobel Prize in Mathematics. The award was founded by the government of Norway in 2002, the first prize was presented in 2003. All 16 previous award winners have been men. Among them are the famous mathematicians John Nash (also the 1994 Nobel laureate in economics) and Andrew Wiles, who proved Fermat’s Last Theorem, as well as our former compatriots Jacob Sinai and Mikhail Gromov.

The fact that Karen Uhlenbeck has become the new Abelian laureate, the Norwegian Academy of Sciences announced today, March 19. The prize itself, the cash equivalent of which is six million Norwegian kroner (more than 45 million rubles), will be presented by King Harald V of Norway on May 21 in Oslo.

Karen Uhlenbeck was born in Cleveland (Ohio, USA) in 1942. She graduated from the University of Michigan in 1964. Initially, Karen, inspired by the books of Fred Hoyle and Georgy Gamow, planned to study physics and what is multiples, but later switched to mathematics.

Uhlenbeck’s research interests are nonlinear partial differential equations, differential geometry, calculus of variations, topological quantum field theory and the theory of integrable systems. During her scientific career, she developed the tools and methods of analysis that every geometer and analyst now uses, according to a press release from the award. Her work also lays the foundations for modern geometric models in mathematics and physics. Another area of ​​research in which Uhlenbeck worked was gauge invariance theory. It is used in theoretical physics and in particular for the mathematical understanding of models in particle physics, string theory, and general relativity.

“Her theories have revolutionized our understanding of minimal surfaces, such as those that form bubbles, and more general minimization problems,” said Hans Munte-Kas, chairman of the Abel Committee that determines the winner. In addition to him, the committee also includes Irene Fonseca (Carnegie Mellon University), Alice Chan Sun-Yun (Princeton University), Gil Kalai (Hebrew University of Jerusalem) and François Laboury (University of Nice).

Karen Uhlenbeck is known not only as a scientist, but also as a supporter of gender equality. In particular, she was one of the co-founders of the Women and Mathematics (WAM) program. The goal of this program is to overcome the imbalance between women and men studying mathematics.

Last year, Canadian Robert Langlands received the award “for creating a visionary program linking representation theory with number theory.”

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Sleep is difficult for cancer patients. More than half of all cancer sufferers sleep poorly during treatment and rehabilitation, according to the US National Cancer Institute. But a new study from the University of Texas found that breast cancer patients who received chemotherapy and simultaneously practiced yoga just 2 times a week saw qualitative and quantitative improvements in sleep.

“Previous studies found yoga including to reduce sleep disorders in cancer patients, but they did not involve active control groups and did not follow-up the study participants. This study aims to fill in the gaps of the previous ones, ”said the lead author of the experiment, PhD, professor of palliative, rehabilitation and integral medicine and head of the integral medical program Lorenzo Cohen.

The importance of sleep in cancer treatment

A couple of nights without sleep will do no good to anyone, much less people undergoing cancer treatment. Lack of sleep is directly related to the weak activity of NK cells (natural killer cells that are cytotoxic against tumor cells), and they play a leading role in the functioning of our immune system.

Yoga relaxes and calms our body, thus making it easier to fall asleep and making sleep more sound. Yoga nidra and tonic yoga are especially good for these purposes.

Other methods are also suggested to promote healthy sleep. Often we sit at the computer until the moment we go to bed, and this, of course, interferes with restful sleep. It is necessary to put aside your electronic devices an hour before bed and prepare yourself in a certain way, for example, take a soothing bath, do stretching or yoga. It’s also good to go for a daytime walk while sunbathing to help you sleep at night.

But what can cancer patients actually do to improve the quality of their sleep?

Science is one thing, but life is completely different. How can cancer patients really help their bodies fall asleep?

Many cancer survivors recommend yoga. According to them, yoga is able to restore the connection between body and mind, which is often broken during treatment. Breathing exercises, flowing movements, and meditation – necessarily of a calming, relaxing type – help find this missing link. Having given up asanas, but choosing exercises for concentration, practicing pranayama and writing down their observations in a personal diary, many people note easier falling asleep and improved quality of sleep.

Another great way is to combine outdoor activities and yoga practice. However, some note that asanas are difficult for patients in the early stages of recovery from illness and may even cause physical pain. Therefore, the first thing to do before starting classes is to consult with your doctor about whether you can already start practicing yoga. If so, warn your yoga teacher that you are recovering from an illness. And then do not chase after some results, listen to your body and try to absorb all the love and positive that yoga carries.

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Meditation Posture

As mentioned above, a stable meditative posture during meditation and is very important. Why is this so? The fact is that there are five energy currents in the human body: apana-vaya, prana-vaya, udana-vaya, samana-vaya and vyana-vaya. And one of them, which is called “apana-vayu” is in the lower part of the body and it moves downwards.

Woman meditating and drinking tea on beach

You’d think you could let it go, but it’s not that simple. Having a downward direction this stream of energy “drags” our consciousness down to the lower chakras, and we start manifesting ourselves through these energy centers, which, as experience shows, usually does not lead to anything good. Therefore, in order to cut off the energy going down, it is necessary to clamp the energy channels. This is done by practicing meditative asana. Also, meditative asana allows you to keep your back straight, which is extremely important during meditation because only then can the energy of the body flow properly. The spine is the physical projection of the central energy channel, the sushumna. Adequate movement of energy in the sushumna is the key not only to successful practice, but also to an adequate life. Therefore a straight spine during meditation or pranayama is 50% of the success of your practice. Otherwise, meditation will be ineffective. This is why regular hatha yoga practice is important, because opening the hips and maintaining a proper level of stretching in this area is necessary in order to sit painlessly in Padmasana for at least 30-40 minutes. If it’s not possible to sit for that long, don’t go into extreme ascesis; simply reduce the meditation time, or, better yet, choose a simpler meditative pose.

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Namaste (Sanskrit: नमस्ते, namaste) is a Sanskrit greeting and gesture of reverence, respect, and gratitude, performed with two palms folded together before one another. The gesture is used in everyday life to say goodbye and hello, and in worship, prayer, and spiritual practices. It is also an expression of one of the most commonly used mudras in hatha-yoga and

The polite greeting and sacred gesture of Namaste
With this gesture, we salute the divine in man. God is omnipresent, he is the essence of the whole universe and is manifested in all things. Therefore, when we greet another person with this gesture, we honor the source of divine light, the spark of which is his soul that has shone forth in the manifested world in human incarnation. Thus, by making this gesture to a person, we honor the Higher Consciousness of which we are all a part. All being dwells in unity, which is the original source that gave rise to the entire manifested world, and to which all will return at the end of the cycle of Creation. Outwardly we are separate, but in essence we are one. This is what Namaste means: “The divine in me recognizes the divine in you. Therefore, by the gesture of Namaste, we honor the one divine essence in ourselves as well as in the other person.

Anjali mudra is a gesture of greeting and reverence that has the same meaning as the Namaste greeting. Mudras are used in spiritual practice to activate certain energies or help establish a connection with the deity, they are a kind of sacred formulas that form the realization of a certain intention, be it protection, healing, blessing, etc. That is why mudras are used not only during ritual sacred acts, but also in classical traditional dances and yoga. Like any mudra, the Anjali mudra is accompanied by a mantra or sacred phrase, in this case, Namaste. “Anjali” (अञ्जलि, añjali) means ‘divine offering’ or ‘blessing’ and “mudra” (मुद्रा, mudrā) means ‘sign’ or ‘seal’. Thus, Anjali mudra means ‘sign of divine veneration’.

It is believed that when you put your hands together, it balances the activity of the right and left hemispheres of the brain, so this gesture sets us in a harmonious state, balances, calms the mind, increases concentration, concentration, which brings us back to the moment of “here and now”. Therefore, any hatha yoga practice should begin with this gesture.

In India, Nepal, Bhutan, Burma, Thailand, Laos, Cambodia, Indonesia and Sri Lanka, people greet each other with the Namaste gesture regardless of age, gender or social status.

Namaste is performed with folded palms, fingers held together (not apart), pointing upward at the level of the heart in the center. The gesture is accompanied by a slight bow, in which you can say the word Namaste. The wrists should be at or above the level of the rib cage, but not below. The hand positions above the level of the heart chakra may be as follows: at the level of the Vishuddha chakra, when the hands are joined in the gesture in front of the throat center; at the level of the Ajna chakra, when the thumbs slightly touch the “third eye”; or the palms joined above the head (at the level of the Sahasra chakra).

What Namaste means when performed from the heart (at the level of the Anahata-Chakra) is the activation of the heart center energy, which manifests itself in compassion, love, devotion, service, and friendliness. By the gesture of Namaste at the heart you address to people who are on the same level with you. This is how we develop the qualities of humility and gratitude.

At the level of Vishuddha Chakra, the energies of the throat center are activated, which manifests in service and active compassion. By folding the palms of the hands in Namaste at the throat chakra, one expresses reverence for elders.

From the Ajna chakra, the center of Knowledge, where the mind reaches the state of non-dividedness and the Truth of Being is grasped, we salute the wisdom of the teacher and instructor, the Guru, who guides us on the Path of spiritual development with the gesture of Namaste.

What Namaste means is performed by joining hands above the head (at the level of the crown chakra), where the connection with the Higher Consciousness is activated and pure bliss in the Spirit is attained – this is an expression of the highest reverence for God.

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Matsyasana. Pose for Lung Health

This technique and has helped me a lot with lung inflammation.

A variation of Matsyasana. Case in point.

While climbing Kilimanjaro, I got pneumonia. Quite acute form with cough with blood.

The MRI picture was discouraging. The doctor said that the case was serious, home treatment was very desirable, and recovery would take at least a month.

A week later, when I saw the second scan, the pulmonologist was a bit dumbfounded and said that this was the first time he had seen such a rapid recovery. What were you doing? The lungs still need to be worked on, but he is calm about the pneumonia now.

Of course, there was not only “yoga”, pharmaceuticals were also present, but the techniques of yoga, or rather even their right combination, have played a significant role in the recovery (soon I will publish an interview where I will talk about this case and practice in detail).

And now back to the effects of Matsyasana.

As a result of the flexion, the lungs are well stretched and open. It’s possible to activate the alveoli by full breathing, to “ventilate” the lung tissue better.

This technique is very useful for city dwellers, because it compensates for shallow stress breathing, helps to get rid of stagnant gases in the lungs.

It is also recommended that people who sit a lot at the gadgets should do Matsyasana regularly to get rid of stooping. To have a stable effect, practice at least three times a week. Hold the position for 1 to 5 minutes. Breathing is full from the belly to the collarbones, and as smoothly as possible. Focus on the center of the chest or on the smoothness of the breath.

Above are two useful actions of this technique in terms of physiology. But you should always remember that in hatha yoga, physiology is secondary! It’s always your mental attitude toward life and energy that counts.

Asanas, mudras and pranayamas primarily affect the movement of internal “currents” in the body. And the functioning of these “currents” and the chakra system determines the state of the physical body. But this is a separate topic.

P.S. But most of all the support and love of friends contributed to the rapid recovery. So, despite the fact that yoga makes us self-sufficient, it is very important to communicate and appreciate friends!

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Why do we need math?

Posted by August 8, 2021

There are million-dollar rewards for solving some mathematical problems including to go math 1st grade. And yet they remain unsolved, no matter how hard the best minds fight over them. How does mathematical thought evolve-isn’t everything still proven? And why invest huge sums of money working on abstract problems? Let’s look into the complicated world of mathematics!

Tells the guest of the program “Question of Science”, candidate of physical and mathematical sciences, head of the laboratory of popularization and propaganda of mathematics at the Steklov Mathematical Institute of the Russian Academy of Sciences Nikolai Andreev.

Mathematics is a way to study our world
Mathematics including go math second grade studies our world in its maximum abstraction. Physicists are used to working with manifestations, but mathematicians work with abstraction: they look for those properties of our world purified from manifestations, which can then be studied and applied. There is a terrific article by our great mathematician Manin, entitled “Mathematics as a Language for Describing Possibilities. At the end of his article Yury Manin gives the example of a shaman, who didn’t decide what to do with the tribe, but gave advice to the leader of the tribe on the best way to proceed. Yuri concludes with the following words: “Mathematics describes the phase space of the real world, the space of possibilities. It studies the laws that determine possible trajectories in this phase space, as well as the conditions, the set of information that is required to choose a particular phase trajectory. So it basically tells society that if you do this, you will go this way; if you do that, you will evolve that way. And this is such a higher view of mathematics.

[The Russian mathematician] Alexei Nikolayevich Krylov compares mathematics to a workshop: mathematicians prepare all kinds of tools for various needs. And when mankind has to solve some problem, professors of mathematics – scientists-experts of these tools – give mankind the tool it needs to solve the next problem: sometimes a rough file, sometimes a fine file…

New fields are constantly emerging; mathematics expands a little, and from the new frontiers one can see further fields that can continue to be explored, which will then result in benefits for all of humanity. Here we can give a very simple example – the article “From Lobachevsky’s “crazy” geometry to GPS-navigators”. Here Lobachevsky invented his geometry. He died without even seeing a single working model. And of course, he could not think that later Riemannian geometry would be created. And later, on the basis of Riemannian geometry, Einstein would create his theories of relativity, special and general relativity. And now we, every day using a GPS navigator, use including the general and special theories of relativity. Because if we did not take into account the effects that they give, the error in determining the coordinates on the ground would be huge and the GPS would not be needed. The power of mathematics is that always one way or another these seemingly abstract studies then find their application to mankind.

Does the world only seem mathematical or is it a property of nature?
It is, of course, a property of nature. In general, all mathematicians are Platonists. There’s a little book called “Proof from the Book” – it’s about the fact that somewhere there are written proofs, and we humans can get close to them. And indeed, sometimes you take a proof, it can be very short, but it proves something very important. Actually, the book Proof from the Book itself begins with Euclid’s proof of the existence of an infinite number of prime numbers. A proof that has endured for centuries, and yet is really very beautiful, powerful, and interesting! And sometimes you look and see that there’s still something we don’t know and that’s why the proof is so complicated.

Mathematics dispels the fog – it explores what we haven’t yet explored in our world. It reveals the essence by formalizing and abstracting from something that is already attuned to that essence. And then it remains to apply the resulting laws either to the subject or to the task – it already depends on what humanity needs.

Mathematics Around Us
Let’s talk about those manifestations of mathematics in our surrounding world that everyone understands, and on the other hand, reveal a mathematical approach, a mathematical component.

For example, the cycloidal curve made it possible to create the first isochronous pendulum clock, in which the period of oscillation did not depend on the amplitude. These were the first clocks. Obviously, there’s a lot of math in engineering. Here at school, everyone went over the parabola. But there is an optical property of a parabola, namely, that rays of light passing parallel to the axis of the parabola, after being reflected from it, fall into focus. Parabolic dishes, satellite dishes that look at a satellite, work according to this principle. And here’s a clear, simple example that has to do with high school math.

Or let’s take colors. The way computers provide us with color, how it adds up, it’s all based on mathematics. Just recently we were celebrating the 50th anniversary of the moon landing, the moonwalker’s delivery to the moon. And in 1970, our Soviet lunar program began, and there was a device that we all know, which is a cataphot, an angle reflector. Three planes are mutually perpendicular to each other, and if you shine a beam on them, then after reflection from all three mirrors, the beam goes parallel to the direction from which it came. And it does not matter where you shine from, the beam will come back to you.

Exactly, so out of these little corners, mutually perpendicular to the three planes, make cataphot on a bicycle. And when you car headlights illuminate some cyclist, you see the glow from the cataphot, although there are no bulbs. But you have to be careful – those coming from the right and the left may not see the reflections from the cataphot because the light comes back to you. The same idea of reflection is used at stoplights when the road turns. The exact same cataphot, the same set of corner reflectors, was installed on the moonwalker. And so far, this ongoing experiment in laser locating of the moon allows us to measure the ever-changing distance to the moon to within a few centimeters, maybe even millimeters. And there are only three planes! There is a law in physics: “The angle of incidence equals the angle of reflection” and then comes the math.

Or here is, say, an example we pass by all the time: why are cups made in the shape of a cone? A piece of paper can be made into a cylinder or a cone, so why choose a cone? Because cups of this shape can be inserted into each other, while cylindrical buckets or cups could not be inserted, they would have to be transported separately. It’s the same thing we see when road services come in and set up cones. But a cone is a bendable surface, and just a tapered cup would have been inconvenient to use. It was necessary to give it rigidity. This requires a bagel surface – in mathematics it is called a “torus”. As it turned out, even a small piece of torus, which contains a circle, is a rigid surface. Being made of the same thin plastic, it is an inflexible surface. The question, “Which surfaces are bendable and which are not?” – was an important topic, and it is still an ongoing one for mathematicians. Of course, it wasn’t studied for the sake of cups. But once we learn a property, we can apply mathematical knowledge to different areas.

Proofs as the basis of mathematics
When you’re trying to solve a problem, you have to go through a lot of different options. And mathematics tells you right away that these are the options you can’t even consider. Rigor is the power of mathematics.

In fact, even within mathematics, the concept of proof has varied. In ancient India it was enough to draw a picture with a proof and write, “Look,” and they considered that a proof. It is clear that now such a proof is not even considered, well, only as an illustration. One of the big, important breakthroughs came in the time of Hilbert, who began to systematize proofs.

In 1900, the International Mathematical Congress heard Hilbert’s famous paper in which he posed his famous “Hilbert Problems,” which were very prestigious to solve. And it should be noted that more than half of those solved were solved by members of our institute. And here I will refer readers to an article by Lev Dmitrievich Beklemishev, our academician, it is interestingly called: “Mathematics and Logic.” Indeed, logic is sometimes called the basis of mathematics, because proof is the subject of the study of logic. A lot of things were born out of logic that, again, we don’t even realize are based on logic. For example, databases are one branch of this science. Linguistics also draws heavily on logic.

As we say goodbye, I want to take you back to your childhood. I take in my hands the book “The Mathematical Component”, where we collected all kinds of plots of manifestations of mathematics in our lives, and open it on the page “Arithmetic tricks”.

Guess a number. Have you guessed it?

Add five to it. Did you add it?

Now multiply the result by 2. Have you multiplied it?

Subtract the number you guessed from the result. Have you subtracted it?

Subtract the number again. How much did you get?

I’ll tell you: 10.

Mathematics actually simplifies the way we look at life. If you write down the formula for our trick, it looks very simple. And that makes the point very clear. If schoolchildren were reading us, I would tell them: if you put in a little effort now and learn math, it will be much easier to live later. When formulas were introduced instead of words in Viet’s time, everything became clear and transparent.

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