perhaps, as you expected, I was very happy to learn that you were considering studying mathematics. Not only did you spend several weeks reading A Wrinkle in Time several summers ago, but I also took the trouble to explain hypercube and high-dimensional space to you. I will answer your questions in no order, and first answer the most practical question: who really lives by mathematics besides me?
the answer to this question is different from what many people think. The university I served conducted a survey of alumni a few years ago and found that among all kinds of degrees, people with mathematics degrees have the highest average income. I need to remind you that although this happened before they set up medical college, it at least refutes a fallacy: people who study mathematics can't have high-paying jobs.
the fact is that we can meet mathematicians everywhere every day, but it's hard to detect them. Some of my former students managed wineries, founded their own electronic companies, designed cars, wrote computer programs and traded futures in the stock market. We almost never realized that bank managers might have a degree in mathematics, many people who invented and manufactured DVD and MP3 players were mathematicians, or that the technology of transmitting amazing photos of Jupiter satellites to the earth also contained a lot of mathematics. We know that doctors have medical degrees and lawyers have law degrees, because those are special and clearly defined professions that require the same professional training. However, it is impossible for people to find the name of a licensed mathematician on the copper nameplate of a building and advertise for the mathematician: the mathematician can help solve any mathematical problem he wants after receiving a large sum of money.
Our society consumes a lot of mathematics, but everything is only done behind the scenes. The reason is quite direct: mathematics belongs behind the scenes. When driving a car, you absolutely don't want to think about all those complicated mechanical things, just want to get into the car and drive it away. Knowing the basic condition of the car machinery will certainly help you to become a good driver, but it is by no means necessary. Similarly, so is mathematics. You want the car's navigation system to guide the direction, without having to calculate the mathematics involved yourself. Besides, you want your phone to work even if you don't know the signal processing and error correction code. However, some of us need to know how to do math, otherwise the above-mentioned cars and telephones will not work. It would be a good thing if others could understand how our daily life depends on mathematics. Why put mathematics far behind the scenes? This is because many people have no idea that mathematics is hiding behind the scenes.
I sometimes think that the best way to change people's attitude towards mathematics is to put a red label on anything that uses mathematics that says "contains mathematics". Of course, one will be posted on every computer, and if it is interpreted literally, we should also post one on every math teacher. We should also put red math labels on every plane, every phone, every car, every traffic sign, every vegetable ...
Vegetables?
yes. The days when farmers only cultivated according to the patterns handed down by their fathers and ancestors are long gone. Almost all the vegetables you can buy are the result of long-term and complicated commercial cultivation plans. The whole theme of "experimental design" in the mathematical sense was developed in the early 2th century to provide a systematic way to evaluate new species of plants, let alone a relatively new method of gene modification.
wait, isn't this biology?
biology, of course, but it's also math. Genetics first used mathematics in biology. The success of the Human Genome Project is not only because biologists have done a lot of wise work, but also because they have developed powerful mathematical methods to analyze experimental results and reconstruct accurate gene sequences from very broken data.
So, vegetables get a red label. Like vegetables, other things should also be labeled with a red label.
do you watch movies? Do you like special effects? Is there math in Star Wars and Lord of the Rings? The earliest fully computer-animated movie "Tory Story" contributed to the publication of about 2 mathematical papers. "Computer painting" is not only the use of computers to draw pictures, but also a mathematical method to make pictures look more realistic. In order to achieve these effects, we need the knowledge of solid geometry, the mathematics of light, and a series of smooth movements between the starting and finished images, and so on. "Interpolation" is a mathematical idea. Without a lot of clever mathematics, interpolation will not work-another red label!
Of course, it also includes the Internet, which is entirely a mathematical operation. At present, Google, the most important search engine, uses mathematical methods to find the web pages that are most likely to contain the information users need according to the combinatorial mathematics of matrix algebra, probability theory and network.
But the mathematics on the Internet is more basic than these. Telephone network depends on mathematics, unlike in the old days, when operators had to manually plug telephone lines into the switchboard, but today these telephone lines have to transmit millions of messages at the same time. There are so many people who want to talk to friends, fax or surf the Internet that we have to share telephone lines, submarine cables and satellite repeaters, otherwise the network can't bear such busy traffic. So every conversation is broken down into thousands of small segments, only about 1% of the small segments are actually transmitted, and the remaining 99% are recovered as much as possible by filling the gaps (it works because the sampling is short but the frequency is very high, so that your voice changes much more slowly than the sampling interval). Oh! The whole signal is encoded so that any transmission error can not only be detected, but also be relocated to the correct receiving position.
without a lot of mathematics, the modern communication system will not work. Coding theory, Fourier analysis, signal processing ...
In short, you buy air tickets online, make reservations, go to the airport, get on the plane and fly to other places. Aircraft can fly because engineers use the mathematics of fluid flow and aerodynamics to design and ensure that the aircraft can fly in the sky. Planes use the Global Positioning System (GPS for short) to navigate. After mathematical analysis, satellite signals can tell you the position of the plane within a few feet. Every flight must be included in the timetable, so that every plane can be in the right position, which requires mathematics in other fields.
dear Meg, this is the way mathematics works. You asked me whether all mathematicians are isolated from the university, or whether some mathematicians' work is related to real life. In fact, all your real life is like a boat wandering in the ocean of mathematics, swinging up and down.
but few people pay attention to this. Avoiding mathematics will make us feel comfortable, but it will belittle mathematics. It's a shame. In this way, people think that mathematics is useless and doesn't need to care. Mathematics is just an intellectual game and has no real importance. That's why I want to see those red labels. In fact, the best reason not to use red labels is that most of the earth will be covered with red labels.
your third question is the most important and sad. You asked me if I had to give up my feeling of beauty to study mathematics, and if everything would become just numbers, equations, theorems and formulas. Meg, please relax. I won't blame you for asking this question. Unfortunately, this is a very common but wildly wrong idea, just the opposite of the truth.
Mathematics is as follows for me: It makes me feel the world I live in in a brand-new way, opens my eyes to the laws and patterns of nature, and provides a brand-new experience about beauty. For example, when I see a rainbow, I not only see a bright and colorful arc, but also see the influence of raindrops on sunlight. Raindrops restore white sunlight to its color components. I find rainbows beautiful and inspiring, and I am grateful that rainbows are not just the refraction of light. These colors are like red (and green and blue) herring. The shape and brightness of the rainbow need to be explained: why is it arc-shaped? Why is the light so bright?
maybe you haven't thought about these questions yet. You already know that when sunlight is refracted by raindrops, rainbows will appear, because each color of sunlight will turn to a slightly different angle and reflect from raindrops into our eyes. But things are not so simple. Why don't tens of thousands of colored rays generated by the refraction of tens of thousands of raindrops overlap and blur?
the answer lies in the geometry of the rainbow. When the light is reflected inside the raindrop, the spherical shape of the raindrop causes the light to focus in a certain direction. Each raindrop emits bright conical light, or each color of light forms its own cone, and the angle of the cone formed by each color is slightly different. When we look at the rainbow, our eyes can only detect the cone in a specific direction, and the direction of each color forms an arc in the sky. So we see many concentric circles, and each color forms a concentric circle. The rainbow you see and what I see are formed by different raindrops. Our eyes are located in different positions, so we detect different cones produced by different raindrops.
the rainbow is a personal experience.
Some people think that this kind of understanding will "destroy" the emotional experience, because it will produce some repression of aesthetic satisfaction, but I think this is a boring idea. People who make such statements usually like to pretend that they are poetic and open to the wonders of the world, but in fact they are seriously lacking in curiosity: they refuse to admit that the world is more wonderful than their own limited imagination. Nature is always deeper, richer and more interesting than you think. Mathematics provides you with a very useful way to appreciate the beauty of nature. The ability to understand is the biggest difference between human beings and other animals, and we should cherish it. Many animals have emotions, but only humans can think rationally. I must say that my understanding of rainbow geometry has added new luster to its beauty, but the emotional experience will not be less.
the rainbow is just an example. I also observe animals from different angles, because I notice the corresponding mathematical patterns when animals move. When I look at a crystal, I notice the beauty of atomic lattice and external color. I can see math in waves, sand dunes, the sun rising and falling, rain dripping in puddles and rippling, and even birds parked on telephone lines. In addition, like looking at the foggy ocean, I vaguely understand that these wonderful things are full of infinite unknowns.
The inherent beauty of mathematics should not be underestimated or ignored. Mathematical research itself is very beautiful and elegant. The inner beauty of mathematics is not the "addition" we use in school. Although the general principles behind addition have their own beauty, they are mostly ugly and amorphous. The inherent beauty of mathematics lies in: ideas, universality, inspiration that suddenly flashes by, and trying to bisect an angle with a ruler and compasses is equivalent to proving that 3 is an even number; We cannot construct an equilateral heptagon, but we can construct an equilateral heptagon. There is no way to untie a single knot; Why some infinitesimals are bigger than others, while some infinitesimals that should be bigger have equal results; The unique square number equal to the sum of continuous squares (1+4+9…) is 49 (except 1).
Meg, because you have a logical and inquiring mind, you have the potential to be an excellent mathematician. You are not satisfied with vague arguments. You want to see the details and check for yourself. You want to know not only how to make things work, but also why things work. Besides, your letter made me hope that in the future, you can see the fun and beauty of mathematics as I do now-in a unique way of looking at the world.
I hope this has established the necessary background for you to study mathematics.