Mathematics is very good at making two compasses: one called "practical" and one called "elegant".
Understanding the world is a great and magical ability of human beings. Each of us has this kind of understanding since childhood. Five or six-year-old children can answer who runs faster between trains and bicycles, who is taller between giraffes and elephants, who has more stars in the sky and children in kindergarten classes, and why apples thrown into the sky fall down... The real world is very complicated, and the problems will become more complicated: there are many stars in the sky, and what is the order of magnitude of their number? The child may not be able to answer, but he/she may guess "several million" "hundreds of millions"... any large number, but he/she will not say "tens" or "hundreds".
Look, in fact, everyone has a "mathematical brain" to understand the world.
Unfortunately, most of our "mathematical brain" may stay at this stage. We can't help asking: How could scientists like Newton, Galileo, and Einstein be so smart? How did they figure out the concepts of "acceleration" and "gravity"? Why did they invent something as powerful as calculus? How did they come up with the idea of using relativity to describe the fascinating space-time?
Under the Umbrella of Math: The Joy of Understanding the World
Author: [France] Mickaël Launay
Translator: Ou Yu
1
How do mathematicians solve small problems?
What is the difference between the brains of mathematicians and ordinary people? Mikael Launay, the author of "Under the Umbrella of Mathematics" and "Everything Is Number" tells a story:
I remember a remark made by a mathematician friend with whom I often collaborate a few years ago. We were both about to say goodbye, and we decided to see each other on the same day, at the same time, two weeks later. As she pulled out her notepad to jot down the date of the meeting, I heard her murmur something, more to herself than to me: "Today is April 20th, so 14 days later is 34th, which is 34 minus 30—May 4th." The calculation made me laugh.
I thought long and hard on the subway back, and she invented a date that didn't exist: April 34th. This way of thinking is both natural and typical for a person with mathematical training!
That evening, I posed the question to a few friends who were not math majors: "What date is 14 days from now?" I found that each of them deduced the date in a different way. Some people say that April 30th is 10 days later, so May 1st is 11 days later, and May 4th is 14 days later. The transition from April to May breaks the rules of arithmetic because 30 is followed by 1, and the transition seems to limit them to one extra-mathematical step for the month conversion.
Since the natural growth of numbers is interrupted, this thinking has to be intentionally interrupted. And I must admit that if someone asked this question to me, I would probably derive dates in the same way. In contrast, my mathematician friend did not stop at these all-too-practical obstacles. The last date in April did not stand in the way of her addition. Since 20 plus 14 equals 34, the date would be April 34. And April 34 is equal to May 4, nothing more. She invents a date that doesn't exist in order to get her derivation straight to the point. And that didn't stop her from getting the right result in the slightest!
Challenges in reality and problems in learning are sometimes like a sudden heavy rain, which makes you feel overwhelmed. This is where mathematics plays an important role. Mathematics, which can use things that don't exist, allows us to think properly. Just like an umbrella, it can open up a virtual world and let us walk in the heavy rain.
In fact, thinking about things that don't exist can be said to be a property of mathematics. What does not exist is abstract. The "concrete" in reality becomes a kind of "idea" in mathematics, which appears in the middle link of thinking as a kind of imaginary thing.
Mathematics is very good at making two compasses: one called "practical" and one called "elegant". Just like the mathematician mentioned above, she invented "April 34" out of thin air to derive the date. Not only is it simple and convenient, but more importantly, this method is more in line with her own thinking habits and can quickly help her sort out the situation and solve problems. So, does this approach work when it comes to bigger, more complex problems? works just as well.
2
Newton's "Umbrella"
A "simple" question: why does an apple thrown into the sky fall down?
Some people say, um, because the earth is below. Kepler was the first to guess that this attraction is not unique to the earth, but a universal property of matter. Of course you do, he guessed right. For ordinary people like us, the problem may end here, but scientific judgments must be expressed in accurate and verifiable terms-then Newton came.
Did the apple tree in Newton's yard grow apples that changed the history of human science? Who knows. In the 17th century, Newton proposed on the basis of Kepler's conjecture that any two objects in the universe, no matter what they are or where they are, will constantly attract each other. In this way, we can explain various seemingly irrelevant phenomena of everything in the world: why do apples fall to the earth? Why are there tides? The moon and planets that are constantly rotating in the sky have the same principle as the movement of other objects on land?
A principle explains the operation of the world-this is incredible for people in the 17th century; for people today, whenever we look back on this period of history, we will also feel that Newton is really too "beautiful"!
However, as good as it sounds, a lot of it is actually ambiguous. Newton realized this, so he held up the umbrella of mathematics without hesitation: Newton made a comprehensive mathematical treatment of gravity, quantified the phenomena he described, and compared them with reality.
In 1687, Newton published one of the most influential works in the history of science, The Mathematical Principles of Natural Philosophy, commonly known as Principia. This book brings about many "turning points" for science. Among them, the elaboration of universal gravity and the mathematical formulation of physical concepts are particularly eye-catching.
In Principia, the scientific genius wrote that gravity depends on two things—the mass of objects and the distance between them. Armed with this information, you can calculate this force through a mathematical equation. At the same time, the equation clearly shows that the greater the mass of the objects and the closer the distance between them, the stronger the force; conversely, the smaller the mass of the objects and the farther the distance between them, the weaker the force.
Some historians of science say that the reason why Newton was able to ridicule his "dead enemy" Hooke without any doubt that his achievements were higher than any of his contemporary opponents was because he invented tools for making tools—he played with mathematics. Centuries later, the scientific community named the unit of measurement of "force" after the British scientist.
Starting from gravity, human beings move towards dimensions, time and space, and black holes step by step... The most intelligent people among humans began to explore the most basic principles of everything in the universe, proposing, proving, and perfecting these great scientific concepts. Most of what they did followed Newton's old method:
1. Create a mathematical world in which to model problems;
2. Solve problems in the mathematical world;
3. Translate the results back into the real world.
Mathematics is beautiful and powerful, but not perfect. Newton suffered, and so did Einstein. Gauss had discovered (at least he said so) new geometries long ago, but dared not do anything. Sometimes math can feel awkward when faced with reality. But mathematics as a tool to understand, analyze, prove, and clarify this physical, realistic, and real world has never been soft. The key is that abstracting and modeling reality is the fundamental scientific method to clear the sky; bringing abstract and modeled things back to real life is the extraordinary ability of applied science. The methods and abilities of scientists can also be our methods. This ability can and should be learned.
In this heavy rain, we no longer flinch.
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Author: [France] Mickaël Launay
Translator: Ou Yu
surprise! is the starting point of thinking;
Mathematics is a tool to understand the nature of the world and the relationship between all things!
Take mathematics as the starting point and think for joy!
A masterpiece of popular science by the winner of the "D'Alembert Prize" of the French Mathematical Society.
Mathematics is a tool for understanding the nature of the world and the relationship between all things. It can create two compass: one is called "practical" and the other is called "elegant". If you don't understand the meaning of mathematics, you can't really learn and understand mathematics.
Why are scientists so smart? Because they have extraordinary ways of thinking.
Use mathematics as a tool, take thinking as joy; cultivate your own thinking and observation skills, and become a real thinker.
【About the Author】
[France] Mickaël Launay
Doctor of Probability from Ecole Normale Supérieure in Paris, France. After graduation, he participated in many mathematics promotion activities for the public and is a member of the French "Cultural and Mathematical Game Salon". His online math show "Micmaths" has more than 500,000 subscribers. Won the "D'Alembert Award" of the French Mathematical Society and the book award of the French popular science magazine "Tangente". He is the author of the best-selling mathematics science book "Everything is Number".
