Doing math problems just for fun? A review of some of the most famous “useless” mathematical studies in history

Editor’s Note:

In the 2024 "Ali Mathematics Competition", Jiang Ping from Lianshui Secondary Vocational School in Jiangsu Province ranked 12th in the world with a high score of 93. Overnight, this 17-year-old technical secondary school girl from Jiangsu Province swept the entire Internet, and discussions about mathematics also became lively.

For most people, mathematics is a love-hate relationship. So, what is the mathematical world like in the eyes of mathematicians? What is the use of studying mathematics? Especially for those basic mathematicians who mainly focus on theoretical research, are they doing "useless" research?

Today, let us follow the gaze of a mathematician, look back at the two thousand years of mathematical history, recall the mathematicians in the long river of time, and feel the charm of mathematics together.

I believe that everyone has been frequently taught about “useful mathematics” recently (we have also written about it), but there are also some mathematicians who seem to have been doing “useless” research, such as basic mathematicians who mainly focus on theoretical research .

Part 1

Those famous "useless" studies

The world of basic mathematicians is a paradise full of rationality and innocence, a palace built with thoughts and talents. Every mathematician who devotes himself to basic research is building his own spiritual edifice with his life. However, although the door to this paradise is open to everyone, such a beautiful spiritual world does not attract most people. On the one hand, the threshold for admission to basic mathematics is high, which can easily make visitors feel deeply frustrated; at the same time, the separation between the ideal kingdom where basic mathematics resides and the real world is also affecting the choices of generations. If we take a close look at the terms used by basic mathematicians to communicate with each other and the problems they are deeply fascinated by, it is not difficult to find that they and the public seem to live in two parallel worlds.

1. Goldbach's conjecture

"Any even number greater than 2 can be written as the sum of two prime numbers..." This conjecture, proposed by German mathematician Goldbach in 1742, has haunted countless outstanding mathematicians in later generations like a ghost. So far, the Chinese mathematician Chen Jingrun, who has made the most significant achievements on this issue, has become famous for Xu Chi's reportage "Goldbach's Conjecture". But unexpectedly, Chen Jingrun's personal image has also been regarded as a label of a mathematician by the world.

Bronze statue of Chen Jingrun at Xiamen University (Source: Wikipedia)

As a typical scientist who bravely climbed to the top, Chen Jingrun's story has touched generations of Chinese people, but it is difficult for the public to understand his research results and significance. We often see questions like "What's the use of proving '1+1=2'?" Many people even misunderstand the "Goldbach conjecture" as a question like "Prove '1+1=2'". People talk about Chen Jingrun's contradictions. He has extraordinary wisdom to study top mathematical problems but lacks basic common sense in life. He still strives hard in a harsh and difficult environment without forgetting his original intention.

However, in this process, the public unconsciously formed a deep misunderstanding of Chen Jingrun, the Goldbach conjecture, and even the mathematician community. The rich wisdom and indomitable fighting spirit of mathematicians have been partially condensed into such a label: mathematicians are a group of people who use their top intelligence and talents to waste their talents on useless or even boring research. At the same time, they are detached from the mundane world and do not care about worldly affairs.

A hot knowledge

Goldbach's conjecture: Any even number greater than 2 can be decomposed into the sum of two numbers, one of which is prime and the other is also prime.

2. "42"

In 2019, "42", known as the ultimate code of the universe, was deciphered by mathematicians. This is a relay race similar to a math game. Mathematicians conjecture: except for natural numbers of the type 9n±4, all natural numbers within 100 can be written as the sum of the cubes of 3 integers. For example:

Figure: 1, 2, and 96 can all be written as the sum of the cubes of three integers

9n±4 type natural numbers refer to numbers like 4, 5, 13, and 14, and there are 22 of them within 100. After arduous efforts, mathematicians have found that all non-9n±4 type natural numbers except 42 satisfy the above conjecture. Only 42 has never been able to get a positive or negative answer. Some people even think that 42 is the ultimate code that cannot be deciphered, so it is jokingly called "the truth of the universe" by mathematicians. It was not until 2019 that mathematicians finally discovered that 42 can really be written as the sum of the cubes of three integers.

Figure: 42 can be written as the sum of the cubes of 3 integers

The public's fragile nerves were once again struck by surprise, puzzlement, confusion and even dizziness. Is the problem that has so many mathematicians racking their brains and losing their appetite really just a mathematical game? Is their intelligence and talent just to provide people with after-dinner talk? Are mathematicians innocent or bored?

3. Others

Russian mathematician Perelman became a genius in the media because he solved the Poincare conjecture, one of the world's seven major mathematical problems. In 2010, this genius once again showed the unruly side of mathematicians: he announced to the public that he would give up the Fields Medal, the highest honor in mathematics, and the $1 million reward. What is difficult to understand is that when Perelman made this decision, he was still living a "tight" life. This unusual move by Perelman seems to have once again confirmed the impression of mathematicians in the public's mind.

In history, many mathematicians have indeed made similar moves as Perelman. French mathematician Grossendick, known as the "Pope of Algebraic Geometry", once retired to the mountains when he was young and strong. The stories of mathematicians always seem to be useless, weird, paranoid and unfathomable. Why do so many mathematicians choose a path that is willing to be misunderstood by the public and "keep themselves out"? Why are they obsessed with the so-called "useless mathematics"?

Image source: veer gallery

Part 2

Mathematics: From understanding the world to understanding the universe

Israeli historian Yuval Noah Harari put forward a view in "Sapiens: A Brief History of Humankind": Humans are the only creatures with imagination. It is imagination that allows humans to continuously iterate their civilization process. In the early days of human civilization, survival was the first element. Astronomy, calendar, sowing and harvesting of crops, counting of objects, and accumulation of wealth all require the help of mathematics. Therefore, mathematics is naturally integrated with the first step of human understanding and transformation of the world. It was also from that era that the mathematical edifice of mankind started from scratch and raised its height little by little. In this process, mathematics has gradually developed from meeting simple counting and calculation, which are daily needs of human beings, to the understanding of the basic principles of the universe: in ancient times, people believed that the sky was round and the earth was square, and in ancient Greece, people believed in the "geocentric theory", partly because the mathematician Ptolemy's great circle method almost perfectly explained the movement trajectory of the planets in the solar system.

More than a thousand years later, astronomical observations during the Renaissance first discovered that the "geocentric theory" led to observational deviations in planetary motion, and Copernicus proposed the "heliocentric theory". Based on Kepler's stargazing data and his "three planetary laws", Newton developed the "absolute space-time view" with the birth of calculus and extremely accurate theoretical predictions, which consolidated the view of the "heliocentric theory".

More than two hundred years later, in order to solve the inference of the constant speed of light revealed by Maxwell's equations, Einstein proposed the groundbreaking "Theory of Relativity", which gave birth to the monster of the universe under the "Theory of Relativity": the black hole under the big bang of the singularity. At the same time, in order to dispel the dark clouds of black body radiation, Planck broke into the ghost door of "quantum mechanics". To this day, the concepts of multidimensional universe and parallel universe triggered by string theory have opened the "Pandora's Box" of the technological kingdom.

Image source: veer gallery

Humans have always been trying to uncover the mysteries of the universe, and have come to this day step by step. Every progress and subversion of the cosmic view has brought humans closer to the ultimate truth of the universe. Mathematics has played a vital role in every leap of ideas. But if humans travel back to the eve of any great technological change, people will still think that mathematics at that time was just a useless game for the nobles. Until one day, these mathematical tools and ideas created by the great thinkers of the past opened new doors for humans to understand the world.

One of the interests and missions of mathematicians is to explore the most basic order and laws of the world, interpret the principles behind them, and find the simplest and most essential response strategies for the complex phenomena in the real world. Surprisingly, the rapid progress of natural sciences often benefits from the prophetic predictions of mathematical theories.

Part 3

"Useless" will eventually become "useful"

Entering the 21st century, more and more mathematical theoretical achievements have spread, and many branches that were considered "useless" in the early days have become the most powerful tools of modern science and technology and the driving force of modern scientific and technological development. Mathematical research, which was once shelved and isolated, is now becoming a sharp tool in people's hands, breaking through obstacles in the exploration of the material world and providing people with more and more ideological motivation and creative sources. The birth of calculus opened the grand era of Newton's mechanical universe view.

People were surprised to find that the vast universe was not mysterious at all. The matter and phenomena in the outer space were closely related to the planets we live on at the deepest level. Even the celestial bodies hidden in the deep space followed the same laws of motion. Since then, Newtonian mechanics has become the key to unlocking the treasures of science and technology. The steam engine and engine invented based on its principles directly ignited the flames of the first industrial revolution.

Image source: veer gallery

The information age civilization we enjoy today, such as computers, mobile phones and the Internet, has benefited greatly from the development of quantum mechanics. This science, which has completely changed people's lives, is derived from the gifts of many basic mathematical theories: from linear algebra, matrix analysis, statistics, to group theory invented by mathematicians to solve the problem of solving quintic equations. Based on general relativity, people invented satellites that break through the gravitational constraints of the earth, which makes space-to-earth communication possible and lays the foundation for deep space exploration and land and sea navigation.

People are traveling more and more frequently, and GPS navigation based on geographic location is providing unprecedented convenience for our lives. The mathematical principle of Einstein's general theory of relativity, which made him famous for thousands of years, is the application of non-Euclidean geometry (especially Riemannian geometry) and tensor analysis. Since the late 1980s, string theory, which has been a standout in physical theory, has been favored by scientists at the time because of its bold and avant-garde ideas. This grand unified theory, which is expected to solve relativity and quantum mechanics, has gradually aroused huge waves in the mainstream scientific community. On the road to the booming development of string theory, it is not difficult to see the firm back of differential geometry.

In 2016, three physicists shared the highest honor - the Nobel Prize. They won the prize for discovering topological phases of matter and their outstanding contributions to the theory of topological phase transitions. For the first time, the difficult and abstract mathematical topological theory has found its place. Physicists used the topological tools invented by mathematicians to theoretically predict the existence of a special material. In it, people can observe the incredible anomalous quantum Hall effect. The material discovered based on this effect can spontaneously show the characteristic of zero resistance in a certain direction at room temperature without the assistance of a super strong magnetic field.

This provides an infinite space for the development of computer chips, and the dream of quantum computers and micro-supercomputers is one step closer to us. In the eyes of theoretical mathematicians, number theory is the most basic and important branch. This mainstream branch of mathematics, which has lasted for more than two thousand years, has accumulated a lot of valuable spiritual wealth. Whether it is the Goldbach conjecture or Fermat's Last Theorem, the process of deciphering them has given birth to extremely creative ideas. Their conclusions are so concise and clear, but the road to exploration is so tortuous. People also don't know when these crystallizations of wisdom can break through the possible predicaments in the real world.

Fortunately, the RSA algorithm based on number theory has built a solid security line in the financial system, Internet security, online shopping, mobile payment and even national defense, and the elliptic curve invented based on Fermat's Last Theorem has continued to benefit the world through blockchain technology, etc. Once the key technologies in quantum computers and quantum cryptography are successfully introduced, mankind will surely enter a new era.

IBM Q System One, the world's first commercial quantum computer (Image source: Wikipedia)

In this paradise where basic mathematicians live, there are countless inventions and benefits that may have a profound impact on future generations. History has proven that many major inventions and creations originated from the theoretical wealth that mathematicians inadvertently left in history. Mathematicians' breakthroughs and innovations at the theoretical level are precisely providing original ideological tools for future generations to transform the real world.

Mathematicians are not as solitary as people imagine. They generally have calm inner qualities, such as purity, humility, perseverance, diligence, and awe of science. They are willing to live in a corner of the noisy feast of wealth and continue to contribute their lifelong wisdom to the search for universal truth in the world.

Produced by: Science Popularization China

Author: Huang Yiwen

Producer: China Science Expo

This article only represents the author's views and does not represent the position of China Science Expo

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