The Gift of a Small Universe—From Linnaeus to Poincaré

"The most important things are invisible…"

"Just like flowers. If you fall in love with a flower that grows on a star, then at night, when you look at the sky, you feel sweet and joyful. It seems as though flowers bloom on all the stars."

"At night, when you look up at the stars, my star is too small, I cannot show you which one it is. It's better that way. You can think that my star is among these stars. Then, you will like to watch all of the stars... and all these stars will become your friends. And I am going to give you a gift..."

"At night, when you look at the sky, since I live on one of those stars and since I am laughing on one of those stars, it will seem to you that all the stars are laughing. The stars you will see will be laughing stars!"

Please forgive me for quoting such a long passage from "The Little Prince." It's a story that Young Li dearly loves, and its childlike innocence and purity touch me every time I read it. Young Li never wants to grow up, because children view the world with such curiosity and hope. However, although we can't stop the arrow of time, we can always maintain the beautiful heart of a child. So today, let's see the world through the eyes of a child!

What attracted me the most during my childhood was fantasy novels. They created new worlds independent of our reality, like the animal kingdom in "The Chronicles of Narnia," the wizarding school in "Harry Potter," the Middle-earth in "The Lord of the Rings," and the continent of Westeros in "A Song of Ice and Fire." These worlds had different rules from ours, filled with wonders like magic and strangely shaped creatures. Immersing myself in these worlds, imagining myself as one of their inhabitants, was simply thrilling.

It wasn't until I lifted my eyes from the pages and looked out at the street lights coming on outside the window that I snapped back to reality: all those worlds were fictional, hmm, maybe they really exist, but at least no one but their authors has truly been there. But why can we imagine such worlds so vividly in our minds, just from symbols on a white page, as if we had been there ourselves?

Perhaps it's because we can think and imagine through language, and every symbol in language represents something that really exists. We cleverly name everything in the world, using a sound or a symbol to represent it, so when we ask someone to do something, we just need to speak, without having to do it ourselves first to demonstrate. Names are indeed a magical thing. When we see the words "Saint-Exupéry," we think of that French pilot with goggles and a touch of melancholy, and the North African desert he often visited, not someone else. With such clear correspondences, our communication becomes much more efficient, and we can imagine places we've never been and make up stories that have never happened.

But how should we name things so that everyone will accept and use these "codes"? People can tell others their names, but inanimate objects can't speak for themselves. When naming, we hope to give it a "good name," so that upon hearing it, others will involuntarily use their lovely imagination to guess some characteristics of the thing. For example, knowing what "elementary school" and "middle school" are, one can guess there's also a "university," where the knowledge studied must be more difficult than in "elementary" and "middle school." Moreover, this also includes a kind of "classification," dividing all the schools in the world into three categories: "large, medium, and small."

It seems not so difficult to come up with names, but the world is so wonderful that when it comes to naming the countless plants and animals, people start to feel overwhelmed. Not to mention finding a good name, even ensuring that names are not repeated is a problem. At this time, a botanist named "Linnaeus" appeared.

"I have passed through the clouds

And I have reached the ends of the earth

I have witnessed a solar eclipse

And in one year, I traveled 6000 miles across the land"

— Excerpt from Linnaeus's "Journey to Lapland"

In 1732, Linnaeus single-handedly surveyed much of the Nordic region, producing a 1200-page "Species Plantarum." Not only was he incredibly diligent, but he was also exceptionally astute, classifying plants based on primary characteristics such as the pistils, roots, and shapes of the leaves, covering more than 5900 species and 1098 genera. The names he assigned to plants were binomials: the genus and the species name, these Latin words depicted the main characteristics of the plants. This was a vast and effective system; when encountering a new plant, by examining its pistils, roots, leaves, etc., one could determine what name it should be given. Linnaeus later became the president of the Royal Swedish Academy of Sciences, and during his tenure as rector, Uppsala University in Sweden reached a "world-class" level.

I climbed the towering peaks of Lapland,

Ventured deep into the endless valleys of the Falun mines,

Witnessed the perpetual daylight of Lapland,

And saw the unending darkness of Falun,

The endless cold of Lapland,

The ceaseless heat of the Falun mines,

In one day in Lapland, I experienced the change of four seasons,

Yet found no trace of seasons in the Falun mines,

I went through life and death in Lapland but always remained safe,

Faced various crises in Falun but emerged unscathed,

Praising the Lord's every creation,

From beginning to end.

Excerpt from the "Linnaeus Diary"

The itenery of Linnaeus's scientific expedition in Lapland

Diagram of Linnaeus's botanical classification system

Often, something that might seem "trivial" can hold profound significance. At the time, Linnaeus's work might have primarily impressed people with its astonishing complexity. However, later on, when Darwin proposed the theory of evolution, it became a hot topic discussed worldwide. At that point, Linnaeus's classification and nomenclature became very important: the dynamic laws of species evolution proposed by the theory of evolution are built upon the classification of species. With a systematic classification of species, we can discuss the evolutionary connections between different species and genera. Thus, Linnaeus's system gained new significance. From this point, with the efforts of many later naturalists and theorists, modern systematic biology was developed. People began to view themselves and the world from a macroscopic perspective of connections, seeing all things as an indivisible whole, with the light of Mother Nature shining into the human spirit.

"Once names are correctly established, reality can be clearly understood; once the Dao is practiced, aspirations are achieved" — Xunzi

“名定而实辨，道行而志通” —《荀子》

From this, it appears that classification and naming may be important steps in understanding nature and knowing the world, the significance of which we may not yet fully comprehend. However, the spirit of Linnaeus continues and is constantly reinterpreted and recreated by later generations.

By the end of the 19th century, the brightest minds had begun to transcend the natural systems of Earth, attempting to establish the order of the entire universe. One issue that perplexed people was the three-body problem, concerning the motion of a system formed by three celestial bodies. People established differential equations to describe their motion but could not solve them due to their complexity. At this point, the French mathematician Henri Poincaré had a different idea: perhaps we cannot express the solution to these equations with mathematical symbols, but we can think of the solution as a surface in a higher-dimensional space and investigate the properties of this surface to find answers to the problem. With this idea, Poincaré began to think about how to qualitatively study the characteristics of spatial geometric bodies, that is, those properties that remain invariant under continuous transformations of the geometric body (such as stretching and shrinking). Poincaré aimed to classify all these geometric bodies and discover their patterns, which would profoundly affect our understanding of the entire universe.

In fact, Poincaré's vision extended far beyond the three-body problem. As he developed the abstract theory of algebraic topology, his mind was filled with profound thoughts. Poincaré and Einstein simultaneously came up with ideas about relativity, sharing similar understandings of the relativity of space and time. One issue Poincaré focused on was how a smart ant could determine whether it was crawling on a spherical surface or a toroidal surface (the surface of a doughnut). Furthermore, for humans, how can we know the shape of the spacetime we inhabit? Poincaré thought of an answer: the ant could keep going around in circles, and if it finds that its path could shrink to a point, then it is on a sphere; however, we know that a circle on a torus cannot be contracted to a point. From this starting point, Poincaré established the theory of homotopy, classifying geometric bodies based on whether "circles" can be contracted to points, which became the cornerstone of algebraic topology.

"For me, in exploring the universe, all the paths I have taken continually lead me to situs analysis (topology)." — Poincaré

The essence of algebraic topology lies in "naming" all geometric bodies, but this name is a mathematical one. To standardize and extract the main features, Poincaré first approximated surfaces: like a sculptor with a chisel, he sliced the surface into many interconnected triangles, sparkling like diamonds, called "triangulation."

Thus, a surface becomes a collection of many points, lines, and planes. The numbers of these points, lines, and planes can vary with different triangulations, but the relationship among these numbers remains constant (this is the Euler characteristic formula learned in high school). It's not hard to guess that this numerical relationship contains important properties of the surface and can be expressed by coefficients of these three quantities. Consequently, these coefficients become the "name" of the surface; for example, the name of a sphere is "1-0-1," while a torus is "1-2-1." For surfaces in higher-dimensional spaces, similarly, a string of numbers can represent them.

Euler's formula

Thus, numbers can represent not only the quantity, length, weight, etc., of an object, but in Poincaré's view, a sequence of numbers can characterize the important features that influence the shape of a geometric body. Poincaré even discovered intrinsic rules within these numbers (known as Poincaré duality), further refining the measures that depict the surface's deeper qualities. In essence, knowing the "name" tells us many important characteristics of a geometric body; on the other hand, to what extent surfaces with the same "name" are similar, and whether we should assign longer names to surfaces to further distinguish them, are questions that mathematicians continue to explore today.

It's worth noting that the subsequent great mathematician Emmy Noether devised a new way of naming geometric bodies: not with a string of numbers, but with a string of abstract mathematical structures called groups. A group can be seen as a set, within which elements have a binary relationship, and some special elements in the group called "generators" can generate all other elements through relationships with themselves. The number of generators in each group within the "new name" of a surface corresponds to the numbers in the "old name." This new name appears more complex and abstract, and might not be popular (much like Daenerys's long list of titles in "Game of Thrones"), but it has profoundly influenced topology: more complex mathematical structures began to be used to "name" surfaces, which in turn solved many important problems. Mathematicians shifted their focus from specific computational techniques to abstract concepts and relationships. The entire field of mathematics underwent continuous transformation.

From this, it appears that naming is indeed a crucial task: a good name can greatly assist our research and thinking, and a new naming method can even revolutionize a discipline. Our human language naming system is like a "small universe"; it is independent of the world we live in, has its own rules, and many interesting things happen in this universe.

Each language is a universe, and those who speak it are like voyaging within it. In this universe, we don't need to travel far to grasp all its mysteries. We can attend magic classes at Hogwarts, converse with the beautiful elves in Rivendell, and look back on the history of Westeros from beneath the weirwood of Winterfell.

Modern linguists increasingly deepen their understanding of language, focusing on how using language affects our thinking and cognition. Studies of language may reveal secrets of intelligence. Like in the movie "Arrival," where the aliens who use a unique non-linear language can foresee future events. To explore this issue, scientists began to study and analyze the structure of language. Each word can be composed of several fixed symbols and considered as a point in high-dimensional space, while a piece of text is a geometric body in high-dimensional space. At this point, people once again thought of algebraic topology. We know that some changes in words do not alter the semantics of a sentence, that is, geometric bodies with different structures but similar shapes can correspond to the same semantics. So, what exactly is semantics? It likely corresponds to some features that algebraic topology is concerned with, perhaps based on the names topologists give to those types of geometric bodies. A philosophical speculation is that our brains inherently understand mathematical theories, or operate according to certain mathematical theories, to decode the meanings of sentences, to understand the world through language. Perhaps our brains have an innate internal naming system, and understanding this system will unveil the secrets of intelligence.

By now, perhaps everyone can understand that all our knowledge is essentially the "name" of something: knowledge of an object is not the object itself, but a depiction and abstraction of its main features, a model. In psychology and neuroscience, this is known as "representation." Its significance lies in our ability to see the connections between things through the model, thereby revealing the secrets hidden deep within nature. As George Santayana said in his "The Sense of Beauty":

"An artist changes the relationships of different parts always in a consistent direction and intentionally, so that a certain 'main feature'—which is the principal idea the artist has of the object—becomes especially clear. Gentlemen, we must remember the term 'main feature.' This feature is what philosophers call the 'essence' of a thing, so they say the purpose of art is to express the essence of things."

I can immediately reply that a main feature is a property; all other properties, or at least many other properties, are derived from the main feature based on certain relationships.

In the sci-fi novel "The Three-Body Problem," at the end, the surviving humans wander to a corner of the universe. Cheng Xin's friend gives her a "miniature universe" in which they can survive the "Big Crunch" and welcome a new birth of the universe after the Big Bang. Perhaps, the ultimate mystery that humanity longs for is hidden within language. The method of abstract thinking through "representation" has profoundly impacted human society's development, from Linnaeus's naming system to Poincaré's establishment of algebraic topology, helping us understand the natural world and the entire universe, and helping us understand ourselves. "This little reed flute, you shall carry it over mountains and hills, and from its tube blow ever-new music." (from "Gitanjali" by Rabindranath Tagore)

Let us conclude with these words Alexander Garden wrote in 1761 from America to Linnaeus:

"Even though oceans and hills separate us, we can still be certain that before the curtain of death falls, the father of wisdom and science will lift the veil for us and clear the obstacles. Perhaps, I will find myself sitting on a meadow where Linnaeus is explaining the wonders of the world to many pure and sincere souls, praising the Creator's magical shaping of their minds."

Linnaeus's favorite flower during his lifetime—the Arctic Flower, also known as Linnaea borealis.

References:

Consilience: the unity of knowledge

《荀子·正名篇》

Systematics and the Origin of Species

Poincaré’s stated motivations for topology

Emmy Noether and algebraic topology 

Lecture Notes on Algebraic Topology (Miller)

Algebraic Topology (Hatcher)

他竟如此自负，幻想给全世界制定标准公众号：商务印书馆

逆袭的硬核祖师爷— 林奈公众号：林奈实验室

人类认知的边界在哪里？—乔姆斯基的回答公众号：BrainQuake 大脑激荡