The beautiful Harz Mountains sit right at the center of the modern Germany. Its calm ancient forests, its dense trees, its silky fogs and its altitude so close to the sky make it a paradise to clear the mind and cleanse the soul. City dwellers, tired of the endless bellow of machines, escaped here for their vacations to enjoy the air and the silent, yet mighty, landscape, where mountains after mountains unfold in front of the eyes.
In the Mindscape, the world of thought, the Harz Mountains are, too, a spectacular wonder. Over one hundred years ago, great thinkers come here, discussing in little hotels of little villages, drinking beer, eating Wurst, and climbing mountains after mountains. The long hikes fatigue the body, strengthen the sprit and purify the mind. As you will surely notice if you climb a mountain, when almost all the energy is washed up, the only questions that remain are the most fundamental ones.
This is definitely true for the young mathematician Georg Cantor.
One day in the year 1872, during a hike that looked infinite, counting the steps that he took, Cantor asked himself: In the eternal journey towards larger and larger numbers, how far are we able to go? Is there an infinity beyond infinity? Is there a sky beyond the sky?
For thousands of years, infinity was believed to be the ultimate, the unimaginable, the highest mystery belonging only to the realm of gods. Galileo found a relation between natural numbers and square numbers that looked ordinary, but becomes extremely bizarre when you think about it.
Every natural number corresponds to its square, and conversely, every square number corresponds to its square root. But shouldn’t there be much fewer square numbers than there are natural numbers, since the former is only a small part of the latter? Imagine walking in an autumn forest, where leaves come in all colors: green, yellow, red, brown, how is it possible that the number of red leaves is equal to the total number of leaves? Galileo concluded that infinity can’t be grasped by human comprehension. Better leave this monster alone!
But if you have competed with your friends about who can come up with the largest number, you know that infinity is not the end of the game. If you shout, “Infinity!”, your friend will respond you, “Infinity plus one!”, and then you say, “Infinity plus two!”… and it goes on and on. For Cantor, infinity is not the end of human imagination. If infinity exists in some way, it is not to be feared, but to be awed directly in the mind’s eye.
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Thinking about infinity may be daunting to anyone, but not for Cantor. In his early years, he was exposed to different cultures and their stories. He was born in Russia, but moved to Germany when as was 11 years old. His parents had Jewish, catholic and protestant backgrounds. Little Cantor was educated as protestant but loved to think freely about spirituality and the world of gods. At one point. this lonely soul in the mountains roared: you mighty wind, take me to the highest realm of the divine!
In his hike, Cantor wondered, what if the correspondence found by Galileo is not a sign saying “danger”, but instead a key to understand infinity itself?
After all, the history of mathematics is full of blessings in disguise. Subtracting a number by a larger number: danger! But from there come the negative numbers, that help us describe temperatures below freezing. Dividing 7 by 3, there is a remainder that won’t go away: danger! But from there come the fractions that help us cut cakes. This time, Cantor believed that he would not be scared away by the danger of infinity.
Perhaps what discovered by Galileo is not an anomaly, but a truth about infinity in itself? Perhaps the one-to-one correspondence of natural numbers and square numbers means exactly that there are as many natural numbers as there are square numbers. But how is it possible? Sure, rabbits can’t be as many as all animals, red leaves in a forest can’t be as many as all leaves, and if a football team gets too many red cards that it has only 5 players, it probably can’t play as well as when it had 11 players. But animals, leaves and football teams are finite: part can’t be as large as whole for finite things: there is no reason to carry this assumption over when we’re talking about infinity! For infinite things, maybe part can be as big as whole: nobody said they cannot.
Though an excellent student at school, Cantor was not satisfied with how the schools taught mathematics. In schools, we are taught that math is about what is correct, that math is objective and cold. This is not true for Cantor. For him. math is freedom and creativity, and nothing is too scary to think about. Little by little, he became convinced that infinity can be compared, and that the one-to-one correspondence that we saw just now is how we compare infinities.
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According to Cantor, some infinities are “countable”, that is, they are exactly as large as all the natural numbers considered together. How to determine if an infinity is countable? Well, imagine that you have infinite amount of time to discover a huge magic kingdom with an infinite number of cities, but you must visit only one city every day. Can you write up a travel plan that allows you to visit every city once and only once? That is, given any city, can you look at your plan and say “Okay, I will visit it on the X-th day, and I’m sure I will not visit it twice!” If you can come up with a plan like that, then you say that the number of cities is countably infinite.
Here are some examples of countable infinities, and maybe you will be surprised.
The positive even numbers are countable. If every city is named by an even number, like City 2, City 4, City 6, …, we just visit City 2 on Day 1, City 4 on Day 2, City 6 on Day 3, and City X on the Day X/2. Every city will be visited once and only once.
The integers (containing natural numbers, zero, and the negatives of natural numbers) are countable. The integers can be thought of as points lying on a straight line, each point is a city named by an integer. Our travel plan is that we visit City 0 on Day 1, City 1 on Day 2, City -1 on Day 3, City 2 on Day 4, City -2 on Day 5, and so on, back and forth…, and we are able to visit every city once and only once.
The points on an infinite square grid are countable. Imagine that every point is a city. Starting from any city, we are able to visit all the cities in the spiral travel plan shown below.
So, in short, Cantor thinks that all the infinities above are equal! Natural numbers are as many as positive even numbers, which are as many as integers, which are as many as points on a square grid, which are, as Cantor can show, also as many as fractions!… They all look so different, but they are equal.
Do you believe what Cantor says? If not, don’t worry: Cantor didn’t believe it either. He wrote to his friend Dedekind one day that “I see it, but I don’t believe it!” He and Dedekind were both German, but Cantor wrote this sentence in French, to show his surprise. We don’t believe these relations between infinities, because we live in a finite world. We tend to think that infinity just means very, very large. But it doesn’t. Infinity has weird properties foreign to any finite numbers, not matter how large. The Gospel of Matthew calls it a miracle when Jesus was said to feed 5000 people with 5 loaves and 2 fish. Every 1000 people have a single loaf: it is not possible to feed everyone in real life, unless we have a loaf as large as the football playground. But if we have infinitely many people, and every 1000 people have only a single loaf, then it is possible to redistribute the food, such that everyone has a loaf.
How to do that? Well, we give everybody a number: 1, 2, 3, … Now, because only one in a thousand has a loaf, we let the 1000th, the 2000th, the 3000th, … be the people with a loaf. Then we ask Person 1 to take the loaf of Person 1000, Person 2 to take the loaf of Person 2000, … and so on. How about Person 1000, who has donated their bread to Person 1? Well, they take their bread from Person 1000000… If you think about it, everyone has a bread. This is never possible if the number of people is finite.
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However, is there ever an uncountable infinity? Or can we think of a magic kingdom, such that no matter how we make our travel plan, there is always a secret city that is missed out? Cantor began to study an infinity that looks like a perfect candidate for being uncountable: the continuum, that is, the points on a perfect line segment.
In daily life, “continuum” is used to describe the property of water, or heat, or the variation in colors, as opposed to flowers, trees and sands. Continuum doesn’t have smallest units: A drop of water is small, but half a drop is still smaller. However, the continuity of water and sunlight is really nothing more than a trick to the naked eye. Water is made of little Mickey Mouse shaped molecules, and light too, transmits in little bags of energy called phonons. Looking at Nature with a magnifying glass, it is in fact particulate and abhors continuum. However, it is never difficult to imagine a perfect ruler in the mind: smooth, simple, homogeneous, complete.
If the ruler is 10 centimeters in total, starting at 0 cm and ending at 10 cm, then the points on the rulers can be described by numbers that are greater than 0 and smaller than 10: every possible number that you can or cannot think of between 0 and 10 is on the ruler. These numbers contain everything that can be written as a fraction of integers, like 0.5=1/2, or 0.66666…=2/3, or 5=5/1, but also everything that can’t be written as such. A fraction can be written into finite decimals or recurring decimals, but the continuum also contains those numbers that have nonrecurring decimals, like π=3.14159265… Integers, finite decimals, recurring decimals, and nonrecurring decimals are together called real numbers. The question “How many points are there on our perfect ruler?” is same as “How many real numbers are there between 0 and 10?” Infinity, for sure, but how infinite exactly? Countable or not?
When Cantor wrote to his friend Dedekind about this question, Dedekind was extremely skeptical about what Cantor is up to: Even if we find the answer, so what? Will math become easier? Why should anyone even care about it?
But Cantor cares, in the pristine Harz mountains. the place closest to the secrets of heaven.
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If you have felt the answer in your intuition that the continuum in not countable, bravo! In 1873, Cantor was able to prove your intuition: there are strictly more real numbers than there are natural numbers. You might think that Cantor’s demonstration must be impossible to understand. But no: the greatest philosophies are those that are simplest. And Cantor’s demonstration is one of the simplest of all time: so simple that it is striking.
In the kingdom of real numbers, every city is named by a real number, say, between 0 and 1 for simplicity. There are City 1, City 0.5, City 2/3, City π/10=0.31415926…, and much more. Like before, a tourist comes up with a travel plan about which city to visit every day. That is, they have constructed a correspondence between natural numbers and the real numbers. Here is a random example
Cantor’s magic touch is, then, to show that no matter how the list looks like, there is always a real number not in the list. There is a lost city, no matter how you choose to tour the magic kingdom of real numbers.
This number, let’s call it 😄, can be found like this: to get its x-th decimal digit, simply look at the x-th decimal digit of the x-th number on the list. That is, we look at the diagonal of the list above. If that digit is a 2, then we write 1 for the x-th decimal of 😊, if that’s not a 2, then we write 2 as the x-th decimal of 😊. So, for the list above. we would have
And this will be our lost city a number not on the list. Why not? Suppose someone claims that this number is the 2020th on the list, we can be sure that this is impossible. Because if the number on the list has its 2020th digit equal to 2, then the 2020th digit of 😄 is not 2. And if the number on the list has its 2020th digit not equal to 2, then the 2020th digit of 😄 is a 2. But we can change 2020 to any other number, and the same argument follows. The conclusion: 😄 can be nowhere on the list, and this means there are absolutely more real numbers than there are natural numbers, because you are never able to count the real members without missing something.
Therefore, Cantor says, he discovered two infinities, one (the continuum) absolutely larger than the other (the countable infinity). In fact, the number of mathematical points in one centimeter is much larger than the number of seconds in the life of the universe. And this is true, even if the universe has no beginning and no end. In Essay on Harmony of Everything（《齐物论》）, it is written “The universe is not larger than the tip of a hair”. Maybe it’s just mathematically true that every flower contains a world. A poem summarizing Cantor’s work would be
格物始知可齐物
Observing all, he realizes the equality in all
登天惊得另有天
Reaching the sky, he discovers a new sky
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I can share with you what I think of Cantor! When I first heard of Cantor’s discovery, I was utterly shocked. This is a soul directly communicating with the highest edifice of the spirit: Only an immense mind can think of immense things, let alone infinity! This is a human being extremely pure, who doesn’t have to care about anything physical that restrict us on earth. The proof is simple, and the fact is simple, but it only arrived at the mind who is willing to receive it. I was so enchanted that everything in the world became less important, because the greatest secret of heaven is right in front of my eyes. Cantor, I thought. is the ultimate bird of infinity.
But as I write the book, as I retell to you this story that I know so well, I begin to doubt what is it that Cantor really discovered. His argument that there are leftover real numbers is beautiful, but it means nothing more than what it says. A complete list of real numbers is impossible, and that’s all. You may feel that this is a far cry from the claim that the reals are more “numerous” than the natural numbers, and I feel it too. The former is an observation of correspondence, and the latter, a statement about the mysterious infinity. In other words, both natural numbers and real numbers are artifacts of humans to help organize objects like apples or water. No matter what bizarre relation you find from those, it is a relation between human-made concepts. Are you allowed to talk about infinities as if they really exist, and even start a sport to see who is larger? Not sure.
In his time, mathematicians were divided into two camps of views of Cantor’s ideas. One camp, mostly Dedekind, thought that infinity is not a taboo to be studied in mathematics. Zero and negative numbers, which were feared and resented for many centuries, ended up being used as a staple not only to mathematicians but to everyone. After all, nobody has said what is allowed and what is not allowed in math: math is a freestyle dance of the mind. The other camp, most importantly Kronecker, believed that taking infinity seriously is like witch-hunting. They think that infinity is just a way of speaking, and can’t be studied as if it really exists. It’s in the same way that we don’t study unicorns in zoology! Furthermore, a mathematician called Feferman said Cantor’s theories are simply not relevant to everyday mathematics.
Kronecker also said: “I don’t know what dominates Cantor’s theory- philosophy or theology, but I am sure there is no mathematics there.” This is of course a fierce criticism, but it is fair to say that Cantor’s work contains deep philosophical or theological thoughts. Some of these thoughts even motivated his work.
For many who believe in some kind of deity, or some spirit larger than the physical world, the mathematical nature of Cantor’s theory is a less important issue. The more important is what shock it brings into their beliefs. Infinity has always been considered as a property of the deity, an omnipresent, omnipotent, ultimately benevolent being. In the religions where God is unique, it is just evident that infinity is unique. So if there are larger and larger infinities, where is the place for God? (Cantor later showed that there are much more than two infinities. For any infinity, you can create an infinity still larger.) We are like the little prince, who loved a rose on his planet, who suddenly discovered a field full of roses. Humans have worshipped infinity all the time, and suddenly Cantor throws to us a plethora of infinity. So what is so special about infinity at all? The disturbance of Cantor’s theory to philosophy is profound. Cantor himself, deeply religious, believed that his theory was not the problem but the solution. The larger and larger infinities, he said, are the ladder to God.
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It would be great if all the discussions are in the Mindscape, that mathematicians are friends even if they disagree in mathematics. However, the world of ideas and the world of matters are never separated. The dispute between Cantors and mathematicians like Kronecker grew to become enmity. Kronecker refused to publish Cantor’s work in the journal where he was editor. He said that Cantor was “a corrupter of youth”: Cantor was rejected by the mainstream researchers in Berlin, and had to teach in the University of Halle, a good but isolated school. He tried to invite Dedekind to Halle, to start a new form of mathematics. However, Dedekind declined, saying that he had to take care of his old mother. All these stumbles in career made Cantor’s spirit fragile, which has always been burning in all the searches of infinity. He developed mental disorder and sometimes devoted his energy to unreasonable projects, like proving that Francis Bacon wrote the work of Shakespeare. It is not known if his mental problems arose mainly because of academic enmity, or more attributable to his research, or to physical reasons like heredity. Maybe there is no single “main reason”! Cantor, a passionate, proud and even explosive free mind, who looks at the sky even if he reaches the mountaintop, just isn’t capable to withstand the competition and hierarchy that he experienced. For me, this freedom is the most important. Whatever is the value of his theory, Cantor made it clear that it is possible for human beings to seriously think about infinity.
And exactly because of this, we don’t have to dwell on the sadness in Cantor’s life. It is his free, courageous mind that was most important. Finally, let’s imagine another day of hiking, where Cantor walked with family through the fog in the mountains, and asked his children what they thought of infinity.
One of the children said: “I imagine infinity as the endless fog, immense, mysterious.”
They climbed further, until the sky cleared and the fog receded into an endless cloud beneath their feet.
Cantor said, with a firm voice: “I imagine infinity as this cloud: we can see it, but only when we stand high.”