The other day on Instagram I mentioned struggling with a piece of writing that I still wasn’t sure about. In fact, it revolves around the works of Anish Kapoor at the recent Venice Biennale. I went there with Charlotte, a mathematics professor of Chinese origin. She’s thirty-six this year , about ten years older than me.
While looking at Kapoor’s works, I started thinking about writing something on AI , more specifically, on large language models and the current condition of displacement and loss of inhabitable ground. The whole thing actually began from a joke Charlotte made after getting frustrated with AI algorithms during her research on conservation laws near black holes in relativistic physics. At one point she became so irritated that she jumped down from a short flight of concrete steps, landed badly, and hurt her ass. Afterwards she laughed and said that maybe AI could never replace “the romance of gravity” , the ability to look at the stars and still feel pain in your butt at the same time. To contemplate the cosmos while also bruising yourself on the earth.
But in the end, the text is not really about AI, or black holes, or any of those things. It’s about the fact that I met Charlotte, and spent time with her.
She’s a single mother with a five-year-old son. She let me stay with her for a while because of my visa situation, and she also helped create opportunities for me to collaborate with some of her publications. Yet most of our time was simply spent watching over the child together. We talked a lot about numbers, about looking at gardens, about small observations that somehow kept unfolding into larger questions.
Most of the people who have pushed me to grow, or who have become mirrors through which I compare and understand my own life, have been women. And honestly, I find it exhausting listening to men complain endlessly about their lives haha.
Anyway, these are a few pages I wrote while spending/learning time with the two of them.
The Development of Logic
Before bed, while sitting among piles of scattered Lego pieces, Charlotte’s five-year-old son suddenly asked me, “Are you beautiful and famous?”
I was already half asleep, but the question woke me up a little. I asked him, “That’s a strangely interesting question. So what do you think?” He replied, “You have to be both beautiful and famous. If you’re beautiful but not famous, then it doesn’t count.”
I added, “And if you’re famous but not beautiful, that doesn’t count either.”
Logically, it was a fascinating question. In order for the answer to be yes, both conditions had to be satisfied at the same time. Fulfilling only one of them would never be enough. Mathematically speaking, it was already touching on Boolean algebra.
I’ve always found the development of logic in children incredibly interesting. Concepts like “and,” “or,” and “not” are entirely human inventions — they were only formalized quite late in the history of mathematics, yet children grow up inside the logical structure of the adult world, and without any formal training, their cognition somehow arrives there naturally.
He will start primary school this September. Which means this wild stage of development, guided purely by curiosity and instincts, is already nearing its end. Soon he will learn a much more difficult kind of mathematics, one that exists within systems and curricula. That is, of course, a good thing. But he will also inevitably lose the ability to spend days, months, or even a whole year slowly thinking through a single question.
Counting and Arithmetic
Lately, Charlotte’s son has become obsessed with doing arithmetic problems that seem completely meaningless to adults, things like 2048 + 2048 = 4096. Part of it is probably just his desire to beat his mother at calculations. From the perspective of modern computing, it almost feels as though there is no practical use in the human brain performing calculations like these anymore.
But the real significance lies elsewhere. The process through which a child moves from counting on their fingers to performing abstract calculations mentally is far less effortless than it appears on the surface.
If you observe a three-year-old learning numbers, you notice that they often need extremely concrete references: one apple, two hands, three candies. If you ask them what “3” is, they probably will not say, “3 is a natural number.” Instead, they might hold up three fingers or point at three cookies on the table. To them, three fingers and three cookies may not yet belong to the same category at all.
Only much later ,after endless repetition , does a moment suddenly arrive when they understand: three cookies are “3,” three cars are also “3,” and three bird calls are still “3.” The label “3” finally detaches itself from the objects and becomes the abstract number itself. That moment of detachment — when the label separates from the thing — is the mathematical movement from the concrete to the abstract. The process itself feels profoundly meaningful.
Of course, sometimes his calculations are driven by more immediate concerns. One winter night before bed, he suddenly said, “When I’m thirty, Mom will already be sixty-five.”
I asked him, “How did you calculate that?”
He answered, “Mom is thirty-five this year. So when I’m thirty, she’ll be sixty-five.”
I told him, “Not quite. Try again.”
He recalculated several times and still arrived at sixty-five. I originally wanted to let him figure it out on his own, but I couldn’t tolerate being mysteriously aged five extra years, so I finally tried to explain it carefully.
I said, “When you were one, your mom was thirty-one. When you were two, she was thirty-two…”
Before I could finish, he interrupted me: “I got it!”
One day this January, C suddenly realized he had learned negative numbers. The reason was probably the weather , temperatures had dropped below zero, and he had seen the minus sign while checking the forecast.
At home, screen time works on a point system. Completing certain tasks earns points, and each point can be exchanged for three minutes of cartoons. One day he asked C, “Can I dump all my toys out and then put them back again?”
C told him, “Cleaning up your toys gives you three points, but throwing them around costs six points.”
He immediately replied, “Then I’d have negative three points!”
For him, the concept did not seem burdensome at all.
There’s another arithmetic question C once asked him that I still love very much:
“Imagine there are two plates with the same number of apples. Ý eats apples from the first plate, and you eat apples from the second plate. In the end, there are three apples left on the first plate and five left on the second plate. Who ate more apples?”
To my surprise, he answered correctly.
I had fully expected him to ask me how many apples there were in the beginning.
Infinity
I had already tried explaining the idea of infinity to Doudou when he was four years old.
One day he suddenly asked me, “What is infinity?”
I told him, “If you start counting from 1 and keep going forever without ever reaching the end, that’s infinity.”
He thought about it for a moment and then asked, “Then what is infinity plus one?”
I said, “Still infinity.”
He paused again before asking, “Why does one hundred plus one become one hundred and one, but infinity plus one is still infinity?”
At that age, he still lived in a world where everything had boundaries. No matter how much candy his parents bought, eventually it would all be eaten. Even a cartoon with three hundred episodes would someday end.
At the time, i explained infinity by saying, “It’s something you can never finish counting.” But what exactly does “never finish counting” really mean? Looking back, I do not think I explained it very well.
Then one day, at five years old, he asked C, “What is infinity minus one?”
C answered, “Still infinity.”
Naturally, that made very little sense to him.
This Easter, he found a large number of chocolate eggs hidden around the yard. He counted them aloud for me:
“One, two… twenty-five.”
Then he stopped.
Whether there are twenty-five chocolates or twenty is obviously important, it determines how much chocolate we get to eat. But more importantly, twenty-five is exactly twenty-five. If you remove three chocolates, then there are three fewer chocolates. It can never remain the same amount as before.
And yet, strangely enough, this impossibility is precisely what gave mathematicians the idea for defining infinity: if there exists a collection of things that can be the same size as one of its own parts, then that collection is infinite.
Why define it that way? Because neither chocolates nor cartoon episodes , nothing finite , can ever behave like that. Only infinity can remain equal to a part of itself.
She told him this story, and his little CPU seemed to overheat for quite a while. He never said whether he understood or not.
Maybe infinity can wait for now.
For the moment, we should stay a little longer inside the finite world and continue playing there.
Simple Graph Theory and Programming
Sometimes, through play, Doudou and I also drift into things resembling graph theory.
We invented a small train game where I placed pieces of paper on the floor as train stations. Each station came with its own task: go to Station A to collect honey, Station B to collect apples, and eventually deliver everything to a bear waiting at another station. The point of the game was to figure out how to minimize the route and avoid unnecessary backtracking.
But he eventually admitted that he did not really like the game, and that he could not quite understand why it mattered.
We have also been playing small programming games together. My own attitude toward programming is fairly detached; I tend to feel that as long as someone’s logical instructions are clear enough, the code itself is almost secondary.
Recently, he finally grasped the syntax of “do… until…”. Because of that, he managed to solve an exercise ,“keep walking until you reach the finish line” , using only a single instruction. Without loops, the same task would have required repeating the “move forward” command five separate times.
later i aske for C's help. C wanted to introduce conditional logic next , things like, “if there is a path on the left, then turn left.” But she still cannot quite understand why, despite fully grasping “do… until…”, the structure of “if… then…” somehow exceeds his cognition for now.
I wonder whether this relates to what some people call the three-layer ladder of cognition:
Observation: recognizing associations and patterns.
“X and Y often happen together.”Intervention: understanding actions and consequences.
“If I do X, will Y happen?”Counterfactual reasoning: imagining worlds that never occurred.
“What would have happened if I had not turned left?”
At the moment, he already understands the idea of repeatedly performing an action until a condition is satisfied. This belongs to active control of a process: “if I keep doing A, eventually I will reach state B.” It does not require imagining alternative realities or missed possibilities.
But “if there is a road on the left, then turn left” , although it sounds simple, already touches the boundary of counterfactual thinking. In order to use the instruction properly, he may need the hidden ability to imagine several possibilities at once: turning left, continuing straight, turning right , and then comparing the outcomes in order to decide that left is the correct choice at that moment.
So, for now, we stopped there.
A Digression That Isn’t Really a Digression
One day, while wandering through a secondhand bookstall, I came across an old magazine. Inside it was a page showing multiplication methods completely unfamiliar to someone like me, who grew up memorizing the standard multiplication table.
One example explained that (7 \times 7) could be calculated through something like (4 \times 10 + 9).
If you look carefully, the logic behind it is perfectly understandable. C was talking about the enormous difference between systems of elementary mathematical education across cultures.
I imagine the intention behind methods like the one in the image is to transform unfamiliar calculations into structures that are already known and easier for children to manipulate. The underlying idea seems to be: instead of memorizing answers directly, try to derive them through relationships between numbers themselves, break difficult problems into smaller parts you already understand.
In some sense, the school system itself resembles a multiplication table. A child will almost certainly be taught how to reach the answer through the shortest possible route, but rarely allowed to spend an absurdly long time thinking about a single question. And yet real mathematicians are often closer to the latter type.
The shortest path matters, of course. But I also hope he can preserve some of the curiosity he has now , the willingness to wander down roads that were never written in the textbook. It does not really matter whether those roads lead very far.
I do not know how many more days like this I will have with they two, talking about mathematics in such small and ordinary ways. But when I watch him and Charlotte now, every day feels like witnessing mathematics growing naturally inside a human mind.
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