The Making of How to Cook Real Numbers #1

In the last week, I uploaded my first YouTube video, The Peano Axioms, which is the first episode of an in-progress video series called How to Cook Real Numbers.

https://www.youtube.com/watch?v=Qvpv8POP3WQ

This series aims to introduce to the audience how the field of real numbers is constructed within the context of set theory. I have compiled all the necessary definitions, propositions, and their corresponding proofs for the videos, and have carefully selected which proofs require rigorous statements and which ones only require intuitive explanations.

The process of writing the content was delightful, as was creating the necessary illustrations. However, the audio recording was quite agonizing. Although my English is relatively proficient, it is limited to reading and writing. I struggle with maintaining the tone and coherence in spoken English, and consequently, the recording process was often interrupted. Even when reading from the script, I frequently mispronounced words, singular or plural, and tenses. Eventually, I had to mark every pause and highlight the details that I frequently misread on the script. After several attempts, I achieved the current result, although I still found some errors in it later. This recording experience made me admire the documentary hosts of the BBC, who achieve such perfection even when speaking in Chinese, which I cannot do.

Why make videos on mathematics?

In fact, How to Cook Real Numbers is just one of the videos that I have compiled on mathematical knowledge. In my spare time, I have also organized many other topics, such as general topology, cardinal numbers, and measure theory, and have contemplated how to explain these concepts better to people.

I hope that through my presentations, I can inspire people to start exploring mathematics.

From a young age, my passion for mathematics was extinguished by school exams. I struggled with algebraic equations, and so exams that focused on them were torture for me. I soon came to believe that I did not have a mathematical brain, so did teachers tell me. After entering art collage, I exercised my right to not take any math course, and I never touched the subject again for 15 years.

However, despite this frustration lingering for many years, at 33 years old, my passion for the design of printed publications led me to buy Elements of Euclid by Taschen. As I had learned about axiomatic systems while studying the history of philosophy, I quickly noticed the use of axioms in Elements. The underlying connections between these propositions made me overlook the book’s beautiful design and focus on its content. To encourage me, my wife, Lu Shu, bought me a compass on my birthday. From that moment on, I began to relearn mathematics and discovered a subject that was entirely different from what I had been taught in school.

This is also one of the reasons why I decided to start making videos on YouTube. I want to use my videos to tell people (who think they don’t have a talent for mathematics or have been told so) what kind of discipline mathematics really is. No doubt, math is difficult, and sometimes, when faced with a mathematical problem, you may feel like staring into an abyss. However, this difficulty does not stem from complex calculations or even formal logical deductions but rather from our limited understanding of rationality. The beauty of mathematics lies in the moment when intuition is described in mathematical language — perhaps not entirely, but currently, that’s all I can think of.

Focusing on Fundamental Questions

Personally, I never considered how to use mathematical knowledge to improve our lives from the very beginning. What I was obsessed with was only the “missing details” in theory. Some of them, I don’t think, are mathematical problems at all, such as what various levels of infinity means in the physical world. Others are obviously some mathematical problems, such as how to explain limits more generally, how to explain integrable functions, how to define real numbers, whether mathematical analysis can be described without the involvement of real numbers, etc. The latter questions are more specific, and thus, I have also come into contact with many interesting theories.

Many people may think that these questions are useless. To be honest, I can’t find any way to refute this viewpoint, although emotionally, I dislike it. However, I think that such questions have always existed before they are truly resolved, and they will make generations of people entangled.

I believe that these questions often fly through people’s minds in a non-mathematical appearance, making people wonder, “How big is infinity?” This doesn’t have to be a mathematical problem, and through mathematical methods, we can only discuss a partial aspect of this problem. The advantage of this discussion is that compared with other “ambiguous” discussions, we can more surely know what we are talking about. But on the other hand, this rigor has indeed frightened many people, especially when they see a topology textbook full of mathematical symbols without a single illustration. I know that mathematics is a subject about form; with finite symbols, we can describe a mathematical theory that has no physical meaning. But I also know that understanding itself is not a process that can be described by formal logic. In this process, what we still need is intuition, which is often inspired by experience in the world.

Therefore, I thought of using more illustrations to represent those abstract problems. Of course, unlike abstract definitions, the possibilities that these illustrations can contain are quite limited. But on the one hand, the intuitive expressive power of illustrations is far superior to abstract definitions. On the other hand, we are humans, not machines, and humans associate with infinite possibilities starting from limited examples. These limited examples, like seeds in our minds, will grow into a tree. Abstract definitions are our imagination of the whole picture of this tree.

Starting by asking questions

Just as the history of mathematics, we always start by asking questions about a particular problem, often a paradox. Then, we may try to explained the problem in the language of mathematics. For example, the development of calculus was clearly stimulated by the paradox of instantaneous velocity; the motivation for measure theory comes from some paradoxes arising from the definition of integrable functions, and the development of axiomatic set theory was largely driven by Russell’s paradox. There are many such examples, and I will not list them one by one.

In my view, the history of mathematics reflects the process of how we learn mathematics. This process is not only effective for the process of acquiring knowledge, but also intuitively shows us what we have mastered, making mathematics so abstract in resent days. That’s why, in the last year, when I shared my knowledge of math with people, I would focus on describing the problems rather than the explanation of the definitions and propositions.

Although the these theories are presented as a formal logical picture, they have never been abstract to me. Because I attach great importance (perhaps too much importance) to speculating, investigating, and thinking about the initial motivation of the theory, the initial hypothesis in these theories were initially generalized from the the objects mentioned in those paradoxes — in other words, the objects in those paradoxes are actually motivating examples of something more abstract.

When learning mathematics, an example of motivation shows you a more pure idea object in a less pure way, allowing you to better enter a theory. For example, counting apples can help children understand the concept of natural numbers, while set operations between real intervals can help students more intuitively understand ring theory. That’s why I tirelessly think about the motivation of the theory and examples when I try to explain the theory.

I know that some people may not think highly of this learning method. Because they think that although this method lowers the threshold for entering a certain mathematical theory, the cost is that we need to spend more time discussing clues rather than direct derivation processes — you know, the process of moving from these intuitive clues to the abstract field often relies not on logic, but on intuition and imagination. But isn’t this the charm of mathematical theory?

These things were also understood in the detours I took. I once hoped to introduce everything from a strong theory, and the weak theory subsumed in this strong theory was only a special case. But after wasting a few years, I realized that if you want to introduce a theory to someone, this “unified” narrative method will only confuse the audience. Moreover, theoretically, we can never find a strongest theory as the unique starting point of a theory through formal logic.

Playing as a child

Most of the mathematical problems I focus on are not understandable by children, despite of those genius. Indeed, my videos are mainly for adults. However, I use a more childlike style to express my videos, and there are two main reasons for this.

Firstly, I don’t want my audience to be scared off by the visuals in my videos. I hope that adults can face these abstract mathematical problems like children playing with building blocks, rather than being educated in a depressed mood. (If time permits, I even hope to draw some children’s picture books for these mathematical problems for adults, just like my favorite book 100 Steps to Science.)

On the other hand, I firmly believe that the childlike heart of adults is also something that should be taken seriously. Adults are pressed down by too many adult things, but as long as they still pursue beautiful things, their childlike heart will not be completely extinguished. In my opinion, a childlike heart is nothing more than a whimsical attitude towards play — it is not unique to children. However, there are too many adults around me who feel guilty when facing their childlike heart. I know that this sense of guilt may come from the intimidation caused by not-so-good things. Therefore, I hope that through my videos, they can see that the adult world is not entirely worthless. At least, in my opinion, what supports the mathematical world is not formal logic, but a childlike heart of asking why — formal logic is a product of this passion, not the place where the passion is generated.

Soon, I will upload the second video of the How to Bake Real Numbers series about Natural Numbers. I hope you will like it!