Liu Danting/Wen
One
First of all, I must confess that what prompted me to open David Foster Wallace's mathematical monograph, The Way Through a Street: A Brief History of Infinity, was not a passion for mathematics (which I had nothing to do with me), but the psychology of gossip—a mathematical treatise written by a legendary writer! David Foster Wallace, a blessed man of god, mastered both the symbology of words and numbers, and was both "one of the greatest American writers of our time" and an intellectual elite trained in philosophy, mathematics, and modal logic.
Undoubtedly, words and numbers are the cornerstones of Wallace's mysterious spiritual world. In his childhood memories, his father, who taught philosophy, and his mother, who was an English teacher, always maintained a sensitivity to knowledge and language: his mother would create her own words to express them when she was "poor"; during a summer trip, his parents and Wallace agreed that whenever the point of sending such a snack was mentioned, the value of the π would be 3.1415926... to replace.
π is an irrational number, and in The Way To Cross a Street Wallace has a special discussion of such real numbers that arrive at infinity in a wireless, non-cyclic way. For him, mathematics was like a ubiquitous π, always intertwined with his life and growth. He once recalled his past as a juvenile tennis player in the work "String Theory", he has a high talent for tennis, which is also due to his super computing ability, he will calculate the movement route of tennis in the game, regard the game as a process of cracking the equation of the quadratic function, and even take the wind factor into account. Mathematics, in a way, is a mirror image of reality that has been highly abstracted and presented in Wallace's eyes.
Also in String Theory, he wrote: "Before knowing that infinitesimal symbols represent railroad tracks and that integrals are a schema, I could find an area at the junction of heaven and earth on the edges of these wide curves with the naked eye alone. Learning mathematics in the eastern hills is an epiphany that shatters memories and brings them back to mind. Calculus is indeed a lot like a childhood game. While reconstructing personal memories, he mixed mathematical theorems and knowledge to participate in the narrative. They are not only as rational knowledge carriers, but also carry personal perceptions, feelings and subconscious, and together with words, they construct a dual-track narrative mode.
This dual-track model is also one of his signature styles, giving his work infinite tension, as well as difficult-to-decipher complexity. He not only captures images in reality, but also uses mathematics to present the pure, formal connections behind reality. They construct a existence beyond language, an endless and disorderly wasteland where reason and nothingness intersect.
Whoever dares to set foot in that wasteland is bound to be lost in it. Wallace himself was not immune.
A Way to Cross a Street: A Brief History of Infinity
David Foster Wallace / by
Gravity / Guangdong People's Publishing House
Hu Kaiheng / Translation
November 2021
Two
At the age of 9, Wallace showed symptoms of depression and anxiety for the first time. This question, which his mother called a "black hole in the teeth," has since repeatedly haunted him. He once borrowed the words of a character and said: "Dullness is connected to the sore spot of the soul." Perhaps because of this, he indulged in the world of mathematics and philosophy, and decided that only thinking could free himself from the "special buzz" in his head. In college, he read Pynchon's works and saw his own path to writing in it.
At the age of 25, he completed his first work, Systematic Broomstick, inspired by Wittgenstein's thought, a jumble of works. But it's far less tedious and difficult than Wallace's next work, Endless Jokes. According to Wallace, "Endless Jokes" was deleted by the editors nearly a third, but it still had a volume of nearly 500,000 words, so much so that at the book launch, someone jokingly asked if anyone had really read the whole book. The book was selected for Time magazine's 100 Best English Novels in the World Since 1923. Some critics praised Wallace as "one of the greatest writers of our time, with the talent to write anything".
After Endless Jokes, Wallace also published several collections of short stories and essays. And "The Way To Cross a Street" is another jewel in his heavy and magnificent laurel crown, a unique mathematical treatise. Given the difficulty of reading Wallace's novels, perhaps this mathematical treatise is more popular.
The title "Crossing a Street" comes from the dichotomy of the ancient Greek philosopher and mathematician Zeno: "You stand on a street corner and you try to cross the street." Notice the word 'try', because you obviously have to go through half of the street before you exhaust all your means, and you have to go through half of the street before you go through half. And before you can go through half of the half, you obviously have to go through half of half..." The essence of this paradox is to break down the activity of crossing the street, which we have to complete many times a day, into an infinite number of actions— since it is "infinite", it means that the number of these actions has no end. Thus, a terrible conclusion emerged: man is logically unable to cross the street.
The abomination of the dichotomy paradox is that it succeeds in leading man into an infinite path. In daily life, the word infinite appears every time, always accompanied by people's praise and worship of it, it is vast, vast, unrestricted, and close to perfection, which is impossible for human beings to reach. However, the infinity reflected in the eyes of mathematicians is completely contrary to this. As Aristotle said, "The essence of infinity is absence, not perfection but finite absence." "How do you find and prove the existence of the missing?" This seems like an impossible task. What Wallace sought to reveal in The Way Through a Street is precisely how mathematicians discovered (or created, in their own limited way, which has long been debated) infinite history in mathematics.
To trace infinity in the history of mathematics, we must follow Wallace back to ancient Greece. Surprisingly, the ancient Greeks who loved mathematics not only did not appreciate and worship infinity, but also had a disgusted and skeptical attitude towards it. The ancient Babylonian-Egyptians achieved high mathematical success before the ancient Greeks were involved in mathematical research, but their interest in mathematics stemmed entirely from the practice of life, and mathematics was a tool for solving real-world problems. The Greeks abstracted mathematics, believing that abstract mathematics was completely different from the empirical reality with which mankind was familiar, and that mathematics demonstrated a world that had nothing to do with the perceptible reality of man and the familiar category of experience, although there were pure, formal mathematical relations behind the phenomena of the real world. We can understand this: when faced with the problem of adding and subtracting 5 oranges, the predecessors were interested in oranges, while the ancient Greeks were interested in 5. They don't care about what is tactile and sensible, and strip numbers from specific characteristics, perceptual experiences. At this point, mathematics acquires the essential property of abstraction.
Three
However, even for the ancient Greeks who loved abstract thinking, infinity was an extremely troublesome existence, and as Aristotle pointed out, infinity is an abstraction of abstraction, and one needs to abstract all finites in his mind in order to obtain this symbolic ∞ representing infinity. But when we try to define the concept of infinity, we find that this is a bigger problem.
Of course, neither Wallace nor we, the average reader of this book, are the first to realize the problem. Wallace mentions in his book that Aristotle had discussed the dichotomy as early as The Sixth Book of Physics (here we go again!). Noticing the ambiguity of the concept of infinity, he points out: "Length, time, and anything continuous can be called 'infinite' in two senses, i.e., in the sense of division and size." That is to say, infinity can mean both infinity, infinitesimal, infinite length, infinity shortness, or infinite divisibility of finite things.
But do not think that the tormenting infinity has spared us so far, for Aristotle then divides the infinity into "real infinity" and "latent infinity." So, once again, we have to return to the dichotomy paradox – if we divide the streets continuously, there will be an infinite number of dividing points that exist as a complete entity, and the set of these points is the real infinity. If we think of the "6:54" that appears every morning as a set, we must admit that all the "6:54" have never coexisted, and that is the infinity...
Continuing to explore these kinds of questions will irritate those who are reading this article, but compared to the vast universe of "infinity" that "The Way Through a Street" shows us, all the troubles have only just begun. Although the ancient Greeks had long given abstraction to mathematics, it was not until the 17th century that mathematics fundamentally became a formal system derived from abstraction rather than from the real world. With the help of the mechanics of this system, mathematicians can and dare to actually touch ∞.
In the face of such human trespassing, ∞ must fight back. This freak with a peculiar shape that makes people blink and blink, shows more grotesque and incomprehensible faces, and wants to lead people astray. Its counterattack includes the fact that a subset of an infinite set contains as many elements as this set. Any sane person may feel the collapse of reason when confronted with this fact. ∞ torture inflicted on humans was not over, because Newton and Leibniz discovered calculus one after the other. The existence of infinitesimals is indispensable to calculus, and its fickleness is intolerable, it is 0 when the computation needs 0, and it is greater than 0 when it is not needed. Has anyone ever seen a chameleon or wall-headed grass that has no bottom line than it? To salvage calculus, Newton made a "powerful" argument for it: it was not an infinitesimal, but a number of streams, a rate of change based on a time variable. But that only raises more doubts, and more justifications...
With that in mind, reading The Way Through a Street is a magical experience of a momentary epiphany intertwined with pain for someone like me who has never been allowed to step into the Temple of "High Numbers," and a true spiritual adventure. Guided by Wallace, a knowledgeable and sharp-minded guide, I traveled into completely unknown territory, with so many difficulties that I didn't know how far I could go—and indeed not too far. When I came to Cantor, whom Wallace most admired and admired, my mind was so confused that I couldn't figure it out. But I am glad that I have plucked up the courage to embark on this journey, and thus have been able to peek into the forbidden scenes of the past and the present, where only mathematicians are allowed to gaze.
I even realized that the avenue laid in the book is not only a collection of dichotomous paradox points, but also an invisible path opened up by the brilliant intellect of mathematicians. It is like a number axis with a beginning and an end, and a mathematician who appears briefly in the book is a small coordinate point, they cannot exhaust the number of axes, nor can they reach the end of the number axis, but they extend the length of human cognition, digging out a magnificent potential world from the back of reality.
Four
The Way To Cross a Street is a story of the underlying mathematical world, an overview of a purely rational world. But from this book, which is not intended to be involved with individual life and private experience, we can still find some fragments of Wallace's life.
At the beginning of the book, Wallace solemnly introduces Cantor and his theory of super-poverty, and inevitably mentions the tragic end of mathematicians' fate - Cantor's personality is complex and changeable, and the mental hospital is his lifelong home; Gödel, who died of mental illness; Boltzmann, died of suicide... Wallace, quoting Chesterton, concluded: "Poets don't go crazy, but chess players do; mathematicians and tellers go crazy, but creative artists rarely do." I'm not attacking logic – I'm just saying that the danger is not imaginary, but does exist in logic. Wallace immediately corrected, however, that what is dangerous is not logic, but a crushing "abstraction," the act of reducing everything to the most basic level, which means trying to think about things that most people can't think hard about. And that's unbearable.
Perhaps most people are content with a "limited" and muddy life because of the physiological self-protection mechanism. Wallace knew the dangers of mathematics, the dangers of thinking, and pointed them out with ease, as if he were in control of everything. But this wisdom did not prevent his own life from ending in tragedy, but rather fulfilled prophecies and proverbs. Wallace is known to hang himself at home in 2008 at the age of 46. He was not as isolated and desperate as the tragic hero Cantor whom he admired—Wallace's reputation was booming, and his newlywed wife, who kept a close eye on his mental condition. (On marriage, Wallace writes in a commentary in this book: "Strange fact: Almost all the great philosophers of history were unmarried.") Heidegger is the only exception. Great mathematicians are about half to half, and the marriage rate is still lower than the average person's level. There is no convincing explanation for this, and everyone is free to play. I believe that his thinking on this issue is far less relaxed than this breezy note suggests. And his life failed to follow Chesterton's rhetoric, reversing course by blending in with literature, which is generally considered to be low-threshold and illogically weak.
Wallace's tragedy warns us that the greatest danger posed by mathematics is not to make up for the exam, but to make people lose themselves in the obsession with abstraction and never find a way out. However, "The Way Through a Street" also reassured me that my deepest understanding of mathematics was only in antiquity, and the misguided modern mathematics separated the insurmountable barrier between Achilles and the tortoise—although Wallace used explicit and implicit in the book, with great fanfare, grass snake gray line to predict and lay out the epoch-making mathematical genius Cantor's theory of super-poverty, but when I finally came to that chapter, I regrettably found that only Superman, who was super-poor, was the same as Super-Poor in order to understand the true meaning of these words and symbols.
At least I'm safe. This adventure once again confirmed that not everyone is fortunate enough to be chosen by the doom of mathematics. Most unfortunate and fortunate people will never enter the secret realm of mathematics that is completely independent of reality in their lifetime, and will not be infinitely harmed.
Closing the book, I can continue to live the unspeakable happy life of summer worms, leaving the terrible infinity behind me, and snickering at the finiteness of my own perfection.