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The Wisdom Condensed in Words (I): Structuralism

Nov 20, 2021·16 min read

We write in order to think better.

None of us is a sage. The very fact that a single person's ideas always have holes in them is exactly why they are worth bringing out into the open. Having someone point out a flaw is where the room to improve lives. An idea you keep hidden away can never grow up.

I like to think, and some thoughts, if I do not write them down in time, simply slip away, which feels like a real waste. So I am starting a new series, which I am calling "Stray Thoughts." The point of it is to untangle my own reasoning, to keep a record of ideas, and to make both self-reflection and conversation with others a little easier.

figure 1 El Lissitzky, 1923

Ready? Let us begin with a strange claim made by a mathematician.

Starting with Hilbert

The mathematician Hilbert once said something like this. In every geometric proposition, "we must always be able to replace 'points, lines, planes' with 'tables, chairs, and beer mugs'." In other words, the first axiom of geometry, "any two points can be joined by a straight line," could be rewritten as "any two beer mugs can be set down on one table."

The claim sounds a bit baffling, doesn't it. "Point, line, plane" are concepts so deeply rooted that it is hard even to picture a geometric proposition without them. I can maybe imagine "two points lie on one line" as the image of "two beer mugs set down on one table," but the moment I hit a harder sentence (say, "two straight lines divide a plane into at most four parts"), the whole translation machinery seems to crash on the spot. Two chairs set on one table? Or two tables stacked up on top of one chair? And where on earth do all those beer mugs go? The picture in my head turns into chaos, a little seaside bar that a tornado has just torn through, tables and chairs heaped into a hill, the floor a field of broken glass. You want to drag our dear Professor Hilbert over and say to his face, "Please, just tell me, how exactly does this replacing work, because I really cannot make sense of it!"

As a fan of Professor Hilbert, I will be so bold as to come to his rescue. Reader, you do not have to cling to the mug-chair-table picture. Open the frame a little wider. How about swapping "point, line, plane" for "rice noodle rolls, rice vermicelli, soup dumplings"? Or "happiness, sadness, fear"? Or just drop the words altogether and use "X, Y, Z"? Any of them works, because the only thing we need to care about is the relationships between concepts, not the words we use to name those concepts. And so, as long as X, Y, Z satisfy the handful of rules of geometry, as long as the rules and relations among them are pinned down, it makes no difference at all what words we use to call them. This way of thinking is what I call the "axiomatic structuralist approach."

Professor Hilbert was called a leader of world mathematics in the early twentieth century precisely because he raised the banner and pushed forward an enterprise called "the axiomatization of mathematics." In his ideal, there exists a "list of rules." This list contains all the most basic rules that any piece of mathematical research could ever need, rules that are self-evident and also impossible to prove (these are called "axioms" or "postulates"). From these rules one can derive, in principle, every mathematical truth (every true proposition) there is. Later mathematicians would only have to follow this set of rules honestly, and sooner or later the field would be able to prove every objective truth, and humanity's dream of absolute truth, a dream running back to the ancient Greeks, would at last come true. Hilbert's ringing line still echoes in my ears: "We must know. We will know!" Two short sentences, and yet they carry the same vast spirit as the old Chinese vow to "establish a heart for heaven and earth, secure life for the people, carry on the lost learning of past sages, and open up peace for ten thousand generations."

It is a pity that the lovely dream of axiomatization had several buckets of cold water poured over it, hemmed in by Turing's "halting problem" (there is no algorithm that can decide whether a given proposition can be proved from the axioms in a finite number of steps) and Gödel's "incompleteness theorem" (any formal system that contains the Peano axioms will contain true propositions that cannot be proved). The "third crisis in mathematics" is a fascinating story, but in this essay let us keep our attention trained on the "axiomatic structuralist approach." I will save the tale from the history of mathematics to tell you properly another time.

Words and structuralism

Here I want to turn the reader's gaze toward "structuralism" and think through, together with me, a single question: what exactly is a word?

We are all human, and the ancient Greek sage Aristotle called us "the animal that speaks." Whether you study philosophy or mathematics, whether you work in sales or in engineering, our working lives all seem to come down to "talking" and "writing," that is, to using language. But language alone is not quite enough. An artist pouring talent onto a building or a canvas does not seem to need language, and a religious figure preparing a space and the materials for a ritual can apparently do without it too, so language does not make up the whole of our lives. And yet these people are certainly using "symbols." They have to convey their thoughts and feelings through shape, color, form, and gesture. So we can push this one step further and say that every kind of human intellectual activity is "the use of symbols," that our knowledge and our ideas form a "system of symbols," and that social rules and logical reasoning both belong to a single "symbolic order."

Nonverbal symbols cannot be explained to you through words alone, so for the sake of the writing let us stick to "language" and "words." Wittgenstein, one of the founders of analytic philosophy, used chess as a metaphor for language in his late work Philosophical Investigations. Words are the chess pieces, and the rules for using language are the rules of the game of chess. Suppose a chess "rook" rolls onto the floor and goes missing. I put a small stone of about the right size on the board, and you and I both know that this stone is the "rook" (because we use the stone according to the game rules for the "rook" in chess), and so the two of us can happily finish the game.

Following this line of thought, a great many puzzles in science suddenly dissolve. A physicist can create the concept of the "electron" without having to labor over questions like "what is an electron? does the electron exist?" To use the word "electron," all we need is for it to sit comfortably alongside other physical concepts and to obey a series of definitions and rules in physics. Whether it is a little ball orbiting the nucleus or a cloud of probability surrounding the nucleus, these questions are of course worth investigating, but they do not stop me from using the concept "electron" at a higher level, say in a nuclear reaction equation or in chemistry.

I used to be deeply, deeply confused. "If the electron is a cloud of probability, then how do I make sense of a sentence like 'there are two electrons outside the nucleus,' and what does 'gaining and losing electrons' in chemistry even mean? I know we can redescribe it more precisely with the overlap and the shift of electron clouds, but is this still the same 'electron' we set out to study? The definitions of these terms have changed so completely. Can we still be sure that the thing we are describing with the term is the same object?" Think these questions over and they really can make your skin crawl. They surface every so often while you are poring over a textbook or buried in a stack of problem sets. Sadly, almost nobody raises them, and even when a student works up the courage and the words to ask a teacher, it is hard to get an answer that truly sets the mind at rest.

Taking up the "structuralist" way of thinking, we finally get our hands on a theoretical tool that can answer these questions. As long as we speak according to the agreed-upon rules, we can perfectly legitimately use a word within a system, without having to fret too much over what the "essence" of the concept that word points to actually is.

Letting go of the fixation on essence

The thoughtful reader is bound to object: how can you say "essence" does not matter? Isn't science precisely the study of essences? But as I see it, letting go of the fixation on "essence" is something we do because we have no other choice. Here are my reasons.

Physics has now reached a point where we have to admit the limits of our imagination. Just as even the most gifted illustrator of fantasy tales cannot create an animal with no features at all drawn from real-world creatures (look closely at things as outlandish as the chimera and the Chinese dragon, and they too are pieced together from the templates of real animals), we cannot use imagination to "see" a physical entity. Scientists argued themselves hoarse over whether "light" is a wave or a particle, and in my view the source of their dispute is simply an unwillingness to admit that "light" is something we cannot picture using the everyday objects we see around us.

For the concept of "light," whether we call it a probability cloud, a solid little ball, or a wave function, we can write down equations and even draw diagrams, but we have to admit that these things are at best a reflection of, or an approximation to, the "light" of reality. They cannot possibly be, one to one, the thing called "light" itself. So we are driven to say "light has wave-particle duality." This does not mean that "light" is at once a "wave" and a "particle," because saying it that way commits an enormous error. We would be assuming in advance that the reality of matter is something similar to the "waves" and "particles" we see in our daily lives, instead of daring to imagine that the reality of matter is something wholly cut off from common sense, something that cannot be grasped by direct intuition.

This is exactly the moment for our "structuralist approach" to make its grand entrance. How do we use the structuralist methodology to understand the "wave-particle duality" of "light"?

In physics, when we say X is a "particle," all we are really saying is that this X satisfies the rules for using the word "particle" (for instance that it is indivisible, that it bounces back, that it occupies a certain amount of space, all in line with the definition of a particle in physics). The same goes for "wave" (it can spread out across the whole of space, in line with the definition of a wave in physics). So when we say "light" has wave-particle duality, all we are really saying is that "light" can satisfy a little of the usage rules for "wave" and a little of the usage rules for "particle." As long as this whole system of rules can run in a self-consistent and stable way, we can know perfectly well what properties this thing called "light" will display in experiments, and how "light" can be put to use in engineering. And the great aims of science, to be confirmable by experiment, to hold universally, to explain the past and predict the future, have at this point all been met.

The attentive reader may not be satisfied with the explanation above and will keep pressing, stubbornly: "Hold on. You have just talked your way in a circle around 'light' and still have not told me what the essence of light actually is."

But if I ask you what "essence" means, would you not also be stuck for a moment, unable to answer? Yes. In the context of "structuralism," the question "what is the essence of a thing?" (in philosophy this is called the "ontological question" or the "metaphysical question") turns out instead to be an illegitimate one, because everything our talk can reach has been reassigned to the use of linguistic rules, with no regard at all for the words in themselves. A noun standing all alone has no meaning whatsoever. Only when it is placed inside a web of meaning made up of the mutual relations among words (the usage rules) can a word be understood and felt.

A short summary

In Ersilia, to establish the relationships that sustain the city's life, the inhabitants stretch strings from the corners of the houses, white or black or gray or black-and-white according to whether they mark a relationship of blood, of trade, of authority, of agency. When the strings become so numerous that you can no longer pass among them, the inhabitants leave: the houses are dismantled; only the strings and their supports remain.

From a mountainside, camping with their household goods, Ersilia's refugees look at the labyrinth of taut strings and poles that rise in the plain. That is the city of Ersilia still, and they are nothing.

They rebuild Ersilia elsewhere. They weave a similar pattern of strings which they would like to be more complex and at the same time more regular than the other. Then they abandon it and take themselves and their houses still farther away.

And so, traveling in the territory of Ersilia, you come upon the ruins of abandoned cities, without the walls which do not last, without the bones of the dead which the wind rolls away: spiderwebs of intricate relationships seeking a form.

from Invisible Cities by Italo Calvino

"Without the walls which do not last, without the bones of the dead which the wind rolls away: spiderwebs of intricate relationships seeking a form." This little fable of a city is Calvino's wildly romantic literary expression of structuralism. The words we use from day to day are no more than crumbling, perishable ruins. What truly matters, what alone matters, is not the "things" but the "form."

"Structuralism" holds an important place in the history of philosophy and linguistics, but exactly what kind of claim this "ism" stands for has never been clearly settled in the academic world. So it is fair to say that the "structuralism" I lay out in this essay is not an interpretation or a retelling of some existing body of thought. It contains my own original take and understanding.

We are better off understanding "structuralism" as a method rather than a position. It holds that the essence and the meaning of a thing live inside its "structure." "Structure" here is a synonym for "form," "rule," and "system." It means the network woven out of the wide-ranging connections among countless things. I often feel that the word "formalism" would fit better and be easier to grasp, but "formalism" has already been widely used to mock bureaucratic red tape and pointless showmanship, so I had no choice but to drop the idea, a little reluctantly.

I said above that "structuralism" does not care about "essence." That is correct, and yet not entirely correct. It would be better to say that structuralism is pursuing the "essence" of things by a different road. It holds that the "essence" of a thing exists in, and shows itself only through, the structure among things. It holds that anything has meaning only inside a system, and that pulling it out of the system to talk about it on its own has no meaning at all. So it denies the "essence" of any single thing. Whether it is "vector," "atom," "fairness," or "freedom," no single word has any meaning. A word has meaning, and can therefore be understood, only when it is placed in context.

To sum up structuralism in one line: a word has meaning only in context. And so the sum of the parts is greater than the whole, while the whole is what gives the parts their meaning.

Then why did I write "wisdom condenses inside words" in the title? I will leave that hanging, as a bit of suspense and a question to chew on. In the next piece of this series, I will explain it to you in detail.

figure 2 Malevich, 1915

I am Ning Ning Ning Jing Hai. Thank you for reading to the end.

References:

Books

  • Philosophical Investigations by Wittgenstein
  • Science, Philosophy, Common Sense by Chen Jiaying
  • Invisible Cities by Italo Calvino

Web

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