‘Oh, “philosophy”. You know. When you try to imagine a mirok [small pink potato] without the least reference to any you have eaten or will eat.’ (Vladimir Nabokov)

A concept is a tool for thinking about things. In fact, when we “conceptualize” something we think of it as an “object” — not just any old thing but a thing that can be known. When we imagine facts, we’re actually imagining things in objective relation to each other, things arranged in ways that can be known, not just by ourselves, i.e., subjectively, but by any similarly qualified peer, i.e., objectively. We will usually gain this knowledge by means of some sort of observation, and an observation is just an experience that contains a judgment. We see something that might be a big lizard, but we observe that it has wings and breathes fire. To do this — to make an observation — we needed concepts (wings, fire, breath) and to make sense of it we need further concepts (dragon, fiction, fantasy). We can’t observe a dragon, i.e., experience something as a dragon, i.e., see it and deem it a dragon, without the relevant concept of a dragon and its dragon parts and the fantastical universe to which it belongs. We bring them together in our imagination, along with our experiences, and make up our minds. That’s how concepts work. That’s what thinking is.

But how can we imagine the concepts themselves? How can we think about the tools we use to think about our experiences?

We can say that the aim here is to imagine an object without a thing that lets us experience it. Put a potato on the table in front of you (if you don’t have one handy, at least notice how easy it is to imagine). If you want to imagine the concept of potato, begin by imagining this one to be bigger than it really is, or smaller, or rounder, or lumpier, or browner, or pinker; you can even touch it and imagine it softer or harder, rougher or smoother. In all these variations, it’s still a potato. Imagine cutting it into slices or sticks, or mushing it or baking it or boiling it. Imagine all the things you can do to this potato without changing the fact that it is a potato. It didn’t have to be this particular thing, in this particular state, in order to be a potato. The concept of potato covers all of those variations but, right here and right now, this thing can be only the potato you actually see in front of you. Potatoes can be mashed, but this one, let’s say, isn’t a mashed potato.

This still doesn’t produce an image of the concept of a potato. We’ve just got a bunch of images of potatoes of various sizes in different states of disrepair. To get at the concept we have to imagine the principle that unifies these disparate images. Your potato is of a particular size; you can say “It is a big potato,” or “It is a small potato,” and one of these statements may be true. It’s also a particular color; “It is a pink potato,” or, “It is a brown potato,” may be true. But what is true of all potatoes, no matter what size, shape or color they are? You can actually begin by imagining, sizes, shapes and colors that potatoes can’t be, at least not “normally”. These ranges are part of the concept of a potato. (Norms don’t just apply to people, we might say; potatoes, too, can be normal or abnormal.) Consider, “This is a normal potato,” and, “This is an abnormal potato.” These statements can also be true or false, and they tell us something about the limits of the concept. But even the weirdest potato is a potato. What makes it so?

Consider: “This is a potato like any other potato.” Compare: “This is a potato unlike any potato I have ever seen.” Well, it can’t be completely unlike any other potato because something allowed you to call it a potato. The concept of a potato is, perhaps, the resemblance between all potatoes. And it may actually be the chain of resemblances (what Wittgenstein called “family resemblances”) that links them together, i.e., not some feature they all share, not some “essence of potato” …

There are philosophers who would debate these issues with you as long as you like and I encourage you to seek them out and do so. But if you want to get on with your work, if you want to know something about potatoes, you’re going to have to discipline your imagination at some point. When you imagine a concept you will be imagining a kind of filter, a distinction between potatoes and everything else, a means to judge something a “proper potato” and something else “not a potato”. You’re imagining a kind of machine for sorting the things in the world, for identifying all the potatoes in a scene. Even a machine for counting them. And inside this machine, there may be finer machines that sort all the potatoes into “big” and “small” and “medium” ones. You then imagine pouring a bag full of objects into the top of the machine, and out they come — big, small, and medium potatoes over here, and everything that was never a potato over here. That’s what concepts do. They help you think about things.

As you are doing all of this — and not just with potatoes, but with innovations, and organisations, and managers, and budgets, and assets, and poems, and novels, and revolutions, and atoms and stars, and everything else you might think about in your studies — notice that the big and small machines that help you think are working together and are themselves situated in a world, a world you share with your peers. Dragons populate fictional worlds, but not all fictional worlds, only the “fantastic” ones, let’s say. Potatoes are much more common; they can be found in real life and in fiction, in true stories and in tall tales. Part of the concept indicates the world in which it can be meaningfully applied — a space of possibilities. In fact, the concept makes things meaningful in their respective worlds.

“The essential thing about a poet,” said Ezra Pound, “is that he build us his world.” Imagining your concepts, and especially writing them down, means imagining your world as a kind of building. Your concepts are the structure of that building and your discipline (your “field”, your “science”) has a language made out of the concepts you share with your fellow students and scholars. “Language,” said Heidegger, “is the house of being.” Imagine moving in.

with apologies to Dan Reynolds and Daenerys Targaryen

Back in the middle ages, they were four-legged, fire-breathing snakes with wings. This somewhat nonsensical image of dragons has been replaced in modern times with a more, let’s say, “realistic” one, in which they have two legs, and wings integrated with their arms, like bats. (That’s how they appear in both the latest movie version of The Hobbit and the celebrated television series Game of Thrones.) It is much easier now to imagine these creatures as a kind of animal, a sort of relic from the age of dinosaurs. In any case, allowing for differences between oriental and occidental traditions, everyone knows what a dragon is. If I told you to imagine one, you’d probably know what to do.

But it’s importantly not the same thing as picturing a fact you know or recalling a story about something that has happened to you. Dragons don’t exist except in the imagination. We have only pictures and stories of them to tell us what to do. We can’t go and see one for ourselves “in the wild” (or even in a zoo). You can see what J. R. R. Tolkien or Peter Jackson thinks they look like or how George R. R. Martin and the talented people at Pixomondo imagine them. (And apparently there’s been some discussion about whether Drogon and his brothers are even proper dragons.) But, at the end of the day, there’s no way to decide who is right.

In this sense, they’re a bit like the theoretical objects of our conceptual frameworks. We can’t really draw a picture of an atom, an image of how it looks, but we can draw a model of an atom, with the protons and neutrons in the nucleus and the electrons in orbit around it. This model can be true or false, but not in the same way that a picture of a fact is true or false. Likewise, we can’t draw a picture of an organizational hierarchy, again, if we mean by “picture” a visual representation that is true to how things look. But we can draw a model of an organization that represents an “apex”, a “middle line”, and an “operating core”. There’s no place we can go to see these things stacked on top of each other. But there are ways to determine whether it is true or false of a particular organization. Likewise, we can read Tolkien or Martin and we can form an image of their dragons in our minds. We can draw that image and it can be true or false of the descriptions we find in the books. Imagining fantastic creatures is one way to train your conceptual faculties.

“The best dragon ever shown on film, Vermithrax Pejorative,” Martin reminds us, “has two legs and two wings.” Notice two things there. First, dragons can be modeled well or badly; their cinematic representations can be worthy of our admiration. Second, they have characteristics (two legs and two wings) that can be accurately or inaccurately described and then represented in, for example, their skeletal structure. This takes a lot of careful work; just because something doesn’t exist doesn’t mean anything goes when representing it. You have to get it right so it can serve its function in the larger story. In that way, too, imagining a dragon is like imagining a concept of a theory. It has to work in your analysis. One last thing: like concepts, dragons have essential properties. “Wyverns don’t breathe fire.”

Close your eyes. Lay your hands flat on your desk. Think of something that happened to you last week. Don’t think of something dramatic — a near miss or a broken heart. Think of an ordinary experience, like taking the bus or buying a shirt. What happened? What did you do? Why did you do it? What are some of the things you could have done but did not do? How did you succeed? How did you fail? What did it feel like (what sensations did you experience)? How did it make you feel (what emotions were involved)? What might well have happened instead? What would have been unlikely? What would have been altogether impossible? Since this is your experience you are the best person to answer these questions. You’re the ideal author of a story in which you are the protagonist. You could write a nice little paragraph about it.

If you did it right, your text would evoke a series of images — moving images. And the dignity of their movement, as Hemingway might put it, would depend on the experience that lies beneath the surface of your text. Our experiences are much richer, much deeper, than the stories we tell. Not even Proust could recover every detail in time, and Hemingway made a virtue of this limitation of language. Overcoming it is not easy, as he explained in Death in the Afternoon:

I found the greatest difficulty, aside from knowing truly what you really felt, rather than what you were supposed to feel, and had been taught to feel, was to put down what really happened in action; what the actual things were which produced the emotion that you experienced. In writing for a newspaper you told what happened aided by the element of timeliness which gives a certain emotion to any account of something that has happened on that day; but the real thing, the sequence of motion and fact which made the emotion and which would be as valid in a year or in ten years or, with luck and if your stated it purely enough, always, was beyond me and I was working very hard to try to get it. (p. 10)

What does it mean, here, to be “working very hard”? We imagine Hemingway sitting in front of his typewriter. (But we are wrong about this; he would write standing up.) He struggles to find the right words to evoke the required images in the mind of his reader. After a time, he is satisfied. Somehow the work paid off. How does he know he succeeded?

“A writer’s problem does not change,” he said. “He himself changes, but his problem remains the same. It is always how to write truly and, having found what is true, to project it in such a way that it becomes a part of the experience of the person who reads it.” You can develop this ability by writing about things that have actually happened to you. Whether your “projection” works is something you yourself can judge (though your judgment, too, can improve through training.) Then, when you discover truths (through your research) about the actions of others, you know what it means to get those actions — those sequences of motion and fact — right. You know what would have had to happen, what they must have done and felt, in order for your story to be true. You can imagine it. And that is what you are expecting your reader to imagine too.

Look at your hand. Suppose I asked you to draw a picture of it. You would take out out a piece of paper and pencil and perhaps begin to draw its outline. Or you might have another way; you might begin with the play of light and shadow, or you might map out its surfaces. Notice that you’d be seeing the hand differently in each case and you’d be representing it according to this way of seeing. I don’t suppose you’re already doing this, so the last few sentences will have been acting mainly on your imagination. You may have actually looked at your hand but you have only imagined drawing it. Suppose I had said, “Look at your hand. Now, close your eyes. Imagine your hand.” What is the difference between drawing your hand and imagining it? What is the difference between imagining yourself drawing it and actually drawing it?

Notice that you can neither draw nor imagine your hand as such. You have to approach it as a fact first; you have to see it (imagine it) as a fact. Your hand can be open or closed, it can be relaxed or clenched, it can be palm-up or palm-down, it can be pointing or waving, it can be holding something or it can be empty. I can’t ask you to draw your whole hand in a single picture because a hand can only be seen from a certain point of view. Also, your hand consists of skin and muscle and bone and blood and nerves. You probably weren’t imagining drawing any of that when I asked to imagine drawing your hand. That is the difference between a thing and a fact. A thing (like your hand) can participate in any number of facts (like your closed fist or a wave or holding a pencil). When you imagine a fact you imagine a thing in a particular position, in a particular situation, in a particular arrangement of other things. In an important sense, facts just are the way we imagine things when they’re being something specific. Without the fact it is implicated in, your hand just “is”. Try to imagine that and you will fail. You can’t, said Kant, imagine a thing as such.

Long ago, Chester Barnard made an astute observation. If you take any particular organization, the amount of people who actively desire to be part of it is very small compared to the amount of people there are in the world. There are almost 8 billion people in the world. Only about 20,000 are students at the Copenhagen Business School. (In a strictly formal sense, Barnard would say the fact they study here proves that they are “willing” to do so.) There are probably some people in the world who would like to study here but don’t for one reason or another, but the number is a very tiny fraction of world’s population. The CBS student body is a “thing” (it would be better to call it a “population” of things, a bunch of people); all of humanity is another thing (another population). But it is a fact that there are about 20,000 students at CBS and another fact that there almost 8 billion people in the world. It’s hard to imagine “all of humanity” in any definite way. Even “CBS students” is a difficult thing to imagine. But the fact that some people are not studying at CBS, though they would if they could, is easier to get your mind around. It is a definite population with some definite characteristics. That also allows us to measure it. We can ask, How many of them are there?

One last thing about facts. We can get them wrong. If I say, “My hand!” or “CBS students!” or “Humanity!” I’m not saying anything I can be wrong about. I can form a vague picture in my mind that suggests, vaguely, what the words mean. But I’m not being specific enough to make a comparison with reality meaningful. These things merely exist … and, properly speaking, I’m not even claiming that they do. It is only by imagining something specific, about everything from my own hand, to an organization’s members, to the whole of humanity that the notion of “truth” gains any relevance. That’s what facts are for — for being right or wrong about — for knowing something about. But you really do have to imagine them first. Begin with your own hand. Imagine it. Picture it. Then move on to other people and the situations they find themselves in. Knowing something about them really just means being able to compare them to, yes, the back of your hand.

for Andrew Gelman

Simplifying somewhat, pictures represent facts, stories represent acts, and models represent concepts. Did you see what I did there? What does “simplifying” mean in that sentence? Is it a simplification to say that pictures, stories and models do these three distinct things? Or do they do these things while simplifying something? The sentence doesn’t commit itself to either reading; it is ambiguous about what is being simplified. And it turns out that I mean it in both senses. Pictures are simplified representations of facts and to use this to draw a hard and fast line between pictures and stories and models is itself a simplified picture, story or model of pictures, stories and models. Sometimes a picture tells a story. Sometimes a model represents a fact. The world is a complicated place and the mind is a complicated instrument for making sense of it. Still, simple distinctions can be useful, so I ask you to indulge my simplemindedness for a few paragraphs more.

When I say that a picture represents a fact I mean that it makes an arrangement of things present in your imagination. It’s true that we sometimes also try to imagine what is “going on” in, say, a painting, but we know that this is an extrapolation from the facts it represents. There’s also usually a whole atmosphere or “mood” in a picture, which is hard to reduce to a mere state of affairs. In David Hockney’s “A Bigger Splash,” for example, the fact is a splash of water in a pool with a diving board. We don’t know exactly what made the splash but we assume it is a person. There’s a feeling about the scene that I will leave it to you to experience for yourself, but we can imagine a photograph representing roughly the same facts. It could be made true or false by whether the state of things we imagine when we look at them actually existed at the time the picture was taken. Not just whether the things and people (the pool, the diver) existed, mind you, but whether they were in the state that the picture causes us to imagine. Pictures can misrepresent the facts. Even photographs can be doctored.

When I say that a story represents an act I mean that it gets us to imagine people doing things, or things happening to people. That’s a gross simplification, to be sure. It’s possible to tell a story about a pool freezing over or ducks landing in it. Things happening to things or animals happening to them. But I think we do actually always anthropomorphize these events a little bit when we tell stories, sometimes barely perceptibly. If we didn’t, I want to argue, we wouldn’t be able to tell a story. David Hockney painted “A Bigger Splash” in 1967, with no people visible in it, and then painted “Pool with Two Figures” in 1972. Jack Hazan saw a story unfolding in the intervening years and made a film about it. Like most stories, indeed, it is about what some people were doing at the time, what happened to them, how things affected them. They are true or false according to whether the acts they cause us to imagine actually took place. Was London really like that in 1970s? Did Hockney actually lose a lover? Did it actually inspire a painting? Perhaps he did, but the film is no doubt a simplification of the relationship, just as the paintings are a simplification of the scenes at the pool.

Models are simplifications in perhaps more obvious ways. They will always represent only selected aspects of the reality they are modelling. When I say that a model represents concepts, I mean that they get us to imagine what it is possible to think about a certain population of things or people. Where pictures have to be accurate, and stories have to be plausible, models must be probable. Indeed, they are models of probabilities, which we can actually understand as “pictures” of probability spaces (rather than physical spaces like swimming pools). What is probable or improbable in a model is whether one thing, for example, causes another thing. Was the splash caused by a person jumping off the diving board? That’s certainly the most likely explanation. But there are other possibilities and they are (logically) no less compatible with the concept of a “splash”. How many splashes in swimming pools are caused by something other than a person jumping into them? What percentage? What are the chances? The model lets us know, but this does not tell us what exactly caused this splash. If it was a stone, the model that said it was likely to be a person is not wrong. We would need much more data to decide. Not to mention whether the breakup was caused by the man swimming towards the artist in the pink jacket.

Just as you can look at a picture of a splash in a swimming pool and read a whole story into it, you can hear a story about two artists and extrapolate a model of human behavior, a model of human suffering. It takes imagination to do these things, of course; you don’t just see the picture, or hear the story, or think of the model. You give them a place in your imagination, where the facts (the objective arrangement of things), the acts (the subjective intervention of people), the concepts (the possible arrangements of things) and even the emotions (the possible intervention of people) can meet. You then bring this imagery back to the seeing, to the hearing and the thinking, which allows new facts and acts, as well as new concepts, to enter the picture, to shape the story, to reconstruct the model. Next week, let’s talk about how to write these images down. After all, you don’t have to paint pictures or make films or draw graphs. You can put them into words on the page — a thousand words should do.