On Surfaces and Essences

This is an incredible book, if a bit long. Hofstadter and Sander ask an important, sweeping question — how do we think? — and pursue it through the broad, amorphous subject of concepts. What is a concept, anyway? It isn’t obvious, and it’s easy to fall back on the meaningless “a concept is a concept.” Really, concepts are the categories that order our understanding of reality. The book is intuitively built, and the first chapter treats concepts as words. Take “mother.” To us, grown adults, a mother is any woman with children, biological or adoptive; to a young child the category is far narrower — I have a mother, she’s the woman who takes care of me — and it widens as the child learns that others have mothers too, the single object of “mother” becoming the category “mother.”

For Hofstadter and Sander, concepts are best understood like metropolitan areas. Each has a core, the old downtown, and you can move outward from it, other things fitting the concept with ever-weaker links until you reach the suburbs: the land of metaphor. So “mother” extends to “motherland” and “mother Earth,” which still belong to the concept without belonging to it entirely. And concepts aren’t discrete units; they overlap, which is exactly what makes metaphor work — “motherland” sits inside “mother” but also inside “nation” and “territory.” All of our thinking runs through concepts like these, and there are constant conceptual slippages, which matter because they let us move fluidly among ideas. The same words refer to many concepts: when I say I’m “going out for a coffee,” everyone knows I might end up at a café drinking tea or lemonade. Sometimes we conflate two people or two ideas because they resemble each other, which is a sign that a particular idea inhabits overlapping conceptual space.

One part I found especially useful is the recognition that different languages conceptualize the same things differently. Hofstadter and Sander mostly contrast English, French, and Spanish, but I found it helpful to think in terms of Arabic. In English we “turn on” and “turn off” the lights, which says something about how we picture the action; in Arabic you’d say you open the lights and close them, just as you would a curtain or a window. Arabic is a particularly rewarding contrast, given the richness of its vocabulary and how far its concepts diverge from English: its roots work quite differently from English ones and open conceptual space that English can only reach by combining words. Every idea can be expressed in both, of course, but the way it’s done in Arabic is wholly different, and that’s fascinating. They go on to discuss “naive” analogies — the ones we learn and use to describe things that don’t actually describe what’s happening, and the strengths and limits of doing so. So much computing language is exactly this: “desktops,” “documents,” “folders,” “putting things in the trash,” none of which match the physical world, since what actually happens in a computer is the processing of electrical signals — but the analogies let us use the machine without thinking too hard about the interface. There’s a wonderful case in how we talk about arithmetic. We normally think of division as sharing out: “Sally has thirty candies and splits them evenly among her six children — how many does each get?” Yet we struggle to picture a division whose result is larger than the divisor, even though we know it’s possible in pure mathematics, and the block comes from the sharing analogy. It’s better understood as measurement: if I have four pounds of hamburger and want half-pound patties, how many can I make? Eight, of course — but we don’t intuitively think of division that way.

The book closes with a study of major scientific and mathematical figures who made their biggest discoveries through analogical thinking: imaginary numbers, exponents beyond the fourth power, light as particle, wave functions. Hofstadter and Sander insist that, although we imagine mathematics and physics as matters of logic and formalism, an enormous amount of the thinking happens through analogy. Their argument isn’t subtle: analogy pervades every part of human thought. We tend to treat analogical thinking as the lesser, more childish sibling of mature, logical thought, but they argue — very convincingly — that this is backwards: we formalize ideas with logic, but we generate new ones through analogy, categorical conflation, and conceptual slippage. It should be obvious, and it really isn’t; reading the book, I kept catching how I think and noticing far more often when people reach for analogy, and now it’s impossible to ignore. My only complaint is that it’s far too long — the point could have been made in three hundred pages or fewer. It’s very readable, built from concrete examples rather than airy Platonic abstraction, but the sheer flood of examples can be hard to wade through. That flood does work in the authors’ favor, since it becomes harder and harder to argue against them, but as a reading experience it can be overwhelming.