This book is an incredible work, although it’s a bit long. Hofstedter and Sander ask an important, but broad, question: how do we think? The book then deals with the broad and seemingly amorphous subject of concepts.

What is a concept, anyways? This isn’t entirely clear, and it’s easy to respond to it with the simple, though meaningless: well, a concept is a concept. It’s that thing that we always use when we refer to anything. They’re really categories that order our understanding of reality.

The book is intuitively structured, and the first chapter deals with concepts as words. For instnace, what do we mean by the concept of “mother?” To us, grown adults, a “mother” is any woman who has children–whether biological or adoptive. However, for young children, the category is much narrower: I have a mother. She’s the woman that takes care of me. For the young child, the concept of “mother” expands as s/he learns that other children also have mothers. The discrete object of “mother” becomes the category of “mother.”

To Hofstadter and Sander, concepts are best understood like metropolitan areas. Each of them have a core–the old city, as it were. Or perhaps “downtown.” One can expand outward for the city, and other items can fit into the larger concept, although with weaker linkages, until one hits the suburbs: the land of metaphorical usage. In this case, we can use the term “mother” to refer to “motherland,” “mother Earth,” and other such ideas. They still fit in the concept of “mother,” but are not entirely so.

It’s important to note here that concepts are not discrete units. They overlap with one another, and this is what makes the metaphors work. For instance, the idea of “motherland” fits under the concept/category of “mother,” but also of “nation,” “territory,” and such related items.

In fact, all such thinking happens through the usage of such concepts and, at times, there are conceptual slippages. These slippages are really important, as they allow us to move fluidly between a bunch of different ideas. Moreover, the same terminology can be used to refer to a variety of different concepts. To use an example that Hofstadter and Sander do, when I say that I’m “going out for a coffee,” we all know that does not mean that I literally have to drink coffee–I might be going to a cafe, but instead opt for tea or lemonade. Sometimes, we conflate two people or multiple ideas because they seem alike; this is a sign that a specific idea inhabits overlapping conceptual space.

One part of the book that I found really useful is the correct recognition that different languages conceptualize similar ideas in different ways. Hofstadter and Sander do a lot of contrasting between English, French, and Spanish, but I found it helpful to think about these subjects in terms of Arabic. For example, in English, we talk about “turning on” and “turning off” the lights. This usage says a lot about the way we think about such actions. However, in Arabic, you’d say–and this is obviously a literal translation–that you open the lights, and that you close the lights: just as you would a curtain, a window, or something else of that nature. Actually, I think that Arabic is a particularly useful language to contrast with English here, if for no other reason than the richness of its vocabulary and conceptual divergences from English. Arabic roots function quite differently from English roots, and they open all kinds of conceptual space that–in English–you would need to combine words. Of course, all ideas can be expressed in both languages, but the way things are done in Arabic is totally different from English, and that’s fascinating.

Hofstadter and Sander go on to discuss the importance of “naive” analogies–that is, analogies that we learn and use to describe things that don’t actually describe what’s taking place, and both the strengths and limitations of doing so. For instance, so much computing language is made up of these kinds of naive analogies: we talk about “desktops” and “documents” and “folders” and “putting things in the trash” and so on. But, these concepts are not the same as those in the physical world. In fact, what happens in a computer is merely the processing of electrical signals. Even so, these analogies allow us to use a computer without thinking to much about how we engage with the interface.

Another important case that Hofstadter and Sander discuss is how we talk about basic arithmatic functions. While the subject of multiplication is insightful, this was really illustrated well by the authors’ discussion of division. We normally think about division as a way of sharing or dividing. For instance, this word problem is quite obviously about division: “Sally has 30 candies and evenly distributes them to her 6 children, how many candies does each child receive?” Yet, we have a really difficult time visualizing how to divide in a way that causes a final sum that is larger than the divisor. This is curious, as we know that it can be done, at least in the world of pure mathematics. The reason why we have this blockage is because of the analogy we draw about division. In fact, it’s best understood in terms of measurement. For example: imagine that I have four pounds of hamburger and want to divide them into 0.5 pound patties, how many patties can I make? The answer, of course, is eight! But we don’t tend to intuitively think about division in this way.

The book concludes with a study of major scientific and mathematical figures who make their largest discoveries via analogical thinking: the development of imaginary number, exponents to the fourth-power and beyond, light as particle, wave functions, and so on. Hofstadter and Sander are at pains to argue that, while we tend to think of mathematics and physics in terms of logic and formalisms, the reality is that so much thinking happens through analogical comparisons.

Indeed, the argument of this book isn’t at all subtle: it is that analogy pervades every aspect of human thought. Analogical thinking is frequently viewed as the lesser, more childish sibling of mature, logical thinking. But, Hofstadter and Sander argue–very convincingly, I think–that this isn’t actually the case. We formalize ideas through logic and other formalisms, but the way new concepts are ordinarily generated is through analogy, categorical conflation, and conceptual slippage.

That this is true seems like it should be so obvious, but it really isn’t. As I’ve read this book, I’ve been thinking about the way I think and picking up much more when people use analogy. That is to say, it’s impossible to ignore now that I notice it.

My only issue with this book is that it’s far too long. I think the point could have been made in 300 pages or less. It’s a very readable book, as most of the text consists of concrete examples rather than some airy Platonic abstraction, but there is such an overwhelming unloading of examples that it can be hard to sift through it all. The amount of content does seem to work in the authors’ favor: by flooding us with examples, it becomes increasingly difficult to argue against them. But, in terms of actually experiencing the book, it can be overwhelming.