mathematician Alan Turing showed how to make a “universal” machine that can read the descriptions of other machines—and then, by switching among those descriptions, it can do all the things that those machines can do.[151]

All modern computers use this trick, so today we can use the same machine to arrange our appointments, edit our texts, or help us send messages to our friends. Furthermore, once we store those descriptions inside the machine, then those programs can change themselves—so that the machine can keep extending its own abilities. This showed that the limits which Descartes observed were not inherent in machines, but resulted from our old-fashioned ways to build or to program them. For each machine that we built in the past had only way to accomplish each particular task—whereas each person, when stuck, has alternatives.

Nevertheless, many thinkers still maintain that machines can never achieve such feats as composing great theories or symphonies. Instead, they prefer to attribute such feats to inexplicable ‘talents’ or ‘gifts.’ However, those abilities will seem less mysterious, once we see how our resourcefulness could result from having such diverse ways to think. Indeed, each previous chapter of this book discussed some way in which our minds provide such alternatives:

§1. We are born with many kinds of resources.

§2. We learn from our Imprimers and friends.

§3. We also learn what we ought not to do.

§4. We can reflect upon what we are thinking about.

§5. We can predict the effects of imagined actions.

§6. We use huge stores of commonsense knowledge.

§7. We can switch among different Ways to Think.

This chapter discusses yet additional features that make human minds so versatile.

§82. We can see things from many points of view.

§83. We have special ways to rapidly switch among these.

§84. We have developed special ways to learn very quickly. Move the

§85. We have efficient ways to recognize which knowledge is relevant.

§86. We can keep extending the range of our ways to think.

§87. We have many different ways to represent things.

At the start of this book, we noted that it is hard to conceive of ourselves as machines, because no machine that we’ve seen in the past seemed to understand the meanings of things, but could only react to the simple commands that we designed them to execute. Some philosophers argue that this must be because machines are merely material things, whereas meanings exist in the world of ideas, which lies outside the realm of physical things. However, Chapter §1 suggested that we, ourselves have constrained our machines by defining those meanings so narrowly that we fail to express their diversity:

If you ‘understand’ something in only one way then you scarcely understand it at all—because when something goes wrong, you’ll have no place to go. But if you represent something in several ways, then when one method fails, you can switch to another. That way, you can turn things around in your mind to see them from different points of view —until you find one that works for you!

To show how this kind of diversity makes human thinking so versatile, we’ll start with examples of the multiple ways we use to estimate our distance from things.

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§8-2. Estimating Distances

Why has not man a microscopic eye? For this plain reason, man is not a fly. Say what the use, were finer optics giv’n, T’ inspect a mite, not comprehend the heav’n? —Alexander Pope (in Essay on Man)

When you’re thirsty, you look for something to drink—and if you notice a nearby cup, you can simply reach out to pick it up—but if that cup lies further away, then you will have to move over to it. But how do you know which things you can reach? A naive person sees this as no problem at all because, “You just look at a thing and you see where it is.” But when Joan detected that oncoming car in §4-2 or grasped that book in §6-1, how did she know its distance from her?

In primeval times we had to guess how near our predators were to us; today we only need to judge if we have enough time to cross the street—but, still, our lives depend on this. Fortunately, we each have many different ways to estimate the distance to things.

For example, you know that a typical cup is about as large as your hand. So if a cup fills as much of the scene as does your outstretched hand , then you can reach it from where you stand. Similarly, you can judge how far you are from a typical chair, because you know its approximate size.

However, even when you don’t know an object’s size, you still have ways to estimate its distance from you. For example, if you can assume that two things are of similar size, then the one that looks smaller is further away. Of course, that assumption may be wrong, if one of those objects is a small model or toy. And also, whenever two objects overlap, then the one in front must be closer to you, regardless of its apparent size.

You can also get spatial information from how the parts of a surface are lighted or shaded, and from an object’s perspective and context. Again, such clues are sometimes misleading; the images of the two blocks below are identical, but the context suggests that they have different sizes.

If you assume that two objects lie on the same level surface, then the one that looks higher lies further away. Also, finer-grained textures look further away, and so do things that look hazier.

You can also judge a distance to an object by difference in its images in your two eyes—either by the angles between those two images or by the small ‘stereoscopic’ disparities between those slightly different images.

Also, if an object is moving, then the closer it is to you, the faster it will appear to move. You also can

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