We can understand a word like "cube" without placing it in a sentence. We may, for instance, have a mental picture of a cube. But how do we apply that picture? There is no reason we shouldn't take it to be a picture of a prism, and yet we would say that one has misunderstood the word "cube" if one points to a triangular prism and says, "cube." We normally apply a picture of a cube only to what we call "cube" and not to what we call "prism," but this tells us only how we normally behave, and not how a certain picture must be applied.
Is understanding a mental state? We want to say someone has understood the series one, five, eleven, nineteen, twenty-nine because he has a certain mental state. We also want to say that writing down the next three terms of the series correctly or uttering the formula for the series are only manifestations of this mental state. But we do not say of understanding, as we would of other mental states like pain or depression, that it has been continuous for a few days, or that it went away yesterday evening. Understanding is determined by the particular circumstances in each case that justify us in saying we understand.
In this sense, "understanding" is much like "reading." There are a variety of activities we would call "reading," ranging from spelling out words letter by letter, to skimming entire paragraphs, and there is no distinctive mental state or mechanism that is common to all these different activities. In sections 156–178, Wittgenstein considers three possible general definitions of "reading." First, reading is accompanied by a distinctive set of sensations, different from when I am only pretending to read. Second, reading is a matter of deriving sounds from the rule provided by the alphabet. Third, sounds come to me, or my speech is guided when looking at words in a way that I am not when looking at random squiggles.
If, as in the first objection, reading were a matter of certain mental sensations, couldn't a drug make us simulate those sensations when we are not looking at a page? We would not call it reading unless these sensations were accompanied by certain forms of behavior. With regard to the second objection, there is more than one way we can interpret a rule, and it is not even clear when we are deriving from a rule and when acting at random. With regard to the third objection, there are just as many different uses of the phrase "being guided" as there are of "reading" or "understanding."
When we try to formulate definitions of "reading," we are generally trying to identify what differentiates reading from certain non-reading activities such as pretending to read or uttering sounds while looking at random squiggles. And indeed, these features that we identify are often characteristic of reading; but we cannot say these features alone define reading. What we call "reading" depends on a wider context of behavior and circumstances, and cannot be reduced to a particular sensation or mental state.
One of Wittgenstein's main preoccupations is to undercut commonly held ideas about the mind and mental states that are most present in so-called "scientific" theories of psychology. Just as physics takes as its object of study the workings of natural phenomena, psychology takes as its object of study the workings of mental phenomena. However, Wittgenstein suggests that psychology, unlike physics, bases its investigations on unwarranted assumptions that we take from everyday speech about the mind.
There are a number of very different surface features that we might call manifestations of understanding a series like one, five, nine, eleven, nineteen, twenty-nine. A person may utter the algebraic formula for the series, or remark that the difference between each successive term increases by two, or simply say, "that's easy!" and write down the next five numbers. However, we will also be tempted to say that there must be something in common in all these cases. In each case, the person understands the series, but that understanding expresses itself in different ways. Because this commonality is not evident in the person's behavior in each of the three different cases, we will be tempted to say that the understanding is a hidden, mental phenomenon or state.
In discussing the mind, Wittgenstein frequently returns to the theme that when our grammar suggests that there must be something common in a number of unlike phenomena, we say that commonality exists in the mind. Because we call these three different reactions "understanding the series," we feel the compulsion of a "must": they must have something in common if we call them all by the same name. Because we cannot find any direct evidence for this assumed commonality, we conclude that it must be hidden in the complexities of the mind.
Psychological investigations begin after this assumption has been made. Firm in the knowledge that there must be a mental mechanism of some sort that constitutes understanding, a psychological investigation then probes the mind to identify exactly what this mechanism is and how it works.
Wittgenstein argues that there is no such mental mechanism. It is perfectly acceptable to talk about understanding as a state, and to say "Jane understands the series, but John does not." The problem arises when we take this talk of a state of understanding to point to some underlying psychological truth.
Wittgenstein tries to simplify the discussion to make his position clearer. For instance, people generally don't think that understanding a word consists of the word's summoning up a mental image, which is the point that Wittgenstein is trying to make with the cube example. However, if we want to talk about understanding as a mental state that somehow causes the different manifestations of understanding, a mental image is the clearest picture of such a cause. If the example of "cube" as a mental image does not work, then more complex theories will probably not work either.
The discussion of reading is a similar simplification. There are many complications in the discussion of understanding that do not come up in the case of reading. "Reading" covers a more limited range of phenomena. It is a distinctive act that we can recognize easily, and there is less of a distinction between reading and the various manifestations of reading. If, despite all these simplifications, we still cannot identify a single state or mechanism that we call "reading," we are unlikely to find one in the more complex case of "understanding."
In recognizing that Wittgenstein denies that there is a mental state or mechanism of understanding that underlies the various manifestations of what we call "understanding," we should be careful to avoid two possible misreadings. The first reads Wittgenstein as a behaviorist, saying that there are no hidden mechanisms underlying different forms of behavior, and that the behavior itself (the various manifestations) is understanding. The second reads Wittgenstein as claiming we are mistaken to talk about understanding at all. Wittgenstein denies that understanding is a thing we can identify and talk about, whether it be a mental mechanism or certain forms of behavior. However, this move is meant to counteract attempts to turn "understanding" into a technical term and include it in our psychological vocabulary. Wittgenstein has no problem with talking about understanding in ordinary contexts.