We asked to consider when we are justified in saying that a pupil understands some system, or to consider when we ourselves are justified in saying, “Ah, now I understand the system”. That is, what is it that goes on in these situations when one is credited with understanding. Consider four pupils each examining the same series of numbers: 1, 5, 11, 19 …. We say that a pupil understands the series if he can correctly produce the next term of the series; for if a pupil did not understand the series, then he would not be able to correctly continue the series. So we say one understands when they can “go on” continuing the series correctly. The question, then, is what does this understanding consist in?
Suppose each pupil can correctly carry on the series, establishing that the fifth term is 29. What is going on in the pupil’s head when he realizes he can correctly continue the series? It seems there are many processes that could have been at work. For instance, it might “occur” to the first pupil that the first four terms can be united under the formula: 
. The formula did not, in contrast, occur to the second pupil. Instead the second pupil notices a progressive series of differences: 4, 6, 8 …, and infers that the next difference is 10 so the fifth term should be 29. For the third pupil, it could be the case that this series is simply as familiar to him as the ordinary series of natural numbers, and from this familiarity can “go on”. The fourth pupil might have some immediate intuition that the fifth term is 29. Regardless of what particular process occurred in each pupil’s head, we will still credit each pupil with having understood, for they can continue to correctly carry on the series. The moral is that there is not one unique “occurrence” in one’s head that constitutes understanding. Understanding cannot consist merely in having the appropriate formula occur to you; for the pupil who notices the differences and yet doesn’t having a formula occur to him is still credited with understanding. Because of this we cannot say that “Now I understand the series” means the same thing as “the formula occurs to me”. To emphasize this point, consider a pupil to whom the appropriate formula does in fact occur. It could still be the case that they misapply the formula, and fail to correctly carry on the series with 29. Consequently we will say that this pupil does not understand the series, even when the correct formula occurs to him. So understanding must be something besides merely having the appropriate formula occur to you.
. The formula did not, in contrast, occur to the second pupil. Instead the second pupil notices a progressive series of differences: 4, 6, 8 …, and infers that the next difference is 10 so the fifth term should be 29. For the third pupil, it could be the case that this series is simply as familiar to him as the ordinary series of natural numbers, and from this familiarity can “go on”. The fourth pupil might have some immediate intuition that the fifth term is 29. Regardless of what particular process occurred in each pupil’s head, we will still credit each pupil with having understood, for they can continue to correctly carry on the series. The moral is that there is not one unique “occurrence” in one’s head that constitutes understanding. Understanding cannot consist merely in having the appropriate formula occur to you; for the pupil who notices the differences and yet doesn’t having a formula occur to him is still credited with understanding. Because of this we cannot say that “Now I understand the series” means the same thing as “the formula occurs to me”. To emphasize this point, consider a pupil to whom the appropriate formula does in fact occur. It could still be the case that they misapply the formula, and fail to correctly carry on the series with 29. Consequently we will say that this pupil does not understand the series, even when the correct formula occurs to him. So understanding must be something besides merely having the appropriate formula occur to you. If when I say “Now I understand” I have not said “the formula occurs” to me, what then does it mean to say “Now I understand”? You might think that understanding is a (presumably mental) process which somehow occurs behind or along with the occurrence or utterance of the formula. How are we to think of mental processes? Wittgenstein suggests that a pain’s increasing or decreasing, or the listening to a tune or sentence are mental processes. The pain experience or the auditory experience are mental processes insofar as they are particular occurrences “in one’s head”, so to speak. These processes may be interrupted; for instance, I may be in pain and then fall asleep. When I fall asleep, we do not continue to attribute the mental process of being in pain to me. Or if Barry Stroud falls asleep at the opera, his mental process of listening to the tune has been interrupted; we no longer attribute the listening of a tune to him. In a similar way, the occurrence in your head of the appropriate formula is a kind of mental process. You may be representing the appropriate formula, fall asleep, and so cease to be in a state of representing the formula (or of having it occur to you).
Understanding does not seem to be a mental process in the same sense as the listening to a tune. Sam, grandmaster of chess, we attribute understanding of chess to. When Sam falls asleep, we do not say that his understanding is interrupted. When Sam is asleep we still say he understands chess. Sam doesn’t understand chess merely when the appropriate chess move occurs to him during a game (as when a particular formula may occur to you during a math problem). Sam’s being in a state of sleep does not strip him of his ability to play chess. In this way, Sam’s understanding cannot be identified with the occurrence of some mental process that happens alongside his action.
So when a pupil thinks “Now I can go on” and utters the correct formula, what can we point to which actually justifies the pupil’s thinking “Now I can go on”? We’ve seen that we cannot point to some unique occurrence in his head, for there were many various occurrences which accompanied each pupil’s ability to go on (e.g. representing the appropriate formula or seeing the sequence of differences). Nor can we point to some mental process, for mental processes can be interrupted in ways that the understanding seemingly cannot. We cannot point to something inside the pupil’s head and say that that’s what the understanding consists in; consequently, we should look at the circumstances of the situation outside of just what is in one’s head. This leads Wittgenstein to suggest that if there is something which justifies the pupil’s thinking “Now I can go on”, it is the particular circumstances which underly the utterance of the formula (or the noticing of the series of differences). There is something about the pupil and the external situation he’s in which determines whether we are justified in saying that he understood. It is not enough for the the formula to occur to the pupil, but the formula must occur to the pupil in the right circumstances, where the pupil reacts in the right way to the given external stimulus (e.g. the series). If the external stimulus had been different but the pupil reacted the same way we would not credit him with understanding. To see that someone has understood, we must look at more than what goes on in their heads, but also what the external stimulus was and what the external reaction on the part of the pupil is. In this way, we see understanding as more of an external process – an ability or a “can-do” – than any particular mental process.
Reflecting Light 