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Is there a Rule-Following Problem?

If we are asked to count, or to follow someone's directions, or to do A as soon as someone else does B, we seem to understand exactly what is being asked of us. Better still, we seem to have a totally unshakeable understanding - these sorts of requests are so ingrained in us from a very early age, that for an adult to be unable to comply with one would generally seem very bizarre indeed. It seems, however, that Wittgenstein pointed out some very important limitations for such attempts at provoking similar behaviour; in fact is seems likely that Wittgenstein showed that we can never be certain that someone will behave as we expect, even when we ask them to perform the simplest of tasks.

Wittgenstein's rule-following considerations are particularly staggering for their simplicity - affecting not just complicated tasks, but everyday, frequently practised ones. Let's take (again for simplicity's sake) the simple example of counting, imagine a teacher showing a primary-school pupil how to count: the teacher counts to ten in front of the child, and invites the child to follow suit. If the child correctly counts to ten, she will be congratulated (and perhaps even rewarded); if not, corrected (or even chastised); and regardless the process will be repeated until the child is able to count correctly to ten 'every time' she is asked two. Regardless, however, of the teacher's success, it is clear that no amount of repetition or practice will result in the child being able to count - since to do so would require the child to go beyond the point to which she has reached with the teacher holding her hand.

We might argue that, of course, if we have only taught the child to count to ten, she will not be able to continue into multiple digits, and that additional prompting is necessary. This clearly missed the point - initially by mistakenly taking the way we divide and write numbers to be essential to their understanding, when it is not clear that this should be so - with minimal thought it seems quite reasonable to question how we should ever come to a point where a child might continue without prompting. I suspect that this is not as extraordinary a claim as it might first seem - we no doubt will run into some significant trouble attempting to understand how one is to go about counting beyond numbers for which we have no names: given that it is rule-following that is being discussed, the easiest solution, that we can just invent a new name, does not seem to be a terribly attractive approach. Furthermore, we cannot even guarantee that the child does not believe that the rule goes 'When the teacher asks you to count, say '1, 2, 3, 4, 5, 6, 7, 8, 9, 10'; at all other times, say whatever you like', thus convincing the teacher that she can count, but failing miserably to apply it to the world.

There are a number of other points that immediately emerge from this account. The first is that of the question of the teacher 'inviting the child to follow suit'. It is unclear how such an invitation might work, unless there was already a successful convention used to indicate 'I want you now to do what I just did', and this assumes just the problem - of the inability to form successful modes of practice, and of being certain that other people are playing the same game as you are - that we are debating. It seems intuitive that a child might be taught to follow suit: she might be congratulated when she mimics the teacher when he says 'Now you try it', or some such, or corrected if she fails to do so. Even this seems a remarkably brave claim to be making: how would a child learn, when a teacher says 'Now you try it' that they were suppose to produce the same action as the teacher, if the teacher merely 'corrects' them by repeating his action when they fail to do so? Perhaps the most plausible explanation will be that children learn what 'Now you try it' means through their teacher, say, picking up their hands and guiding them when they are playing with bricks, for example, though it isn't clear how with this they wouldn't simply come to believe that 'Now you try it' meant something different from 'Now you do it with me'.

The second major obvious problem for this account is that of the use of 'every time'. Clearly there will never come a time when the child has counted correctly 'every time', even if her initial mistakes soon become a quickly diminishing percentage of her overall attempts. If, however, there never comes a time when all her attempts have been successful, we must either lay down a number of success necessary to declare her as correctly understanding, or (presumably) arbitrarily declare a moment at which the tally is reset to zero, and after which if every attempt to count is successful, we declare that she understands the rule. Wittgenstein suggests that the second option might seem the most intuitive. He points out that we generally view understanding as something that happens 'in a flash' - that there is a single moment were suddenly you understand how a rule works.

Kripke considers the comparison of two mathematic operators, plus and (an invented one) 'qus'. Whilst we might believe that we understand how we go about addition, he points out that we can never know that the rule we have been using all along has been 'plus' and not 'qus', which is exactly the same as plus except that when it uses a certain pair of numbers, the result is 5. His example is that of 58 + 67, though clearly it might be easier to consider an addition that has not previously been performed. So, whilst 1 quadded to 2 would be 3, and 10 quadded to 5 would be 15, 67 quadded to 58 would be 5. As ludicrous as such a suggestion might at first seem, there doesn't seem to be anything in our previous usage of our mathematical practices that could mean that we were using plus and not qus all along.

One possible response to the qus problem is to suggest that when we talk of the meaning of 'plus' we are not simply talking about our previous uses of the function, but of a disposition. So it is the disposition to make certain utterances that means that we are using plus and not qus, and that we can say that, were we to perform the calculation '58 + 67' we would produce the answer 125. Kripke attacks this view on a number of counts - principally because this would seem to rule out the possibility of mistakes. Clearly is it possible that were I to add 67 to 58 I might come up with the answer 115 (or perhaps even 5!), and we do not want to say that this means that 58 plus 67 is 115. The only way around this seems to be to say something that might be unpacked as 'We have a disposition to correctly appraise 58 + 67 as 125', and this clearly won't do at all.

A better response, proposed by Kripke (but following closely ideas of Wittgenstein's) is what he calls a 'sceptical solution'. It is sceptical in that he denies that there really is essentially a problem, suggesting that it is the philosophers usage of the term 'meaning' that is confusing the issue, not the normal way we go about communicating. His solution is to deny that words should have an independent meaning, and instead to ascribe them meaning simply as a comparison to how someone uses them. So to say that someone counts correctly, he argues, is simply to suggest that they count in just the same way that we (or more precisely, if I were to say it, then in just the same way that I do) do; for me to say that they have understood the 'rule' is simply to suggest that they will continue to count in just the same way that I would. This account, which makes meaning indexical and relational, seems very attractive to me.

One of the main reasons to support this response is that, whilst it undermines the metaphysical question, and fails to explain how a word comes to mean something, simply because there isn't any fact about what a word means, it will be entirely pragmatic, and there will be no worries about applying it to the world. Furthermore, it seems to account perfectly for communication, given that it aligns successful usage of a word with the word communicating effectively: if I go into a grocery store (to use an example of Wittgenstein's), I need to know nothing more to know that the grocer has understood me, but that when I ask for three red apples he gives me three red apples. There is no essential rule that he has grasp in providing me with what I wanted, he has just successfully used the words in the way that I use them, and thus has assessed correctly its (indexical and relational) meaning.

The essence of this response is (to use Wittgenstein's words) that 'to use a word without justification is not to use it wrongly' - simply because there is no essential meaning of a word is not to say that we should never use it. To return to the question at hand, therefore, is there a rule-following problem? Well, yes, it seems to me that there is, in the sense that we can never assess whether we have the same rules for, say, counting as anyone else. This problem for rule-following, however, need not be a problem for us as we go about our everyday life: we need simply accept that such rules do not matter (if they exist in any serious sense at all, which is a question for another day), since to talk of someone successfully following in our footsteps might simply be to suggest that they have 'grasped our meaning', in a new, indexical sense - they have successfully followed suit purely in virtue of continuing just as we would have done. Whilst in a sense the plus-user might gasp at our use of qus, so long as we are consistent (and occupy the same mathematical culture), we need not worry about strange looks.