Let us return to our example (§143). Now – judged by the
usual criteria – the pupil has mastered the series of natural numbers. Next we
teach him to write down other series of cardinal numbers and get him to the
point of writing down series of the form
0, n, 2n, 3n, etc.
at an order of the form “+ n”; so at the order “+ 1” he writes down the series of natural numbers. – Let us suppose we have done exercises and given him tests up to 1000.
Now we get the pupil to continue a series (say + 2) beyond 1000 – and he writes 1000, 1004, 1008, 1012.
We say to him: “Look what you’ve done!” – He doesn’t understand. We say: “You were meant to add two: look how you began the series!” – He answers: “Yes, isn’t it right? I thought that was how I was meant to do it.” – Or suppose he pointed to the series and said: “But I went on in the same way.” – It would now be no use to say: “But can’t you see....?” – and repeat the old examples and explanations. – In such a case we might say, perhaps: It comes natural to this person to understand our order with our explanations as we should understand the order: “Add 2 up to 1000, 4 up to 2000, 6 up to 3000 and so on.” [Wittgenstein 1953 §185]
0, n, 2n, 3n, etc.
at an order of the form “+ n”; so at the order “+ 1” he writes down the series of natural numbers. – Let us suppose we have done exercises and given him tests up to 1000.
Now we get the pupil to continue a series (say + 2) beyond 1000 – and he writes 1000, 1004, 1008, 1012.
We say to him: “Look what you’ve done!” – He doesn’t understand. We say: “You were meant to add two: look how you began the series!” – He answers: “Yes, isn’t it right? I thought that was how I was meant to do it.” – Or suppose he pointed to the series and said: “But I went on in the same way.” – It would now be no use to say: “But can’t you see....?” – and repeat the old examples and explanations. – In such a case we might say, perhaps: It comes natural to this person to understand our order with our explanations as we should understand the order: “Add 2 up to 1000, 4 up to 2000, 6 up to 3000 and so on.” [Wittgenstein 1953 §185]
One response is to think that this passage points to a
genuine, empirical possibility – a possibility for us – which results from a
gap between explanation and understanding. That is how Collins takes it in Changing Order . He concludes both that
the notion of ‘sameness’ is ambiguous and that it is not possible fully to
specify a rule (unless a limited range of responses is defined in advance). But
‘since in spite of this we all know the correct way to go on, there must be
something more to a rule than its specifiability’ [Collins 1980: 14]. The extra
element is described as ‘social entrenchment’ or a ‘shared form of life’ and
also ‘tacit knowledge’. Thus it is tacit knowledge that underpins the ‘mysterious abilities that enable us to
know when to continue ‘2,4,6,8’ with ’10,12,14,16’ and when with ‘who do we
appreciate?’’ [ibid: 22 italics added].
So there is a gap between explanation and understanding
and the gap is filled by tacit knowledge. Various problems, however, follow
from this. But, further, it conflicts with what Wittgenstein himself says a
little later.
But do you really explain to the other person what you
yourself understand? Don’t you get him to guess the essential thing? You
give him examples, – but he has to guess their drift, to guess your
intention.” – Every explanation which I can give myself I give to him too. –
“He guesses what I intend” would mean: various interpretations of my
explanation come to his mind, and he lights on one of them. So in this case he
could ask; and I could and should answer him. [Wittgenstein 1953 §210]
“But this initial segment of a series obviously admitted
of various interpretations (e.g. by means of algebraic expressions) and so you
must first have chosen one such interpretation.”–Not at all. A doubt was
possible in certain circumstances. But that is not to say that I did doubt, or
even could doubt… [ibid §213]
So, empirically at least, Wittgenstein thinks that there’s
no gap. And this picks up again the naturalistic tone from the discussion of
signposts that starts earlier in the Investigations but then returns in the mid
point climax of the rule following discussion.
A rule stands there like a sign-post. – Does the
sign-post leave no doubt open about the way I have to go? Does it shew which
direction I am to take when I have passed it; whether along the road or the
footpath or cross-country? But where is it said which way I am to follow it;
whether in the direction of its finger or (e.g.) in the opposite one? – And if
there were, not a single sign-post, but a chain of adjacent ones or of chalk
marks on the ground – is there only one way of interpreting them?... So
I can say, the sign-post does after all leave no room for doubt. Or rather: it
sometimes leaves room for doubt and sometimes not. And now this is no longer a
philosophical proposition, but an empirical one. [ibid §85]
“Then can whatever I do be brought into accord with the
rule?” – Let me ask this: what has the expression of a rule – say a sign-post –
got to do with my actions? What sort of connexion is there here? – Well,
perhaps this one: I have been trained to react to this sign in a particular
way, and now I do so react to it.
But that is only to give a causal connexion; to tell how it has come about that we now go by the sign-post; not what this going-by-the sign really consists in. On the contrary; I have further indicated that a person goes by a sign-post only in so far as there exists a regular use of sign-posts, a custom. [ibid §198]
But that is only to give a causal connexion; to tell how it has come about that we now go by the sign-post; not what this going-by-the sign really consists in. On the contrary; I have further indicated that a person goes by a sign-post only in so far as there exists a regular use of sign-posts, a custom. [ibid §198]
So in the context of a custom, there’s no gap between
explanation and understanding and the problems I’ve alluded to can be avoided.
But there is a potential niggling worry here. Have we earned the right to dismiss the ‘gap’? It’s true
that, naturalistically, we have no problem following the signpost. We do not
act as though there’s a gap to be bridged. But once Wittgenstein has deployed
the considerations of §185 perhaps we should realise that there is a gap, in
principle, after all? That is, that we do not so act needs some sort of
explanation akin to the invocation of tacit knowledge to plug the gap in
principle even if we do not notice it in practice?
I think that the niggling worry is misplaced but is
maintained as a potential concern because of a yen for a further explanation of our ability. Against the
background of a mechanical explanation of understanding, the ‘gap’ is a real
issue. But with the reminder of how an explanation works in the context of a
custom, there’s no need for the hypothetical explanation. And without the request
for mechanical explanation, we have all we need. So we need to stress the almost
Strawsonian comment above: ‘I have further indicated that a person goes by a
sign-post only in so far as there exists a regular use of sign-posts, a custom’.
Of course, there are other views available. Charles Travis
thinks that there is a gap which has two elements (the relation between which I
still have not got clear). First, any prior understanding of a rule is distinct
from the novel understanding one has at the cutting edge of application because
the latter is an object depending thought whilst the former is descriptive.
Second, understanding is always a context-sensitive matter.
