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This question seems to either be at the forefront or the background of countless philosophical enquiries. Much has been written on Wittgenstein's rule paradox (e.g. Kirke's Wittgenstein: On Rules and Private Language," as well as many others). Here is Wittgenstein's rule paradox:

§201. This was our paradox: no course of action could be deter- mined by a rule, because every course of action can be made out to accord with the rule. The answer was: if everything can be made out to accord with the rule, then it can also be made out to conflict with it. And so there would be neither accord nor conflict here.

It can be seen that there is a misunderstanding here from the mere fact that in the course of our argument we give one interpretation after another; as if each one contented us at least for a moment, until we thought of yet another standing behind it. What this shews is that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" and "going against it" in actual cases.

Hence there is an inclination to say: every action according to the rule is an interpretation. But we ought to restrict the term "interpretation" to the substitution of one expression of the rule for another.

§202. And hence also 'obeying a rule' is a practice. And to think one is obeying a rule is not to obey a rule. Hence it is not possible to obey a rule 'privately': otherwise thinking one was obeying a rule would be the same thing as obeying it. (Wittgenstein, Philosophical Investigations)

I guess the question is: how do we know how to apply some rule we learned to something unknown. Why are we certain that 13 x 13 = 169 and not 196 (if we’ve never gone past 12 x 12 before)? --As in, how can past experience help us if what we are applying the rule to is something unexperienced? It could seem as though we need to follow a rule to follow a rule (an infinite regression). That is, there would be a certain rule for applying rules by some similarity of circumstances; of course, it gets somewhat tricky for something like mathematics--where we often change the elements of the system to build foundations (e.g. the change in the concept of number that Dedekind and Cantor brought about), and thus, are still expected to follow certain rules in new (but perhaps analogous) circumstances. (To identify situational analogs seems to be another sort of rule following.) The point is that it seems our minds "fill in the blanks," so to speak, of what is unknown by what we know. But it's also unclear why there is certainty when we apply a rule to new sets of problems.

The question is not only "how do we know how to follow a rule," but also how do we know that when we follow a rule we are using it correctly--or even if we are using it correctly, in the same sense that we learned the rule, how do we know that what we are calculating behaves (consistently) according to the rule? There are some analogous situations outside of mathematics in which we seem to follow either a learned, experienced, or innate, rule of perception: some colors that the human eyes sees are not spectral colors (e.g. pink, tan, greys etc.), and thus, don't correspond to a single wavelength on the color spectrum.

  • How do we know that we are certain that (some big number we've never seen before) squared = some other big number? Because we are certain about the rule. How do we know that 8 squared is 64? you can do the caluclation following a rule but you can also use memory to corroborate. – Mitch Nov 9 '11 at 1:27
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I see two parts in your question. Firstly, what did Wittgenstein think about rule-following and what is his critique meant to teach us? Secondly, how can we, in the light of this paradox, ever know we are following a rule. I take them in turn - my answer to the second question will be informed by the Wittgensteinian analysis that grows out of the first.

There is something one always needs to keep in mind when reading the Investigations, which is that Wittgenstein's principal aim is to criticize philosophical parlance, not philosophical arguments per se. What Wittgenstein is doing here is not, as Kripke would have it, presenting us with a skeptical paradox, namely that there is no fact of the matter that makes it true that at any given instance I am following a particular rule. What he is criticizing instead is the philosophical temptation that leads us into thinking that the concept of a rule can only be approached through the concept of "interpreting a rule."

It is, according to Wittgenstein, a paradigmatically philosophical conceit to put distance between a rule (viewed in abstraction) and the understanding of a rule. Only in philosophy are we tempted to talk like this: in reality (i.e. in language games that arise naturally in our form of life) rules are followed in the same way that commands are carried out, viz. as a reaction to prompts and ostensive definitions carried out in a shared form of life (what in our case we may call a shared 'linguistic environment'.) This is what Wittgenstein means when he says that by "interpretation" we should understand simply alternative ways of expressing (as opposed to understanding) the same rule.

The fundamental grammatical mistake (there is always one of those in the later Wittgenstein) therefore is that of adopting too wide a scope for the term "interpretation" when we are doing philosophy. It is this that leads us into temptation and it is this that we must get rid of if we are to clear the 'misunderstanding' and realize that there really is no problem with rule-following at all: the fly, as it were, was never in the bottle to begin with.

So, to get to your second question: "How do we know we are following a rule?" This is a question that ONLY becomes meaningful if you accept there is a separation between grasping a rule and the rule itself. Think of the situations where it becomes useful to exclaim: "I know I am following a rule!" Perhaps in a maths class, an innumerate classmate challenges you to prove to him that the scribbles in your notebook are not arbitrary doodles. When he remains unconvinced, the epistemic exclamation: "I know I am following a mathematical rule!" is meant to convey to him a certain regularity in your behaviour, perhaps in an attempt to convince him not to dismiss mathematics altogether. Here the exclamations make sense.

What is crucial is that we must not let ourselves take this proclamation literally. There is no fact by virtue of which I can know that I am following a rule. To say that I do is just a manner of speaking. There is a feeling of being 'guided' by the rule, of being led to the next step as though by an invisible hand. That is the feeling we try to convey when we say: 'I have grasped this rule! I now know how to apply it in all future applications!' The mistake is to understand this invisible guidance in terms of an interpretation that somehow, as it is grasped, magically fits all the facts, past present and future. The feeling is no more mysterious than the feeling of guidance one gets when one sees an arrow pointing right. One knows where the arrow is pointing at simply by virtue of being a member of a linguistic community (a form of life) that employs arrows in such a manner.

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I'll begin with a few general remarks. I might suggest one way to look at this is to think about language. In order to become a fluent speaker in the language, you really need to be immersed. Why? Because just knowing the rules themselves really doesn't tell you as much as you might think.

Rules demand meta-rules, in a potentially infinite regress: where to apply them? How strictly? How will we know if our application has been effective? And so on. In other words, for every written rule, there are a number of unwritten errata; and further unwritten codices for when and how to apply it.

At an even higher level of abstractions we can consider heuristics -- strategies suitable for broad classes of problems, which don't even specify an explicit set of rules but rather offer a kind of recipe, or more or less formal 'story' or 'theory' about the solution process.

I might suggest that human beings are generally fairly adept at navigating among these 'clouds' of meta-law around the explicit framework and the general guidelines or heuristics describing ways of using the framework to resolve questions, because we are trained to do it.

We have to learn all of this; and our teachers have to be patient with us as we learn. We are gradually 'subjectivized' into ever more complex and abstract rule-dominated processes. They are linked strongly to judicial as well as disciplinary institutions.

Wittgenstein himself seems to indicate that rule-following is a praxis, though perhaps not an unparadoxical one -- from the selection you asked after:

...[T]o think one is obeying a rule is not to obey a rule. Hence it is not possible to obey a rule 'privately': otherwise thinking one was obeying a rule would be the same thing as obeying it.

I happen to think this is pretty interesting and definitely agree there is something to this. Zizek might perhaps suggest that there is a kind of terrifying excess when one really 'follows' the rules, without respecting the various meta-rules which make them workable.

In passing, I am reminded of Deleuze at the beginning of Difference and Repetition, who talks about the two ways of overturning the law: either with irony, by challenging the law as itself secondary or derived; or on the other hand, through humor -- by offering submission in a "too perfect" way so that you effectively evade it. From there:

There are two known ways to overturn moral law. One is by ascending towards the principles: challenging a law as secondary, derived, borrowed or 'general'; denouncing it as involving a second-hand principle which diverts an original force or usurps an original power. The other way, by contrast, is to overturn the law by descending towards the consequences, to which one submits with a too-perfect attention to detail. By adopting the law, a falsely submissive soul manages to evade it and to taste pleasures it was supposed to forbid. We can see this in demonstration by absurdity and working to rule, but also in some forms of masochistic behaviour which mock by submission... (Gilles Deleuze, Difference and Repetition [p. 6 in the Continuum 2004 edition])

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