Wittgenstein: Rule-following Paradox, Forms of Life and Education

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Wittgenstein: Rule-following Paradox, Forms of Life and Education

Kenny HUEN

Abstract

This chapter explains the view of the later Wittgenstein, in particular his rule-following considerations and their implications for education. Many people suppose that grasping a rule amounts to getting at the right interpretation of the rule. Through the rule-following paradox, this conception is shown to be deeply misconceived. We are led back to a plain view of practices in which rules are embedded, a view that the criteria which determine the right or wrong of the use of a rule lie in the application, not in some mental entity. Meaning or the extension of a rule is settled on “agreement in form of life”, according to Wittgenstein. But this notion can only be appreciated when we note its dynamic nature and the role the rule-follower plays as an agent.

Introduction

Ludwig Wittgenstein (born 1889 in Vienna, died 1951 in Cambridge, England) is widely regarded as one of the most influential philosophers of the twentieth century. Following the advice of Gottlob Frege, Wittgenstein became Bertrand Russell's student at Cambridge in 1911. His passionate pursuit of the philosophical foundation of logic culminated, in this early period of his life, in the Tractatus Logico-Philosophicus (TLP). The book (much broader in its sweep than an investigation of logic) was later intensely studied and discussed by members of the Vienna Circle. Wittgenstein advocated in this work a kind of harmony between language and reality, such that the constituents of the former correspond to the constituents of the latter in a one-to-one relation. He argued that the world (i.e. the represented) and propositions (i.e. what does the representing) share a fixed logical form (see, in particular, TLP 2.18 & 3.25). Having completed the Tractatus, Wittgenstein thought that he had no further basic problems of philosophy to deal with. He then put aside philosophy and worked for six years (1920–26) as a primary-school teacher in some villages in the mountains of Lower Austria, some forty miles south of Vienna. His interest in philosophy was revived thanks largely to Frank Ramsey, who had reviewed the Tractatus and, on a visit to Wittgenstein, raised some queries about the text. Also inspired by L. E. J. Brouwer's lecture on the foundation of mathematics in 1928, Wittgenstein was determined to return to Cambridge and engage in philosophy again. He was beginning to have grave doubts about his early book and was developing an entirely different philosophical method, as shown in his posthumous works including Philosophical Investigations (PI) and On Certainty (OC). This chapter is an attempt to open a window on the later Wittgenstein by elaborating on what has come to be known as his “rule-following paradox” as well as his positive view on normativity. The considerations of that paradox played a crucial part in motivating the emphasis he came to place on our “ungrounded way of acting” (OC §110). Above all, it is the aim here to show what lesson Wittgenstein has given to education from this new perspective.

A Recalcitrant Student

In Philosophical Investigations, Wittgenstein provides a scenario involving a set of deep misconceptions about rules, mastering rules, following rules and their relations (PI § 185). A student is taught how to continue a series of natural numbers in accordance with the rule “add 2”: 0, 2, 4, 6, 8, 10, 12 and so on. It first appears that he has grasped the rule. But from 1000 onwards, he progresses in an unexpected manner: 1000, 1004, 1008, 1012 … It is normal for students to make mistakes during the learning process and have to be corrected from time to time before they succeed in learning a rule. The interesting thing in this case, however, is that no matter how patiently and clearly the teacher explains the rule again and again, the student still insists that “I'm just following the same rule as you have shown me.” In his mind, he employs the same “+2” rule as his teacher. Yet, from his answers, we might well say that he actually understands the rule as we would understand this instruction: “Add 2 for each step up to 1000, then add 4 up to 2000, after that add 6 up to 3000, and so on.” For him, such an interpretation of the “+2” rule indeed matches all the examples, exercises and corrections the teacher gave him before, involving numbers below 1000.

The teacher reminds him that the progression of the “+2” series is parallel to that of the numbers uttered when we count objects two at a time, but that his response now (1000, 1004 …) is inconsistent with the counting-in-twos method. However, the student thinks his “+2” series extends in just the same way as do the numbers uttered when counting objects in twos. To him, counting objects in twos means the same as how we would understand the following rule: “Point to one pair of objects and say ‘2’, then point to another pair and say ‘4’, then ‘6’, ‘8’ and so on; and when 1000 is reached, point to the next pair of objects and say ‘1004’, then ‘1008’ and so on.”

The teacher then calls attention to a schema previously introduced which illustrates the action of each step in counting objects in twos, indicating alternately with the left hand pointing to the first (and third, fifth, seventh, ninth and so on) object and the right hand pointing to the second (and fourth, sixth, eighth, tenth and so on) object. And he instructs the student to utter 2, 4, 6, 8, 10 and so on according to the ordinal numbers of the objects pointed to with the right hand. However, the student thinks his response (1000, 1004 …) is entirely in accord with this schema and instruction. He takes them in the same way as we would understand the requirement that “when the ordinal number of the object being pointed to is over 1000th, double the action of pointing to a subsequent pair of objects” (so he points first to the 1001st and 1002nd objects and then to the 1003rd and 1004th objects and says “1004”).

Following Rules: Paradox and Diagnosis

Even a very simple schema (or operational rule) may be read in a non-standard way. On what grounds can we say that the student, after counting 1000 objects, wrongly doubles his action of picking out the next pair of objects? Shall we depend on a second-order schema (i.e. a schemafor determining whether the use of a given schema is correct or not) to forbid that interpretation? If yes, then shall we expect a still higher-order schema to govern the application of the second-order schema? An infinite regress will result. The fact is that no highest schema exists, since no schema can fix its meaning by itself. ¹ As Wittgenstein remarks:

This was our paradox: no course of action could be determined 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 (PI §201).

It is wrongly supposed that grasping a rule consists in getting at the right interpretation of it. The problem, as we have shown above, is that under this conception a rule (“rule” is used here in a broad sense to refer also to a schema or an operational rule) may fit any action. For any interpretation can be reinterpreted in a different way without failing to match all the previous applications of the rule in question. “However many rules you give me—I give a rule which justifies my employment of your rules” (Remarks on the Foundations of Mathematics [RFM] I-113).

Wittgenstein does not deny that at times we interpret some rules in our own way when applying them in concrete circumstances. Actually, making interpretations is common in many practices. What Wittgenstein attempts to overcome is the idea that “the right interpretation” of a rule dictates the whole range of correct instances in all future circumstances. He asks, “How was it possible for the rule to have been given an interpretation during instruction, an interpretation which reaches as far as any arbitrary step?” (RFM VI-38). In teaching the “+2” rule, it is not possible to explain all the steps in advance, of course. (Similarly, in teaching a word [a semantic rule], normally we cannot show all the possible circumstances of its correct usage. Imagine a rural resident who comes to a city being perplexed when encountering the terms “microwave cooking” and “laser knife” the first time.) It is by no means possible for any interpretation of a rule which occurs at some moment in the learning process to dictate responses in all other cases yet to be encountered, of which we are not yet aware.

We are invited to see the significance of rules, not in a detached manner, but in their “games” (or customary practices in life),² in which the rules' applications and functions are salient. The cognition-based interpretational picture of rule-following is overturned: instead of conceiving practice as merely some manifestation of grasping (the essence of) a rule, whether one really understands the rule depends on whether one responds in the same way as we are used to doing in various contexts in ordinary life.³ Grasping a rule is “exhibited in what we call ‘obeying the rule’ and ‘going against it’ in actual cases” (PI §201).

Agreement in Form of Life

Wait a moment! Does Wittgenstein suggest that it is our agreement that settles the rule-following paradox, either by matching an application with the ongoing custom or with the judgment that we collectively make case by case? Aren't such regularity and social concord somehow finite and fallible? If yes, wouldn't the community experience a similar thing that happens to the recalcitrant child? These questions are urgent, and we need to make clear what agreement is meant by Wittgenstein.

Wittgenstein actually does not suggest any supreme rule to the effect that it is the community which plays the role of the final arbiter. Nor does he contend that following a rule rests upon a convergence of dispositions to distinguish right from wrong (see RFM VII-40). His idea is not that now we have come up with an explanation of rule-following with reference to social agreement. Without the “+2” practice, we would not have acknowledged that 1000 is followed by 1002.⁴ Do not suppose that the correctness of following the rule is grounded on our collective judgment as if the latter is lying on a more basic level, but note: “Following according to the rule is FUNDAMENTAL to our language-game” (RFM VI-28).

The shared response (1000, 1002, 1004 …) in the practice of continuing the “+2” series is not a matter of “agreement in opinion” but “agreement in form of life” (PI §241). We just act in this way. Normally we do not doubt: “Disputes do not break out (among mathematicians, say) over the question whether a rule has been obeyed or not” (PI §240; see also RFM VI-21). To query further about the community's finitude is to ignore this givenness. “What has to be accepted, the given, is … forms of life” (PI, part II, p. 226).⁵ That means, “no reason can be given why you should act (or should have acted) like this, except that by doing so you bring about such and such a situation, which again has to be an aim you accept” (Culture and Value, 16e).

As maintained by Baker and Hacker (1984), “[A response] is correct if it accords with the rule. We can only speak of accord with a rule in the context of a regular use of a rule as a measure of correctness” (p. 46). “A regular use of a rule” or custom, in their view, need not be social in character.⁶ However, following this line of argument, how can we account for the right or wrong applications of the same rule in novel circumstances?⁷ Our lives are characteristically changing and developing (see OC §256). In new circumstances, we are not so sure of what to say or do. Can we call a cloned copy, which does not have parents, a human? As our world becomes more and more globalised, our idea of citizenship has undergone a radical change. In the past, for instance, we held without doubt that citizenship must pertain to bounded territories. Now we have begun accepting and even espousing the idea of global citizenship. How can the preexisting regular use of a rule or concept provide us “a measure of correctness” for unforeseen cases? Wittgenstein asks, “But what do we count as the use?” (RFM VII-10).

Saul Kripke's (1982) exposition, unlike the reading of Baker and Hacker, highlights our practices of attributing rules and concepts.⁸ But it does not follow, in Kripke's view, that normativity is defined by social licence or dispositions, as if every actual case is subsequent to some endorsement action from the community (see Kripke, 1982, pp. 111–112). The attribution games, in which we judge whether a rule-follower's performance is correct or not, play significant roles in life,⁹ and are especially important at times of crisis. A community's occasional errors are imaginable. But it is not possible for rule-following to take place if the whole community always commits mistakes. For, in that case, checking would become unthinkable.

While these points of Kripke's reading are insightful, the agency part of rule-following is unfortunately marginalised. This will be clear when we scrutinise the notion of education according to what Wittgenstein's perspective of forms of life suggests.

A Further Puzzle: Education as Initiating a Child into a Form of Life

The rule-following paradox has far-reaching implications. Its considerations have enabled us to remove misconceptions on normativity, not only of rules, but also of concepts and meanings, many of which are rule-governed. Moreover, prior to knowing that p is true, we have to understand (the meaning of) p; and in order to know, we apply concepts in most cases. So the paradox also sheds some light on our conception of knowledge—and, of course, of education.

Teachers should help students appreciate the significance of words, concepts, rules and principles by noting their use in relevant practices in life. For instance, “What ‘determining the length’ means is not learned by learning what length and determining are; the meaning of the word ‘length’ is learned by learning, among other things, what it is to determine length” (PI, part II, p. 225). Inspired by Wittgenstein's form-of-life standpoint, teachers will be aware that (critical) thinking is not to be “reified” as certain inner processes but that it is linked to context-sensitive practices (see Bailin et al., 1999a, 1999b).¹⁰

Education is not simply a domain where Wittgenstein's philosophy can be applied. The natural responses we acquire (this naturalness pertains to our “second nature”) have an important bearing on our upbringing (apart from our customs) (see PI, part II, xi, p. 201). Education, in Wittgenstein's view, is actually the foundation of our normativity. He frequently refers to our ways of learning. We can only figure out a different practice through imagining a different educational process behind it (see OC §§387–388).¹¹

Before learning how to continue the “+2” series, children must be able to identify Arabic numerals, to count (by twos), to master the technique of writing down or saying a series of natural numbers, and so on. “For only through a technique can we grasp a regularity” (RFM VI-2). The criterion of understanding a rule is, precisely, being able to follow the rule. Children are trained and taught the relevant skills, methods and concepts so as to “play the game”. But note, they should find these meaningful, not something alien to them. When they become practitioners of the practice, they have learnt some accepted ways to proceed, not in the sense of merely getting some information about the rule, but of being able to follow it spontaneously and with trust. What is noteworthy is that they subject themselves to the public standard of correctly applying the rule and that their judgments, experience (seeing), and other capacities and attitudes have been shaped (both empowered and constrained) through a certain course of education. This is still true for the moral life into which students are initiated. They are guided by, and respond to, such principles as, say, do not impose on others what you yourself do not desire; respect others as persons; be responsible; be sincere; and so on. They trust (and feel and are even committed to) these principles. They really think that these rules and principles are indispensable for a good life.

How can a person “formatted” through training and the community's criteria be a genuine agent or rule-follower? Wittgenstein affirms that the rule-follower's being able to choose to follow a rule is a fact as important as that of the compellingness of a rule (see RFM VII-66). “But I don't want to say,” he goes on, “that the rule compels me to act like this; but that it makes it possible for me to hold by it and let it compel me” (ibid., my emphasis). The resolution is not simply that I accept the rule and I would like to comply with it. In order that there is a relation between the rule and myself, there must be such a rule, the scope (or “extension”) of which is stable and the normative force of which is established.

Dynamic Life: Agency and Compellingness

At some point in my life, I begin to reflect upon my actions, my relations and my roles. I accept the form of life I am engaged in (see OC §344). Does this mean that I surrender my responsibilities to the state to which I belong? No, I may be critical of the agreed notion of good life in the community where I live. Critical thinking is indeed a crucial practice in the form of life I share (see Huen, 2005). More importantly, this suggests that the system of values and of the right and wrong involved are not taken as closed: “We learn first the stability of things as the norm, which is then subject to alterations” (OC §473), “the river-bed of thoughts may shift” (OC §97). To let the rule compel me (recall Wittgenstein's phrase above) does not amount to letting myself always be a passive conformer. As a participant in the practice of good life, I am also a maker—I not only judge with the relevant criteria that I learn from my teachers and my experiences, but in some critical moments I have to decide upon what to do in order to fulfil the requirements of these criteria and, thus, update their content (maybe with other participants with whom I interact).¹²

Notes

1 See The Blue and Brown Books, pp. 33–34. Think also about the related example Wittgenstein offers in PI §86.

2 Rules are contrasted with games, the former being embedded in the latter. References to various games, especially “language-games”, are abundant in Wittgenstein's investigations of concepts and rule-following. He also uses the word game to illustrate his notion of “family resemblances” (PI §67).

3 Interpretation of rules constitutes a large part of Wittgenstein's scrutiny in his rule-following considerations. But he also deals with, in particular, the idea that mastering a rule is a kind of non-interpretive cognitive state, namely, knowledge (see PI §§147–151 and RFM VI-38).

4 See RFM IV-8, as well as related remarks in Zettel §430 and §431.

5 Practices of counting and calculating are indispensable for many other practices in ordinary life, like buying and selling, bookkeeping, and resource distribution. The former are part of the framework of the latter. Likewise, we cannot but agree on some ways of giving descriptions or raising questions in order to communicate in language.

6 There are two kinds of reading of the later Wittgenstein. Well-known representatives of the individualist view include Gordon Baker and Peter Hacker, David Pears, and Colin McGinn; of the community view, Norman Malcolm, Saul Kripke, John McDowell, Eike von Savigny, and David Bloor. Malcolm (1989) writes, “There must be a use of a sign that is independent of what an individual speaker does with it, in order for the latter's use of the sign to be correct or incorrect”, and the independence condition can only be fulfilled by “a community of speakers who use the sign in a customary way” (p. 28). To the individualist expositors, the misconception that interpretation mediates rules and applications is largely due to the problematic supposition that there is some mediator in between. They think that Wittgenstein would not allow social agreement as the intermediary explanatory ground to take the place of interpretation. And, in their view, the distinction between obeying a rule and thinking that one is obeying a rule can also apply to the community.

7 Admittedly, Baker and Hacker recognise that, as Wittgenstein has claimed, “rules are human creations, made not found” and that “it must always make sense to modify and annul a rule in practice”. However, they wrote at once in a footnote that “a change in the rule will imply that following the (new) rule will no longer count as falling under the concept defined by reference to the (old) rule. Note that following such a rule would not bring us into conflict with any truths, we would just be engaged in a very different activity, a different language-game—which might have its uses” (1985, p. 63). This shows their reservation about attributing to Wittgenstein a thoroughly dynamic notion of rule-following such that the content of a rule is being maintained and updated during the process of human natural history. But while logical rules, like modus ponens, are adamant, they are still taken by Wittgenstein as a kind of grammar in our life. (See Glock, 1996, “Logic”, for a brief discussion on the development of Wittgenstein's thoughts on logic, and Stern, 1995, p. 120.)

8 I should mention that Kripke's interpretation has come under much harsh criticism and attack (e.g. from Baker and Hacker and from McGinn). Many commentators think that what Kripke presents is not the historical Wittgenstein. Some prefers “Kripkenstein” to Kripke's Wittgenstein. Yet, it is widely agreed that Kripke's book on Wittgenstein contains good insights and contributes to contemporary philosophy (see Wright, 1989). But I share the view of Paul Boghossian (1989), among others, that Kripke's exposition is very inspiring and helpful in showing “the central thread of Wittgenstein's later work on the philosophy of language and the philosophy of mathematics” (Kripke, 1982, p. vii). However, I suggest not to focus on the issue of whether to describe Wittgenstein as individualistic or communitarian. The key word in the philosophy of the later Wittgenstein is “life”. A form of life is developed and updated from time to time by participants collectively as well as individually. In fact, Wittgenstein talks about two kinds of objectivity, pertaining to “I” and “we” respectively (see Huen, 2005).

9 Actually, in most aspects of ordinary life (not only in law courts), we license rule-following. Say, when you pay a bill, implicitly you judge that the shopkeeper follows some arithmetic rule.

10 Though Bailin et al. do not claim that their standpoint is Wittgensteinian, I think it is indeed in many respects akin to Wittgenstein's (see also Huen, 2005).

11 Consider also the example about colour description in Zettel §368.

12 I discuss this important point in more details in Huen (2005), a point concerning Wittgenstein's sophisticated notion of the compelled rule-follower. Although the ideas germinated in that work are still taking shape, I hope they will help. Wright (1989) has noted this theme of Wittgenstein (referring to PI §186, among others): “It might be preferable, in describing one's most basic rule-governed responses, to think of them as informed not by an intuition (of the requirement of the rule) but a kind of decision” (p. 240). But, as Wittgenstein notes, talking about acts of decision in following a rule (say, in the use of language) may be misleading (Blue and Brown Books, p. 143). For in many practices (e.g. in speaking or writing), we just act spontaneously. However, think about this very crucial suggestion of Wittgenstein: “Following a rule is a human activity. I give the rule an extension” (RFM VI-29). (Compare Finkelstein, 2000, who criticises Wright's ideas on decision. A discussion on the place of decision, as opposed to mere intuition, in the development of Wittgenstein's thought can be found in Stern, 1995, §§4.3–4.4.)

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