At around remark 211 Wittgenstein begins a specific discussion on rule-following that has connections to mathematics in a number of places. In the time prior to putting together drafts of the Investigations—around 1944—Wittgenstein was engaged in a concentrated study of mathematics. Many of the passages in this section of the Investigations appear in the Remarks on the Foundations of Mathematics.1 Wittgenstein’s interest in mathematics deserves its own treatment, but there are some clear similarities to what Wittgenstein is addressing in other parts of the Investigations to what he says on math. In general Wittgenstein’s investigations into mathematics are conceptual or logical, not mathematical. That is, just as Wittgenstein is not producing any scientific theories in the Investigations, so, too, his mathematical investigations produce no mathematical theorems. However, he is anxious to rid mathematical concepts of any psychological or metaphysical justifications for their truth or usefulness. We can see a relationship here to the investigation of psychological concepts discussed above. In general Wittgenstein rejects the notion that our psychological vocabulary is meaningful because it refers to an inner process of some sort or an external logical system. Rather, meaning is found solely within the language itself. In the same way, he argued that math should appeal to nothing outside itself. Again, this is quite a large topic which we can only sketch.
To fully appreciate what Wittgenstein says here, we should draw some connections lurking in what was said above. We discussed some ideas connected with behaviorism and the mechanical nature of thought. Let us try to be a little clear about mechanism and draw out some consequences of this idea. We often think of a machine as operating according to a fixed plan or a design, and consequently we think that all the actions of a machine are predetermined by this design and are therefore predictable. Often this mechanical notion, as we saw above, is applied to the nature of ideas. Sometimes ideas are thought to be like designs or formulas, and therefore everything that follows from an idea is predetermined, as is the case with the blueprint or design of a machine. Just as, we think, all the possible actions of a machine are exhausted by an examination of the design, so too, everything we need to know about something is contained in the idea of that thing. Whatever we want to know about something, then, is merely a matter of unraveling the idea of the thing.
It may seem to be the same with a scientific law, such as the law of gravity. We might think that the law sets everything in the universe up on some sort of railroad tracks, and all the motions of each particle follow the law and then the particles fall precisely into a predetermined position.
But there is a very subtle flaw with the above description of a scientific law, which is important to see because of the close connection that is often noted between the nature of scientific law and the nature of ideas. The above description tends to impart a causal or ordering power to the law. It is important to note and bear in mind for the discussion that follows that really the law determines nothing in the sense that it doesn’t cause anything to happen. A law—such as Newton’s law of gravity—is really only a tool for making predictions about, say, the path of heavenly bodies, if we know some of the data involved. The “force” of gravity—or whatever it is—acts between massive bodies in a way that can be described by Newton’s inverse square law. We might say that the force causes the planets to behave in such a way, but not the law. Hence the law on its own determines very little about the actual motion of particles, etc.; that is, until we put it into practice by, e.g., using the law to calculate the orbit or position of a planet.
As we said, many theories of ideas in philosophy have connected the nature of ideas with the nature of scientific law. Thus, the concept humanity, for example, can be compared to the rule or rules that define the human species. If this is true for all ideas, then an idea, like a scientific law, can seem like a machine in which all the possibilities of “movement” are given in its structure. That is, just as the design of the machine restricts the actions of a machine, so too, what can be said or done with humanity is restricted by the concept. The concept tells us what human beings are and what they do. It seems, then, that the concept tells us what is possible for human beings.
We should examine the notion that a law, design, or idea “restricts the possibilities with regard to x” a little more closely, following Wittgenstein’s remark at 194. For example, we say that a machine has a particular possibility of movement. We can think of a variety of machines, but a simple one such as a lever constructed for moving a stone will suffice. We want to ask about the possibility of movement for the lever and, of course, the actual lever may move in a variety of ways—up and down, side to side—it may even break or not move at all. All these things are possible. But this is not what we are after—we want to know something like all the inherent limitations in the design because we are looking for the laws associated with the design. We are looking for the essence of the motion, and so we wish to idealize the machine and think of it in the abstract as something super-rigid. Because of this need to idealize the design, the “possibilities of movement” become ideal as well and function as a limiting factor in the design. The fact that the design specifies just these and no other possibilities of motion is what gives the design its ideal character. Once we have looked at the machine in an abstract ideal state and delineated the possibility of motion in this sense, then we think we will know everything that follows from the design and so we know everything that can happen with the machine. It now appears as if we have captured the essence of the machine. Just as there is no possibility that two plus two can equal anything other than four, in the same way the lever can only rise to a specified height, and so on. These restrictions or “possibilities of movement” look like something inherent in the design.
But Wittgenstein wants us to note that “possibility” described in this way, as the limiting factor of a design, is a superfluous notion. Notice that when possibility is discussed in this way, it is nothing like what we would ordinarily associate with what could possibly happen with our machine. We have created some sort of “superpossibility.” It cannot be seen or calculated, but it must be there. It is an ineffable “something.” It is not the movement of the machine, but a ghost or shadow of the movement.
For Wittgenstein, we have just talked ourselves into a metaphysical entity—possibility—a substance that can neither be felt, seen, or described; a substance of no use to the machine or ultimately to us, yet it must somehow be there. A notion of this type can even acquire a kind of superimportance. We might think this “possibility” directs the machine, and since it cannot be found in the blueprint or the machine, but only in our language, that philosophy should replace engineering.
But as Wittgenstein says, “the waves subside” as soon as we look at how the phrase “the possibility of movement” is used. This metaphysical idea of possibility developed out of our need to idealize the design in order to uncover its lawlike “essence.” Of course the possibility of motion is not a shadow of motion but is dictated by the placement of, for example, the gears, levers, rods, and pulleys—the laws of mechanics and physics and so forth. Apart from this, some sort of metapossibility adds nothing to our understanding of the machine. In the same way, as we will see, the “mathematical impossibility” that two plus two cannot equal three is not a metaphysical or natural barrier but simply a function of the logic of mathematics.
A rule or a concept can appear in much the same guise and suffer from a similar misunderstanding. The steps to be taken given by a formula may appear as a shadowy accompaniment of the formula like the “possibility” of movement for the machine. As an example, think of the recipe for a cake. The recipe is a list of instructions. A good deal else must happen in order for us to get from the list of instructions to the baked cake, to do with time, temperature, and chemistry, to name a few things.
Of course no one would expect to find all these “extras” in the recipe, or make these “extras” dependent on the recipe or formula in any way. After all, the cake is not in the recipe but is a result of applying the recipe. However, it often happens when philosophers discuss the nature of ideas, particularly with regard to mathematics, the idea and the consequences of the idea are often identified. That is, all results must be “hardwired” somehow to certain basic principles. In mathematics, this idea has led to a search for the absolute foundations of mathematics. For example, a math teacher tells a student to add a column of numbers and multiply the result by two. Obviously, there are numerous other things the student might do in order to arrive at the same number—he may multiply by four and then divide by two, or multiply by eight and divide by four, and so on. Clearly all these results are connected, and they are all equally true. But the question that has occupied philosophers of mathematics is the nature of this connection and truth. What gives mathematics its absolute certainty? What are the foundations of mathematics? Is all this consistency to be found in the mathematical idea itself? Are all the possible moves to be made in the above example somehow present in the original rule—add the numbers and multiply by two—in the way that we thought the “possibilities of movement” are somehow present in the design of the machine? Or is there some other source? There is, of course, another possibility—perhaps there are no “foundations of mathematics” in the traditional sense.
Wittgenstein spent much of his working life, since before the Tractatus, trying to answer the question as to the foundations of mathematics. As I said, this work occupied so much of Wittgenstein’s working life that it clearly deserves a separate treatment. However, for our purposes here and now we can say that in the end, I think, Wittgenstein came to believe that mathematics has no foundations in the way that he once thought. There is no “superstructure” that supports mathematics. In what follows we shall explore a little of what Wittgenstein says on this subject as it relates to the Investigations.
During his lifetime, Wittgenstein became embroiled in a controversy between two different approaches to the foundations of mathematics called intuitionism and logicism. Logicism is the thesis that all mathematical statements are statements of logic and logic is the foundation of mathematics. The work of Frege and Russell was dedicated to showing how mathematics is derived from logic. A large part of Wittgenstein’s Tractatus intended to refine and further this idea. Logicism was embraced by a number of mathematicians in Russell and Wittgenstein’s circle at Cambridge. Intuitionism, which we will discuss shortly, offers a challenge to the logicism of Russell and Frege, and some commentators have argued that this was a primary reason for Wittgenstein’s return to Cambridge and philosophy in 1929. Apparently Frank Ramsey, a Cambridge mathematician, had discussions with Wittgenstein on the controversy between logicism and intuitionism while the latter was a schoolteacher in Austria after WWI following the publication of the Tractatus. Wittgenstein also heard L. E. J. Brouwer, the founder of intuitionism, lecture in Vienna just prior to his return to Cambridge. A number of scholars have pointed to this event as the turning point in Wittgenstein’s later career, prompting his renewed interest in philosophy. The story goes that after the lecture Wittgenstein was quite animated and expressed the idea that he might still do good work in philosophy again.
While intuitionist mathematics and its debate with other schools of thought would take us well beyond the present volume, some information on the topic is important. I think Wittgenstein was very much concerned with the ideas of intuitionism, but I do not think he subscribed to any of them. In fact, from what I understand on the topic, I would argue that Wittgenstein is sharply critical of intuitionism.
The intuitionist program is in part designed to rid mathematics of any metaphysical speculations. On this point I think Wittgenstein and the intuitionists would agree. To pursue this agenda, intuitionists claim that mathematics is based on an immediate “intuition” of the natural numbers. According to intuitionism, the natural numbers 1, 2, 3, . . . can be empirically—albeit mentally—grasped and verified as true, and they are known to everyone. Anything else said in mathematics is a mental construct—a finite set of operations following this fundamental intuition—that can be, again, empirically verified by introspection. In order for a statement to be mathematically “meaningful” there must be a means of calculating whatever statement is made. Thus Intuitionists deny the law of the excluded middle. In order to see this, let’s look at an example of what appears to be a valid mathematical statement: x is the greatest prime number such that x - 2 is also prime.2 A lot of numbers satisfy these requirements: 3, 5, 7, 13, etc., but we are looking for the greatest number that satisfies the statement. Now if we think of the set of natural numbers as really existing in the Platonic sense, we might think, according to the law of the excluded middle, that either such a thing must exist or it must not. Either such a thing will be found if the set of primes is finite—or it will not be found if it is not. One case or the other must be true. However, since no method is given of determining which case is true—we only have the supposition that either one or the other must be the case based on the law of the excluded middle—then intuitionists reject the above statement as a valid mathematical proposition.
Part of this Wittgenstein would agree with: a mathematical statement requires a mathematical proof in order for it to be accepted as true. We cannot assume the statement must be true based on the law of the excluded middle or any other logical law. Demonstration is necessary, and we will have more to say on this in the final chapter. However, one idea here that I think would be particularly bothersome to Wittgenstein is that a mathematical idea is a mental construct that is “intuited” and verified by introspection, that the rules for the application of the formula are known because the formula has “appeared” in the mind. In other words, according to the intuitionists, the intuition of the number line 1, 2, 3 . . . is sufficient to understand what these symbols mean and what follows from them. On this model, it appears as if you can simply see introspectively how the rule or formula is meant, you just grasp it. True, as we saw before when someone understands a rule it seems to make sense to say that he has grasped or intuited something. The difficulty is how we understand the concept of grasping. One tendency Wittgenstein is addressing here is that we often want to describe what is being grasped about a rule in terms similar to the shadow of “possibility” discussed above—as an accompanying mental phenomena. In other words, we “intuit” or mentally inspect a formula and because all that follows from a formula, that is, what it means and how it is to be used, is not present in what we see introspectively, we then want to say that what follows from the formula is something that accompanies or surrounds the formula. Of course, since the object under discussion is “mental” we want to locate these accompaniments in the “mental” as well—traditionally, in a psychological act of intending or meaning.
But this is where we have taken a wrong turn. What I think Wittgenstein wants to show is that what is grasped is not some psychological accompaniments of a rule but the practice of using a rule—or at any rate that this is what we mean when we say “he has understood” or that “he now knows the rule.” We say a person has learned to calculate x formula not by finding the intuited something, but when he has calculated as we all do. He is justified in saying he knows how to go on in certain circumstances—not because of any mental process. Thus a “mental construct” is not required to follow a rule—this is an unnecessary intermediate step. Knowing how to use or apply the rule is what is important.
When we make the mistake of thinking of a rule as a mental construct that we grasp by introspection, then we need to explain how this would occur, and at this point we are often led down several philosophically blind alleys involving words such as meaning and intending. The mistake is failing to distinguish between the logically determined and causally determined where meaning is concerned, and this tendency gives rise to epistemologies of various types, such as psychologism.3 In what follows, I will try to explore this idea in some depth.
Say I want someone to develop a series, and so I give him the formula for the series, and I explain what I mean or intend for him to do. He then goes and does what he has been told to do. If we don’t try to explain what happened here with any more depth, then this scenario presents little problem. However, if we ask what psychological or mental occurrence is responsible for his grasping the rule, then we might think that what happened here is that the student had a mental construct come before his mind and that this construct is responsible for his behavior—that the construct guides him in how he is to follow the rule. We may think of this as the explanation of meaning or intending: my conveying the meaning of the formula or intending him to act in such a way means that this mental construct came before his mind as the result of my instruction. But it would be natural here to ask how did this occur and how is the mental construct involved in determining what the student is supposed to do? Notice we have entered the realm of the psychological or epistemological, and now we need a theory to explain what happened. We have interpreted understanding the rule or meaning and intending as grasping a mental construct, and thereby we have painted ourselves into the corner of a room where only psychology or epistemology will count as understanding what has happened. Now we must explain how the mental construct determines the actions of the student. Again we have limited our inquiry to the “mental” or “inner,” and so we seek to explain this by appealing to some sort of inner guidance or intuition. Since the mental construct is somehow the determining factor in the student’s ability to follow the rule, we must discover (or invent) the mental mechanisms that accomplish this task.
We think that the steps to be taken have to be predetermined by the formula or construct. Again this means more than the answer is found in an analysis of the equation. We have made intuiting the formula responsible for the student’s behavior. So it seems we are mired in a mechanical or causal explanation of meaning, intending, or understanding. The idea of the series that is supposed to be developed appears, says Wittgenstein, as if the expansion of the series were set on rails stretching out to infinity, negating all choice.4 It is as if following a rule was like being given the order to “move out” by a drill sergeant of whom I am terrified. I do it blindly, without thinking, without choice or reasons.5
But is this really a good understanding of the way we operate with rules? If I give someone the formula y = x + 3 and tell him x = 2, must he involuntarily write 5 if he had understood the formula? Couldn’t he write 3 + 2, 10 - 5, or 2 × 2 + 1 or any number of other constructions and still be correct? Or perhaps nothing occurs to him. Isn’t it possible to understand a rule—let us say a very complicated one—and still get the answer wrong? In other words, even if it were true that all the steps were determined by the rule and we had this picture of the answers as set in stone, does this help clarify “understanding the rule”? One must still get the answers or write them down—one must do what the rule specifies—that is, apply it successfully. Correct application is what determines the meaning of “grasping the rule.” Without at least something like the above set of circumstances, the fact that the rule occurs to an individual means little. Again, someone might know the rule and still not know how to apply it. I may tell someone who is learning to drive to apply the brake when he sees a stop sign. He may successfully repeat the rule to me, but it is not much good if he doesn’t know what the brake is.
If it is the case that applying the rule is crucial to grasping the rule, then the causal and mechanical explanation of grasping a rule becomes superfluous. This psychologically oriented explanation held that understanding a rule or a formula was grasping a mental construct that was somehow connected to the steps that needed to be taken to accomplish the task. This construct determined how someone was to proceed in the presence of the rule. But here, as we have mentioned, we have substituted an ineffective causal explanation where a logical one is required. The mere statement of a rule does not contain its application, and it is the application that is of paramount importance. That the results of the equation or formula are predetermined, even if this were the case, does not guarantee that they will be applied by our learner—that he will do the correct thing—unless we include the notion that the rule forces its application on us. But it doesn’t follow from a series being written in stone that it is forced on someone. Even though the English alphabet is something that is “predetermined,” it doesn’t follow from this that a learner will automatically remember it correctly and be able to recite it properly. Unless he is able to do this we cannot say that he knows the alphabet. If we agree that the rule does not contain its own application then we are inclined to posit some psychological apparatus to explain why we follow a rule in a particular way. But clearly following or understanding a rule means adopting a particular practice—doing something as we all do it. And the concept of my “meaning” the rule or “intending” that the student should follow it is clearly explained as getting the student to adopt a particular practice. Psychology or epistemology need not enter into the picture in order to explain meaning, intending, or grasping a rule. And here we are just brought around in a circle to the mental construct. Its connection to following a rule is an “unnecessary shuffle.”
Again the important consideration is the logical one—understanding how the phrase following a rule is used in our everyday lives. What counts is doing what we all do under these circumstances. After all, a rule is simply a rule—“drive on the right” or “drive on the left.” Following a rule is seen in what we do under these circumstances. The idea of a rule being dependent on a mental construct adds nothing at all here and screens the real problem from view.
We started this discussion with mathematical concerns—specifically intuitionism. We noted that a key idea in intuitionism is that a formula or mathematical theorem is similar to the mental construct as we described it above—in the sense that we determine the correctness of the formula through a type of introspection. I think what Wittgenstein is trying to show here is that this is an unnecessary step. A formula need not be thought of as a mental construct in order to operate as a formula or a rule. However, we should also note that there is something correct in what the intuitionist says: a formula need not be thought of as metaphysically expanded to be of any use. It may be that Wittgenstein is trying to show the inconsistencies of both classical mathematics and intuitionism. As was mentioned, philosophers have been searching for foundational concepts in mathematics, the ultimate ideas from which everything else may be deduced. These ideas have often seemed to many thinkers to be something like ultimate metaphysical rules, and the law of contradiction and the law of the excluded middle have always been candidates for such “super-rules.” But if we deny these ideas this metaphysical status, mathematics remains unaffected. It loses nothing in generality or consistency. We would simply carry on calculating as before. Although we will return to the idea of contradiction when we examine Moore’s paradox later, unfortunately a more complete treatment of this deep and fascinating topic is beyond the scope of what we are doing here.
To finish this section and transition to the next we should look at Wittgenstein’s remark 237. Here Wittgenstein asks us to imagine the following situation: there is a line on a page and someone has a compass (the kind you use in geometry—not the kind that points north) with which he traces the line with the point, following it as if it were a rule. While the person traces the line with the point, he very deliberately studies the line and opens and closes the compass, making a figure with the pencil end. Since there is nothing we see that guides him in this process, and since we see no regularity in what he does, we can’t learn his method of drawing the figure. Here we might say that the line intimates to him the way to go or that he is following an inner voice. In either case, clearly the line can’t function in the way a rule ordinarily does.
But can we say that it is a rule but it is a rule only for him? Does it make sense to say that there can be private rules? Here we have raised an issue that Wittgenstein spends a good deal of time and an issue that has received a good deal of intention: the so-called private language argument (discussion, analysis, whatever). Some commentators think this question is interesting in its own right—i.e., can there be a private language? But I think the discussion follows the above thread and has an impact on questions that vexed Wittgenstein since his teenage years when he was an avid reader of Schopenhauer: skepticism and solipsism. We should recall that these ideas figure very prominently in the Tractatus.
1. Ludwig Wittgenstein, Remarks on the Foundations of Mathematics, eds. G. H. von Wright, R. Rhees, G. E. M. Anscombe, trans. G. E. M Anscombe (Cambridge, MA: The MIT Press, 1978), 413-422.
2. Cf. A. Heyting, Intuitionism: An Introduction (Amsterdam: North-Holland Publishing Co., 1971), 2-12.
3. Cf. PI 204ff.
4. PI 218-219.
5. PI 212.