International Catholic University
In this lecture I will be discussing with you the liberal arts, their history and philosophy. We have considered the linguistic arts of the Trivium, grammar, logic and rhetoric. This time we are beginning to discuss the mathematical arts of the Quadrivium. The signs used in mathematics constitute a language which is in many respects extraordinary -- so extraordinary there are times when mathematics itself appears to be no more than a language. These times are transient, since the knowledge achieved in mathematics can be distinguished from the means used to achieve and express it.
What the object of this knowledge is constitutes a difficult question. Fortunately it need not detain us since it is possible and useful to consider the signs used in mathematics without raising the question of the nature and status of its objects. Mathematical signs are especially important to any consideration of the liberal arts as arts of signs. They compel us in the first place to distinguish among various kinds of signs. So far in discussing the general nature of signs and the arts of their use it has been possible to assume the distinction and leave it more or less implicit. But in comparing the mathematical art with ordinary language some distinctions between the kinds of signs become imperative, and this can further the understanding of signs themselves and the arts of their use. Such a consideration also provides us with an occasion for raising the question of the historical inclusion of mathematics among the liberal arts.
As we have seen in considering their history, the arts of the Quadrivium are the mathematical arts as distinguished from the linguistic arts of the Trivium. I've been arguing that there must be at least three liberal arts when understood as arts of signs and that these three correspond to the linguistic arts of the Trivium. The question can now be raised whether there are more than these three and particularly whether mathematics makes it necessary to extend the list as a tradition holds. On a theory of the liberal arts as the arts of signs, I see no need for any extension outside the arts of signs themselves. I hope to show that the arts of signs employed in mathematics are fundamentally the same as linguistic arts that we have already met. But what differences there are arise from the differences in the kinds of signs that are used and in the objects on which they are used.
Mathematics is a language in a sense that it is a structure of signs and symbols. It is also much more than that, but it is at least this much. Indeed, one of the difficulties in learning mathematics lies in mastering the language. To a beginner a mathematical paper appears as an array of strange and meaningless signs that provide a shock in which there is no shock of recognition for a long while. Many after discouraging and futile attempts to come to grips with it have abandoned the effort with the excuse that they do not possess a mathematical mind. Yet the perfection, even the beauty of its language constitutes one of the highest claims advanced for mathematics.
Such extremes of disgust and delight raise suspicion that there is something peculiar and special about this language. Something of this peculiarity becomes evident for even a cursory inspection of a simple example. In classical plane geometry the theory named after Pythagoras as its discoverer and placed by Euclid at the end of his first book of Elements states that in right angle triangles the square on the side sustaining the right angle is equal is to the squares on the sides containing the right angle. In the proof of it we employ a diagram of a right-angle triangle with squares on its sides. The parts of it may be designated with letters. If we call the side opposite right angle C and the other two sides A and B we can with some further supposition express the theorem in the form that
Considering only this much and laying aside for the moment the assumption that makes possible this former expression, we at once note certain differences from the practice of our ordinary speech. We have diagrams, letters, instead of names, numbers written in Arabic notation in the expression C2, the squares being the numbers. A sign for expressing the operation of addition and another for expressing equality.
None of these signs or even words for them appear with much frequency in ordinary speech, yet each of these is a sign since it makes known something other than itself. And furthermore they not only differ from ordinary speech but they differ from one another. They are different kinds of signs, the diagram being one kind of sign, the letters and the addition of equal signs being other kinds of signs. Now of these various signs the least complex is the letter designating the side of the triangle, A, B or C. As a sign it functions very differently from any of the others. The diagram, numerals, the signs required in addition present an object, they have a meaning. The letters only refer or denote and as such do not mean or connote anything. We fasten them to the sides of the triangle and use them as proper names to indicate or refer to its sides, A, B and C. A direct connection is established as it were between the sign and the object which functions in much the same way that pointing does. I'll point to the triangle. The letters accomplish the pointing too.
For this reason it's good to call this kind of sign a pointing sign or an index, indexical sign. What is the advantage of an indexical sign? Consider economy in expression. C2 = A2 + B2. Very few -- eight -- signs, whereas if you put it in words you've got a lot more than that. Perhaps even more important, they eliminate ambiguity in reference. A, B, and C, each of them designates one side and only one side of that triangle. Designating the two legs of the triangle by different letters, we can always refer to one of them without any doubt about which one is being indicated. The letters serve as proper names or relative pronouns, establish an unambiguous connection with their objects. Because of this virtue letters are frequently used to clarify ambiguities that arise in verbal discourse. C. S. Peirce, the greatest of American philosophers, though he died in 1914, cites the following example from a grammarian as a case of a verbal ambiguity difficulty to avoid with words. Here it is in words:
One man said to another that he thought his brother more unjust to himself than to his own friend.
Now where is the ambiguity? It arises because the pronoun his may refer to any one of three persons. But such ambiguity is readily avoided by assigning letters to the three persons and using the letters instead of the pronouns. A said to B that A thought A's brother C more unjust to to C than to C's own friend. Now you see the advantage of the use of letters.
Of course the use of letters is not confined to mathematics, it's also used by lawyers. It may be used in discourse whenever it is necessary to indicate unmistakably which one of several things is being referred to. Of course this is not the only use of letters in mathematics, but this is one of the clearest examples of the indexical sign. Other uses involve different kinds of signs even when an indexical element is present. Sometimes a letter may be no more than an abbreviation for a longer expression. So in the diagram of the triangle we might use H as the abbreviation for sides subtending the right angle which in Greek is hypotenuse.
Most important of all, letters are used for variables. Thus to express the algebraic identity that the product of the difference in the sum of any two integers is equal to the differences of their squares, to write it in letters is very, very easy, it makes it very, very clear. We've been considering the indexical sign, the signs that indicate or point to the objects they are about. Take the algebraic expression that
The x's and the y's are indexical signs, but you have to know something more. They stand for integers, for whole numbers. In this case there is more than a simple indication of an element for the letter to function as the sign of a variable this additional knowledge must be available. It is produced by the sign through the sign because x and y are used to stand for integers, for whole numbers.
Now this is a conventional association. It's agreed among us mathematicians that these letters, x and y, as used in this expression are to be filled in by whole numbers. That reveals the presence of another kind of sign. The A, B and C of the triangle indicate the sides of the triangle. x and y indicate numbers, but we must know that it's limited to whole numbers. Making the diagram of the triangle and labeling the sides A, B and C indicate, identify at once, the letters with the triangle. That x and y stand for numbers we have to know in addition something else: it's a conventional sign in that case. We agree now in this case to use the x and y's.
And the diagrams, of course, are signs, but the first and most obvious characteristic of the diagram as a sign lies in the fact that it contains a likeness of the object it signifies, A, B, C, x, y -- nothing in the signs themselves is like what they are signs of. The triangle drawn on the board is an image of the triangle itself. A street made with pencil or chalk is as a length, it's a likeness of geometrical line. The triangle drawn on the board is a likeness of the geometrical triangle. It's only a likeness of course. A geometrical triangle is a two dimensional figure but there is a layer of chalk on the board there making it a three dimensional figure.
Some words obviously attempt to present an auditory likeness of their object. The buzzing of bees in English is an example. Bees are supposed to sound like the sound buzz. And yet this verbal sign succeeds in functioning only because an association has also been established between the sound that we make and the sound of the insect. This association is the work of human convention or agreement. Bees do not buzz in all languages. In German they don't buzz: the German word is summen, in French bourdonner. It's only in English that they buzz.
In possessing some likeness with their object this kind of sign shares with a mathematical diagram a certain character as signs. Peirce's name of them as icon is a good one: an icon is a picture, a likeness. And it's a likeness now with its object. This fact that there is a likeness with its object distinguishes the iconic sign from the index which just points. It establishes as it were a dynamic connection with the object that it's about, also from these signs in which the connection between sign thing and object is conventional. English, German and French words used for the sound of bees differ in belonging to different linguistic conventions, belonging to different languages. The association of certain sounds as sign things with certain objects is a work of human imposition. It's the English speaking group that called that buzzing insect a bee.
Most words are this sort. We need a name for this kind of sign now, a third kind of sign where the meaning is contained in the imposition by human agreement that in English the insect is a bee and that a bee buzzes. That's by human agreement. And so this third kind of sign is well called a symbol. Unfortunately the word symbol has come to have many different uses and so its value is somewhat questionable. However if you give it a distinct and separate meaning from an indexical sign that points, a iconic sign that images, and think of a symbolic sign or a symbol as a conceptualizing sign, you will have no trouble.
The special utility of the iconic sign lies in the way in which it can be taken for the object itself and subjected to observation and experimentation. You work on the sign things now. With words no amount of juggling will yield any information that you don't already possess of the things that they signify. Analysis of the word man offering its parts, its letters, shows nothing about the rational animal that it names. But from the icon there is much to be learned about the object itself, as is evident from the use of the diagram in mathematics. The Pythagorean theorem provides a simple illustration. To prove it we start with a right angle triangle with the sides A, B, and C. And C is the hypotenuse. To prove that they are equivalent, we draw the squares on the sides and then start experimenting. It's not necessary that the figures drawn on the three sides of the triangle always be squared as long as they are any similar kind of figure it's going to make the point.
Suppose we take a much simpler case now, another right angle triangle, and draw the diagonal so that we have two right angle triangles, and now draw equivalent triangles on all three sides. Here we have a right angle triangle and we draw the whole triangle over here on this side this going down the diagonal over here on side b and the smaller one down here below. Now how can we experiment with this? Well we can, the whole triangle over here equal to the whole triangle, it's the same so we have pulled it out of the way and now what can we do with these other triangles drawn on the other sides? Suppose that we have the one triangle drawn on that side and a little triangle at the bottom. Now, how can we show that those two triangles are equal to the whole triangle? Well, we can invert so as to have that triangle there and turn around and have that triangle there and we've recovered our original triangle.
However awkward this illustration is, the point should be clear. We are operating upon the sign thing itself. This is an iconic sign of that triangle on which we draw the triangles on the sides, and then we can experiment as we have done in this very crude and simple fashion. From this simplified version we can see that a hypoteneuse is equal to the sum of those on the two sides if we did it with our original one. This doesn't constitute a proof; it does provide visual or intuitive evidence for the truth of the theorem. For rational proof of it we would need to prove the similarity of the triangles which will in turn involve proving many other theorems.
This is a good example of a mathematical reasoning in illustrating generalization, specialization, and analogy. We have generalized in passing from the squares to strive on the original triangle to any kind of similar figures on the sides as we do triangles on the triangle in the second case. We then also specialized in reducing the problem to the triangles alone moving from squares to triangles. And then we have an analogy, since we can go from that and say that any similar figures drawn on the sides of the triangle will be equal. In all of this of course we are reasoning, but we are reasoning with the help of diagrams, or iconic signs. Furthermore the icons have been different kinds. Geometrical diagrams are iconic. So too are the algebraic expressions used to designate the areas of the figures in the expressions of the Pythagorean theorem in its generalized form.
An icon: the drawing of a triangle on the board, the squares drawn on the sides. Icons because they're likenesses of what we are talking about, triangles and squares and their relations. But the algebraic expression C2 = A2 + B2, that's also an icon, but we usually don't think of it as icon because the symbolic or conventional character is so strong in it. The notion of squaring, for example, is a conventional sign. We have to know what we mean by squaring when it's applied not to drawing a square on the side of a triangle. But you're to express a relation that holds in this case between the squares that are drawn.
But what's iconic about that algebraic expression apart from the diagram? It manifests the relation in the way that the diagram does not. The diagram drawn as such does not exhibit the equality of the squares on the sides being equal to the square on the hypotenuse. In the algebraic expression of it, you have an equal side that divides the whole expression into two parts, saying that when the operation of addition is performed on A and B then the result is equal to that operation of squaring upon C The equal sign itself is an icon, two equal lines. The algebraic expression then manifests more in this respect, more about the relation between the parts in the triangle, than the diagram of the triangle does. Now the algebraic expression is symbolic and indexical as well as iconic. But this is only to be expected, since it is rare to find only one kind of sign at work. The combination helps to increase the power of signs. However the greatest power of signs especially in the generalizing power lies in the symbolic sign.
Now let's see how that's the case. In locating and isolating the iconic and the indexical sign, we have found it necessary to consider only that diagram which accompanies the statement of the Pythagorean theorem. Even so we have not been able to avoid entirely reference to the symbolic or conventional sign. This kind of sign, the symbolic, becomes a predominant concern in making a general statement. The Pythagorean theorem makes such a statement. The square on the hypotenuse is equal to the sum of the squares of the other two sides of a right angle triangle. A statement in English. However that statement holds not only for this diagram on the board, it holds for all right angle triangles. In other words, it expresses a generality, a universal, and that is expressed through the sign, is the work of the symbol and has neither index nor icon. The diagram is a picture of one particular triangle; the algebraic expression in which the letters appears indices of the triangles sides is a description of that triangle. But we take both of them as standing for or representing any triangle of this sort.
Furthermore with only a little algebra we might not restrict the equation to geometry but allow the A, B, and C to be indices for numbers, and then it expresses the property that when you square a number the square of one number can equal the square of two other numbers. This expresses a different thing from the Pythagorean theorem, a theorem in arithmetic. In this sense the algebraic expression possesses greater generality than the geometrical diagram because the algebraic statement expresses a truth both of geometry and of arithmetic. Yet in both cases the sign stands for any element of a certain sort, for any triangle, for any whole numbers.
That fact is the work neither of icon nor index but of the symbol, the symbol being a result of what its users have agreed to mean by it. It has been established so to be, it's not given by either the index or the icon. At most the indexical sign functions as a proper name, the iconic sign is a particular picture having some likeness with its object, each as a particular. That the particular can present the general constitutes a further achievement which is a work of a distinct kind of sign that is a symbol or sign with conventional imposition. An icon in other words is an illustration, it's not just a likeness. Any illustration that we provide in the course of trying to manifest something is presenting a likeness that enables us to establish a relation to what it is that we are talking about. Of course any kind of a sign is a class capable of having many, many members.
The diagram of the Pythagorean theorem is a picture of one triangle, the letter A is the sign of a particular vowel sound. Yet each of these can be copied and multiplied many times, there are as many copies of the diagram in figure 1 as there are presentations of this lecture. The vowel sound A can be represented as well in as many ways as there are of saying the letter A or forming the written letter A. In both cases the many instances are only so many different tokens of one sign we might say that this A of singular or particular is a token of a type. What's the type? The letter A, so that every copy of it are only so many different tokens of that type, so that we are in the use of letters.
In the use of our words we are dealing with generalities, with universals all the time -- even when we tie ourselves down to a single diagram. The greatest generality of the symbolic sign appears most clearly when we move it from geometry, from arithmetic, and we move it from any particular interpretation and consider only certain elements and operations upon those elements. C2 = A2 + B2. There are three elements: A, B and C. There is an operation performed upon them indicated here by the twos. There is a relation established; there is another operation performed upon two of them that establishes a relation to the third one. Now, we applied the C, A and B, or interpreted them as applying to the triangles or the whole numbers. Let's free them: they can stand for anything you want.
What we are going to talk about and what we are concerned with is the operations and the relations that are established. So we are going to analyze those relations and see what operations we can perform upon them and see what results we can achieve from performing those operations. Now the operations themselves will have to be defined, but the elements need not be defined until we want an interpretation of them. So in that way we construct a formal system. No material elements are identified until you come to interpret it. But what you do establish is a formal structure with many interrelations that can be formulated and proven, and thus you get a general symbolic structure capable of many different interpretations which you can draw, many different models.
There is the very greatest generality that can be achieved and it is the achievement possible, for us anyway in this earth now, only in mathematics. This kind of formal system in logic and mathematics explains why it can be said that mathematics never knows what it's talking about -- meaning that A, B and C can be anything you want. What mathematics is concerned with is with the relations among such elements, and it's the relations in their patterns they perform that are the important things. The expression, such as this algebraic expression, is a complex sign in which indexical, iconic and symbolic elements may be distinguished. The letters are indices of the elements on which the operations are performed; the operational and relational are conventional or symbolic, meaning they are defined; and the whole expression is an icon of the relation that results when we perform these two operations upon our elements so as grouped here in the third. However the equation as a whole, including the letters in it, constitutes a general statement of what may be done with any elements of the sort to be susceptible to the operations indicated whatever they may be, and thus in this it is a symbolic sign, or better a symbolic structure.
We distinguish the indexical sign, the iconic and symbolical sign. In many cases in mathematics in the use of their signs it is the iconic that is dominant. Mathematicians sometimes claim that their language is superior to other languages, and that it saves thought by doing their thinking for them. Whitehead, for example, the mathematician and philosopher, contrasts what he calls substitutive signs with words, and adopts as his own this statement of a previous author, "A word is an instrument for thinking of the meaning which it expresses." A substituted sign is a means of not thinking of the meaning which it symbolizes. Whitehead explains a substituted sign as being such that in thought it takes the place of that for which it is substituted with the result that we think about the sign itself and not its object.
In the use of ordinary verbal language we usually think about the object signified, not about the words themselves. Thus there is always the danger of connecting the words with the wrong object. This danger would no longer exist if it were possible to consider only the sign and not bother about its object, only the sign thing. The substituted sign would thus be able to save that and serve it not only by economizing it but also by eliminating the great danger of ambiguity and equivocation. This is acknowledged by mathematicians when they praise their language as intuitive. By this they mean that their language or sign structure manifests of itself the property that they are talking about. It possesses a likeness with its subject.
There is one mathematician, for example, who claims that the most basic departure of mathematics from ordinary language consists in the use of parentheses to indicate groupings and of variables to make cross references. Parentheses, of course, are iconic signs. And when we state a law such as our Pythagorean Law that C2 = (A2 + B2), we could, and do in fact, put parentheses around the addition of B squared and C squared to indicate that they are to be added together, and then the equal sign indicates the relation to C. Parentheses, in other words, provide an actual likeness of groupings. They group and hold together their elements separate from each other. The variables, in this case C2 = (A2 + B2), are indices indicating the elements being associated, and the whole expression is a substitutive sign, as Whitehead says, or an icon as we've been calling it, since the law is fully manifested in the expression. A and B are first taken together or associated on one side and then equated as equal to the C on the other.
Mathematics has developed and perfected the use of the iconic sign above all others. Of course it has to use other signs as well. Without the index, indexical sign, it could not particularize. Without the symbol it could not generalize. Of the three the symbol is undoubtedly the most important, even in mathematics. But with these qualifications it can still be said that mathematics differs from other languages in the use and exploitation of the iconic sign.
The importance of the iconic sign in mathematics helps us to explain why mathematical knowledge is sometimes identified with imagination and the constructable. Its signs are constructs that may be readily imagined, as we can imagine a triangle and picture it by drawing a diagram of it. A great French mathematician claimed that in thinking about numbers he always imagined patterns of points. Peirce, the American philosopher thought such a procedure so characteristic that he named mathematics diagrammatic thinking, since for many purposes nothing more than the signs as things needs to be thought of; it may even be claimed that there is nothing more.
Mathematics is then identified with the character of its signs and becomes a game of signs. However this constitutes an interpretation of the object of mathematics, and it is by no means the only one compatible with recognition of the iconic character of its language. It is this interpretation that is the most radical, but it makes the likeness between sign and object one of identity with the result that distinction between the two becomes trivial. However there is no need to go to such lengths to recognize the importance of the iconic in mathematical thinking. The philosophical problem involved is whether the objects, the sign objects, of mathematics are merely the sign things that they work with, in which case mathematics is interpreted as nothing but a game of signs and is not about anything but the signs and tells us nothing about reality. That's the positive or strongly limited character of mathematical objects. On the platonic side, the realm of mathematics is an ideal role of reality, and more real than this moving world, this contingent world in which we live. However those are matters of the philosophy of mathematics, the interpretation of the objects that it works upon and with.
The dominance of the iconic sign, however, also illuminates the claim that mathematics is somehow more natural than our ordinary conventional language. This claim, as we have seen, underlines the traditional division between Trivium and Quadrivium. Thus we've seen that the linguistic arts of the Trivium, grammar, logic and rhetoric, can be thought of as merely verbal arts, whereas the arts of the Quadrivium are dealing with the nature of things, and thus they are more natural. They are not merely verbal. They are reaching beyond words and signs to the realities beyond them.
The determined quality of the iconic sign, however, which differentiates it from all others, consists in the presence in the sign thing of some character that is the same as its object. The drawing of a triangle is itself a three sided figure, whereas the word triangle possesses nothing in shape like its object, though in sound it has three syllables, tri-an-gle. But we don't think of that when we say triangle, do we? But as you picture the diagram of a triangle you see it is three sided. Parentheses or brackets of themselves group together certain marks on a page, whereas the sounds, although grouped together, do not of themselves perform the function of grouping that we describe with the word group.
The words accomplished through sign function through a conventional association which they have received. Mathematical icons do so through the likeness that they possess with their objects. Mathematical icons however are not natural signs such as smoke is a natural sign of fire, or the footprint of the animal that made it. In these cases there is a natural connection between the sign and its object: it's fire that makes the smoke; it's an animal that left its footprint in the mud. But in the diagram of the triangle or in the groupings of the algebraic equation the object does not make the sign. Rather one character is selected from the object and a likeness of it has produced another matter. In chalk marks on the board for example, those marks then being used as a sign of the object under consideration. Furthermore the interpretation of the sign differs in the two cases. In the case of a natural sign we cannot read, we interpret the footprint precisely, unless we know the animal that makes such a mark. So what is needed is knowledge of the world. A fossilized footprint in a mountainside in a stone is a sign of the animal that made it, but it's only a virtual sign until someone comes along who actually knows what animal it is a footprint of. We can't read it or interpret it until we know the animal that made the mark, and that demands knowledge of the world.
Interpretation of mathematical signs depends on the way that they are used, and this use is conventional. An algebraic expression is an icon of a general structure, the elements of which may be interpreted as either real numbers or lines. And so you have a theorem in arithmetic or geometry which results depend on the interpretation of the elements. If in possessing a likeness with their object icons are more natural than any other artificial sign, this likeness provides a common bond, for in finding the number of an aggregate of things we do find a property that is more natural than the artificial name we use for it. We discover that our flower has five petals: that's a natural fact about the flower. The ordinal relation among numbers corresponds in some way to a property of the things. Counting can be defined as establishing a one-to-one correspondence between the elements of the aggregate and the natural integers. The names of things in ordinary language tell us more about the world, but they do so because they are associations with our experience. In mathematics, the iconic character of its language enables the sign things themselves to speak and to manifest their object. It demands relatively little experience of the world since the language can provide much of the relative experience by itself, and in this mathematics resembles poetry.