International Catholic University
I've been discussing with you the history and philosophy of the liberal arts. In fact this is the final and eighth of the lectures on the subject that you will be hearing from me. In this final lecture I want to consider certain crossovers among the liberal arts. By this I mean certain respects in which the linguistic and mathematical arts can be seen to be interrelated. For this purpose I will examine three particular topics as follows: first, poetry and mathematics; second, the rhetoric of science; and third, the humanistic study of science. And then to close the series I will present a summary statement of what I hope I have accomplished in these lectures. On then to the subject of poetry and mathematics.
You will recall, I trust, that when discussing the language of mathematics we found that that science has a distinct partiality for the iconic sign. That is, for those signs whose very matter in organization presents a picture of their object. So to analyze triangles and their properties we can draw a picture of a triangle to show that the Pythagorean theorem holds true that the sum of the squares on the sides of a right angle triangle equals the square on the hypotenuse. We can draw squares on each side of the triangle. You get in a diagram an icon, an image, a likeness of the thing that is being studied. Mathematics even constructs an artificial language of its own that is more iconic than our ordinary native language when it wants to show the way items can be grouped and regrouped by the use of parenthesis and brackets particularly. It achieves a language that images the very grouping that it's talking about.
This preference and partiality of mathematics for the iconic sign provides the link that connects poetry and mathematics. For among all the uses of ordinary language it is the poetic use that leans most heavily upon the iconic sign. So much is this the case that poetry has been described as producing a verbal icon as its distinguishing achievement from other uses of language. But in such a description it is usually not the material signs that are meant but the whole imaged world that the poem or the novel creates. Such images are the work of signs but using more of symbols than of icons in the narrow sense in which I have been using the term. It is possible, for example, to conceive of a green sun, and a poet might create a world in which a green sun was the natural and normal thing. The poem might then be described as a verbal icon of such a world. The poem, like the combination of green and sun produces an image, and hence a likeness of an imagined world, but it does so primarily through the symbolic associations of the signs and not through any iconic character as material things. There is no colored picture of a green sun when you just say the words green and sun. That says the words themselves are not iconic. The word could be iconic you see the word green -- could become iconic if you wrote it in green ink or green crayon. All this is only to say that poetry depends heavily, as all language does, on the symbolical sign where the meaning is a conventional imposition not a material likeness with the thing signified.
Yet poetry differs from ordinary verbal usage in also seeking the icon in the sign as a material thing. Consider, to take a simple example, Milton's description of the fall of the devil, Mulciber:
Men called him Mulciber; and how he fell
From Heaven they fabled, thrown by angry Jove
Sheer o'er the crystal battlements: from morn
To noon he fell, from noon to dewy eve,
A summer's day, and with the setting sun
Dropt from the zenith, like a falling star,
On Lemnos, th' Aegaean isle.
Now it's frequently misleading to consider sound apart from sense, since the sense contributes to the very sound. But granting this, the sound of these lines possesses a similarity with what they describe. The combination of sounds in the word sheer reflects the magnitude of Jove's throw. The long sounds in noon, which is repeated ("From morn to noon he fell, from noon to dewy eve") -- prolonged in dewy eve as it is in summer's day -- compell by their slowness the feeling of the passing of time, thus catching in a little over a line the length of a summer's day. The sounds and dropped like a falling star themselves compose a drop. And in voicing Lemnos, the Aegean isle we are brought to a stop as Mulciber was himself in his fall.
The pattern of sounds thus traces out the path of the fall. The sounds construct an auditory diagram such as the geometrical drawing produces a visual diagram. In short the sounds constitute an iconic sign. It may function as such only with the accompaniment of a conventional association of the meaning of the words. So that without such an association the same sounds would not produce the same effect. But it is no less true of the geometrical diagram that its full use also depends upon a conventional interpretation, especially upon the generalization that is achieved. A singular particular diagram of a triangle is used as an image to stand for all triangles of that sort, and that's a symbolic or conventional association of understanding.
As is apparent, the poetic differs from the mathematical icon in being predominantly auditory rather than visual. Some poets have sought to make visual icons as well out of their poems. George Herbert, for example, in the poem entitled "The Altar", arranges his lines in the poem so as to produce the figure of an altar. And e.e. cummings in our own time has experimented much with the visual effects possible through typographical arrangement. But the fact that poets have sought different kinds of icons only emphasizes more strongly the difference.
Since mathematics cares only for the visual, not for the auditory icon, mathematics may even use signs that have no verbal equivalent whatsoever and are never pronounced. In this respect it might even be claimed that as revealed in their use of signs poetry proceeds by auditory and mathematics by visual imagination. Psychologically it would be interesting to investigate to what extent this affects the scope of their achievement. One obvious result is the need of the auditory icon to employ other devices to achieve definition or presentation of its object. The drawing of a triangle produces a triangle, which is all the geometrician wants. But the falling sounds of Milton's lines are a likeness of any fall and not specifically of the fall of Mulciber, which is what the poet wants. To connect them with Mulciber it is necessary to draw upon the symbolic signs in which they are incorporated. That is, the likeness with its object is recognized only in conjunction with the conventional association with the meanings that his words have. It's not in the sign itself as in mathematics. The connectives have to be supplied so that the falling sounds may be interpreted as Mulciber's fall. Whereas in a mathematical icon the connectives may be manifest in the sign thing itself sufficiently for the purposes of mathematics.
Since in poetry the connectives have to be supplied from outside, as it were, the possibility of ambiguity is always present. Poetry in fact often makes a virtue out of this so as to achieve an added richness. Mulciber's fall, for example, is a fall from grace as well as a fall from heaven, and one enhances the other. Thus in the verbal icon the symbolic association is always present and often explicit and exploited. Whereas the mathematical icon is stripped bare as far as possible of any such association and allowed to stand by itself. In being so closely joined with the symbolic the poetic icon is obviously more imaginative than the mathematical icon in the sense that it is capable of producing more images. But since it accomplishes this through the symbolic sign it is to this extent less purely iconic.
Even as icons the signs of poetry and mathematics serve different purposes. Milton's few lines give the feel of his fall in a way that no mathematical icon ever could. At most he could present the curve that Mulciber described in his fall. The poem has an analog of this in the sound of its lines, which is what makes it an icon in the literal and material sense of the sign. The auditory is not the only or even the most important iconic aspect of poetry however. Nor is it the only basis for comparing poetry with mathematics. Through its interest in rhythm poetry has always been associated with number and as the music of sounding numbers it possesses properties which place it in the music of the ancient Quadrivium. In rhyme poetry also establishes likenesses and ratios within the poem itself. All these however depend for their function more upon the symbolic association with their object than any literal iconic likeness with it. The image constructed by the poem results from the work of many devices among which the auditory icon is no more than one element. However to the extent that it illustrates how poetry aims to produce a sign that can stand by itself and on its own it manifests a character that is shared also with mathematics.
We have just considered poetry and mathematics. Now I want to look for awhile at the rhetoric of science -- whereby you see we move again from the Trivium to the Quadrivium. Science especially makes the claim that rhetoric is unnecessary for its work. Aristotle, for instance, declares that style is no concern of the geometer, and there is certainly some truth to the remark. The distinction between discourse in terms of things and discourse in terms of the hearer certainly indicates why it is so. The geometer, as it were, has an eye only for his object, and once he succeeds in seeing it the words will take care of themselves. He may be almost completely unaware that he is constructing with signs and need never be as much aware of it as the public speaker is, or the poet. Euclid is typical of science in making no effort to win the reader to the importance, delight and usefulness of geometry. He confines himself strictly presenting his objects with the utmost brevity and clarity.
A scientific discourse, however, is still a sign structure, and it is possible to ask valid questions about it as such without going outside the sign to its purpose. Granting the scientific purpose, it can be asked whether the discourse is well or poorly written. So doing we will find ourselves concerned with elements that lie in a rhetorical dimension, in the broad sense that I've given the term rhetoric, as a concern with something more than a minimum of grammar and logic.
As it cannot be said truly, for example, that Euclid has no style, nor do I think that Aristotle himself is saying so. His remark is to be understood it seems to me as saying that the geometer is not interested in style in itself and for its own sake. Yet a style there is in Euclid, and it may well have been a work of much labor on Euclid's part. His work produces a very definite effect, establishes his objects in a very definite way -- that, namely, of making known the properties of geometrical figures as developed according to an axiomatic system. Much is already taken for granted -- how much is known to anyone who has ever struggled to teach a beginner what a proof is.
Euclid does not tell us what he is trying to do: he does it, and leaves it to us to see what it is. He presupposes our wanting to learn his subject. He states definitions, postulates, axioms, proceeds to use them in proving certain properties about geometrical figures. However, he does not state all the principles that are absolutely needed, nor does he prove everything that his principles enable him to prove. Furthermore he says nothing at all about the order in which his propositions are presented. Yet there is a definite order in them, and as soon as we ask why this one order rather than another we become involved in a question which is properly rhetorical more so than anything else. It might even be described as a question of style since it concerns the mode of presentation.
In many cases the order it propositions in Euclid is determined by the relations among the objects of his proofs. One property cannot be proven within his system until another property has been proven. Such an order is one of logical dependence that derives from the objects under investigation and their relation to one another. However there are cases in which a logic offers the possibility of proving any one of several propositions, and a choice has to be made of what proposition to prove next after the one that has been proven.
In the first book, for example, there are 48 propositions, the last two of which are the Pythagorean Theorem about relations between the sides of a right angle triangle. No proposition is needed for the proof of this proposition which could have been given several propositions earlier by a little rearranging. He could have proven the Pythagorean Theorem by as much as ten propositions earlier so that the Pythagorean Theorem could have been first presented in the thirty-seventh place instead of the forty-seventh. In other words, judged solely with reference to the logical demands of this system, Euclid was presented with the choice of several ways of ordering the last six propositions of his first book.
What then determined the choice that he made? Obviously not a grammatical concern with the sign things he was using, consisting of the Greek language and his geometrical diagrams. Nor have we seen what was his logical demand since the logic offered several possibilities. The reason then for the order that he chose lies somewhere else -- in what can be properly described as a rhetorical concern. The reason it seems to me is that he wanted the Pythagorean theorem to come in the last place in the first book, just as he wanted the consideration of the five regular solids of the platonic solids, as they are called, reserved for the very last, the thirteenth book of his work. It's to say that he ordered his materials and objects so as to present them one way rather than another, to provide a context in which to achieve a certain effect.
This mode of presentation, the production of this effect is not wholly a measure of logic or mathematics but involves the judgment that one presentation and effect is superior to another for achieving his scientific purpose. It involves a judgment concerning the sign use, not just the sign things and sign objects. It is an exercise of an art distinct from either grammar or logic. It is an exercise in rhetoric. This much suffices, it seems to me, to show that the sign use that is the concern of rhetoric is not absent in scientific discourse. There may be less concern with it than in other types of discourse. It is present, however, not only in the general context in the choice of language that aims at presenting an object to be known, but also in the details of the presentation. All this may be more or less taken for granted.
Coming to the Elements for the first time, you've come there because you want to learn some mathematics, see what his mathematical work is about, what shape it takes. All this matter that is taken for granted is especially true, and especially large, in any well developed science. For the well developed science possesses its own special terminology, its own methods of procedure and testing its results. It even has its own special audience consisting of the signs it's trained to operate within it. The element of sign use is not lacking in their discourse: it is assumed, taken for granted. At the beginning of a science we find often more conscious concern for the determination of the sign use.
Even questions of literary style arise. A glance of the forms under which the results of modern physics have been presented shows the difference. At the beginning of modern physics we find Galileo presenting his science in the form of a dialog, a dramatic form, a highly rhetorical form. Yet it's purpose is a scientific one of investing the nature of falling bodies for example. Newton at his first great synthesis of celestial and terrestrial mechanics writes in the geometrical form of a axiomatized system. In these latter days an Einstein writes a specialized monograph which only a physicist at the frontier of his science can understand.
Einstein may preface his work, as he does, with the remark that he will not bother about rhetoric. But this in itself of course is a rhetorical device. It's the same kind of device that Socrates used in making his Apology, saying he wasn't going to make any speech about it which he then prefers to do. But if Einstein has no concern with rhetoric it is because of the context that is supplied by the state of contemporary physics. The sign use, the purpose that he is fulfilling, is determined by the tradition within which he is working. So far we have been considering the sign use in science only as it helps to determine an object to be known rather than pursued and contemplated. These are different purposes. It is not surprising that the difference should show up within the signs to achieve them.
However there are also many different ways in which an object may be known, and it might be asked whether the sign use is not differently invoked in presenting objects for the different sciences. Peirce claimed that this is primarily the work of what he called the interpretant determination of the purpose, at least in one of its aspects, and proposed calling the consideration of this work by the name of speculative rhetoric. In other words the rhetoric of science or methodotic. If so, the scope and import of the element I am calling the sign use is greatly extended. Even without this extension it still appears as a distinct element in scientific discourse which demands an art of its own, the art of the sign use, the art necessary to show within the sign the affect it is to achieve.
In science no less than elsewhere there are always several ways of saying one and the same thing, each of which is grammatically and logically correct. It may make little difference in a very short discourse which way is selected. But the longer the discourse the greater the difference, and one way may clearly be better than another for achieving a given purpose. The criterion in this case is not provided by grammar or logic, though both may be involved, nor even by the science itself, but by another art which concerns primarily the mode of presentation or the means of organizing your material, your signs, so as to achieve a certain effect. The longer the discourse the more choices there are, the more occasions for elaborating different contexts. And this, as we have seen, is the work of rhetoric. So much then for the rhetoric of science. We have considered the crossover between poetry and mathematics, and with this a crossover between rhetoric and science.
I want to consider now the sciences as humanities. The somewhat queer subtitle for this, or the title for it is queer because there is no adjective to do for the humanities what scientific does for the sciences. For this purpose humanistic is sometimes employed, but the choice is not a happy one since the term humanistic has too many connotations that are wrong and are misleading for our purposes. The humanities cannot be identified with the activity of the Renaissance humanists or with the secular religion that goes by the name of humanism today. Nor can religion and much of theology be denied to the humanities on the ground that they concern God and not man. For the concern remains one that is centrally and fundamentally human in the case of theology and philosophy.
Now, however, my purpose is to consider the way in which the sciences too can enter into the humanities. Science is a great human achievement, a work of the mind, a product of culture. Its practical effects are present everywhere about us and are so overwhelming that we sometimes tend to forget that science is first of all a great theoretical achievement of knowing. More than any other kind of knowledge it is cumulative and progressive. One discovery constitutes a truth achieved that then serves as starting point on the way to new discoveries in a process that continues seemingly without end.
Science is a work of mind and of culture, is imminently worth studying in and for itself quite apart from its practical or even its theoretical results. Its history is the story of one of the greatest adventures of ideas, a laboratory for the study of many ideas and a source in record of the development of the mind. It shows perhaps more clearly and dramatically than anywhere else the revolutionary changes that occur with the appearance of New World views such as the shifts in the conception of man and the universe that are associated with the work of Copernicus, Darwin and Freud. If we would understand what man has made of man we must take science into account. Indeed it provides many privileged cases of the effect of theory upon the way in which we approach the world and attempt to order and understand our experience of it.
The history of science is thus obviously a great interest and concern to the humanities. The history of science, it can be objected, is not science but history, and so it is. But science of science is itself a work of art, and it is in this respect that science has close similarities with the humanities. Yet the artistic aspect of science is frequently overlooked and neglected but is not entirely denied. Freedom and creativity are thought of as belonging to the poet, the painter, the musician, contrasted with the patient, meticulous, and often tedious observation and experimentation of the scientist. Scientists frequently express complete disdain for expression and composition and much of their writing too frequently lacks verbal nicety and grace. It remains a fact nonetheless that science as it is presented, as it is expounded, is a structure of words and reasons and hence a work of liberal art. As such it has many similarities with the work of the humanities. To appreciate this fact it is useful to distinguish between science as an ongoing inquiry, that is science in process, and science as an achieved result and finished product. These two sides of science are strikingly different.
Science considered in terms of achieved result has up to now been almost our sole concern in considering science. It exhibits all the differences that have been noted and emphasized as setting it apart as distinct and different from and even opposed to the humanities. But when we consider science as an ongoing inquiry and look at the scientist actually at work we find many of the same features and qualities that are associated with the humanities. Science's inquiry does not consist merely in the accumulation of facts and strict adherence to emperical observation and mathematical reasoning. A fact in the sense of a mere occurrence or event has little meaning until it is brought together with other facts and caught up and interpreted by a theory or hypothesis.
Indeed it is often the theory that initiates the search for the facts. The theory at first often only a vague hunch, or even a suggestion from a dream, poses questions for which answers are sought by ascertaining the facts through observation and experimentation. But as a discovery the theory is a work of creative imagination and has behind it all the mysterious powers and feelings that motivate and impel the activity of discovery. There is no logic of discovery, no heuristic as there is the logic of deduction. Creative inspiration, as Shelley pointed out, is not at the beck and call of conscious reason and will.
Science as a creative achievement is thus in its source no different from any of the most creative works of the humanities and the arts. The difference between these two aspects of science is an important one, yet it should not be allowed to obscure the fact that in both respects science as a structure of words and meanings, of reason and arguments, is a work of liberal art. In achieving this work science employs all the ways of knowing that we possess. Experience, ideas, imagination, memory, reasoning, insight, as well as all the arts of signs and of learning. Even the most formal sciences such as those represented in the work of Euclid, or Russell and Whitehead, use grammar and rhetoric as well as logic. Logic alone is unable to determine unequivocally or uniquely either beginning, middle or end -- that is the propositions with which to begin, the order in which the theorems should be presented, or the conclusions at which to aim. Thus even a deductive system where logical sequence exercises its greatest power still requires the help of the other liberal arts.
Science enters into the humanities when it is considered for its historical development or its achievement as a work of liberal and linguistic art. Its presence is perhaps even more evident within philosophy where it has manifested itself in a variety of ways. The most obvious is the direct influence that is exercised upon philosophical inquiry at the very beginning of its career in the work of the pre-Socratics, Plato and Aristotle, and continuing thereafter through the Middle Ages and the modern world from Descartes and Leibnitz through Locke, Berkeley, Hume, Kant, Mill, Bergson, James, Whitehead, and Dewey, down to Wittgenstein, Husserl, Sartre and on to the present. In the course of this career science has influenced philosophy by providing new approaches, new problems, new methods. But especially by unburdening and taking upon itself tasks previously borne by philosophy, aiding it thereby to become clearer about its own mission -- aiding philosophy as sciences take over burdens that philosophy once had to bear.
Philosophy itself becomes clearer about its own purposes. Within scientific inquiry proper there are problems of great philosophical import that lie beyond solution within the science itself. Among the most interesting are those problems concerning the fundamental preconceptions of science that cannot be decided by observation and rational analysis alone, and that are often assumed more or less unconsciously. Sciences are beginning to take cognizance of these principles especially as they become increasingly aware of their importance in scientific discovery. Gerald Holton, for example, at Harvard, calls these things themata or thematic presuppositions, since he views them as general themes that have guided the process of scientific discovery. As examples of such themata he lists the following. The thematic dyad of constancy and change. The efficacy of mathematical forms versus the efficacy of materialistic or mechanistic models. Simplicity, order, and symmetry. The primacy of experience versus that of symbolic formalism. Reductionism versus holism. Discontinuity versus the continuum. Hierarchical structure versus unity. The animate versus the inanimate. The use of mechanisms versus theological or anthropographic modes of approach. Themata, the presence of which are often unacknowledged by the scientist or philosophers in their work.
Now these themata of Holton's provide a rather mixed bag, but it is significant that so many of the themata appear as alternatives and rivals for position of privilege. From this it seems clear that the efficacy of understanding versus empirical methods could still count as another thema. With this we are brought back to the question of relation between the specialized experience of science and the fundamental human experience of the humanities. Nothing shows us directly and immediately the impact of science upon the humanities, the extent to which it demands their attention and concern, as the fact that it is impossible to develop a philosophy of the humanities without constantly addressing the question of their relation to one another and of their differences even more than their similarities. Although some may feel that laboring the differences become a bore, it is only by becoming clear about their differences as well as their similarities that understanding can be achieved.
In this concluding segment I want to try to summarize what we have accomplished in the seven or eight lectures that deal with the subject of the liberal arts. Now in the first lecture we saw reasons for believing that the liberal arts are primarily the arts of learning. To see why that is so we considered two kinds of learning. Learning to play the game of tennis, learning to acquire a certain form of physical activity. Secondly, learning a language, learning a form of sign and symbolic activity. And we saw in the case of both games there were three elements involved that we had to learn.
In the case of playing a game of tennis we had to learn to perform the natural motions of our arms and legs, the motions of the racquet and ball, and doing so according to the conventional rules of the game that determines what a game of tennis is. It is played upon a court, it's a certain form of procedure. In learning a language we saw that we also had to learn how to produce certain vocal sounds in order to make the words of the language. We had to acquire an understanding of the meanings that those words have and also the conventional rules of constructing in a given language since rules of organization of syntax as well as the vocabulary differ from one language to another. It could be said in a general way that in both kinds of learning we have to learn how to observe certain rules of the game. And in both cases, whether it's a game of tennis or learning a language, we have to learn how to observe the rules that have been handed down to us from before. They are not things -- playing a game of tennis or the English language -- that the individual learner invents.
In other words to learn in either case is to participate in something that has been handed down to us, is to participate in a tradition. It's a structure in both cases possessing conventional elements, elements that have been agreed upon among the members of the social group in which the game or the language is played or spoken, something we inherit. And in the case of speakers of English or other languages it's a very long tradition that we inherit in speaking, learning the language that we do. The great difference of learning a language than that of learning to play a game such as tennis is that we learn how to deal with signs and meanings.
Since there is this long tradition we look next at the tradition of the liberal arts. Now that tradition, as I'm going to consider them, is longest in the one that extends from classical antiquity through the Middle Ages right down to the sixteenth century. In that tradition two groups of arts, of liberal arts, are distinguished. The arts of the Trivium, the threefold way of the linguistic arts consisting of grammar, logic, and rhetoric; then the fourfold way the Quadrivium, the mathematical arts of geometry, arithmetic, "music" in a special sense of the word, and astronomy.
In this tradition it was generally agreed that arts, the liberal arts, should function as means not as ends. You can study them for themselves, but that's only in order to learn them so that you can then put them to use to learn, to study something else. There is a fight among the arts -- two ways in which a fight can occur. The arts as means are directed and ordered to a further end so there may be differences about the ends and then those differences will affect the arts themselves.
The difference between Aristotle and Cicero, for example, is that for Aristotle the arts were ordered ultimately to the work of philosophy. In the case of Cicero they were ultimately ordered to the work of the statesman and the orator. In doing the work of science such as Aristotle wants logic assumes a prominence and a predominance in fact, whereas in Cicero it's rhetoric that takes the lead. This model occurred in the ancient world between the followers of Aristotle and the followers of Cicero, it shows up in the later Middle Ages in the fight between grammar and logic; again that appears with the rise of scholastic philosophy in its contention with the previous predominance of patristic theology.
But of course the biggest battle between the arts over which should be predominant comes with the Renaissance and the beginning of modern science. For then the mathematical arts of the Quadrivium take over, since science, modern science as we know it, uses primarily the language of mathematics.
Having looked at the tradition of the arts and the distinction between the linguistic arts of the Trivium and the mathematical arts of the Quadrivium, we then turn to consider the linguistic arts of the Trivium. For that purpose we had to look first at the nature of the sign and the arts of the use of signs. So considering a group of words making a statement we were able to distinguish three different elements in the working of a sign. In the case of spoken language there is first the spoken sounds, material things, percussions of air. As a material thing we referred to this element as the sign thing. Then there is a relation to something other than the sign thing, something more than these puffs of air that we make when we speak. There is a reference or a meaning to something else. So the second element we call the sign object. But the sign thing not only presents a sign object it also presents it in a certain respect so as to indicate how the sign object is to be taken. Whether it's to be taken as being asserted, as something asked about, as something commanded.
By definition then a sign is a triadic relation in which a thing presents an object other than itself in some respect or capacity. All three elements, sign thing, sign object, sign use, are always present in any use of the sign, in any sign process that is. But any one of the three elements may predominate over the others although all three are present. Concern for each one as predominant gives rise to three different arts of their use, grammar, logic, and rhetoric. To locate them what we did was to hold fixed for awhile one of the three elements and then see that each of those elements call for a distinct art. The art for dealing with sign things calls for the art of grammar. The art for dealing with the sign object calls for the art of logic. The art for the sign use calls for the art of rhetoric. So much for the Trivium of the linguistic arts.
We then looked to the mathematical arts, and there again we started out at the beginning to look at the kind of signs that mathematics employs to consider mathematics as a language as it were. And we could distinguish three distinct types of signs. An indexical sign is a sign that achieves its purpose by pointing, it's an index. A diagram of a triangle put before us on paper or on the blackboard is an index of a triangle. Diagrams are any kind of pointing. An iconic sign is a sign that has a likeness within itself of what it signifies. If, for example, we wanted to speak of the color red, if we painted the word red it would indicate what it is. Any kind of illustration or icon is an iconic sign. Symbols take us into the realm of the conventional meaning. With that we have the basis for the two main divisions of the arts. That's the end of our discussion of the liberal arts, their history and philosophy.