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Abstraction in art Grants, Arvid John 1963

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ABSTRACTION IN ART by ARV3D JOHN GRANTS B.A., The University of Bri t i s h Columbia, 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN FINE ARTS (AESTHETICS) in the Department of FINE ARTS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Ap r i l , 1963 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s - I t i s understood t h a t copying, or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be. a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of 9 ^ The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. ABSTRACTION IN ART. The Abstract. (1) I. Statement of the problem. The subject of abstraction i s of very great importance not only i n science, in logic and mathematics, or i n the theory of know-ledge, but also i n art. It might be said that abstraction i s a process i n which con-sideration i s given to some aspect or feature of a complex whole to the neglect of the remainder; but this statement i s both too vague and too restrictive to cover a l l the cases i n which abstraction i s commonly said to occur. In order to avoid here the mistake involved i n the "paradox of analysis", exemplified by saying that "centaur" and "medusa" mean the same thing (since there never was either the one nor the other), i t i s necessary to recognize that the abstractness of a work of art i s some-thing quite different from that of science, mathematics, logic or epis-temology. This difference does not l i e i n the meaning of "abstraction", but i n the purpose for which abstraction i s used. In both art and science abstraction i s the recognition of a (l) A l l acknowledgements of indebtedness to sources are to be found in the text. i i . relational structure or "form" apart from the specific thing in which i t is exemplified. But the word "form" has different meanings in various fields. A logician or mathematician may question what sense i t makes to call anything "form" except the logical form of discourse, the structure of propositions expressed either in ordinary language or in the refined symbolism of the pure sciences; an artist, in his turn, may ask how one can speak of the "form" as something invisible and in-tangible, as for example, the series of natural numbers, or elaborate mathematical equations, when for him "form" must be sensible. The problem of abstraction in art i s complicated further by the fact that, although, (speaking of painting and sculpture) abstrac-tion is achieved by the use of the technical device of schematized shapes (usually called "abstractions"), contemporary art critics and artists are divided as to how abstraction "works". Roughly speaking, they are divided into two camps, and the defence of each position rests on a different view of the world. One theory of art experience presupposes a world consisting of mysterious entities which, by interaction with another equally mys-terious entity, the ego, produce sensations. Configurations of these sensations result in artistic form in two basic varieties: one, the naturalistic, the product of an "affirmative" attitude toward the world, is bound to concrete experiences, in fact, to physical things) the other, the non-naturalistic or "abstract", the consequence of "negative" i i i . attitude toward the world, essentially a r t i f i c i a l and difficult, re-sults from modern man's "dread of space" or "fear of nothingness", etc. I shall call this theory in its two aspects the "existentia-l i s t theory" of art experience. Another theory of art experience takes a rational view of the universe (recognizing, however, certain limits to man's reason) and pro poses that that experience is closely bound up with sense perception and cognition analyzable in logical terms. In this tradition abstrac-tion is viewed as an intellectual process, capable of being discussed in terms other than "positive" and "negative". Accordingly, the work of art is so constructed that its categorical elements are in common with reality, and combined to represent a coherent structure. Abstrac-tion in art is defined as the constant experimenting with syntactical combinations of the language elements of that special type of language which is the actual work of art. I shall call this the "rational the-ory" of art experience. Discussion of this view of abstraction will involve the dis-cussion of what is meant by "work of art", "internal" and "external" logic of the structure of works of art, what I mean by such banalities as "reality" and "truth". In a word, the discussion will turn upon the meaning and purpose of a l l artistic activity. i v . II. Method of investigation. In discussing the problem outlined above I am not going to de-clare which of the two views of the world i s right, which i s wrong, but I w i l l support the theory which appears to me the more exciting and which seems to afford the more satisfactory explanation of what i s meant by abstraction i n art. In doing i t I shall use as supporting arguments my own conclu-sions about works of recent and contemporary masters along with quota-tions from their manifestos, autobiographies and other writings, as well as the ideas of c r i t i c s and historians about them. Also I shall not shrink from using class i c a l sources, because, to my mind, the phenomenon of abstraction cannot be reserved for contemporary art alone. It cannot be denied, however, that the use of the adjective "ab-stract", synonymous with "modern", "non-objective", "non-representational", etc., has become quite frequent and has acquired special importance since the turn of the century. The term "abstract art" i s generally understood as denoting cubism, futurism and expressionism, and demonstrating a com-mon effort on the part of contemporary ar t i s t s , musicians and writers on the one hand, and scientists, philosophers and mathematicians on the other, to solve similar or identical problems. Therefore some attention w i l l be directed to contemporary developments in these disciplines along with art. V. In ray treatment of the subject I s h a l l stress the philosophi-c a l implications rather than the h i s t o r i c a l significance. Above a l l , ray approach w i l l be determined by the influence upon me of the Cam-bridge philosophers and the adherents of philosophical movements re-lated to them, but es p e c i a l l y the writings of Ludwig Wittgenstein w i l l be seen to have inspired the concepts developed i n t h i s thesis. I I I . General conclusions. The general conclusions arrived at i n t h i s thesis might be stated i n the form of the following p r i n c i p l e s : (a) A l l art i s abstract. (b) In art the aim i s to depict r e a l i t y . (c) A work of art i s constructed so that i t s categorical elements are i n common with r e a l i t y and combined to represent a coherent structure (correspondence theory). (d) A work of art i s an interpreted f a c t , and thus repre-sents a prototype t r u l y or f a l s e l y . Truth i s arrived at when inquiry i s stopped. (e) Since the elements of a work of art are combined i n a d e f i n i t e way, i t s truth value w i l l be determined upon the examination of the " i n t e r n a l " l o g i c of i t s structure, together with the "key of interpretation." (f) I t makes sense to say that there are many r e a l i t i e s and one way of interpreting them, just as i t makes sense to v i . say that there is one reality and many ways of interpreting i t . (g) Abstraction in art i s the constant experimenting with syntactical combinations of the "language elements" of that special type of language which is the actual work of art. C O N T E N T S Page Chapter I Introduction 1 Chapter II Percept and Concept in Art 11 Chapter III Abstraction i n Contemporary Music . . . 25 - > Chapter IV Abstraction in the Visual Arts . . . . 40 Chapter V The Truth Function in Abstraction . . . 51 Appendix Feeling and Intellect in Art 65 Bibliography 78 - 1 -Chapter I. Introduction. Sense perception and cognition are closely bound up with art experience. We find that the way these concepts are analyzed by the logical positivists or by the philosophical analysts (1) makes sense in aesthetics. In short, such notions as abstraction i n art, when analyzed, on the one hand, in the logical positivist tradition, seem to afford a satisfactory explanation as against other accounts of how abstraction i n art i s achieved which present us with conceptual muddles or seem to us less complete. (l) The ideas expressed i n this thesis are derived largely from the writings of Ludwig Wittgenstein (1889-1951), and especially from his Tractatus Logico-Philosophicus, London: Routledge & Kegan Paul, 1922 (Yth impression, 1958J. Although his writings and teaching at Cambridge, with some interruptions from 1930-1947, gave impetus to the "logical p o s i t i v i s t " and "logical analyst" movements, i t i s debatable whether Wittgenstein properly be-longs to these philosophical movements. I am using the term "logical p o s i t i v i s t s " widely to refer to the "Cambridge Philosophers" (Bertrand Russell, G. E. Moore and Wittgenstein), the "Vienna Circle" - a group of philosophers which included Moritz Schlick, Rudolf Carnap, and others, as well as disciples of Wittgenstein. The logical positivist movement i s well described in A. J. Ayer, ed., Logical P o s i t i - vism, Glencoe, 111.: Free Press, 1959. See especially pp. "Editors Introduction", and pp. 381-446 for bibliography on the movement. For the biography of Wittgenstein, see: Norman Malcolm, Ludwig Wittgenstein, a Memoir, London: Oxford U. Press, 1958. - 2 -We w i l l take as the point of departure for our discussion a definition of abstraction in art, then we shall construct a theory of how abstraction i n art i s achieved; an unavoidable digression, or so i t might seem, into logic w i l l lead us to a discussion of problems con-nected with contemporary art which w i l l occupy the next chapter of this work. It makes sense to say that i n art "we make to ourselves pic-tures of facts. "(2) To think is to form pictures - this, we assume, i s how a r t i s t s (and here we include musicians, sculptors, architects, actors, dancers, etc.) understand the world. Just as the composer thinks of the world in terms of ordered and interrelated tonal events that unfold i n time, so the painter thinks of the world in structures of interrelated lines and colour. But just as the composer need not in his music imitate the sounds of nature by using, for example, the flute to sound like a bird, so also the painter need not have a thought of "red" and "green" as elements in a picture of red flowers and green leavesj there must be only a structural similarity between a picture and what i t claims to represent. It i s in this sense that we define abstraction i n art as the recognition of a relational structure apart from the specific thing i n which i t i s exemplified. (3) For the purpose of exemplifying structural similarity between (2) Ludwig Wittgenstein, Tractatus Logico-Philosophicus, 2.11. (3) S. K. Langer, "Abstraction in Science and Abstraction in ArtV, in P. Henle, H. M. Kallen, S. K. Langer, eds., Structure, Method and  Meaning (Essays in Honour of Henry M. Sheffer), New York: the Liberal Arts Press, 1951, pp. 171-183. - 3 -"/ a picture and what i t represents,I am using the following simile(4) j Let us imagine two paral l e l planes, I and II. On plane I figures are drawn, say, ellipses and rectangles of different sizes and shapes, and i t i s our task to produce images of these figures on plane II. Then we can imagine two ways, amongst others, of doing this. We can, f i r s t , lay down a law of projection - say that of orthogonal .projection or any other - and then pro-ceed to project a l l figures from I into II, according to this law. Or, secondly, we could proceed thus: We lay down the rule that every ellipse on plane I i s to appear as a c i r c l e in plane II and every rectangle as a square in II. Such a way of representation may be convenient for us i f for some reason we prefer to draw only ci r c l e s and squares on plane II. Of course, from these images the exact shapes of the original figures on plane I cannot be immediately inferred. We can only gather from them that the original was an e l -lipse or a rectangle. In order to get in a single i n -stance at the determinate shape of the original we would have to know the individual method by which, e.g., a particular ellipse i s projected into the c i r c l e be-fore me. It i s my view that the case in art is quite analogous. ^ ) The elements of the picture - the circles and squares i n plane II - deputize, as i t were, for the facts of r e a l i t y - the ellipses and rectangles on plane I. The second projection method (described i n the simile) i s found to be characteristic of the method of abstraction i n , e.g., cubist (4) Ludwig Wittgenstein, "Some Remarks on Logical Form", i n Aristo- telian Society Supplement, vol. IX, 1929, p. 164. (5) Wittgenstein says: **the case of ordinary language i s quite ana-logous". While paraphrasing Wittgenstein, I am not equating "ordinary language" with "art". My point i s , as w i l l be seen later, that some art can be compared with the case i n "ordinary language", whereas some other i s as involved conceptually as, say, "pure" mathematics. - 4 -or neoplastieist art, but the orthogonal projection may very well re-present a les3 intellectual way to describe visual data, such as that of the lens of the eye or the camera recording the scene. On the strength of what was said i n the simile, we w i l l suggest that every picture (composition, sculpture, etc.) presupposes a key of interpre-tation, and that i t i s the function of the p i c t o r i a l (musical, sculp-tured, etc.) elements to furnish i t with such a key, i f we are to under-stand the creation (painting, etc.) "in a single instance". The impli-cations of this suggestion w i l l have to be scrutinized in due course. So far we have been talking about abstraction manifest speci-f i c a l l y in art. How i s our concept related to what i s generally under-stood by "abstraction"? An explicit description or analysis of what the term means i s not an easy task; i t presupposes an oversimple con-ception of how thinking proceeds, and i t requires an investigation of the purpose of the process of abstraction i n each specific case; i n mathematics and logic, i n classificatory sciences, in lexicology, etc., which would take us far from our topic. But to disregard the various applications of the term would be like saying that "centaur" and "medusa" mean the same thing, since there was neither the one nor the other. "Ab-straction" has not always the same meaning, nor i s i t ambiguous: there i s one generic meaning of "abstraction" and n specific meanings. But definitions w i l l get us nowhere; l e t us not trouble ourselves about a definition which w i l l give abstraction "an essence". In order to get a better understanding of i t s significance, we shall consider b r i e f l y , as - 5 -suggested earlier, mathematics and logic, sciences and lexicology, be-fore resuming discussion of abstraction in art. Pure mathematics, i t has been said, i s the most abstract of sciences i n that i t makes no reference to the actual world, and i n that the truths of mathematics can be known by thought alone. When we think of mathematics, we have in mind not only a science devoted to explora-tion of number, quantity and geometry, but also including investigation of yet more abstract concepts of order and into analogous types of purely l o g i c a l relations. Sciences are said to be less abstract than mathe-matics in that they take notice of particular occasions in order to v e r i -f y general propositions (and general propositions involve abstraction). Classificatory sciences involve abstractions that are expressible i n general propositions which are related to, or are given i n , sensible ex-perience, and thus are abstract i n lesser degree than mathematics. His-tory also involves abstraction, "for whatever can be communicated i s ab-stract. But the historian who i s concerned with what has happened may be said i n a sense to avoid this inevitable abstraction by a description which accumulates details so as to be relevant to one given occasion only. Thus history i s the least abstract form of knowledge. n(6) To come down towards the concrete, take as an instance the notions i n mathematics of "number" and "same number". Professor A. N. Whitehead says: (6) L. S. Stebbing, A Modem Introduction to Logic. London: Methuen & Go. Ltd.; (1930;, 7th edition, 1953, pp. 445-6. The i t a l i c s are Miss Stebbing* s. - 6 -We think of the number 'five' as applying to appro-priate groups of any entities whatsoever - to five fishes, five children, five apples, five days ... During a long period, groups of fishes w i l l have been compared to each other i n respect to their multiplicity, and groups of days to each other. But the f i r s t man who noticed the analogy between a group of five fishes and a group of five days made a notable advance in the history of thought. He was the f i r s t man who entertained a con-cept belonging to the science of pure mathematics.^'^ The abstraction of "number" from the notion of any particular set of entities led to an equally important conceptual feat, that of abstraction of the notion of symbols i n form of algebraic or "unspeci-fie d numbers" from the notion of any particular numbers. The notions of numbers "one", "two", "three", etc., can be written i n so many ways. We use Roman numerals (I, II, III . . . ) , Arabic numerals ( l , 2, 3 . . . ) , i n Hebrew they are written as the f i r s t three letters of the alphabet, and there i s no end to various possible number systems of the same struc-ture one can invent. The recognition that these individual systems have the same structure has suggested that they are interchangeable, i.e., they are isomorphic, which means that each number of one system has a counterpart i n the other system and that the results of computa-tions i n one system w i l l correspond to the results of computations in the other system.(8) Obviously, there are number systems of structures different from the ones just presented in that they are not interchange-(7) A. N. Whitehead, Science and the Modern World, N.Y.: Mentor Books, (1925), 9th ed. 1959, pp. 25-26. This i s based on the famous de-f i n i t i o n of number by Gottlob Frege. (8) Adler, Irwing, The New Mathematics, N.Y.: Signet Science Books, 1958, p. 26. - 7 -able. We shall return to this point when discussing various "systems" in contemporary art. Presently i t is only to be noticed that the notion of the "same number" has led to the idea of sameness of structure in different systems and thus to the idea of algebraic symbols, which are nothing else than the general form of number^) represented by substitute signs. Such signs can only occur when there are fixed rules of manipulation or ope-ration together with a key of interpretation of the signs. Moreover, the importance of number in man's conceptual achievement lies in that the study of number has led to the discovery of the nature of reality. Mathe-matical structures are sometimes said to be "pure" structures. When appre-hending in mathematics, for example, a finite series of natural numbers, there is no predicative element involved: i t is apprehended as an unspe-cified "picture", and the attention is paid only to the relational struc-ture of the elements. So also one point of space, considered by itself, is indistinguishable from another, and so is a straight line, or a plane, or one of a number of congruent bodies or areas or line-segments: they are distinguishable only when conjoined as elements in a single total in-tuition. Now, on the one hand, one could consider arithmetic as concerned only with the rules governing the manipulation of the arithmetical signs, and not with the reference of these signs called "numbers" or "quantita-tive ratios"; this kind of arithmetic we could term "formal" arithmetic. (9) L. Wittgenstein, Tractatus, 6.02. - 8 -In formal arithmetic we need no basis for the rules of the game; we simply stipulate them by introducing figures with rules for their mani-pulation. On the other hand, one could adopt a standpoint of a meaning-f u l arithmetic i n which the numerical signs have references, i.e., they have sense (these signs signify different things in different contexts) and therefore the "meaningful" arithmetic has a bearing on l i f e . As Gottlob Frege has shown,(10) formal arithmetic proved un-able to define the irr a t i o n a l (for i t had only a f i n i t e number of numeri-cal figures at i t s disposal), and although the formalists regard their approach to arithmetic as relieved of the need to supply reference for the signs, they unconsciously borrow terms and expressions from meaning-f u l arithmetic without explanation. The signs, whether in mathematics and logic or i n art, i s the part of the symbol perceptible by the senses (ID. and i t i s arbitrary u n t i l we a f f i x the rule that we are not to apply the same sign i n d i f -ferent symbols, and that we are not to apply signs in the same way which signify in different ways in what to the eye appears as the same way. Such symbolism i s said to obey the rules of logical grammar - of logical syntax (l«0. if a certain system (number system, language, art form) possesses a "logical syntax" i t i s a "logical system"; i f not, let us (10) Gottlob Frege, Grundgesetze der Arithmetik, 1903; vol. II, Sections 86-137. Translated by Max Black in Translations from the Philoso- phical Writings of Gottlob Frege, Oxford: Blackwell, 1960 (1952). (11) Wittgenstein, tractatus, 3.32. (12) i b i d . , 3.325. - S l -oaU i t an "ordinary system". The term "ordinary system" here i s not used to designate an i l l o g i c a l system, i.e., not one that i s diagonally opposed to a logical system, but one into which we project i n ever so many different ways ever so many different logical forms. And for this very reason we can draw no conclusions - except very vague ones, from the use of these norms as to the logical form of the phenomena described(l3). These considerations w i l l have a bearing upon the forthcoming examination of the process of abstraction involved, for example, in the De S t i j l move-ment which in i t s tendency toward a logical extreme of mathematical ab-straction i s most strikingly represented by the works of Piet Mondrian, i.e., neoplasticism, and which has influenced the architecture of Walter Gropius and Le Corbusier. If i t i s true that in art to think i s to form pictures, and i f this i s how a r t i s t s understand the world (which one w i l l not deny at least for some a r t i s t s ) , the problem facing us is as follows: in our superficial examination of man's conceptual effort we saw that progress i n forming ideas depended entirely on man's capacity to abstract and gene-r a l i z e . We saw also that mathematics and logic called into play a higher level of abstraction than othsr sciences, because, i t has been said, "no reference to the actual world is involved in any mathematical proposi-tion", ( l * ) But i f we distinguish i n art between "logical systems" and "ordinary systems", and i f we assert that, e.g., the De S t i j l movement (13) L. Wittgenstein, "Logical Form", op. c i t . , pp. 164-165. (14) L. S. Stebbing, op. c i t . , p. 446. - 10 -tends toward a logical extreme of mathematical abstraction, then are we not saying that such movements in art involve no reference to the actual world? This would contradict strangely the statement that artists form pictures of the world. One way of solving this problem would be to show by the recur-sive argument that even though i t i s said that mathematics as a pure science does not refer back to any given set of particular occasions, according to the experts on the subject we have quoted this cannot be so. We may ask, what does pure mathematics refer to? - The answer i s , to antecedent mathematical knowledge which i n i t s e l f i s an abstrac-tion from a given set of particular occasions, for examples number "five", which i n i t s turn refers back to given sets of particular occa-sions, such as five fishes, five children, five apples, and five days. Another way of solving this problem would be to probe into the aims and ideas of the ar t i s t s bent on making what are generally known as "abstract" pictures. This w i l l be the subject of the third and fourth chapters. - 11 -Chapter II. Percept and Concept in Art. Modern times find themselves with an immense system of institutions, established facts, accredited dogmas, customs, rules, which have come to them from times not modern. In this system their l i f e has to be carried forward; yet they have a sense that this system i s not of their own creation, that i t by no means corresponds exactly with the wants of their actual l i f e , that for them, i t i s customary, not r a t i -onal. The awakening of this sense i s the awakening of the Since Matthew Arnold wrote these words the modern s p i r i t has demonstrated i t s e l f in the visual arts i n impressionism, pointillism, neoimpressionism, fauvism, expressionism, cubism, futurism, suprematism, neoplasticism, da-daism, surrealism, constructivism, nonobjectivism, etc.; and this l i s t of modern art-isms can be easily expanded. If the vocabulary of a people affords direct testimony as to i t s interest in what i t observes, then the modern a r t i s t undoubtedly is endowed with a multiple vision. In a l l ages and periods of Western art we distinguish various schools and styles, but the homogeneity of works of any one time preceding the modern becomes s t r i -king i n comparison with the phenomenon of multiplicity. The phenomenon of multiplicity seems to be the distinguishing mark of the modern art. This statement w i l l require treatment further on, but perhaps we need to realize at once that the "modern s p i r i t " spoken of by Matthew Arnold i s di s t i n -( l ) Matthew Arnold, Essays i n Criticism, F i r s t Series. London: Macmillan & Co., 1898 (1865), pp. 159-160. - 12 -guishable from the "not modem" in art primarily i f and only i f i t does not substitute the traditionally eternal and absolute Beauty with an equally eternal and absolute essence of "Modern" or "Abstract Art". The realization, on the one hand, that the world lends i t s e l f to be projected i n so many different structures of interrelated lines and colour, not only in the "customary" forms but i n ever so many d i f -ferent lo g i c a l forms, accounts for this proliferation of art movements and schools. The particular character of the various art-isms proceeds from the various thoughts and ideas i n the artists' minds, i.e., i t has a rational basis. On the other hand, the immense system of institutions, established facts, accredited dogmas, customs and rules organize, so to speak, the f i e l d of perception in such a way that those facts emerge that are of theoretical or practical interest to the person perceiving: the f i e l d of perception i s analyzed so that what could be called "relevant" facts emerge. And what things and predicates are perceived depends on what facts are relevant. Thus i t seems that in the f i n a l analysis the basis for a new art form, school, or movement i s customary and based on tradition, i.e., i t seems to be non-rational. It was already said i n the beginning of this thesis that i n art the world i s expressed as a "fact" and not as a "thing". That the ele-ments of the picture are combined thus or thus represents that objects are combined thus or thus. To perceive a complex means to perceive - 13 -a, that its constituents are combined in such and such a way. To demon-strate: that the figure can be seen in two ways as a cube shows how one and the same ob-ject is really seen as two dif-ferent facts j what appears to be the front plane (for example, "a") in one case, is the back plane in the other, i.e., when the other plane "b" is seen as the front plane. A curious image is created when both the plane "a" and the plane "b" are seen as the front plane (I am capable of seeing the ambiguous figure thus only for a brief moment before i t takes on a definite appearance) -the figure, then, can be seen as "fact" in three ways. Now, since the individual perceives theArorld in such a way that those facts emerge which are of theoretical or practical interest to him (and what things and predicates are perceived depends on what facts are relevant), i t would seem that the individual is tied permanently, so to speak, to his past experience, and his interpretation of the per-ceived data, the analysis of the complex into constituent parts is com-pletely determined by his experience. However, as we have seen by look-ing at the figure, one and the same object can be seen at least as three different facts without any prior experience .( 2) (2) cf. Wittgenstein, Tractatus, 5.5423 and 5.552. The latter entry says: "The 'experience' which we need to understand logic is not that such and such is the case, but that something is; but that is no experience. logic precedes every experience - that some-thing is so. It is before the How, not before the What." - 14 -According to our thesis, to think i s to form pictures. For the composer to understand the world i s to think i n terms of musical structures, for the painter - i n structures of line and colour. Since musicians and painters are contemplating the same phenomena, i.e., the world of facts, i t seems that their discourses are translatable. (3) Moreover, i t seems also that just as there are distinguishable various levels i n conceptual d i f f i c u l t y i n music, there are corresponding con-ceptual levels i n visual art. For example, just as one can imitate i n one's music the sounds of nature by making the flute to represent a bird, or, by the use of certain instruments, mechanical noises such as,a work-ing machine, so also the painter can imitate, for example, the human f i g -ure by drawing i t s silhouette. Let us c a l l this the "imitative" level. Some composers use a symbolic device, leitmotif, a theme associated throughout the work with a particular person, situation or sentiment, so that even out of i t s context we tend to regard a particular theme as a symbol for a particular thing. In the visual arts some symbols, such as the cross, are universally associated with only one idea (event, or person) while others are less universal. Let us c a l l this the "iconolo-gical" level. On another level music might be made to suggest emotion by way of rhythmic configuration and/or clusters of sounds; painting also can be employed to create emotional response (body rhythms and emo-tional states) by certain arrangements of lines and colours. Let us re-cognize this as the "psychological" level. Yet another set of principles (3) Here comes to mind Picasso's aphorism: "To draw, you must close your eyes and sing." - 15 -for musical creation i s derived from the science of accoustics, where-by investigation of overtone series provide a l l sorts of ideas, and these ideas dictate the rules according to which sounds ought to be used; optics likewise is employed i n painting to suggest and regu-late the use of painter's materials. This w i l l be known as the "scien-t i f i c " l evel. Then there are composers who hold that music means noth-ing outside i t s e l f . They say that the fugue, for example, implies the composer's submission to certain musical rules, and that within struc-tures such as these he finds the f u l l flavouring of his freedom as a creator. To mention a few analogical examples in visual art, the rules of linear perspective, or the De S t i j l movement, with Mondrian as i t s purest representative, w i l l exemplify the "tautological" le v e l . Needless to say, i t i s seldom that a work of art w i l l repre-sent one and only one conceptual level, and i n one and the same work various principles can be u t i l i z e d . As an example i n music take Beet-hoven's Pastoral Symphony: although this work is generally recognized as imitative, i t i s d i f f i c u l t to define i t as on a purely imitative level, rather i t might be understood as embodying five principles - the imita-tive, the iconologlcal, the psychological, the s c i e n t i f i c and the tauto-lo g i c a l . Where the flute t r i l l s l i k e a nightingale and where the oboe repeating a note imitates the characteristic rhythm of the quail, while the f a l l i n g third i n the clarinet stands for the cuckoo, the music i s of a conceptually simple kindj the f i r s t movement, "Awakening of serene im-pressions on arriving in the country: Allegro ma non troppo", reminds us - 16 -of the phenomena of f l o r a l growth, where simplicity and charm of surface conceal i n f i n i t e variety and organic intricacy because of the drone-like repetition of fragments from the themes in a sort of murmuring monotony, continuously and subtly changing in tonality, colour and position. We shall assume that the music for which "Allegro ma non troppo" stands ought to be understood as an icon (description) for "Awakening of serene impressions on arriving i n the country". A listener, ignorant of the fact could miss the point, yet be deeply moved by the music. Finally, the whole of this music follows a systematically worked-out pattern capable of expression i n mathematical formulae.(4) Similarly, Marcel Duchamp"s "Nude descending a s t a i r c a s e 0 ^ , 1912 (Philadelphia Museum of Art), an imitative painting on one level, i s imitative of a figure i n descending movement, thus depicting the r e a l i t y of the visual form of a moving figure: and on another le v e l i t i s a work with extensions beyond the imitative level, in that the language elements of the picture which at f i r s t sight have the apparent effect of "an ex-plosion i n a shingle factory", more refl e c t i v e l y present themselves as organized into a pattern of logical order with the aim to present the idea and the values i n question consistently. The nude's motion could be analyzed, to use the phrase, "as a collection of parts that work to-(4) This is an assumption. Prof. £. E. Leimanis t e l l s me he has analyzed and expressed i n mathematical symbols a J. S. Bach fugue. Although Beethoven's 6th Symphony i s not as clear-cut a proposition as i s a Bach fugue, I do not see why i t could not, theoretically, be equally well expressed as a mathematical proposition. - 17 -gether to achieve a common function". (5) It is doubtful whether the "Descending Nude", the happy result of "Duchamp's battle of logic "(6) can be easily expressed in mathematical equations, but, as we have noted earlier, e.g., Mondrian's o i l , "Composition with Red, Yellow and Blue", 1921, (Geraeente Museum, The Hague) could be expressed in a mathe-matical equation. The significance of the realization that art is not shackled only to imitative depiction of the natural world, nor to sym-bolic representation of a Heavenly City, lies in opening up for the artist new horizons, as has been well recognized by most of the modern artists. Perhaps the assertion that art has a logical (or rational) basis is hard to swallowj i t might be more easily acceptable to those who do not see eye to eye with this i f we realize that logic (and mathe-matics, which is a logical method) is not a theory but a reflexion of the world. Logic is transcendental (7) in that i t transcends the imme-diate occasion. And since in art we are dealing with properties which admit of gradation, such properties as the length of an interval, the pitch of a tone, the brightness of a shade of colour, continuous transi-tions, and combinations in various proportions, with forms of space and time, numbers can enter these forms i f we so choose. One shade of colour cannot simultaneously have two different degrees of brightness, (5) Bates Lowry, The Visual Experience, N.Y.: Abrams, Englwood, N.J.: Prentice-Hall, 1961: pp. 222-227. (6) Robert Rosenblum, Cubism and Twentieth Century Art, N.Y.: Abrams, I960; p. 152. (7) L. Wittgenstein, Tractatus, 6.13. - 18 -nor a tone two different strengths. It i s characteristic of these pro-perties that one degree of them excludes any other. (8) The idea of numbers being inherent i n art goes back to the Pythagorean creed that everything i s arranged according to number, from which Plato's mathematical philosophy and his system of aesthetics e v o l v e d . ^ Platonism, i t i s said, developed from the confluence of two streams of inspiration - the Socratic and the Pythagorean. From Socrates Plato learnt that the problems of human l i f e were to be solved by the morality of aspiration and the pursuit of an invariable ideal of perfection: from Pythagoras he discovered how this conception could be extended beyond the f i e l d of human concerns into a system embracing the whole of nature and transforming the scope of science. The germ of Pytha-goras' s mathematical philosophy, curiously enough, was a discovery, not in the f i e l d of arithmetic or geometry, but i n music. This he connected with the doctrine of the immortality of the soul and mathematics. Dryden, keeping s t r i c t l y to an old tradition, wrote i n 1687: (8) I termed the logical approach to art "tautological" because remarks such as these do not express an experience but are i n some sense a r b i t r a r i l y imposed systems. For what follows I am mainly indebted to the sources: (9) Burnet, John, Early Greek Philosophy, 2nd ed., London: Black, 1908. R. C. Lodge, Plato's Theory of Art, London: Routledge & Kegan Paul, 1953; and F. M. Cornfold, From Religion to Philosophy, N.Y.: Harper's Torchbook, 1957, (Arnold, 1912;. As Cornford shows, the Pythagoreans inherited the doctrine con-cerning music from the Orphics and the doctrine of reincarnation from the Dionysians, both of which, together with the Pythagorean procession of numerical series was taken over and made use of by Plato. - 19 -From harmony, from heavenly harmony, This universal frame beganj From harmony to harmony Through a l l the compass of the notes i t ran, The diapason closing f u l l i n man. (10) It i s not at once obvious to the modern mind that there i s any connection between music, the doctrine of immortality of the soul, and mathematics. But the vision of philosophic genius i s a unitary vision. Pythagoras found that the "perfect consonances" (as they s t i l l are called), the intervals of the fourth, the f i f t h and the octave, can be expressed exactly as ratios between the numbers 1, 2, 3, and 4, which, added to-gether, make the "perfect" number "ten". The choice of "ten" as the "perfect" number i s explained by the symbol "tetractys" and i s not, as i s sometimes suggested, a purely biological accident of our having ten f i n -gers on our hand. The original "tetractys" appears to have been obtained by the addition i n the Greek method of 1 + 2 + 3 + 4 • 10: "Ten" i s therefore called triangular number. . . . « • • • The sum of any number of the series of odd numbers beginning with 1 (one) i s similarly seen to be square: . . . etc., and the sum of any number i l ' . oblong number: . . . of the series of even numbers ( I D beginning with 2 (two) makes an , etc. But the "perfectness" of the consonances, although they were arrived at by (10) Quoted in E. M. W. T i l l y a r d , The Elizabethan World Picture, London: Chatto & Windus, 1958, (1943), p. 94. (11) Burnet, John,, op. c i t . - 20 -measuring the length of the strings of the lyre are based on the propor-tions of frequencies of the sounds. The result i s the same: the lengths of strings and numbers of frequencies are inversely proportional, (supported by empirical investigation of the overtone series). The ratio of the octave i s 2:1; the ratio of the f i f t h i s 3:2; the ratio of the fourth i s 4:3. In this discovery Pythagoras divined a principle i l l u m i -nating the order of Nature. For i f the chaotic welter of sounds that besiege our hearing can be reduced by the principle of measure- to the harmonious order of art and f i n a l l y to proportions of number; might not the whole order of Nature, with i t s acknowledged beauty, be framed into a doctrine that the r e a l i t y of things l i e s i n the limiting principle of measure, proportion and number? In astronomy the speeds and distance of the heavenly bodies are ruled by the proportions of harmony: the sun, the moon, and the five planets then known formed a scale cr octave, the inter-vals of which were numerically determined by the distances between their orbits. That octave, as we learn from the Timaeus, has i t s counterpart in the immortal soul of man; the mighty revolutions of the soul of the world which are the planetary orbits are reproduced in a smaller scale i n the soul of man. The forms of surfaces which limit tangible bodies re-present the perfect figures of geometry, and the laws of these figures can be reduced to relations of number. Owing to the Greek mode of re-presenting numbers by patterns of dots, as we have already noticed, the relation between notions of number and of geometrical configuration are more obvious than by the modern method. Pythagoras i s said to have been the f i r s t man to c a l l the universe "cosmos", which meant beauty as well - 21 -as order. Plato believed that, were i t not for the perishable nature of the body, man's soul would make music i n perfect accord with the music of the Cosmos. The body thus forms a barrier between the soul of man and the soul of the world, and the function of music i s to over-come i t . Music can achieve this because, on the one hand, i t can reach the soul through the bodily senses of hearing, while, on the other hand, i t i s i t s e l f attuned to the celes t i a l diapason. It might be that the significance of art i n Pythagoreanism could be explained by the tradition that Pythagoras was said to have been the son of the god A p o l l o , ^ ) and i f i t does not explain, at least i t suggests that the traditions concerning this native son of Samos are, as Bertrand Russell t e l l s us, "an almost inextricable mixture of truth and falsehood".( x3) He is pictured as both a philosopher and a founder of mathematics, but also as a religious prophet and mystic. " A l l the systems that he inspired", Cornford says, "tend to be otherworldly, putting a l l value i n the unseen unity of God, and condemning the visible world as false and i l l u s i v e , a turbid medium in which the rays of heav-enly light are broken and obscured i n mist and darkness". The Pythagorean doctrine,that number l i e s at the base of the real world, i s the chief source of the b e l i e f in eternal and absolute (12) According to legend he was born in a v i r g i n birth. His mother i s called Parthenisj quoted by F. M. Cornford, i n From Religion to Philosophy, N.Y.: Harper's Torchbook, 1957, p. 2"o!n (13) B. Russell, History of Western Philosophy, London: Allen & Unwin, 1955 (1946): p. 49. - 22 -truth as well as i n a super-sensible i n t e l l i g i b l e world. To show how this belief i s arrived at, we can do no better than repeat Bertrand Russell's words: Geometry deals with exact c i r c l e s , but no sensible object i s exactly circular; however carefully we may use our compass, there w i l l be some imperfections and irregu-l a r i t i e s . This suggests the view that a l l exact reason-ing applies to ideal as opposed to sensible objects; i t is natural to go further, and to argue that thought i s nobler than sense, and that objects of thought more real than those of sense perception. Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for mathematical objects, such as numbers, i f real at a l l , are eternal and not i n time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jean's belief that He i s addicted to arithmetic.'^) This account of the Pythagorean tradition w i l l serve a dual purpose; a reminder of the age-old "formalist" approach in Western art from the Greeks through the Middle Ages to modern man, and a reminder also that an idea i s rooted i n l i f e , for abstractions without empirical manifestations are found only i n mythology, although we sometimes like to think they exist as entities. The emphasis i s on structure in the various forms of human act i v i t i e s and suggests that the nature of these various a c t i v i t i e s can be approached through the study of the discourse i n question, whether i t be poetry, painting, music, physics, or chemis-try. Pythagoras counsels us to express quality i n terms of numerically determined quantity, i.e., to measure. In music this means constructing i n time and pitch, i n painting - i n line and colour, and i n sculpture (14) Bertrand Russell, op. c i t . , p. 55. - 23 -and architecture - i n surface and edge, but a l l these can be shown to be manifestations of spatio-temporal properties. Perhaps i t i s due to Pythagoras that we have come to regard such exhibitions of linear rhythm as those exemplified i n the Gothic cathedrals as "frozen music." The spatio-temporal element in the visual arts has u n t i l recently es-caped satisfactory analysis, although i t has always been recognized i n one form or another: as pulse and rhythm known to evoke emotional response; as symbolic pattern (such as i s treated by the "Sacred Geo-metry"); or as design principle. In contemporary painting and sculp-ture, however, since the cubists, expressionists and futurists, space-time has remained foremost in consideration and the outstanding problem. Could i t be that, lik e the scientists since Einstein who construe physi-cal facts such as gravitation as exhibitions of local peculiarities of spatio-temporal properties, the artists and musicians of today are also following the pure Pythagorean tradition? Many signs show that i t might be so. To recapitulate, we have seen that i n art each successive age depends for inspiration, materials and methods on the antecedent ages and periods, yet, i f their art i s to be their own and were to describe the facts of the world as they know i t , a rational process must be involved. The organization of data of experience i n developing vocabulary by constant modification and revision i s a good example. New classes are added, based upon new elements of experience, or upon the recognition of dis-tinctions i n what was previously undifferentiated, and i f they become - 24 -of sufficient interest to a sufficient number, are maintained; old classes, having outlived their interest, become obsolescent and f i n a l l y disappear. It i s clear therefore that, analogically, there never can be entire correspondence, as regards ideational content, between the art of any two people, or between any two periods in the art of one people. On the other hand, there can never be complete difference. Since our capacity to abstract and generalize depends upon our capacity to detect resemblances i n the midst of differences of the data of ex-perience, i n the measure that the resemblances become less numerous, less frequent, less obvious, and more obscured by the increasing v a r i -ety and complexity of difference, the task of abstraction becomes more d i f f i c u l t , and the resulting concept is a mental product of a higher order. It is possible therefore, at least i n theory, to cla s s i f y art works from the point of view of conceptual gradation, that i s , as mental products of lower or higher order, according as an analysis of their meanings ( i f they can be properly deciphered) reveals a lower or higher level of abstraction. We cl a s s i f i e d art, therefore, as "tautological", " s c i e n t i f i c " , "psychological", "iconological" or "imitative", depending on a higher or lower level of abstraction exercised. We may say: what a work of art represents is i t s sense, and i t s sense i s i t s ordinal experience. - 25 -Chapter III. Abstraction i n Contemporary Music. The fact that contemporary art i s often identified with ab-stract art may lead one to assume that of a l l art of periods, the modern, i.e., the contemporary, period, i s uniquely privileged to use abstraction. In this and the following chapters we shall discuss the contemporary art, the art from around 1900, i n order to determine what i t claims to achieve and by what procedure i t does so. However, since in talking about art in general, there i s some danger of losing contact with what painters, sculptors, composers, etc., do, the present chapter w i l l be concentra-ted on one branch of contemporary art only - namely music. There are several reasons for this particular choice. F i r s t , of a l l the branches of art, music i s said to be the most abstract, and due to i t s abstractness, "abstractly minded" visual artis t s are said to have a particular interest i n the musical method. Second, i t so happens that there i s one method of musical composition which seems very suitable for representation i n numerical and lo g i c a l forms, thus permitting us to com-pare i t with the ac t i v i t i e s of the visual artists who use logi c a l repre-sentation as part of their method. SchOnberg's twelve-tone music has been blamed for being "too intellectual" and "ster i l e " , for i t s "emotionalism", or again for i t s absence of beauty". While concentrating the discussion on the twelve-tone music i n - 26 -this chapter, we hope to prepare the ground, so to speak, for discussion of contemporary painting i n the next chapter. Schonberg pointed out that the tonal harmony, which he aban-doned for the new method of composition, had served "not only as a source of beauty, but, more important, as a means of distinguishing the form". (1) It was necessary, then, to find a substitute for what Schonberg called the "structural functions of tonal harmony". It i s important to mention here that Schonberg was f u l l y famil-i a r with the Wagnerian tradition before he entered an experimental period of composition i n which he went back to Beethoven, Bach and the earlier polyphonic writers for inspiration. As a result he abandoned the lengthy Wagnerian style and began to compose extremely short, laconic musical statements i n what he termed "twelve-tone" procedure, or the "Method of Composing with Twelve Tones which are Related only with one another". (2) At that time, around 1910, he was constantly in touch with Kokoschka, Kandinsky and Barlach^), and i t i s clear from SchSnberg* s writings that the ideology of "Der Blaue Reiter" group manifested i t s e l f in his music. (1) Arnold Schttnberg, Style and Idea, N.Y.: Philosophical Library, 1950, Chapter, "Composition with Twelve Tones", p. 105. (2) i b i d . , p. 107. (3) SchOnberg himself was a painter of considerable a b i l i t y and belonged to the Blaue Reiter group; in 1912 he published an ar t i c l e , "The Relationship to the Text" i n Der Blaue Reiter, a symposium edited by Kandinsky and Franz Marc, Munich, 1912; 2nd edition, 1916. Wil l Grohmann in Wassily Kandinsky, N.U.: Abrams, 1948, t e l l s of the friendship between SchOnberg and Kandinsky and includes a photo-graph of both a r t i s t s with their wives sunbathing together at Munich. - 27 -When Schoriberg decided to break with the tradition that nur-tured him, he did not feel that he was creating a revolution, but that his invention was the next logical step i n evolution of music. We must not forget that the method of composing with twelve tones grew out of the development of chromaticism. Here i s what SchOnberg has to say-about the "necessity" to develop this method: In the last hundred years, the concept of harmony has changed tremendously through the development of chroma-ticism. The idea that one basic tone, the root, dominated the construction of chords and regulated their succession -the concept of TONALITY - had to develop f i r s t into the concept of EXTENDED TONALITY. Very soon i t became doubt-f u l whether such a root s t i l l remained the centre to which every harmony and harmonic succession must be referred. Furthermore, i t became doubtful whether a tonic appearing at the beginning, at the end, or at any other point, really had a constructive meaning .... Moods and pictures, though extra-musical, thus became constructive elements, incorporated i n the musical functions; they produced a sort of emotional comprehensibility. In this way, tona-l i t y was already dethroned i n practice, i f not i n theory. This alone would perhaps not have caused a radical change in compositional technique. However, such a change be-came necessary when there occurred simultaneously a develop-ment which ended in what I c a l l the EMANCIPATION OF THE DISSONANCE. The ear had gradually become acquainted with a great number of dissonances, and so had lost the fear of their 'sense-interrupting' effect. One no longer expected preparations of Wagner's dissonances or resolu-tions of Strauss' discords; one was not disturbed by De-bussy' s non-functional harmonies, or by the harsh counter-point of later composers ...(4) Thus i t was the "necessity" to supersede the "structural func-tions of tonal harmony" with the structural functions of the chromatic scale which led to the substitution of the relationship of a l l tones (4) Arnold SchOnberg, op. c i t . , pp. 105-104. The i t a l i c s are SchSriberg's. - 28 -and chords to a central key-note i n tonal harmony with a "note-row" (Tonreihe), or the succession of tones from the twelve notes of the chromatic scale. The twelve tones could be arranged i n any order that the composer thought suitable, but with the rule that every note from the "note-row" which forms the basis for the composition must be sounded before any of them i s employed for the second time. This e l i -minated a common centre on which the tonal scale depended and created a new relationship between the notes. Instead of being related to a common centre,the notes now were related "only to each other". But the note-row was in no way identical with the chromatic scale. The row originally chosen as a basis of a work could be employed in various forms: i t could be played backwards, i t could be turned upside down, and the upside-down version could also be played backwards. This pro-cedure thus had supplied the composer with four note-rows, the original and i t s three variations, for which the terms commonly used are: prime, known also as the "basic set", retrograde, inversion, and retrograde i n -version. Moreover, any of these four rows may be transposed to begin on any of the remaining eleven notes of the scale; and the "basic set" could not only be played forward, backward, right side up, or upside down, but also linearly (horizontally, as a melody), or ve r t i c a l l y (as chords), or mixed, linearly and ver t i c a l l y . But the rule that no note should appear out of sequence was s t r i c t l y obeyed. The basic set of the composition was thus determining the composition completely and every note of the whole composition showed i t s relationship to this fixed fun-damental series. - 29 -This thoroughly rational organization i n Schonberg's method led to severe criticism. Compositions such as Schonberg's were said to be at best "splendid intellectual constructions", at worst they were described as "paper music" capable of being composed by a machine. It i s indeed agreed that e l e c t r i c a l computers, these manufactured brains which do the work of calculating payrolls, keeping track of plane reservations, guiding of missiles, playing games and translating from one language into another, can also be used to compose music. But i t i s also agreed that these machines are s t i l l a long way from supply-ing the note-rows for such compositions, or that they would consistently "write" what can be properly called a musical composition i n that this composition would be a musically coherent whole. Yet, this objection can be overruled by saying that i t i s l o g i c a l l y possible that a machine could supply i t s own note-rows, or impose form on the music i t writes. After a l l , the statement that a machine has not been "taught" how to give form to i t s music should not be taken to imply that composers a l -ways deliberately impose form (5) o n their work. For example, when composers describe the writing of a work, they say such things as, "the (5) I am using the word "form" here i n i t s etymological sense refer-ring i t , as some philologists do, to the Sanskrit root dharman neut., holding, position, order; others compare i t with Latin ferl r e , to strike. - 30 -shape of i t was taken out of my hands, so to speak, as I worked". (6) We understand the phrase as implying that the original plan of the com-poser had to be modified during the operation, not due to the interven-tion of some mysterious entity, but because the composer changed his mind about his original plan, and that th e words "... taken out of my hands ..." i s just a loose way of speaking. The basic principle of a machine-composer capable of composing the twelve-tone music, i t seems,would be the "feedback principle". When a mechanism can detect the results of i t s own actions, and i s guided by the results, engineers say i t has a "feedback". But the "feedback" can operate only i f the machine i s capable of evaluating i t s past actions by using fixed rules that are part of i t s instructions, store the results i n a "memory unit", and deliver on demand the one that is "best" accord-ing to i t s b u i l t - i n standards of judgment. Had the use of the note-rows been enough to give a work the unity which i t would have, we could have been satisfied that the writing (6) Delius, quoted by Eric Fenby i n Delius as I knew him, London, 1936, p. 36; i n this quotation the word "shape" i s used for "form"; this usage goes back to the Scholastic philosophic term "form" which was the essential determinate principle of a thing. "Form" was p r i -marily applied to objects of sense (the v i s i b l e aspect of a thing) but was i n philosophical use extended to "objects of thought": every "thing" or entity was viewed as consisting of two elements, i t s form by virtue of which i t was different from, and i t s matter which i t had i n common with others. In ordinary speech, stuff, a portion of matter, becomes a "thing" by virtue of having a particular "form". By altering the form, the matter remaining unchanged, we make a new "thing". 1 - 31 -of music by means of an appropriately constructed machine would be com-parable to the practice of composers: but, s t r i c t l y speaking, a note-row no more defines the whole of the twelve-note system than the playing of a major scale defines the whole of the harmonic and contrapuntal sys-tem of B e e t h o v e n . T h e difference, then, between the composer's way of writing twelve-tone music and of a machine doing i t l i e s i n the pur-pose of the composer, not i n the external form of the composition. The composer w i l l not only choose according to the rule, his purpose w i l l be to form a musically meaningful sentence i n order to assert something about the world, be i t at any one of the five levels mentioned in the previous chapter. The machine constructed to the writing of twelve-tone music w i l l "know" the rules similarly to the way a grader's apprentice knows the rules of grading. The greater his intelligence and the longer his apprenticeship, the more nearly i n f a l l i b l e he becomes. In this sense machine-composing i s a business done in accordance with principles which the machine can be "taught" to do similarly to the way an apprentice can (7) In the foregoing discussion of the use of a machine for the purpose of composing the twelve-tone music I have drawn heavily from G. H. R. Parkinson, "The Cybernetic Approach to Aesthetics" i n "The Journal  of Philosophy, vol. 36, No. 136, Jan. 1961, pp. 49-61, and from Irving Adler, Thinking Machines, N.Y.: Signet Science Library, 1962. Both writers show how electronic computers can be set up to work out twelve-tone music by means of numbers. - 32 -be taughtj but i f the apprentice were to speak, he would say to his X master: "from your teaching I gain a deeper understanding of what i s involved i n this art, but no greater power to produce i t myself", whereas the machine would gain no understanding at a l l . The appren-tice would learn to compose either i n the sense i n tfiich a parrot can learn to speak English, and thus might realize no more than a parrot of what he is doing; or he might distinguish what is involved i n his master* s art similarly to the person who echoes conventional moral judgments correctly without re a l l y making moral judgments himself. (8) Forming of musically meaningful sentences, then, is quite similar to making of moral judgments and not just echoing them. The "thinking" involved i n the "thinking machines" proceeds according to fixed rules along a predictable course. Behind this "rou-tine thinking" l i e s the creative thinking by which the rules these ma-chines follow are fixed. The creative thinking i s used i n art as i n science, but for different purposes. Science ministers to man's need to be able to determine correctly his expectations, and hence his a c t i -v i t y i n terms of the evidence which he has accumulated, therefore pre-diction i s of utmost importance i n science. Art, on the other hand, is not concerned with accurate prediction, but attempts to picture what human beings value. Since i n art the aim i s not prediction, i t might (8) I owe the simile to J. 0. Urmson, "On Grading" i n A. G. N. Flew, ed. Essays on Logic and Language, 2nd series, Oxford: Basil Blackwell, 1955, pp. 160 to 167. - 33 -be expected that requirements of consistency are non-existent. But this i s not the case at a l l j requirements of consistency take on a special form: the a r t i s t constantly experiments with syntactical com-binations of his language elements to obtain the desired value effects. As we saw from our discussion of the charges against the twelve-tone music claiming that i t i s nothing but "paper music, capable of being composed by a machine", these charges could be brought to bear i f and only i f i t could be shown that a machine can make value judgments and not just echo them. So far as we know,this has not been possible. SchBnberg says: It has been mentioned that for every new composition a special set of twelve tones has to be invented. Some-times a set w i l l not f i t every condition an experienced composer can foresee, especially in those ideal cases where the set appears at once in the form, character, and phrasing of a theme. Rectifications i n the order of tones may then become necessary,(9) which not only confirms what we have been saying about the machine and the apprentice i n contradistinction to the master, but also introduces the notion of "theme" i n a different sense from the "note-row" or "basic set". Thus actually we are dealing with two concepts which Schonberg himself formulated as "Grundgestalt" (basic structure, form, or shape) and "Grundreihe" (basic set, tone-row, or note-row). The "Grundgestalt" i s , i n his words, the " f i r s t creative thought" and includes the basic (9) op. c i t . , p. 114. - 34 -set. (10) Where does this leave the twelve-tone composer? Schonberg quotes Schopenhauer as supplying the answer: The composer reveals the inmost essence of the world and utters the most profound wisdom i n a language which his reason does not understand, just as a magnetic somna-bulist gives disclosures about things which he has no idea of when awake. (H) Identifying himself with a somnabulist he, who "made order ... to resolve the magic essence of music into human reason", now appears to "dissolve human reason i n magic".(x^) Schonberg's belief i n the sub-conscious and other mystical entities, of which there i s plenty of evi-dence in his writings, however, had l i t t l e effect on his music. In the case of the creative idea, the "Grundgestalt", i t i s of no importance whether i t has "taken place" i n the conscious or subconscious; i f the conscious i s equated with intellect and the subconscious with feeling, then i t i s equally true of both that they are capable of sensing and re-cognizing connections or relations, even i f these connections or rela-tions are not obvious. This idea i s so clearly expressed by Schonberg that i t i s worth repeating i n f u l l : After I had completed the work (Kammersymphonie) I worried very much about the apparent absence of any re-(10) See "Translator's Preface", Joseph Rufer, Composition with Twelve  Notes, related only to one another. London: Rockliff, 1954, Translated from the German Edition Die Komposition mit ZwOlf  Tonen (1952), by Humphrey Searle. (11) Quoted in "The Relationship to the Text" i n Der Blaue Reiter, 1912, Style & Idea. (12) Thomas Mann, Doctor Faustus, N.Y.: Alfred A. Knopf, 1948, pp. 193-194. - 35 -lationship between the two themes. Directed only by my sense of form and the stream of ideas, I had not asked sixth questions while composing: but, as usual with me, doubts arose as soon as I had finished. They went so far that I had already raised the sword for the k i l l , taken the red pencil of the censor to cross out the theme 'b'. Fortunately, I stood by my i n -spiration and ignored these mental tortures. About twenty years later I saw the true relationship. It i s of such complicated nature that I doubt whether any composer would have cared deliberately to construct a theme in this way; but our subconscious does i t involuntarily.(13) The creative idea having come to the composer in his "somna-b u l i s t i c " state "in a language which his reason does not understand" i s , after a l l , capable of reduction to the expressible i n a language which reason does understand. "Reason and magic ... may meet and become one i n that which one ca l l s wisdom, i n i t i a t i o n ; i n belief i n the stars, in numbers ... "(14) The idea and technique, then, despite what was said earlier, i s one, i.e., what the composition represents i s i t s sense. Whether the "creative idea" deals with feelings or with thoughts, i t s expression must have an inner logic (consistency). For SchOnberg, the expression of feelings and/or thoughts means a logical expression: laws which regulate our i n t e l l e c t as well as our feelings, function logically, or as i t i s expressed i n the Tractatus, "It used to be said that God could create everything, except what was contrary to the laws of logic. The truth i s , we could not SAY of an 'unlogical' world how^  i t would look.»( i 5) (13) op. c i t . , p. 113. (14) Thomas Mann, op. c i t . , p. 194. (15) L. Wittgenstein, Tractatus, 3.031. - 36 -Theoretically* a well balanced musical composition w i l l con-vey ideas imbued with feeling, or, to put i t the other way, i t w i l l ex-press feelings controlled by i n t e l l e c t : It i s not the heart alone which creates a l l that i s beautiful, emotional, pathetic, affectionate and charming; nor i s i t the brain alone which is able to produce the well-constructed, the soundly organized, the logi c a l , and the complicated. F i r s t , everything of supreme value in art must show heart as well as brain. Secondly, the r e a l creative genius has no d i f f i c u l t y in controlling his feelings mentally; nor must the brain produce only the dry and unappealing while concentrating on correctness and logic. But one might become suspicious of the sin-cerity of works which incessantly exhibit their heart, which demand our pity, which invite us to dream with them of vague and undefined beauty and of unfounded, baseless emotions, which exaggerate because of the ab-sence of reliable yardsticks; whose simplicity i s want, meagreness and dryness, whose sweetness i s a r t i f i c i a l and whose appeal attains only to the surface of the super-f i c i a l . Such works only demonstrate the complete absence of brain and show that this sentimentality has i t s origin i n a very poor heart.(16) Words such as these, although they do not express anything new, coming from a man whose work i s considered of the most abstract kind, de-monstrate once more his concern for the "whole" man. Apparently, Schon-berg no less than Barlach (the sculptor) says, "I must be able to join i n the suffering". But however sincere his wish was to communicate, he never made concessions to the ordinary listener, to the uninitiated, to the "average man i n the street": (16) A. SchOnberg, "Heart & Brain i n Music", i n Idea and Style, p. 179. - 37 -No one should give i n to limitations other than those which are due to the limits of his talent ... No musician ... would degenerate in order to comply with the slogan such as 'Art for A l l ' . Because i f i t i s art, i t i s not for a l l , and i f i t i s for a l l , i t i s not art ... But there are a few composers, l i k e Offenbach, Johann Strauss and Gerschwin, whose f e e l -ings actually coincide with those of the 'average man i n the street 1. To them i t i s no masquerade to ex-press popular feelings i n popular terms, They are natural when they talk thus and about that.(if) SchOnberg's music, then, according to our terms defined i n chapter II, i s never "imitative", seldom, i f ever, "iconological", per-haps "psychological", but, since he explores the elements of music within the strictures of carefully defined rules i n order to make people "understand what music has to say purely through their musical faculties" (18), he can be cl a s s i f i e d rather as being " s c i e n t i f i c " and "tautological". Thus his music reveals i t s e l f most effectively to those endowed with "musical faculties", for whom a piece of music i s not unintelligible or worthless i f i t does not evoke images of some kind. If tiiere are a number of what we could c a l l "extra-musical" ideas perceptible in a piece of music, they should be so connected as not to offend musical logic. This we saw also i n Beethoven's Sixth Symphony ("The Pastoral"), where i n the movement, "Awakening of serene impressions upon arriving i n the country: Allegro ma non troppo", Beethoven uses the minimal number of minor chords, standing for sad-ness; he goes even as far as to leave sections of the melody unharmo-(17) A. SchSnberg, Idea and Style, pp. 50-51. (18) ibid.p.51. - 38 -nized i n order to avoid the use of minor chords where i t was impos-sible not to us them i n respect to the laws of harmony. Before we move on to talk about abstraction i n contemporary visual art and especially i n painting, i t i s necessary to pause here in order to c l a r i f y the main points of the present discussion. C We set out to analyze one branch of contemporary art and chose the twelve-tone music, not because i t i s the chief representative of modern music, but because i t i s said to be the most "abstract". We saw that Schonberg, the inventor of the twelve-tone method, used i t to reduce to the musically recognizable ideas and expressed feelings by means of more restricted musical equipment than the tonal harmony. It was pointed out that some features of this method of musical composi-tion were similar to the methods used by Bach and Beethoven, and later i n this chapter I showed that the dissimilarity between the antecedent methods of musical compositions to Schonberg's l i e s i n the relations between tones. It w i l l be necessary to re-emphasize this point i n a slightly different version because of i t s utmost importance i n the argu-ment. Schonberg and his followers, Webern, Berg, Rankl, Z i l l i g , Eisler, Skalkottas, Hannenheim, Strang, Crerhart and Weiss, refused to c a l l the twelve-tone method "a system" of music, because they were s t i l l using the same musical elements for the same purpose of expressing facts about the world (which includes music); but they expressed these facts - 39 -with a particular "method of projection" which differed from the tonal harmony i n that the tonality as the logical centre relates every note i n the scale to this centre, the tonic (the f i r s t note of the scale): in the twelve-tone method there is no common centre, the association of tones i s regulated by the order of these tones and their inter-re-lationships. That changes the concept of music insofar as i t destroys the impression or appearance of ah absolute "up" or "down", "right" or " l e f t " , "forward" or "backward". Every movement of the tones has to be seen as a relationship of sounds appearing as space-time units.. Let us c a l l this the " r e l a t i v i t y principle" in modern music.^^) (19) Paul Hindemith also speaks of music as having the dimensions of space and time (Elementary Training for Musicians,.N.Y.: Asso-ciated Music Publishers, 1946) where the most primitive spatial action i s the use of tones of different length, and the most primitive spatial action i s expressed in singing or playing tones of different pitch. Some musicians seem to be saying that one of these elements i s more important than the other; but since sound i n music must begin and end i n time, the dis-pute does not make sense. In fact the time and space ele-ments being mutually inclusive present themselves as one, i.e., space-time unit. Thus compositions with twelve tones bring the tone-row to l i f e i n rhythm, just as the compositions i n the tonal method bring a particular scale to l i f e i n rhythm. - 40 -Chapter IV. Abstraction i n the Visual Arts. Generally, music has always been thought of as an abstract art and the visual arts as representational. However, i t has been shown that at least i n some music the composer's intention has been to depict reality, to represent the world, hence the composition i s a model of r e a l i t y j thus i t i s not incorrect to say that music i s , i n a sanse, representational. Speaking about painting, for example, in order to i l l u s t r a t e the fact that a l l good painting i s abstract, we may show that some paintings of some seriously-minded artist s are models of reality, so that the elements of the painting are combined with one another in a definite way, representing that the things are so combined with one another (2 ) : thus the painting consists i n the fact that a set of objects stand i n a certain relation to each other(3). But we a l -ready saw that i n a picture of red flowers and green leaves i t i s not absolutely necessary to have the elements "red" and "green"; there must be only a structural similarity between the picture and what i t stands for. It was i n this sense that we defined abstraction i n art as the "re-cognition of a relational structure apart from the specific thing in which i t i s exemplified." (1) Not a l l a r t i s t s are seriously-minded and not always the ar t i s t s ' intention i s to depict reality. (2) L. Wittgenstein, Tractatus, 2.15. (3) Tractatus, 5.4733T" - 41 -The idea that a picture depicts reality, i f depiction of re-a l i t y has been the artist's aim, then suggests that abstraction has a truth-function: for, as we have seen, there are two distinct features belonging to a painting (sculpture, etc.): f i r s t , the relation between the elements of the painting, and second, the correlation of the ele-ments i n the painting with things outside i t . The f i r s t feature, i n a sense, must belong to a painting before the second feature can, be-cause only i f significant relations hold among the elements of the paint-ing, can they be correlated with objects outside so as to stand for them. The following theory of how the truth-function of abstraction works we shall c a l l the theory of truth-function of abstraction. The general line of thought i n this theory, roughly sketched, i s as follows: i t i s assumed that the artist attempts to depict the world. But i n each and every case of depicting the world special kinds of states of affa i r s exist; thus the world i s an interpreted fact, i.e., facts are formed out of states of a f f a i r s , and thoughts, which i n the case of artists take the form of pictures (or "musical" pictures i n the case of composers), express facts. This i s where the process of abstrac-tion comes i n . A l l pictures are log i c a l because they are formed by the existence of special kinds of states of affairs and these are expressed by certain rules or axioms. Pictures that are the result of what can rightly be called thoughts are meaningful pictures. Meaningful pictures can not only express facts based on existing states of affai r s , but also express facts based on a log i c a l p o s s i b i l i t y of things being thus or thus. - 42 -The logical p o s s i b i l i t y of things being thus or thus consists i n our a b i l i t y to picture (imagine) their being thus or thus. The pictures as an interpreted fact can represent the prototype truly or falsely. Since the elements of the picture are combined i n a definite way, i t s truth-value w i l l be determined upon an examination of the logic of i t s internal structure together with an examination of the "key" of re-presentation or execution, which w i l l t e l l how the elements of the pic-ture are correlated with objects outside so as to stand for them. The p o s s i b i l i t y or impossibility of correlating the elements with the objects for which they stand seems to depend on whether the way in which the painter has arranged the elements contains the possi-b i l i t y of expressing the relation according to his intention. But one could make a nonsensical picture have a perfectly good sense just by changing the kind of reference that some part of the picture had when i t was deemed "nonsensical". Thus i t i s we who give the elements of the picture their reference. Accordingly, i f the painting does not make sense, this i s only because we have given no meaning to some of i t s con-stituent parts. This also explains, i f i t can be called an explanation at a l l , the fascination which the modern abstract art holds for some of usjbecause of i t s wide signification we have a greater freedom to corre-late the elements i n the picture to the objects outside. Assume that an a r t i s t could give a complete description of the "real" meaning of his picture, a description that includes a l l that i s - 43 -or w i l l be the case. What additional interest could there be for us to know what might be, has been, or w i l l come to be as a logical possibility? But due to the wide signification of the elements of his picture the ar t i s t cannot give such a description; and so far as the interpreter (a c r i t i c ) can do so, his description i s always the product of induction, founded on what i s believed to be the case i n his limited part of space and time. Induction occurs, as we have seen, within the presupposed framework. Within this framework we describe such facts as we believe to know and within which we also make predictions as to facts not yet known. Thus the picture as a whole can be analyzed into facts i n d i f -ferent ways, and each way w i l l be a lo g i c a l l y possible interpretation of the picture, just as the picture can be one of the log i c a l l y possible interpretations or depictions of some thing p i c t o r i a l l y . Let us now approach the "theory of truth-function of abstraction" from the hi s t o r i c a l point of view. Earlier i n this thesis(4) we expressed the p o s s i b i l i t y that the contemporary ar t i s t might be following what was described as the Pythagorean tradition, that the reality of things l i e s i n the limiting principle of measure, proportion and number(5). Probably (4) Supra, Chpt. II, pp. 15-18. (5) Lest i t be misunderstood, I do not wish to give the impression that I think the naive metaphor of the Pythagorean cosmology i s taken l i t e r a l l y by the contemporary ar t i s t s , nor do I maintain that the system of Pythagoras, as excogitated by him, was anything lik e the modern conception of what he meant when he reportedly said, "Number i s the guide and master of human thought; without i t s power every-thing would remain obscure and confused." See Philolaos, Frag-ments 4 and 11, i n Diels, Die Fragmente der Vorsokratiker, I, pp. 408, 411. ~ - 44 -i n the minds of the Pythagorean thinkers there was no sharp distinction between symbol and object. Things were not only related to or expressed by numbers, they were numbers. Aristotle(6) says that the Pythagoreans supposed that the motion of the stars must produce sounds because those huge bodies move with such a great speed. As we described earlier, from the observation that their speeds and the distances are i n the same ratios as the concords of the musical scale, they argued that the sounds generated by the heavenly bodies not only are expressed by, but are a "harmonia" or scale. Speaking about the visual phenomena, the extended bodies occupying space have surfaces whose shapes and colours can be measured. From these measurements theorems of geometry were derived which not only could be expressed in numbers but were numbers. In number, and in number alone, they found the i n t e l l i g i b l e world. We have seen that i t was possible to deduce from the Pythago-rean creed, "The nature of things i s Number", that the base of the real world i s a super-sensible i n t e l l i g i b l e world in which order and beauty i s one. In the view of Plato, beauty was a form apprehended by the i n -t e l l e c t which when separated, as i t were, from the senses, the inte l l e c t f i n a l l y was able to grasp the pure ideas as they existed in the realm of forms. In the view of some Church Fathers we have beauty as revealed (6) cf. Aristotle, Methaphysics, I, 5,985 b. - 45 -by way of fait h which led i n art to symbolic representation of the Uni-verse with God as i t s Architect and Geometer. In the view of some ratio-nalists we have beauty as the result of logical intuition: they believe we intuit self-evident principles from which further truths can be de-duced. The Romantics and some of our modern existentialists define beauty i n terms of w i l l and feeling by way of mystical intuition; they t e l l us that the world perceived by the senses i s doubtlessly real, but they also i n s i s t that there i s "something else", and the certainty of this "something else" i s that of a thing experienced. They assure us that they have "seen", they have-"touched" and they "know". But, to use J. L. Austin's phrase, l e t us not ransack every available cupboard to grace our feast. We need not examine a l l the possible meanings of beauty meaning "order" to get at the base of "the real" world i n the hope that that would enable us to work out truth-tables for beauty. If we accept the generic meaning of "truth" as "the end of en-quiry", where any end that i s designated i s the specific meaning of truth(7), we see that any of the c r i t e r i a , enumerated by us as a pa r t i a l l i s t of the specific meanings of the re a l i t y of the world, have served ar t i s t s , c r i -t i c s and connoisseurs for the purpose of working out of truth-tables i n order to judge one work of art "right" and the other "wrong", one painting "true" and the other "not true". Plato's end of enquiry i s reached when the eternal forms are grasped i n their purity; hence the artist who ac-cepts Plato's criterion for truth, as Zeuxis did, w i l l direct his enquiry (7) Barnett Savery, "The Emotive Theory of Truth", i n Mind, vol. 64, No. 256, 1955, pp. 513-521. - 46 -to, and combine the elements of his art so as to grasp the eternal forms i n their purity. Similarly, those who accept f a i t h as a criterion of truth, at once w i l l have accepted a norm, a specific form of representa-tion which their art w i l l show forth i n their form of representation. We can deal similarly with the Romantics and the Existentialists. The method of projection i s the thinking of the sense of the proposition. (8) We shall return to the question of r e a l i t y and truth in art i n the conr eluding chapter; presently i t i s our point that there i s no one and only one proper criterion for the truth-function of abstraction i n art, but that i n each individual case a specific criterion has been used i f and when the artist's aim has been to depict reality. As a result of these reflections i t becomes clear that the ar t i s t ' s projected intention, the work as conceived by the a r t i s t and indicated by the arrangement of his language elements, then, "internally" remains independent of change i n the state of affairs in:the world. Therefore i t i s at least theoretically possible to reconstruct the artist's conception, provided we know the particular method by which the objects (ideas, feelings, events) are projected into the work of art. However, since the elements out of which the work of art i s con-structed have a wide signification, a subjective interpretation of the work of art i s always possible and often inescapable. The fact that we do interpret works of art subjectively i t s e l f suggests that we take the (8) Wittgenstein, Tractatus, 3.11. - 47 -arrangement of the internal elements i n a work of art to stand for the states of affairs in the world. But since the "internal" structure of the elements stands for states of affair s in the world, (9) we may ask whether an imaginary pic-ture can ever be a "true" picture. An imaginary picture would be an interpreted fact of an imagined world. It i s clear that however d i f -ferent from the real one an imagined world may be, i t must have some-thing - a form - in common with the real world. This fixed form con-sists of the objects, and objects contain the po s s i b i l i t y of a l l states of affairs; objects from the substance of the world. (10) And sub-stance, we are told, i s what exists independently of what i s the case; i t i s form and content. (11) Thus i t follows that, i f a picture i s an interpreted fact of an imagined world and this imagined world has a form common with the real world consisting of objects, the truth or f a l s i t y (truth-value) of an imaginary picture w i l l be determined upon an examination of the logic of i t s internal structure together with an examination of i t s "key" of representation or execution. What i s said of a non-imaginary picture depicting the real world, must also be said (9) "State of affairs" stand for the German der Sachverhalt, and i s used here to denote something that could be the case; whereas die Tatsache means f a i t accompli, a fact. Cassell"s Mew German  Dictionary, London: Cassell & Company, 1957. (10) Tractatus, 2.022; 2.023; 2.014; 2.021. (11) ibid., 2\024 and 2.025. - 48 -of an imaginary picture: "What the picture must have i n common with r e a l i t y i n order to be able to represent i t after i t s manner - rightly or falsely - i s i t s form of representation", and further, "what every picture, of whatever form, must have in common with r e a l i t y in order to be able to represent i t at a l l - rightly or falsely, i s the logical form, that i s , the form of reality." "If the form of representation i s the logical form, then the picture i s called a logical picture. Every picture i s also a logical picture. (On the other hand, for example, not every picture i s spatial)",. It i s clear, then, that a picture may have several forms of representation, but one of these must be the logi c a l form. This does not mean that a picture must have the same logical form as what i t pic-tures, but that a l l pictures must have the lo g i c a l form. From what was said can also be deducted that the supposition that the logical form of representation i s common to one picture and real i t y , could afford no ground for supposing that i t could not be represented i n another pic-ture. (13) One more consideration of importance, which applies to every work of art, ancient or modern, but for our purposes very important here when an attempt w i l l be made in the next part to discuss the modern ab-stractionist painters, i s the consideration of the form of objects. (12) Tractatus, 2.17; 2.18: 2.181; 2.182. (13) F. P. Ramsay, "A C r i t i c a l Notice of Tractatus" i n Mind, Oct. 1923, pp. 465-478. - 49 -Objects, such as tables, chairs, books, are complex objects or things, but the world, as we have seen, i s the t o t a l i t y of facts and not of things. Thus when we say that a picture represents that certain objects are combined or related i n a certain way, we mean that the elements of the picture, and not things, are combined in that way. Since the objects (elements) of the picture are correlated with the "objects" ("outside") so as to stand for them, we may ask, what are the forms of these objects? - Wittgenstein gives us the answer, "Space, time and colour (colouredness) are forms of objects".(14) These then are the forms which a "simple object" has. And any "complex object" i s capable of reduction to these "simples". These statements, among others i n the Tractatus, e.g., that there are certain things which cannot be said but only shown, consti-tute the Mystical. The reason why certain things cannot be said, i s that they have to do with the lo g i c a l form which the picture has i n common with re a l i t y . But the logical form of representation cannot be represented. (15) What sort of things the things are which cannot be said but only shown i s explained as follows: We can speak in a certain sense of formal proper-ties of objects and atomic facts, or of properties of the structure of facts, and i n the same sense of formal relations and relations of structures. (instead of property of the structure I also say 1 internal property'j (14) (15) Tractatus, 2.0251. (But ... "There are no 'logical objects'" i b i d . , "4.441 and 5.32: 5.4). cfT" Tractatus, 4.121. - 50 -instead of relation of structures •internal relation'. I introduce these expressions in order to show the reason for the confusion, very widespread among philosophers, between internal relations and proper (external) relations.) The holding of such internal properties and relations cannot, however, be asserted by propositions, but shows itself in the propositions, which present the atomic facts and treat of the ob-jects in question. (16) ibid., 4.123. - 51 -Chapter V. The Truth-Function of Abstraction. In view of what was said about art, reality and the truth-function of abstraction, we hope i t has become clear now that art can admit a unicorn despite of the fact that zoology cannot, and yet art is concerned with the real world as truly as zoology, though with its more abstract and general features. Presently we shall investigate the claims of some of the "non-objective" artists that their aim is not to depict reality but to create i t . First of a l l , i t must be pointed out that what is generally understood as "abstract art" is divided, roughly speaking, into two kinds: (a) an art in which the world of objects is represented by means of schematized shapes called "abstractions", thus involving dis-tortion of the objects represented; and (b) an art in which the ele-ment of representation is rejected and which is known as "non-figura-tive" or "non-objective" art. To the "abstract art" of the first kind belongs, according to the customary division, cubism, futurism, expressionism, dadaism, surrealism, etc.; to the "non-objective" kind, suprematism, constructivism, neoplasticism and abstract expressionism, etc. - 52 -It was already said that in art the world i s pictured as a "fact" and not as a "thing", therefore there i s no sense i n speaking of "objective" art as of necessarily "imitative" i n a narrow sense of the word, although we do not see any reason why i t could not be, i f that was the artist's aim. And, of course, the "non-objective" art could turn out to be as "imitative" of the objective world as any, i f for reasons of his own the art i s t had had in mind to depict the objective world by the use of a very complicated method of projections. (1) The words "imi-tative" and "representational" refer to a level of intellectualization, a conceptual level i n the hierarchy of abstraction. The branching out of art into various movements and schools i s explained, on the one hand, by taking into consideration tradition and custom, i.e., the learned ex-perience, but, on the other hand, also by accepting that, since tradition and learned experience do not actually explain how a new tradition may arise, the visual world such as colours, surfaces, shapes and edges are seen as meaningful " i n themselves".(2) In our view i t i s possible to regard colour as a symbol of some thing (event, feeling or idea) pic-t o r i a l l y only i f one and only one system of presentation of the world p i c t o r i a l l y i s presupposed.(5) As we have already stated(4), there i s (1) Richard Strauss had said, perhaps jokingly, "I can express i n music the moving of a pencil from one place to another". Arnold Schonberg warned the composers of the possi b i l i t y that some day "psychologists and psychoanalysts w i l l have deciphered the langu-age of music. Woe, then, to the incautious who thought his innermost secrets carefully hidden ..."; See Style and Idea, p. 209. ~~ (2) Chapter II, "Percept and Concept i n Art", supra. (3) Supra, Chapter IV. (4) Supra, Chapter II. -53 -yet another explanation how people come to regard the world as different from what either their predecessors or contemporaries thought i t was l i k e , namely, by the exercising of their logical faculty i n arranging the constituents of their world differently,and by a new configuration of the elements "creating" a new reality. In this process of re-arranging we do make use of antecedent conceptual achievements, but logic precedes every experience - that something i s soj but that i s no experience. Logic i s before the .How, not before the What.(5) This i s so not only i n art but i n a l l s c i e n t i f i c and intellectual a c t i v i t i e s . For example, when Einstein developed his general theory of r e l a t i v i t y , X be went back to Riemann's geometry which had been created long before , but which Riemann regarded only as a mere logical possibility. Riemann,s achievement was the hypothesis for Einstein to f i t the description of facts. Similarly, Schdnberg u t i l i z e d the achievements of Wagner, Mahler and Debussy to develop the method of composition with twelve-tones. (6) In this way Cezanne can be regarded as the progenitor of Georges Braquej likewise Monet's paintings are admittedly the "eye-openers" to Kandinsky^?), who after having seen an exhibition of Monet's paintings i n Moscow later wrote: "I had the impression that there painting i t s e l f came to the foreground; I wondered i f i t would not be possible to go further i n this (5) Tractatus, 5.552. (6) Supra, Chapter III, p. 21. f f . (7) Nina Kandinsky, "Some Notes on the Development of Kandinsky's Painting", i n Concerning the Spiritual i n Art, N.T.: George Wittenborn, 1947, p. 10 (Original German ed., 1912). - 54 -direction. From then on I looked at the art of ikons with different eyes; i t meant that I had 'got eyes' for the abstract in art." This was written by Kandinsky in his Notebooks in reference to Monet's paintings of haystacks(Q) i n which the aim was to achieve, i n Monet's own terra, "instantaneity", a point i n time expressed i n colour-light relationships?- Kandinsky's "Improvisations" and the later "geometri-cal explosions i n space" also might be described as "instantaneities", p i c t o r i a l descriptions of space-time. But i n each and every case exemplified here there i s a basic difference between the work of pro-geny and that of his progenitor (or progenitors) as there must also be the p o s s i b i l i t y to pinpoint the moment (or series of moments) when the new logical p o s s i b i l i t y of working out elemental relationships was con-ceived (one can learn as well from an unsuccessful example as from a successful one). We concede that very often attempts to express i n words something that can only be shown result i n failures of expression. Phrases such as Kandinsky's favourite expression that i n ab-stract art "the mystical i s expressed by the mystical" and that "this art creates alongside with the real world a new world which has nothing to do externally with reality. .. It i s subordinate internally to cosmic (8) Exhibited at Durand-Ruel's i n Paris i n 1891. These 15 paintings were a great success. Unfortunately, I have not been able to find out during which periods of Kandinsky's residence i n Moscow the Monet exhibition was held. It must have been during his student days at Moscow University c. 1894-96. (9) W i l l Grohmann, Wassily Kandinsky, N.Y. Harry Abrams, 1958, p. 114. - 55 -laws" (10) are such failures of expression; when taken out of the con-text i n which they were used, and analyzed from a biased point of view, these indeed can be taken as "pathetic bluff" devised to "create a smoke screen around abstract art ...."(11). The truth i s that i n most cases the ar t i s t deserves to be pitied for his verbal incompetency to present his case clearly and to be admired for his courage to try. It i s a truly different task to equally well express i n words what can successfully only be shown. Kandinsky, Georges Braque and Piet Mondrian, to name a few of the highly original Twentieth Century artists who grappled with every new question connected with their art, have l e f t us writings which are d i f f i c u l t to read and to analyze,( x^) partly because of their peculiar l i t e r a r y style, partly due to their use of philosophical expressions (10) In an article i n XX me Siecle, 1935, quoted by Mme Nina Kandinsky, op. c i t . , p. 10. (11) Wagner, Geoffrey, "Organized heresy: abstract art i n the West", in Connoisseur, vol. 39, 1958. (12) Kenneth Lindsay i n an unpublished doctoral dissertation, An Exami- nation of the Fundamental Theories of Wassily Kandinsky, University of Wisconsin, 1951, describes Kandinsky's style as follows: "Characteristic of Kandinsky's writing i s the technique of breaking up the given topic into opposites or alternatives. These opposites or alternatives usually follow directly after the posing of the problem and are numbered. Often they suggest further sets of opposites and alternatives. The sequence of thought i s flexible, sometimes abrupt and cross-tacking, and frequently associative. The dominating r e l a t i v i t y of the thought process contrasts strongly with the conclusions, which are often positively stated." Quoted from Peter Selz, "The Aesthetic Theories of Wassily Kandinsky and their Relationship to the origin of Non-objective Painting", i n Art Bulletin, vol. 39, #2, 1957; pp. 127-136. - 56 -that often reflect only the surface of their basic ideas. One such basic idea is the creation of new realities. We have already stated corresponds or can be reconciled with some of their statements. The most lucid statement, to our knowledge, to the testimony that the modern abstractionist of the "non-objeetivist" kind is bent on creating reality, not on depicting i t , is by the "constructivist" Naum Gabo, who writes: I maintain that knowledge is nothing else but a con-struction of ours and that what we discover with our know-ledge is not something outside us or part of a constant and higher reality, in the absolute sense of the word; but that we discover exactly that which we put into the place where we make the discoveries ... We only know what we do, what we make, what we construct; and a l l that we make, a l l that we construct are realities. I call them 'images', not in Plato's sense ... but I hold that these images are the reality itself and that there is no reality beyond this reality except when in our creative process we change the images: then we have created new There is no desire here to master objects in the "real" world, no wish to compare "constructivism" with scientific experimentation, or gain greater intellectual respectability for the "non-objective" art by likening i t to pure science which also deals with abstractions, as, for example, we find in Franz Marc's diary entry, Christmas 1914: our view of what this idea conveys(13)j let us now see whether i t (13) supra, Chapter IV, 'Parfc-1. (14) Naum Gabo, "A Retrospective View of Constructive Art", in Three  Lectures on Modern Art, N.Y.: Philosophical Library, 1949. - 57 -I am beginning more and more to see behind, or, to put i t better, through things, to see behind them some-thing which they conceal, for the most part cunningly, with their outward appearance by hoodwinking man with a facade which i s quite different from what i t actually covers. Of course, from the point of view of physics this i s an old story ... Scientific interpretation has powerfully transformed the human mind; i t has caused the greatest type-change we have so far lived to see. Art i s indisputably pursuing the same course in i t s own way certainly; and the problem, our prob-lem, i s to discover the way. (15) In fact, Gabo's view comes very close to that expressed i n Wilde's paradox. Since this striking passage not only serves to i l l u s -trate what we would c a l l the "egocentric" view of beauty but also shows that, what Gabo would l i k e us to take for the distinguishing mark of the "non-objective" art from the "objective", the creating of r e a l i t i e s (in-stead of depicting them) i s seen as the privilege of both the "abstract" and the "concrete" art, we quote i n f u l l : ... Paradox though i t may seem - and paradoxes are a l -ways dangerous things - i t i s none the less true that Life imitates Art far more than Art imitates Life ... A great ar t i s t invents a type and Life t r i e s to copy i t , to re-produce i t in a popular form, l i k e an enterprising pub-lisher. Neither Holbein nor Vandyke found in England what they have given us. They brought their types with them, and Li f e with herkeen imitative faculty set herself to supply the master with models ... -Where, i f not for the impressionists, do we get those wonderful brown frogs that come creeping down our streets, blurring the gas-lamps and changing the houses into monstrous shadows? To whom, i f not to them, do we owe the lovely silver mists that brood over our river, and turn to faint (15) Franz Marc, quoted by Peter Thoene (pseud.), Modern German Art, Harmondsworth: Pelican Books, 1938. •=5 - 58 -forms of fading grace curved bridge and swaying barge? The extraordinary change that has taken place i n the climate of London during the last ten years i s entirely due to this particular school of Art. (16) Now, at the risk of flogging the converted, I should l i k e to dwell a l i t t l e longer on the idea of creating r e a l i t i e s i n art.^7) In what sense are a Greek centaur, Gothic or Renaissance Madonna, and a Mondrian's Boogie-Woogie real? Speaking about centaurs, what exists i n art i s not an animal of flesh and blood, but a picture, a representation. To maintain that a Madonna painted, e.g., by Raphael Sanzio, exists i n her own world as truly as Raphael existed i n the "real" world i s to say something delibe-rately confusing. There i s only one world, the "real" world: Raphael's imagination i s part of i t , and the thoughts he had i n painting the pic-ture of the Madonna are real. So are the thoughts we have i n looking at the picture. The thoughts with which I approach Mondrian's "Broadway Boogie-Woogie" (1942/43, New York, Museum of Modern Art) perhaps shows me very clearly what boogiewoogie i s , yet I cannot from the picture de-rive rules how to dance i t . Contrary to what can be deduced from the statement of Gabo and what in his witty way Wilde wished us to believe, (16) Oscar Wilde, "The Decay of Lying", in Intentions and the Soul of Man, London: Methuen, (1891), 1919. (17) An illuminating and entertaining discussion of the use of words "real" and "reality" can be found i n J. L. Austin's Sense and  Sensibilia, ed. G. J. Warnock, Oxford: Clarendon Press, 1962. Especially Chapters VII - IX. - 59 -i t i s the very essence of art that, apart from the material substance of a picture, only the thoughts and feelings of the a r t i s t and his audience are real, and that there i s not, i n addition to them, an ob-jective reality. When you have taken account of a l l the thoughts and feelings roused by Raphael Sanzio i n writers and readers of history, you have not touched the actual man; but i n the case of any one of Raphael* s pictures you would have come to the end of i t . If no one had thought about Raphael's Madonna, there would be nothing l e f t of her: i f no one had thought about Raphael Sanzio, he would have soon seen to i t that someone did.^8) In order to be f a i r to Naum Gabo, we must say that his state-ment should be read as a protest and rejection of the work of art as an imperfect image of an ideal form. Elsewhere Gabo propounds the thesis that art i s a social force, i.e., that i t w i l l be the values i n the cul-ture of the individual a r t i s t that w i l l determine his choice for the particular set of rules he employs i n "constructing" his art. He says: "Abstract art ... involves the whole complex of human relations to l i f e . It i s a mode of thinking, acting, perceiving and l i v i n g . " ^ 9 ) j n their Constructivist Manifesto of 1920 Naum Gabo and Antoine Pevsner^O) formu-lated the rules of their art, i n summary, thus: (18) The foregoing i s modelled on and paraphrased from Bertrand Russell's "Descriptions", i n Introduction to Mathematical Philosophy, 2nd ed., London: Allen & Unwin, 1920. (19) Gabo, N., "Constructive Art: An Exchange of Letters between Naum Gabo and Herbert Read", in Horizon, London, vol. X, #55, July 1944, pp. 57-65. (20) Both men are brothers; In 1914 Naum changed his name to Gabo to distinguish himself from his brother. - 60 -1. To communicate the r e a l i t y of l i f e , art should be based on the two fundamental elements: space and time. 2. Solid volume i s not the only spatial concept. 3. Kinetic and dynamic elements must be used to ex-press the real nature of time; static rhythms are not sufficient. 4. Art should stop being imitative and try instead to discover new forms. (21) The f i r s t and the last of these rules are inclusive general statements, the second and third are exclusive principles. The former j u s t i f y the admission of constructivism i n the realm of art; the latter place i t legitimately i n the realm of sculpture. Space i s conceived not as de-fining sculptural mass but as transparencies. This concept has i t s parallel i n the overlapping planes used by the cubists. The kinetic (or kinaesthetic?) element expressed by the idea that the objects i n a work of art are related as moving parts sometimes i s exemplified by linear means, sometimes by operating mechanisms. If colours are introduced, they are mostly confined to the primary colours, red, blue and yellow, but colour, generally, does not play an important part i n constructivist art. Light, instead of being conceived as colour, i s understood tempo-r a l l y as linear movement or mechanical speed, and spatially as trans-parencies . It i s not at a l l d i f f i c u l t to show that space, time and colour are indeed the fundamental elements, or to use the Wittgenstein phrase, "forms of objects", with which the a r t i s t , any a r t i s t , operates according (21) Herbert Read, in Introduction to N.Y. Museum of Modern Art Cata-logue, Naum Gabo and Antoine Pevsner, N.Y.: 1948, p. 10. - 61 -to some plan. Constructivism i s an attempt to describe i n one par-t i c u l a r way, according to a single plan, what the world i s like} and as such i t does not d i f f e r from any other description according to single plan, such as Cubism, Neo-plasticism, Futurism, etc., from the modern era, or whatever "school" we wish to choose from any other era. This asserts nothing about the world; i t only shows that the world can be described i n that p a r t i c u l a r way i n which as a matter of fact i t i s described. In a sense Kandinsky's "space explosions" do not d i f f e r b a s i c a l l y from EL Greco's "nervously scattered l i g h t s " : there i s the picture plane and format, colours and l i n e s which form s t a t i c and dynamic planes and volumes; these overlap and by overlapping, or by t h e i r a x i a l placement, create movement; there are tensions of bulges and angles which express potential movement; there are colour gradations with movement from l i g h t to dark; i n a word, there are tensions, con-t r a s t s , gradations, movement and other relations between the elements to which we assign meanings. This i s very w e l l expressed by Piet Mondrian: .... Universal beauty does not arise from the p a r t i c u l a r character of the form, but from the dynamic rhythm of i t s inherent relationships, ... from the mutual r e l a t i o n of forms. Art has shown that i t i s a question of determining the r e l a t i o n s . I t has revealed that the forms exi s t only f o r the creation of new relations ... ... The d e f i n i t i o n s (terms) 'figurative' and ,'non-fi g u r a t i v e ' are only approximate and r e l a t i v e . For every form, even every l i n e , represents a figure .... since we need - 62 -words to make our concepts understandable, we must keep to these terms. (22) We have seen the importance of relations between tones i n music. A Beethoven symphony and a SchOnberg composition symbolize, so to speak, human experience through structures of sound, but whereas a Beethoven symphony i s b u i l t upon one type of relationship between struc-tures of sound,whereby each tone i s related to a common centre, and where the contrasts, gradation, tensions and resolutions are worked out by the demands for the tonic, the Schonberg twelve-tone music i s built upon a different interrelationship of structure of sound, based on what we termed " r e l a t i v i t y principle", whereby sounds are interrelated without the employment of a common centre. The relations of contrasts and grada-tions, tensions and resolutions are s t i l l there, but not of the kind i n -herent i n tonal music. As a Beethoven symphony i s f u l l of association references to human l i f e , so also "pure music" w i l l never be disassoci-ated from human actions and desires. It i s our contention that the situation i n painting and sculp-ture i s analogous to the situation i n music: the "non-objective" a r t i s t never r e a l l y thought of his "non-objective" paintings as being divorced from nature. Kandinsky says: (22) Piet Mondrian, "Plastic Art and Pure Plastic Art (Figurative Art and Non-Figurative Art)", in Circle, International Survey of Constructive Art, eds. J.L. Martin, Ben Nicholson, N. Gabo. London: Faber & Faber, 1938 ?, p. 41 and p. 42. - 63 -The isolated line and the isolated f i s h alike are li v i n g beings with forces peculiar to them, though latent. They are forces of expression for these beings and of impression on human beings ... But the voice of these latent forces i s faint and limited. It i s the environment of the line and of the fi s h that brings about a miracle: the latent forces awaken, the expres-sion becomes radiant, the impression profound .... The environment i s composition .... ... The composition i s the organized sum of the inter-i o r function (expressions) of the work.(23) It i s not the object as a "thing" which attracts the a r t i s t : There i s an essential difference between a line and a f i s h . And that i s that the f i s h can swim, can eat and be eaten. It has the capacities of which the line i s deprived. These capacities of the f i s h are necessary extras for the f i s h i t s e l f and for the kitchen, but not for painting. And so, not being necessary they are superfluous. (24) Finding of the "literary" values in a work of art superfluous, at least "extras", was in painting and sculpture a similar logical achieve-ment as the rejection of the tonal relationship i n music. It seems that by accepting the idea that the object as a "thing" plays a similar role i n painting as the tonic in the tonal music, the next step i n what i s usually called "liberation of painting from the objective world" was to institute a new method (or methods) for the interrelation of the elements. As soon as the elements of art were seen as space-time units independent of a "common centre", they could be related to each other i n many ways. With this i n mind, i t i s interesting and amusing to read (23) Kandinsky, "Line and Fish", i n Axis, II, 1935. (24) ibid.,p. 6. - 64 -Grohmann' s perplexity i n his analysis of Kandinsky's Composition VIII (1923, S. R. Guggenheim Gallery, New York): The c i r c l e s , triangles, and checkered figures, the linear elements - straight lines, angles, curves, semi-circles - stand side by side, occasionally over-lapping, seemingly unconnected. Colours are re-duced to rainimumj the ground of the upper right i s whitish, that of the lower l e f t blue; the only ex-ception to primary blues, red, yellows i s the violet inside the black c i r c l e at the upper l e f t , which do-minates the picture. The pink aura around the c i r c l e fixes i t s position i n space. But does the black c i r -cle recede into or protrude from the picture plane? The vermillion red c i r c l e tangential to i t makes i t seem to protrude, while the violet c i r c l e at the centre makes i t recede. The pink aura contributes to the ambiguity. The rest of the composition i s f l a t and i t i s impossible to say whether the variously coloured small circles create depth, or are merely points of rest i n the staccato movement of the predominantly linear design. The colour areas around the linear elements contribute to the airiness of the painting. In the checkered figures blacks and whites predominate counterposing the small vermillion-coloured circles and the light blue triangle. The resulting balance i s nonetheless enigmatic, or perhaps we should say that the angle standing i n the middle makes a l l the elements converge i n harmony?(25) (25) Will Grohmann, Wassily Kandinsky, Life and Work, N.Y.: Harry N. Abrams, 1958, p. 190. - 65 -Appendix. Feeling and Intellect in Art. Our basic theme can be expressed something lik e this: Each work of art exhibits a criterion i n terms of which the elements i n the work of art are arranged i n a definite order. (1) We are never able to give rational grounds for the adoption of one criterion rather than (l) In this thesis I am concerned only with problems connected with works of art and not with artifacts that are not primarily works of art, nor with things such as natural objects. As an answer to the question, "Can a 'work of art' be defined?" I am offer-ing the following which I owe to Professor Barnett Savery: "A work of art i s a sincere, imaginative creation, embodied i n a sensuous form, expressing (conveying, embodying) feeling and/or ideas, which, when contemplated with the aesthetic attitude, gives an aesthetic enjoyment (satisfaction), some of which arises from the sensuous form contemplated." No doubt, we derive many i n -tense aesthetic satisfactions from scenery, from natural objects, from artifacts, foods, drinks, etc., and, of course, also from the "rightness" or "orderliness" of the operations i n any purely intellectual f i e l d such as logic and mathematics. To bring out the f u l l force of the term "aesthetic enjoyment" ("satis-faction"), one would have to show that the aesthetic judgment i s something quite different from moral, economic, intellectual, etc., judgment. Let me offer this simile of Wittgenstein as an example of what i s meant by "aesthetic judgment" i n contradistinction to "mathematical judgment": "Suppose the members of an imaginary savage tribe decorate the walls of their huts by writing on them rows of Arabic numerals - and suppose that what they write i s exactly what would be written by someone doing arithmetical c a l -culations. They do i t exactly right every time, but they never use i t except for decoration - never use i t i n computing how much wood they need to build a hut or how much food they need for a feast, and so on. Would you say they were doing mathematics?" cf.D.A.T.G., A.C.J., "Ludwig Wittgenstein" i n The Australian  Journal of Philosophy, vol. 29, #2, 1951, p. 75. - 66 -another. We must search for the criterion adopted i n the fi e l d s of biology, sociology, economics, p o l i t i c s , psychology, religion, i n short, i n experiencej we do not explain anything by presupposing the inter-vention in art of some mysterious entities which no one understands. Abstract art cannot be explained by saying that an "abstrac-tio n i s t " i s inspired or "possessed" by the "Abstract Muse" and the "con-crete" art by the intervention of the "Concrete Muse" (2). but we come f a i r l y close to saying something similar when we refer to abstraction i n art as "dread of space" or "fear of nothingness" or "negation of l i f e " as opposed to "affirmation of l i f e " i n "concrete"art. Even i f we assume that by these pronouncements the issue has been c l a r i f i e d , we have actu-a l l y done nothing more than tried to explain by a single narrowly de-fined principle something which has a wide signification. We seek to ju s t i f y i t by saying that we have thus found a universal principle which describes the world more simplyj the fact i s , we have formulated a speci-f i c theory, just one of many va l i d value-theories which has no absolute va l i d i t y . We do have a basis for accepting one criterion for descrip-tion of the world rather than the other, and i t w i l l be the values i n our culture that w i l l determine which i t w i l l be; but we should beware (2) cf. Plato, Ion, 534, 536: "The stone Euripides c a l l s magnet does not only attract iron rings, but i t also gives them power of at-tracting other rings as the stone i t s e l f does ... In the same way the Muse herself inspires the a r t i s t s , and through their inspira-tion others are enraptured, and the line of the inspired i s pro-duced ... One poet i s suspended from one Muse, another from anotherj he i s said to be 'possessed'". - 67 -lest we make our criterion the Procrustean bed for one and a l l descrip-tions of the world. We can only formulate one generic meaning and n specific meanings of "abstract art", and among the latte r we may find one defined as "negation of l i f e " . For, indeed, among the "abstract" artists as well as among the "concrete" ones there are some who tend toward what Thomas Mann cal l s the dark and death, and others who re-joice in light and a i r . The confusion i s generated by the widespread belief that ab-straction i s essentially a r t i f i c i a l , "unnatural" and d i f f i c u l t , and that a l l untutored thought i s "natural" and bound to conrete experiences of physical things. The fact i s that reference to the real world does not disappear from "abstract" art as i t ceases to use shapes and colours of the actually existing things any more than objectivity disappears from science when i t ceases to talk i n terms of earth, a i r , f i r e and water. As Susanne K. Langer points out, "If abstraction were really unnatural, no one could have invented i t . If the untutored mind could not perform i t , how did we ever learn i t ? We can develop by training only what i s incipiently given by nature. "(3) In the case of " r e a l i s t i c " or "objec-tive" painting and "program" music, i t seems clear that recognizable ob-jects such as chairs or persons i n the painting, or the tonally "painted" portrait of an object or person i n music, furnish a vocabulary of signs (3) S. K. Langer, Form & Feeling^ London: Routledge & Kegan Paul, 1953, p. 80. - 68 -which are then combined "grammatically" in various ways according to an arbitrary chosen set of rules. (4) In the case of "abstract" art i t is the generality of signification (and, obviously, different sets of rules) that misleads some of us into strange and confused conclusions about i t s meaning. Apart from (a) those who find no meaning i n "abstract" art and (b) those who ascribe n meanings to i t (thus making i t meaningless) there are (c) those for whom departure from the "recognizable" presentation means one and only one thing. As an example of such a case (c) we are offering the view of Wilhelm Worringer(5) whose thesis of abstraction preceded Kandinsky's f i r s t non-objective painting by about two years and whom Kandinsky must have known in Munich. (6) Worringer also profoundly influenced Herbert Read during the latter's studies of philosophy of art i n Munich; i n fact, so much so that he finds i n Worringer's Abstraktion  und Einftihlung " a l l the features which distinguish abstract art as such ... clearly recognized", and thinks that i t i s indeed "... possible that the theory of abstract art not only preceded the practice of i t i n modem (4) Poussin, for example, said: "Painting i s nothing but an image of incorporeal things despite the fact that i t exhibits bodies." (5) Wilhelm Worringer, Abstraktion und Einfuhlung, Munich: R. Piper Verlag, 1908. The English edition, Abstraction and Empathy, London: Routledge & Kegan Paul, 1953"i (6) Herbert Read in The Philosophy of Modern Art, N.Y.: Meridian Books, 1957, a book dedicated to Worringer says that i t is significant that Kandinsky's On the Spiritual i n Art was published by the same house (R. Piper Verlag) in 1912, that had published Worringer's dissertation. The members of the "Blaue Reiter" group, which was also founded i n Munich i n 1912, were a l l philosophically minded and must have followed up Worringer's ideas, cf. pp. 105-106. - 69 -times, but actually inspired and influenced i t s development. "(7) There i s no doubt i n our mind that the writings of Kandinsky bear a strong mark of Worringer' s thesis, but we are not convinced at a l l about the truth of Read* s claims concerning Kandinsky's a r t i s t i c development, or for that matter, the development of any of the "non-objective" or "abstract" painters i n Munich. In Abstraction and Empathy Worringer suggests that the cause for abstraction i s man's inclination to withdraw from the world because of his antagonism toward i t : Now what are the psychological presuppositions of the tendency to abstraction? We have to seek them i n the world feeling, ("Weltgefuhl" or "Weltansichtung") of those peoples, i n their psychological relation to the cosmos. While the tendency of empathy has as i t s condition a happy pantheistic relation of confidence between man and the phenomena of the external world, the tendency to abstraction i s the result of a great inner conflict between man and his surroundings ... This state we might c a l l a prodigious mental fear of space ... A comparison with that physical dread of open spaces, which as a disease a f f l i c t s certain per-sons, w i l l perhaps explain more f u l l y what we under-stand by the psychological fear of space. This physi-cal dread may be popularly regarded as a vestige of a normal stage of human evolution, i n which man, trying to become accustomed to surrounding space, could not re l y on visual impressions alone, but s t i l l needed to be reassured by his sense of touch. As soon as he be-came a biped and thus for the f i r s t time appeared i n human form, a slight feeling of insecurity must have (7) H. Read, op. c i t . , p. 244; i t a l i c s are Read's. (8) Worringer, op. c i t . , p. 16. - 70 -Abstraction i n art i s shown by Worringer as characteristic of the Primitive, the Egyptian, Byzantine, Gothic, and Oriental art as well as of the "dehumanized" Contemporary art i n a c a l l for absolute values, and i s contrasted with the late Greco-Roman, Renaissance, and the "natura-l i s t i c " ("representational") Western art which i s characterized by enfc-pathy.(9) Empathy i s the happy, sympathetic approach of man to the world. Elsewhere Worringer t e l l s us about abstraction: It expresses no joyful affirmation of sensuous v i -t a l i t y , but belongs rather to the other domain, which through a l l the transitoriness and chances of l i f e strives for a higher world, freed from a l l illusions of the senses, from a l l false impressions, a domain i n which inevitabliness and permanency reign.CIO) (9) The word "empathy" i s a rendering of German "die Einftthlung" ("ein" i n & "ftthlung" feeling) by Greek "empathos" (in & feeling) and i s used to denote, in this context, "the power of entering into the experience of or understanding objects or emotions out-side ourselves," (0.E.D.), and was f i r s t made use of by the German psychologist and philosopher Theodor Lipps (1851-1941) i n his Aesthetic, 2 vols., 1903-1906, where he makes a l l a r t i s -t i c appreciation depend upon man's capability of projecting himself into what i s seen, or into the feeling of another. Bodily responses to a situation are very important: a column that i s too slight to support a weight i s unpleasant because i t makes the observer strain undulyj a v e r t i c a l line i s ap-parently longer than a horizontal one because i t induces the observer to stretch up and the movement i s assigned to the li n e . The kinaesthetic sensation and muscular adjustments are the symptoms of emotions and ideas projected into the object. Wilhelm Worringer takes the theory of Lipps as his point of departure. (10) Wilhelm Worringer, Form in Gothic, London: G. P. Putnam's Sons, 1927, p. 37. - 71 -As soon as "yes" and "no" values are assigned to "abstract" and "concrete" art i n an absolutistic theory such as Worringer's, ex-ceptions to i t are found and a l l sorts of ingenious excuses have to be sought to defend i t . Herbert Read, for example, speaking of symbolic art(11) finds the Byzantine art an apparent exception to the rule that symbolic art, "based on the eidetic imagery" eventually loses i t s v i t a -l i t y as i t i s gradually divorced from immediate emotional association of the object symbolized. Yet i n symbolic art man imaginatively trans-cends his "human, a l l to human" nature by contemplating "eternal" forms. It i s not at a l l clear, then, whether symbolic art belongs to the "ab-stract" or "concrete" kind. We noted earlier that i t may be that Kandinsky's writings owe much to his knowledge of Worringer's thesis. (3.2) Whenever he i s philo-sophizing about the aims i n art (to "turn away from the soulless l i f e of the present toward those substances and ideas that give free scope to the non-material strivings of the soul" )(13) ; or when he wishes to point out the difference between "non-objective" and "objective" art ("huma-nity i n general inclines to external beauty and knows nothing of internal (11) Herbert Read, Art and Society. (12) I have not been able to find positive sources of this. The hypo-thetical ones, and they are pretty convincing, I am using, are: Peter Selz, op. c i t . , p. 129 and Herbert Read, op. c i t . , pp. 105 -106; also W. Grohmann, op. c i t . , pp. 85, 87, I'45. Some writers, e.g., Lothar-Gunther Buchheim i n Per Blaue Reiter and die Neue  Ktinstlervereinigung Miinchen, Feldafing: Buchheim Verlag, 1959, makes no mention of Wilhelm Worringer. (13) Wassily Kandinsky, op. c i t . , p. 33. - 72 -b e a u t y " ; " S c h o n b e r g * s . . . m u s i c l e a d s us t o where m u s i c a l e x p e r i e n c e i s a m a t t e r n o t of t h e e a r , b u t o f t h e s o u l . . . " ; " M a t i s s e . . . e n -d e a v o u r s t o r e n d e r t h e d i v i n e " ) ( 1 4 ) h i s p h i l o s o p h y smacks o f W o r r i n g e r ' s a e s t h e t i c s w i t h i t s a n t a g o n i s m t o w a r d t h e w o r l d . P a u l K l e e a l s o w r o t e i n h i s d i a r y l i n e s r e m i n i s c e n t o f W o r r i n g e r : " The more h o r r i b l e t h e w o r l d ( a s i t happens t o be i n o u r d a y ) , t h e more a b s t r a c t a r t becomes , w h i l e a happy w o r l d p r o d u c e s a s e c u l a r a r t . " I n h i s a u t o b i o g r a p h i c a l " T e x t A r t i s t a " K a n d i n s k y g i v e s p s y c h o l o g i c a l r e a s o n s f o r t u r n i n g t o " a b -s t r a c t " e x p r e s s i o n : " T h e naked b o d y , i t s l i n e s and movement somet imes i n t e r e s t e d me, b u t o f t e n m e r e l y r e p e l l e d me. Some p o s e s i n p a r t i c u l a r were r e p u g n a n t t o me, and I had t o f o r c e m y s e l f t o c o p y t h e m . I c o u l d b r e a t h e f r e e l y o n l y when I was o u t o f t h e s t u d i o d o o r and i n t h e s t r e e t once a g a i n . " ( 1 5 ) F r a n z M a r c , t u r n i n g t o w a r d " n o n - o b j e c t i v e " p a i n t i n g s h o r t l y b e f o r e h i s d e a t h , g a v e a s i m i l a r r e a s o n : " V e r y e a r l y i n l i f e I f o u n d man u g l y ; t h e a n i m a l seemed t o me more b e a u t i f u l and c l e a n e r , b u t e ven i n i t I d i s c o v e r e d so much t h a t was r e p e l l i n g and u g l y t h a t my a r t i n s t i n c t i v e l y and b y i n n e r f o r c e became more s c h e m a t i c and a b -s t r a c t . " ( 1 6 ) ( 1 4 ) i b i d . , p . 3 6 . ( 15 ) K a n d i n s k y , " T e x t A r t i s t a " , W a s s i l y K a n d i n s k y M e m o r i a l , N .Y . : So lomon R. Guggenhe im F o u n d a t i o n , 1945 , p . 6 5 . T h i s t e x t b y K a n d i n s k y was p u b l i s h e d i n B e r l i n i n 1913 b y Der S t u r m u n d e r t h e t i t l e R i l c k b l i c k e . ( 1 6 ) F r a n z M a r c , " B r i e f e , A u f z e i c h n u n g e n und A p h o r i s m e n , B e r l i n , 1920 ; l e t t e r , A p r i l 1 2 , 1915 , q u o t e d i n P e t e r S e l z , opT c i t . , p . 129 . - 73 -Worringer's thesis of abstraction and empathy^coupled,with carefully chosen quotations from a r t i s t s beginning with Monet(17^ i s used to show the process of "dehumanization" (18) i n art by abstraction which i s equated with "negation of the world". This theme i s taken up by the Existentialists whose outlook on l i f e , that i t i s incapable of rational account and devoid of senses, i s said to express the mood of the contemporary Western World. In the "existentialist art"(19) ( i f such an art exists) Worringer's "dread of space" i s said to denote the same thing as Heidegger's "fear, of nothingness". The existentia-l i s t philosophers have sought to show that i t i s possible for the i n d i -vidual to react to the world positively or negatively, with despair or with courage, with fear or with confidence, optimistically or pessimis-t i c a l l y . Herbert Read says: (17) Monet i s reportedly to have said to Clemenceau: "I am standing by the bed of a dead person, a woman whom - well, I had loved very much indeed ... and s t i l l love very much. I look at her eyelids. I said to myself: 'There i s a kind of purple ... what kind of blue i s contained i n i t ? And red? And yellow?'" -Which i s to show, according to Max J. Friedlander, i n On Art  and Connoisseurship, Boston: Beacon Press, (1942), 1960, p. 28, that "the absolute visual art was preparing to become inhuman." See also quotations supra, from Franz Marc and Kandinsky (notes 57 & 57). (18) cf. Ortega y Gasset, La dehumanizacion del' arte, Madrid: 1925. (19) Here should, I think appropriately, William Blake's verse i n Annotations to Sir Joshua Reynolds' Discourses be quoted: " A l l pictures that's Painted with Sense and with Thought Are painted by Madmen, as sure as a Groat; For the Greatest the Fool i s the Pencil more blest, And when they are drunk they always paint best." - 74 -In certain cases i t seems possible for an individual to alternate between the extremes presented by this polarity - to tend i n one psychological phase towards an affirmation of the world which results i n naturalis-t i c style, and i n another psychological phase to tend towards a negation of the world, which results i n an painting semi-abstractly? Semi-abstract style ought to result, then, from a simultaneous affirmation and negation of the world, which, clearly, i s a contradiction and could result i n nothing but nonsense. And even i f we could be sympathetic towards such an explanation of semi-abstrac-tion i n art and say, "It i s true that semi-abstract art i s nonsensical or 'meaningless', i n the sense that i t has no clear content", this does not mean that art activity i n which the aim i s semi-abstraction i s mean-ingless i n the sense of "aimless". We find i t d i f f i c u l t to understand, i n what sense Read i s using "affirmation" and "negation" of the world. It could be that he means by "affirmation" that something i s the case, and by "negation1! that some-thing i s not the case; and i n this sense the ar t i s t could affirm the world by depicting what i s the case, and negate or reject the world by depicting what the world i s not, or, to conform to our usage, what i s not the case. Then our ar t i s t would have to hold up the picture i n which he has affirmed the world, or hold up the picture i n which he has negated (20) Herbert Read, Philosophy of Modern Art, p. 24. We wonder what psychological phase would account for an a r t i s t - 75 -the world to make himself understood which of the pictures he affirms and which one he negates. Evidently, he cannot make a picture of the situation as non-existing; i n order to make a picture of a situation as non-existing, instead of i t he would have to make a picture of what did exist. In art the absence of a given quality i s not i t s e l f a qua-l i t y , nor the absence of a given relation i s i t s e l f a relation.(21) Probably, i n Read's terminology, "affirmation" and "negation" are to be understood as "thesis" and "antithesis" of which the "syn-thesis" i s that "particular kind of n u l l i t y " (22) which i s the starting point for the "existentialist-artist" to ensure that art has "freedom". (21) A. J. Ayer i n "Negation", The Journal of Philosophy, vol. 49, No. 26, 1952, (pp. 36-65), says: "If I say that Mediterranean Sea i s blue, I am referring to an individual object and ascribing to i t a quality; my statement, i f i t i s true, states a positive fact. But i n "Atlantic i s not blue" I am not ascribing any quality to an individual, and while my statement i s true, there must be some positive fact which makes i t so. Thus i t would seem that the apparently negative statement i s somehow doing duty for one that i s affirmative, or that i t i s made true, i f i t i s true, by some fact which i t does not state ... (Actually) both "Atlantic i s not blue" and "Mediterranean i s blue" are descriptions only. "A i s not blue" i s relatively uninformative, but this does not say that i t i s not a description at a l l . To say that negative statements do not state positive facts does not lead us to say that, i n the event of their being true, they are not as closely related to the fact as any other true statement. To say what things are not i s i t s e l f a way of saying what they are." A l l statements, then, become positive! In my opinion the expression "negation of the world" i s taken over by Worringer and Read from religious sources without i t being properly applicable to art. I f i t i s "negative judgment" they are talking about, the result may be endless and essentially sterile disputes. (22) Herbert Read, op. c i t . , p. 107. - 76 -Those theorists who explain abstraction i n art by the primi-tive "dread of space" or by the modern "dread of nothingness" commit the "existentialist" fallacy. They point to the diminished earth i n the yawning space between the stars in order to demonstrate that the "dread of space" once again rises i n the hearts of men who must accept i t s immensity, grandeur, and terror. They point out the dread born from the two Great Wars: dread of the Bolsheviks, of the atom bomb, of incurable maladies, and, most of a l l , dread of Dread i t s e l f , the "Angst" of Nothingness. And, as we have seen, they assert that abstrac-tion i n art arises from that dread, and thus explain abstraction i n Modern art. Their fallacy consists i n that they apply a single prin-ciple to the description of art and insist that this principle has an absolute v a l i d i t y . Suppose we grant them that this i s so. Then the question a r i -ses, could the motif of dread be used repeatedly without i t becoming a "dread by convention" and resulting i n nothing but monotony and boredom? Indeed i t seem that some modern artists (including writers and film makers) using the "existentialist" principle have succeeded i n becoming thoroughly boring and monotonous. The "particular kind of nu l l i t y " of Read's i s a phrase that reflects the boredom and monotony of this kind of art, but does not express the mood of the contemporary Western World. If we are to understand the process of abstraction i n our art, then i t i s not by assuming that i t i s incapable of rational account and devoid - 77 -of sense, but rather as I have tr i e d to show i n this thesis,that the process of abstraction i n art i s achieved by making use of man's free-dom to examine, to c r i t i c i z e , and to build imaginatively. It i s the power of thought man possesses by which unthinking nature can be re-fashioned and transformed. The vision of beauty i s attained when by unfettered contemplation the unconscious universe, Time and Space, i s transmuted i n the crucible of the imagination and when the most e v i l material, Death, Pain and Despair yield spectacles of beauty and increase human comradeship. It i s the freedom man possesses to exa-mine, to know and to c r i t i c i z e the hostile universe and to translate i t imaginatively that i s inherent i n the process of abstraction i n art. That i s why art i s always also an intellectual activity. Only by i n -tellectual thought, a r t i s t i c and philosophical, can man succeed in im-posing order on chaos. - 78 -Bibliography 1. Adler, Irwing, The Hew Mathematics, N.Y.: Signet Science Books, 1958. 2. - Thinking Machine, N.Y.: Signet Science Books, 1962. 3. Aristotle, Metaphysics. 4. Arnold, Matthew, Essays i n Criticism, F i r s t Series. London: Macmillan jr. Co., 1865, (1898). 5. Austin, J. L., Sense and Sensibilia, ed. G. J. Warnock; Oxford: Clarendon Press, 1962. 6. Ayer, A. J., ed., Logical Positivism, Glencoe, 111.: Free Press, 1959. 7. - "Negation", in The Journal of Philosophy, vol. 49, No. 26, 1952, pp. 36-65. 8. Buchheim, Lothar-Gunther, Der Blaue Reiter und die "Neue Kunstlervereinigung" Munchen, Feldhafing: Buchheim Verlag, 1959. 9. Burnet, John, Early Greek Philosophy, 2nd ed., London: Black, 1908. 10. Cornford, F. M., From Religion to Philosophy, N. Y.: Harper's Torchbook, 1957. 11. D.A.T.G., A.C.J., "Ludwig Wittgenstein i n The Australian Journal of Philosophy, vol. 29, No. 2, 1951, p. 75. 12. Diels, fl. G., Die Fragmente der Vorsokratiker, '903. 13. Fenby, Eric, Delius as I knew Him, London, 1936. 14. Frege, Gottlob, Grundgesetze der Arithmetik, Vol. II, 1903. 15. Friedlander, M. On Art and Connoisseurship, Boston: Beacon Press, (1942), I960. - 79 -16. Gabo, Naum, "A Retrospective View of Constructive Art", i n Three Lectures on Modern Art, N. 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